Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 289854 9 pageshttpdxdoiorg1011552013289854

Research ArticleThe Monoid Consisting of Kuratowski Operations

Szymon Plewik and Marta WalczyNska

Institute of Mathematics University of Silesia ul Bankowa 14 40-007 Katowice Poland

Correspondence should be addressed to Szymon Plewik plewikmathusedupl

Received 30 August 2012 Accepted 1 November 2012

Academic Editor Nan-Jing Huang

Copyright copy 2013 S Plewik and M WalczynskaThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

The paper fills gaps in knowledge about Kuratowski operations which are already in the literature The Cayley table for theseoperations has been drawn up Techniques using only paper and pencil to point out all semigroups and its isomorphism typesare applied Some results apply only to topology and one cannot bring them out using only properties of the complement and aclosure-like operation The arguments are by systematic study of possibilities

1 Introduction

Let119883 be a topological space Denote by 119860minus closure of the set119860 sube 119883 Let 119860119888 be the complement of 119860 that is 119883 119860 =

119860119888 The aim of this paper is to examine monoids generated

under compositions from the closure and the complement Awidely known fact due to Kuratowski [1] states that at most 14distinct operations can be formed from such compositionsMark them as follows Kuratowski operations

1205900

(119860) = 119860 (the identity)1205901

(119860) = 119860119888 (the complement)

1205902

(119860) = 119860minus (the closure)

1205903

(119860) = 119860119888minus

1205904

(119860) = 119860minus119888

1205905

(119860) = 119860119888minus119888 (the interior)

1205906

(119860) = 119860minus119888minus

1205907

(119860) = 119860119888minus119888minus

1205908

(119860) = 119860minus119888minus119888

1205909

(119860) = 119860119888minus119888minus119888

12059010

(119860) = 119860minus119888minus119888minus

12059011

(119860) = 119860119888minus119888minus119888minus

12059012

(119860) = 119860minus119888minus119888minus119888

12059013

(119860) = 119860119888minus119888minus119888minus119888

The following rules was found in the original paper byKuratowski [1 pages 183-184] In the Engelking book [2] theyare commented by a hint on page 81 Cancellation rules

119860minus119888minus

= 119860minus119888minus119888minus119888minus

119860119888minus119888minus

= 119860119888minus119888minus119888minus119888minus

(1)

Kuratowski operations have been studied by severalauthors for example [3] or [4] A list of some other authorsone can find in the paper [5] by Gardner and Jackson For thefirst time these operations were systematically studied in thedissertation by Kuratowski whose results were published in[1] Tasks relating to these operations are usually resolved atlectures or exercises with general topologyThey are normallyleft to students for independent resolution For exampledetermine how many different ways they convert a given set

This note is organized as follows Kuratowski operationsand their marking are described in the introduction Theirproperties of a much broader context than for topologies arepresented in Section 2The Cayley table for the monoidM ofall Kuratowski operations has been drawn up in Section 3We hope that this table has not yet been published in theliterature Having this table one can create a computerprogram that calculates all the semigroups contained in MHowever in Sections 4ndash8 we present a framework (ietechniques using only paper and pencil) to point out all 118semigroups and 56 isomorphism types of themThe list of 43semigroups which are not monoids is presented in Section 9In this section also isomorphism types are discussed in orderof the number of elements in semigroups Finally we present

2 Journal of Mathematics

cancellation rules (relations) motivated by some topologicalspaces

2 Cancellation Rules

A map 119891 119875(119883) rarr 119875(119883) is called

(i) increasing if 119860 sube 119861 implies 119891(119860) sube 119891(119861)(ii) decreasing if 119860 sube 119861 implies 119891(119861) sube 119891(119860)(iii) an involution if the composition 119891 ∘ 119891 is the identity(iv) an idempotent if 119891 ∘ 119891 = 119891

Assume that 119860 997891rarr 1205900

(119860) is the identity 119860 997891rarr 1205901

(119860) isa decreasing involution and 119860 997891rarr 120590

2

(119860) is an increasingidempotent map Other operations 120590

119894

let be compositions of1205901

and 1205902

as it was with the kuratowski operationsWe get thefollowing cancellation rules

Lemma 1 If 119861 sube 1205902

(119861) then

1205902

∘ 12059012

= 1205906

1205902

∘ 12059013

= 1205907

(2)

Proof For clarity of this proof use designations 1205901

(119860) = 119860119888

and 1205902

(119860) = 119860minus Thus we shall prove

119860minus119888minus119888minus119888minus

= 119860minus119888minus

119860119888minus119888minus119888minus119888minus

= 119860119888minus119888minus

(3)

We start with 119860minus119888minus119888 sube 119860minus119888minus119888minus substituting 119861 = 119860minus119888minus119888 in 119861 sube119861minus This corresponds to 119860minus119888minus119888minus119888 sube 119860minus119888minus119888119888 = 119860minus119888minus since 120590

1

isa decreasing involution Hence 119860minus119888minus119888minus119888minus sube 119860minus119888minus since 120590

2

anincreasing idempotent

Since 1205901

is decreasing 1205902

is increasing and 119861 sube 119861minus wehave

119861minus119888

sube 119861119888

sube 119861119888minus

(4)

Thus119860minus119888minus119888 sube 119860minus119888119888minus = 119860minus if we put119860minus119888 = 119861 Again using that1205902

is increasing and 1205901

is decreasing we obtain 119860minus119888minus119888minus sube 119860minusand then 119860minus119888 sube 119860minus119888minus119888minus119888 Finally we get 119860minus119888minus sube 119860minus119888minus119888minus119888minus

With 119860119888 in the place of 119860 in the rule 119860minus119888minus = 119860minus119888minus119888minus119888minus weget the second rule

In academic textbooks of general topology for example[2 Problem 171] one can find a hint suggested to prove theabove cancellation rules Students go like this steps 119860 sube 119860

minus

and 119860119888 sube 119860119888minus lead to 119860119888minus119888 sube 119860 the special case 119860minus119888minus119888minus119888 sube

119860minus119888minus (of119860119888minus119888 sube 119860) leads to119860minus119888minus119888minus119888minus sube 119860minus119888minus steps119860minus119888minus119888 sube 119860minus

and 119860minus119888minus119888minus sube 119860minus lead to 119860minus119888minus sube 119860minus119888minus119888minus119888minus at the end use thelast step of the proof of Lemma 1 Note that the above proofsdo not use the axioms of topology as follows

(i) 0 = 0minus(ii) (119860 cup 119861)minus = 119860minus cup 119861minus

In the literature there are articles in which Kuratowski oper-ations are replaced by some other mappings For exampleKoenen [6] considered linear spaces and put 120590

2

(119860) to bethe convex hull of 119860 In fact Shum [7] considered 120590

2

as theclosure due to the algebraic operations Add to this that theseoperations can be applied to so-called Frechet (V) spaceswhich were considered in the book [8 pages 3ndash37]

3 The Monoid M

LetM be the monoid consisting of all Kuratowski operationsthat is there are assumed cancellation rules 120590

6

= 1205902

∘12059012

and1205907

= 1205902

∘ 12059013

Fill in the Cayley table for M where the rowand column marked by the identity are omitted Similarly asin [9] the factor that labels the row comes first and that thefactor that labels the column is second For example 120590

119894

∘ 120590119896

isin the row marked by 120590

119894

and the column marked by 120590119896

1205901

1205902

1205903

1205904

1205905

1205906

1205907

1205908

1205909

12059010

12059011

12059012

12059013

1205901

1205900

1205904

1205905

1205902

1205903

1205908

1205909

1205906

1205907

12059012

12059013

12059010

12059011

1205902

1205903

1205902

1205903

1205906

1205907

1205906

1205907

12059010

12059011

12059010

12059011

1205906

1205907

1205903

1205902

1205906

1205907

1205902

1205903

12059010

12059011

1205906

1205907

1205906

1205907

12059010

12059011

1205904

1205905

1205904

1205905

1205908

1205909

1205908

1205909

12059012

12059013

12059012

12059013

1205908

1205909

1205905

1205904

1205908

1205909

1205904

1205905

12059012

12059013

1205908

1205909

1205908

1205909

12059012

12059013

1205906

1205907

1205906

1205907

12059010

12059011

12059010

12059011

1205906

1205907

1205906

1205907

12059010

12059011

1205907

1205906

12059010

12059011

1205906

1205907

1205906

1205907

12059010

12059011

12059010

12059011

1205906

1205907

1205908

1205909

1205908

1205909

12059012

12059013

12059012

12059013

1205908

1205909

1205908

1205909

12059012

12059013

1205909

1205908

12059012

12059013

1205908

1205909

1205908

1205909

12059012

12059013

12059012

12059013

1205908

1205909

12059010

12059011

12059010

12059011

1205906

1205907

1205906

1205907

12059010

12059011

12059010

12059011

1205906

1205907

12059011

12059010

1205906

1205907

12059010

12059011

12059010

12059011

1205906

1205907

1205906

1205907

12059010

12059011

12059012

12059013

12059012

12059013

1205908

1205909

1205908

1205909

12059012

12059013

12059012

12059013

1205908

1205909

12059013

12059012

1205908

1205909

12059012

12059013

12059012

12059013

1205908

1205909

1205908

1205909

12059012

12059013

(5)

It turns out that the above table allows us to describeall semigroups contained inM using pencil-and-paper tech-niques only The argument will be by a systematic study ofpossibilities Preparing the list of all semigroups consisting ofKuratowski operations we used following principles

(i) minimal collection of generators is written using⟨119860 119861 119885⟩ where letters denote generators

(ii) when a semigroup has a few minimal collections ofgenerators then its name is the first collection in thedictionary order

(iii) all minimal collections of generators are written withthe exception of some containing 120590

0

(iv) we leave to the readers verification that our list is

complete sometimes we add hints

4 Semigroups with 1205901

Observe that each semigroup which contains 1205900

is a monoidSince120590

1

∘ 1205901

= 1205900

a semigroupwhich contains1205901

is amonoidtoo

Theorem 2 There are three monoids containing 1205901

(1) ⟨1205901

⟩ = 1205900

1205901

(2) M = ⟨120590

1

120590119894

⟩ = 1205900

1205901

12059013

where 119894 isin 2 3 4 5(3) letM

1

= ⟨1205901

1205906

⟩ If 119895 isin 6 7 13 then

M1

= ⟨1205901

120590119895

⟩ = 1205900

1205901

cup 1205906

1205907

12059013

(6)

Proof The equality ⟨1205901

⟩ = 1205900

1205901

is obviousSince 120590

2

= 1205903

∘ 1205901

= 1205901

∘ 1205904

= 1205901

∘ 1205905

∘ 1205901

we have⟨1205901

1205902

⟩ = ⟨1205901

1205903

⟩ = ⟨1205901

1205904

⟩ = ⟨1205901

1205905

⟩ = MIf 119895 isin 6 7 13 then any composition 120590

119895

∘ 120590119894

or 120590119896

∘ 120590119895

belongs to M1

and so ⟨1205901

120590119895

⟩ sube M1

We have 1205907

= 1205906

∘ 1205901

Journal of Mathematics 3

1205908

= 1205901

∘ 1205906

1205909

= 1205901

∘ 1205907

12059010

= 1205906

∘ 1205906

12059011

= 1205906

∘ 1205907

12059012

= 1205908

∘ 1205906

and 12059013

= 1205908

∘ 1205907

and so M1

= ⟨1205901

1205906

⟩ =

1205900

1205901

cup 1205906

1205907

12059013

Since 120590

6

= 1205907

∘ 1205901

= 1205901

∘ 1205908

= 1205901

∘ 1205909

∘ 1205901

= 12059010

∘ 1205901

∘ 12059010

=

12059011

∘ 12059011

∘ 1205901

= 1205901

∘ 12059012

∘ 12059012

and 1205906

= 1205901

∘ 12059013

∘ 1205901

∘ 12059013

∘ 1205901

we have ⟨120590

1

120590119895

⟩ = M1

for 119895 isin 6 7 13

Consider the permutation

(1205900

1205901

1205902

1205903

1205904

1205905

1205906

1205907

1205908

1205909

12059010

12059011

12059012

12059013

1205900

1205901

1205905

1205904

1205903

1205902

1205909

1205908

1205907

1205906

12059013

12059012

12059011

12059010

)

(7)

It determines an automorphism A M rarr M

Theorem 3 The identity and A are the only automorphismsofM

Proof Delete rows and columns marked by 1205901

in the Cayleytable for M Then check that the operation 120590

3

is in the rowor the column marked by 120590

3

only Also the operation 1205904

isin the row or the column marked by 120590

4

only Therefore thesemigroup

⟨1205903

1205904

⟩ = 1205902

1205903

12059013

(8)

has a unique minimal set of generators 1205903

1205904

The reader isleft to check this with the Cayley table forM

Suppose G is an automorphism of M By Theorem 2 Gtransforms the set 120590

2

1205903

1205904

1205905

onto itself However 1205902

and1205905

are idempotents but 1205903

and 1205904

are not idempotents Sothere are two possibilities G(120590

3

) = 1205903

and G(1205904

) = 1205904

whichimplies that G is the identity G(120590

3

) = 1205904

and G(1205904

) = 1205903

which implies G = A The reader is left to check this with theCayley table forM We offer hints 120590

2

= 1205903

∘ 1205904

1205905

= 1205904

∘ 1205903

1205906

= 1205903

∘ 1205902

1205907

= 1205903

∘ 1205903

1205908

= 1205904

∘ 1205904

1205909

= 1205904

∘ 1205905

12059010

= 1205906

∘ 1205906

12059011

= 1205906

∘ 1205907

12059012

= 1205908

∘ 1205906

and 12059013

= 1205908

∘ 1205907

to verify the details of this proof

5 The Monoid of All Idempotents

The set 1205900

1205902

1205905

1205907

1205908

12059010

12059013

consists of all squares inMThese squares are idempotents and lie on the main diagonalin the Cayley table forM They constitute a monoid and

1205900

1205902

1205905

1205907

1205908

12059010

12059013

= ⟨1205900

1205902

1205905

⟩ (9)

The permutation

(1205900

1205902

1205905

1205907

1205908

12059010

12059013

1205900

1205902

1205905

1205908

1205907

12059010

12059013

) (10)

determines the bijection I ⟨1205900

1205902

1205905

⟩ rarr ⟨1205900

1205902

1205905

⟩ suchthat

I (120572 ∘ 120573) = I (120573) ∘ I (120572) (11)

for any 120572 120573 isin ⟨1205900

1205902

1205905

⟩ To verify this apply equalities1205902

∘ 1205905

= 1205907

1205905

∘ 1205902

= 1205908

1205902

∘ 1205905

∘ 1205902

= 12059010

and1205905

∘ 1205902

∘ 1205905

= 12059013

Any bijection I 119866 rarr 119867 having propertyI(120572∘120573) = I(120573)∘I(120572) transposesCayley tables for semigroups119866

and119867 Several semigroups contained inM have this propertyWe leave the reader to verify this

We shall classify all semigroups contained in the semi-group ⟨120590

2

1205905

⟩ Every such semigroup can be extended toa monoid by attaching 120590

0

to it This gives a completeclassification of all semigroups in ⟨120590

0

1205902

1205905

⟩The semigroup ⟨120590

2

1205905

⟩ contains six groups with exactlyone element

Semigroups ⟨1205902

12059010

⟩ = 1205902

12059010

and ⟨1205905

12059013

⟩ = 1205905

12059013

aremonoids Both consist of exactly two elements and are notgroups so they are isomorphic

Semigroups ⟨1205907

12059010

⟩ and ⟨1205908

12059013

⟩ are isomorphic in par-ticular A[⟨120590

7

12059010

⟩] = ⟨1205908

12059013

⟩ Also semigroups ⟨1205907

12059013

⟩

and ⟨1205908

12059010

⟩ are isomorphic by A Every of these four semi-groups has exactly two elements None of them is a monoidThey form two types of nonisomorphic semigroups becausethe bijections (120590

7

12059010

) (12059010

1205908

) and (1205907

1205908

) (12059010

12059010

) arenot isomorphisms

Semigroups ⟨1205902

1205907

⟩ = 1205902

1205907

12059010

and ⟨1205905

1205908

⟩ =

1205905

1205908

12059013

are isomorphic Also semigroups ⟨1205902

1205908

⟩ =

1205902

1205908

12059010

and ⟨1205905

1205907

⟩ = 1205905

1205907

12059013

are isomorphic Infact A[⟨120590

2

1205907

⟩] = ⟨1205905

1205908

⟩ and A[⟨1205902

1205908

⟩] = ⟨1205905

1205907

⟩None of these semigroups is a monoid They form two typesof nonisomorphic semigroups Indeed any isomorphismbetween ⟨120590

2

1205907

⟩ and ⟨1205902

1205908

⟩ must be the identity on themonoid ⟨120590

2

12059010

⟩ Therefore would have to be the restrictionof I But I restricted to ⟨120590

7

12059010

⟩ is not an isomorphismSemigroups ⟨120590

2

12059013

⟩ = ⟨1205902

1205907

1205908

⟩ = 1205902

1205907

1205908

12059010

12059013

and ⟨1205905

12059010

⟩ = ⟨1205905

1205907

1205908

⟩ = 1205905

1205907

1205908

12059010

12059013

are notmonoids They are isomorphic by A

The semigroup ⟨1205902

1205905

⟩ contains exactly one semigroupwith four elements ⟨120590

7

1205908

⟩ = ⟨12059010

12059013

⟩ = 1205907

1205908

12059010

12059013

which is not a monoidNote that ⟨120590

2

1205905

⟩ contains twenty different semigroupswith nine non-isomorphism types These are six isomorphicgroups with exactly one element ⟨120590

2

⟩ cong ⟨1205905

⟩ cong ⟨1205907

⟩ cong

⟨1205908

⟩ cong ⟨12059010

⟩ cong ⟨12059013

⟩ two isomorphic monoids with exactlytwo elements ⟨120590

2

12059010

⟩ cong ⟨1205905

12059013

⟩ two pairs of isomorphicsemigroups with exactly two elements ⟨120590

7

12059010

⟩ cong ⟨1205908

12059013

⟩

and ⟨1205907

12059013

⟩ cong ⟨1205908

12059010

⟩ two pairs of isomorphic semigroupswith exactly three elements ⟨120590

2

1205907

⟩ cong ⟨1205905

1205908

⟩ and ⟨1205902

1205908

⟩ cong

⟨1205905

1205907

⟩ a semigroup with exactly four elements ⟨1205907

1205908

⟩and two isomorphic semigroups with exactly five elements⟨1205902

12059013

⟩ cong ⟨1205905

12059010

⟩ and also ⟨1205902

1205905

⟩ Thus ⟨1205902

1205905

⟩ containstwenty different semigroups with nine isomorphism typesBut ⟨120590

0

1205902

1205905

⟩ contains forty-one different semigroups withseventeen isomorphism types Indeed adding 120590

0

to semi-groups contained in ⟨120590

2

1205905

⟩ which are not monoids weget twenty monoids with eight non-isomorphism typesAdding 120590

0

to a group contained in ⟨1205902

1205905

⟩ we get a monoidisomorphic to ⟨120590

2

12059010

⟩

6 The Semigroup Consisting of 12059061205907 120590

13

Using the Cayley table forM check thatA [1205906

1205907

12059013

] = 1205906

1205907

12059013

(12)Similarly check that the semigroup ⟨120590

6

1205909

⟩ = 1205906

1205907

12059013

can be represented as ⟨1205906

12059013

⟩ ⟨1205907

12059012

⟩ ⟨1205908

12059011

⟩

4 Journal of Mathematics

⟨1205909

12059010

⟩ or ⟨12059011

12059012

⟩ Also ⟨1205906

1205909

⟩ = ⟨1205906

1205907

1205908

⟩ =

⟨1205907

1205908

1205909

⟩ = ⟨12059010

12059011

12059013

⟩ = ⟨12059010

12059012

12059013

⟩ These rep-resentations exhaust all minimal collections of the Kura-towski operations which generate ⟨120590

6

1205909

⟩ Other semigroupsincluded in ⟨120590

6

1205909

⟩ have one or two minimal collectionof generators One generator has groups ⟨120590

6

⟩ = 1205906

12059010

⟨1205909

⟩ = 1205909

12059013

⟨12059011

⟩ = 1205907

12059011

and ⟨12059012

⟩ = 1205908

12059012

Each of them has exactly two elements so they are isomor-phic Semigroups ⟨120590

7

12059010

⟩ ⟨1205907

12059013

⟩ ⟨1205908

12059010

⟩ ⟨1205908

12059013

⟩ and⟨1205907

1205908

⟩ are discussed in the previous section Contained in⟨1205906

1205909

⟩ and not previously discussed semigroups are ⟨1205906

1205907

⟩⟨1205906

1205908

⟩⟨1205907

1205909

⟩ and ⟨1205908

1205909

⟩ We leave the reader to verifythat the following are all possible pairs of Kuratowski oper-ations which constitute a minimal collection of generatorsfor semigroups contained in ⟨120590

6

1205909

⟩ but different from thewhole One has

(i) ⟨1205906

1205907

⟩ = ⟨1205906

12059011

⟩ = ⟨12059010

12059011

⟩ = 1205906

1205907

12059010

12059011

(ii) ⟨120590

6

1205908

⟩ = ⟨1205906

12059012

⟩ = ⟨12059010

12059012

⟩ = 1205906

1205908

12059010

12059012

(iii) ⟨120590

7

1205909

⟩ = ⟨1205909

12059011

⟩ = ⟨12059011

12059013

⟩ = 1205907

1205909

12059011

12059013

(iv) ⟨120590

8

1205909

⟩ = ⟨1205909

12059012

⟩ = ⟨12059012

12059013

⟩ = 1205908

1205909

12059012

12059013

Proposition 4 Semigroups ⟨1205906

1205907

⟩ and ⟨1205908

1205909

⟩ are isomor-phic and also semigroups ⟨120590

6

1205908

⟩ and ⟨1205907

1205909

⟩ are isomorphicbut semigroups ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ are not isomorphic

Proof Isomorphisms are defined by A Suppose 119869 ⟨1205906

1205907

⟩ rarr ⟨1205906

1205908

⟩ is an isomorphism Thus 119869[⟨1205907

12059010

⟩] =

⟨1205908

12059010

⟩ Given 119869(1205907

) = 1205908

we get 12059010

= 119869(12059010

) = 119869(1205907

∘

12059010

) = 1205908

∘ 12059010

= 1205908

But 119869(1205907

) = 12059010

implies 12059010

= 119869(1205907

) =

119869(12059010

∘ 1205907

) = 1205908

∘ 12059010

= 1205908

Both possibilities lead to acontradiction

So ⟨1205906

1205909

⟩ contains eighteen different semigroups witheight non-isomorphism types Indeed these are four iso-morphic groups with exactly one element ⟨120590

7

⟩ cong ⟨1205908

⟩ cong

⟨12059010

⟩ cong ⟨12059013

⟩ four isomorphic groups with exactly twoelements ⟨120590

6

⟩ cong ⟨1205909

⟩ cong ⟨12059011

⟩ cong ⟨12059012

⟩ two pairs ofisomorphic semigroups with exactly two elements ⟨120590

7

12059010

⟩ cong

⟨1205908

12059013

⟩ and ⟨1205907

12059013

⟩ cong ⟨1205908

12059010

⟩ and five semigroupswith exactly four elements ⟨120590

6

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong

⟨1205907

1205909

⟩ ⟨1205907

1205908

⟩ and also ⟨1205906

1205909

⟩

7 Remaining Semigroups in ⟨1205903 1205904⟩

We have yet to discuss semigroups included in ⟨1205903

1205904

⟩ notincluded in ⟨120590

2

1205905

⟩ and containing at least one of Kuratowskioperation 120590

2

1205903

1205904

or 1205905

It will be discussed up to theisomorphism A Obviously ⟨120590

2

⟩ = 1205902

and ⟨1205905

⟩ = 1205905

aregroups

71 Extensions of ⟨1205902⟩ and ⟨1205905⟩ with Elements of ⟨12059061205909⟩Monoids ⟨120590

2

1205906

⟩ = 1205902

1205906

12059010

and ⟨1205902

12059010

⟩ = 1205902

12059010

havedifferent numbers of elements Also ⟨120590

2

1205906

⟩ is isomorphic to⟨1205900

1205906

⟩ Nonisomorphic semigroups ⟨1205902

1205907

⟩ and ⟨1205902

1205908

⟩ arediscussed above The following three semigroups

(i) ⟨1205902

12059011

⟩ = ⟨1205902

1205906

1205907

⟩ = 1205902

1205906

1205907

12059010

12059011

(ii) ⟨1205902

12059012

⟩ = ⟨1205902

1205906

1205908

⟩ = 1205902

1205906

1205908

12059010

12059012

(iii) ⟨120590

2

12059013

⟩ = ⟨1205902

1205907

1205908

⟩ = 1205902

1205907

1205908

12059010

12059013

are not monoids They are not isomorphic Indeed anyisomorphism between these semigroups would lead an iso-morphism between ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ or ⟨1205907

1205908

⟩ This isimpossible by Proposition 4 and because ⟨120590

7

1205908

⟩ consists ofidempotents but 120590

6

is not an idempotent The nine-elementsemigroup on the set 120590

2

1205906

1205907

12059013

is represented as⟨1205902

1205909

⟩ So we have added four new semigroups which arenot isomorphic with the semigroups previously discussedThese are ⟨120590

2

12059011

⟩ ⟨1205902

12059012

⟩ ⟨1205902

12059013

⟩ and ⟨1205902

1205909

⟩Using A we have described eight semigroupsmdasheach

one isomorphic to a semigroup previously discussedmdashwhichcontains 120590

5

and elements (at least one) of ⟨1205906

1205909

⟩ Collec-tions of generators ⟨120590

2

1205906

12059013

⟩ ⟨1205902

1205907

12059012

⟩ ⟨1205902

1205908

12059011

⟩⟨1205902

12059011

12059012

⟩ ⟨1205902

12059011

12059013

⟩ and ⟨1205902

12059012

12059013

⟩ are minimalin ⟨120590

2

1205909

⟩ Also collections of generators ⟨1205905

1205907

12059012

⟩⟨1205905

1205908

12059011

⟩ ⟨1205905

1205909

12059010

⟩ ⟨1205905

12059010

12059011

⟩ ⟨1205905

12059010

12059012

⟩ and⟨1205905

12059011

12059012

⟩ are minimal in ⟨1205905

1205906

⟩

72 Extensions of ⟨1205903⟩ and ⟨1205904⟩ by Elements from the Semi-group ⟨12059061205909⟩ Semigroups ⟨120590

3

⟩ = 1205903

1205907

12059011

and ⟨1205904

⟩ =

1205904

1205908

12059012

are isomorphic by A They are not monoids Thesemigroup ⟨120590

3

⟩ can be extended using elements of ⟨1205906

1205909

⟩in three following ways

(i) ⟨1205903

1205906

⟩ = ⟨1205903

12059010

⟩ = 1205903

1205906

1205907

12059010

12059011

(ii) ⟨120590

3

1205908

⟩ = ⟨1205903

12059012

⟩ = 1205903

1205906

1205907

12059013

(iii) ⟨120590

3

1205909

⟩ = ⟨1205903

12059013

⟩ = 1205903

1205907

1205909

12059011

12059013

Semigroups ⟨1205903

1205906

⟩ and ⟨1205903

1205909

⟩ are not isomorphic Indeedsuppose 119869 ⟨120590

3

1205906

⟩ rarr ⟨1205903

1205909

⟩ is an isomorphismThus 119869 isthe identity on ⟨120590

3

⟩ 119869(1205906

) = 1205909

and 119869(12059010

) = 12059013

This givesa contradiction since 120590

3

∘ 1205906

= 12059010

and 1205903

∘ 1205909

= 1205907

Neither ⟨120590

3

1205906

⟩ nor ⟨1205903

1205909

⟩ has a minimal collection ofgenerators with three elements so they give new isomor-phism types Also ⟨120590

2

1205909

⟩ is not isomorphic to ⟨1205903

1205908

⟩since ⟨120590

2

1205909

⟩ has a unique pair of generators and ⟨1205903

1205908

⟩ =

⟨1205903

12059012

⟩Using A we getmdashisomorphic to previously discussed

onesmdashsemigroups ⟨1205904

1205909

⟩ = ⟨1205904

12059013

⟩ = 1205904

1205908

1205909

12059012

12059013

⟨1205904

1205906

⟩ = ⟨1205904

12059010

⟩ = 1205904

1205906

1205908

12059010

12059012

and ⟨1205904

1205907

⟩ =

⟨1205904

12059011

⟩ = 1205904

1205906

1205907

12059013

There exist minimal collec-tions of generators such as follows

⟨1205903

1205908

⟩ = ⟨1205903

1205906

1205909

⟩ = ⟨1205903

1205906

12059013

⟩

= ⟨1205903

1205909

12059010

⟩ = ⟨1205903

12059010

12059013

⟩

⟨1205904

1205907

⟩ = ⟨1205904

1205906

1205909

⟩ = ⟨1205904

1205909

12059010

⟩

= ⟨1205904

1205906

12059013

⟩ = ⟨1205904

12059010

12059013

⟩

(13)

73 More Generators from the Set 1205902120590312059041205905 Nowwe check that ⟨120590

2

1205903

⟩ = 1205902

1205903

1205906

1205907

12059010

12059011

and⟨1205904

1205905

⟩ = 1205904

1205905

1205908

1205909

12059012

12059013

= A[⟨1205902

1205903

⟩] andalso ⟨120590

2

1205904

⟩ = 1205902

1205904

1205906

1205908

12059010

12059012

and ⟨1205903

1205905

⟩ =

1205903

1205905

1205907

1205909

12059011

12059013

= A[⟨1205902

1205904

⟩] are two pairs of iso-morphic semigroupswhich give two new isomorphism types

Journal of Mathematics 5

Each of these semigroups has six element so in M thereare five six-element semigroups of three isomorphism typessince ⟨120590

2

1205905

⟩ has 6 elements which are idempotentsIn ⟨1205903

1205904

⟩ there are seven semigroups which have threegenerators and have not two generators These are

(1) ⟨1205902

1205903

1205908

⟩ = ⟨1205902

1205903

1205909

⟩ = ⟨1205902

1205903

12059012

⟩ = ⟨1205902

1205903

12059013

⟩

(2) ⟨1205904

1205905

1205906

⟩ = ⟨1205904

1205905

1205907

⟩ = ⟨1205904

1205905

12059010

⟩ = ⟨1205904

1205905

12059011

⟩

(3) ⟨1205902

1205904

1205907

⟩ = ⟨1205902

1205904

1205909

⟩ = ⟨1205902

1205904

12059011

⟩ = ⟨1205902

1205904

12059013

⟩

(4) ⟨1205903

1205905

1205906

⟩ = ⟨1205903

1205905

1205908

⟩ = ⟨1205903

1205905

12059010

⟩ = ⟨1205903

1205905

12059012

⟩

(5) ⟨1205902

1205905

1205906

⟩ = ⟨1205902

1205905

1205909

⟩ = ⟨1205902

1205905

12059011

⟩ = ⟨1205902

1205905

12059012

⟩

(6) ⟨1205902

1205903

1205905

⟩

(7) ⟨1205902

1205904

1205905

⟩

8 Semigroups which Are Contained in ⟨12059031205904⟩

Groups ⟨1205902

⟩ cong ⟨1205905

⟩ cong ⟨1205907

⟩ cong ⟨1205908

⟩ cong ⟨12059010

⟩ cong ⟨12059013

⟩ have oneelement and are isomorphic

Groups ⟨1205906

⟩ cong ⟨1205909

⟩ cong ⟨12059011

⟩ cong ⟨12059012

⟩ monoids ⟨1205902

12059010

⟩ cong

⟨1205905

12059013

⟩ and also semigroups ⟨1205907

12059010

⟩ cong ⟨1205908

12059013

⟩ and⟨1205907

12059013

⟩ cong ⟨1205908

12059010

⟩ have two elements and Cayley tables asfollows

119860 119861

119860 119861 119860

119861 119860 119861

119860 119861

119860 119860 119861

119861 119861 119860

119860 119861

119860 119860 119861

119861 119860 119861

119860 119861

119860 119860 119860

119861 119861 119861

(14)

Monoids ⟨1205902

1205906

⟩ cong ⟨1205905

1205909

⟩ and also semigroups⟨1205902

1205907

⟩ cong ⟨1205905

1205908

⟩ ⟨1205902

1205908

⟩ cong ⟨1205905

1205907

⟩ and ⟨1205903

⟩ cong ⟨1205904

⟩ havethree elements and Cayley tables as follows

119860 119861 119862

119860 119860 119861 119862

119861 119861 119862 119861

119862 119862 119861 119862

119860 119861 119862

119860 119860 119861 119862

119861 119862 119861 119862

119862 119862 119861 119862

119860 119861 119862

119860 119860 119862 119862

119861 119861 119861 119861

119862 119862 119862 119862

119860 119861 119862

119860 119861 119862 119861

119861 119862 119861 119862

119862 119861 119862 119861

(15)

Semigroups ⟨1205906

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong ⟨1205907

1205909

⟩ and⟨1205907

1205908

⟩ have four elements and Cayley tables as follows

119860 119861 119862 119863

119860 119862 119863 119860 119861

119861 119860 119861 119862 119863

119862 119860 119861 119862 119863

119863 119862 119863 119860 119861

119860 119861 119862 119863

119860 119862 119860 119860 119862

119861 119863 119861 119861 119863

119862 119860 119862 119862 119860

119863 119861 119863 119863 119861

119860 119861 119862 119863

119860 119860 119862 119862 119860

119861 119863 119861 119861 119863

119862 119860 119862 119862 119860

119863 119863 119861 119861 119863

(16)

Semigroups ⟨1205902

12059011

⟩ cong ⟨1205905

12059012

⟩ ⟨1205902

12059012

⟩ cong ⟨1205905

12059011

⟩⟨1205902

12059013

⟩ cong ⟨1205905

12059010

⟩ ⟨1205903

1205906

⟩ cong ⟨1205904

1205909

⟩ and ⟨1205903

1205909

⟩ cong

⟨1205904

1205906

⟩ have five elements Semigroups ⟨1205902

1205903

⟩ cong ⟨1205904

1205905

⟩⟨1205902

1205904

⟩ cong ⟨1205903

1205905

⟩ and ⟨1205902

1205905

⟩ have six elements Construc-tion of Cayley tables for these semigroups as well as othersemigroups we leave it to the readers One can prepare suchtables removing from the Cayley table for M some columnsand rows Then change Kuratowski operations onto lettersof the alphabet For example ⟨120590

0

1205902

1205905

⟩ has the followingCayley table

0 119860 119861 119862 119863 119864 119865

0 0 119860 119861 119862 119863 119864 119865

119860 119860 119860 119862 119862 119864 119864 119862

119861 119861 119863 119861 119865 119863 119863 119865

119862 119862 119864 119862 119862 119864 119864 119862

119863 119863 119863 119865 119865 119863 119863 119865

119864 119864 119864 119862 119862 119864 119864 119862

119865 119865 119863 119865 119865 119863 119863 119865

(17)

Preparing theCayley table for ⟨1205900

1205902

1205905

⟩weput1205900

= 0 1205902

=

119860 1205905

= 119861 1205907

= 119862 1205908

= 119863 12059010

= 119864 and 12059013

= 119865 This tableimmediately shows that semigroups ⟨120590

2

1205905

⟩ and ⟨1205900

1205902

1205905

⟩

have exactly two automorphisms These are identities andrestrictions of A

Semigroup ⟨1205906

1205909

⟩ has eight elements Semigroups⟨1205902

1205909

⟩ cong ⟨1205905

1205906

⟩ and ⟨1205903

1205908

⟩ cong ⟨1205904

1205907

⟩ have nine ele-ments Semigroups ⟨120590

2

1205903

1205908

⟩ cong ⟨1205904

1205905

1205907

⟩ ⟨1205902

1205904

1205907

⟩ cong

⟨1205903

1205905

1205908

⟩ and ⟨1205902

1205905

1205906

⟩ have ten elements Semigroups⟨1205902

1205903

1205905

⟩ cong ⟨1205902

1205904

1205905

⟩ have eleven elements In the endthe semigroup ⟨120590

3

1205904

⟩ has twelve elementsThus the semigroup ⟨120590

3

1205904

⟩ includes fifty-seven semi-groups amongwhich there are ten groups fourteenmonoidsand also forty-three semigroups which are not monoidsThese semigroups consist of twenty-eight types of noniso-morphic semigroups two non-isomorphism types of groupstwo non-isomorphism types of monoids which are notgroups and twenty four non-isomorphism types of semi-groups which are not monoids

9 Viewing Semigroups Contained in M

91 Descriptive Data on Semigroups which Are Contained inM There are one hundred eighteen that is 118 = 2 sdot 57 +

6 Journal of Mathematics

4 semigroups which are contained in M These are fifty-seven semigroups contained in ⟨120590

3

1205904

⟩ fifty-seven monoidsformed by adding 120590

0

to a semigroup contained in ⟨1205903

1205904

⟩groups ⟨120590

0

⟩ ⟨1205901

⟩ and monoids ⟨1205901

1205906

⟩ = M1

and ⟨1205901

1205902

⟩ =

MThere are fifty-six types of nonisomorphic semigroups

in M These are twenty-eight non-isomorphism types ofsemigroups contained in ⟨120590

3

1205904

⟩ twenty-six types of non-isomorphic monoids formed by adding 120590

0

to a semigroupcontained in ⟨120590

3

1205904

⟩ and alsoM1

andM Indeed adding 1205900

to a semigroup which is not a monoid we obtain a monoidIn this way we get twenty-four types of nonisomorphicmonoids Adding 120590

0

to a monoid contained in ⟨1205903

1205904

⟩ weget two new non-isomorphism types ofmonoids But adding1205900

to a group contained in ⟨1205903

1205904

⟩ we get no new type ofmonoid since we get a monoid isomorphic to ⟨120590

2

12059010

⟩ or⟨1205902

1205906

⟩ The other two types areM1

andM

92 Semigroups which Are Not Monoids Below we havereproduced using the smallest number of generators andthe dictionary order a list of all 43 semigroups which areincluded in theM as follows

(1) ⟨1205902

1205903

⟩ = 1205902

1205903

1205906

1205907

12059010

12059011

(2) ⟨120590

2

1205903

1205905

⟩ = 1205902

1205903

1205905

1205906

12059013

(3) ⟨120590

2

1205903

1205908

⟩ = ⟨1205902

1205903

1205909

⟩ = ⟨1205902

1205903

12059012

⟩ = ⟨1205902

1205903

12059013

⟩ = 1205902

1205903

1205906

1205907

12059013

(4) ⟨120590

2

1205904

⟩ = 1205902

1205904

1205906

1205908

12059010

12059012

(5) ⟨120590

2

1205904

1205905

⟩ = 1205902

1205904

1205905

12059013

(6) ⟨120590

2

1205904

1205907

⟩ = ⟨1205902

1205904

1205909

⟩ = ⟨1205902

1205904

12059011

⟩ = ⟨1205902

1205904

12059013

⟩ = 1205902

1205904

1205906

1205907

12059013

(7) ⟨120590

2

1205905

⟩ = 1205902

1205905

1205907

1205908

12059010

12059013

(8) ⟨120590

2

1205905

1205906

⟩ = ⟨1205902

1205905

1205909

⟩ = ⟨1205902

1205905

12059011

⟩ = ⟨1205902

1205905

12059012

⟩ = 1205902

1205905

1205906

12059013

(9) ⟨120590

2

1205907

⟩ = 1205902

1205907

12059010

(10) ⟨120590

2

1205908

⟩ = 1205902

1205908

12059010

(11) ⟨120590

2

1205909

⟩ = ⟨1205902

1205906

12059013

⟩ = ⟨1205902

1205907

12059012

⟩ = ⟨1205902

1205908

12059011

⟩ = ⟨1205902

12059011

12059012

⟩ = ⟨1205902

12059011

12059013

⟩ = ⟨1205902

12059012

12059013

⟩ = 1205902

1205906

1205907

12059013

(12) ⟨120590

2

12059011

⟩ = ⟨1205902

1205906

1205907

⟩ = 1205902

1205906

1205907

12059010

12059011

(13) ⟨120590

2

12059012

⟩ = ⟨1205902

1205906

1205908

⟩ = 1205902

1205906

1205908

12059010

12059012

(14) ⟨120590

2

12059013

⟩ = ⟨1205902

1205907

1205908

⟩ = 1205902

1205907

1205908

12059010

12059013

(15) ⟨120590

3

⟩ = 1205903

1205907

12059011

(16) ⟨120590

3

1205904

⟩ = 1205902

1205903

12059013

(17) ⟨120590

3

1205905

⟩ = 1205903

1205905

1205907

1205909

12059011

12059013

(18) ⟨120590

3

1205905

1205906

⟩ = ⟨1205903

1205905

1205908

⟩ = ⟨1205903

1205905

12059010

⟩ = ⟨1205903

1205905

12059012

⟩ = 1205903

1205905

1205906

12059013

(19) ⟨120590

3

1205906

⟩ = ⟨1205903

12059010

⟩ = 1205903

1205906

1205907

12059010

12059011

(20) ⟨120590

3

1205908

⟩ = ⟨1205903

12059012

⟩ = 1205903

1205906

1205907

12059013

(21) ⟨120590

3

1205909

⟩ = ⟨1205903

12059013

⟩ = 1205903

1205907

1205909

12059011

12059013

(22) ⟨120590

4

⟩ = 1205904

1205908

12059012

(23) ⟨1205904

1205905

⟩ = 1205904

1205905

1205908

1205909

12059012

12059013

(24) ⟨120590

4

1205905

1205906

⟩ = ⟨1205904

1205905

1205907

⟩ = ⟨1205904

1205905

12059010

⟩ = ⟨1205904

1205905

12059011

⟩ = 1205904

1205905

12059013

(25) ⟨120590

4

1205906

⟩ = ⟨1205904

12059010

⟩ = 1205904

1205906

1205908

12059010

12059012

(26) ⟨120590

4

1205907

⟩ = ⟨1205904

12059011

⟩ = 1205904

1205906

1205907

12059013

(27) ⟨120590

4

1205909

⟩ = ⟨1205904

12059013

⟩ = 1205904

1205908

1205909

12059012

12059013

(28) ⟨120590

5

1205906

⟩ = ⟨1205905

1205907

12059012

⟩ = ⟨1205905

1205908

12059011

⟩ = ⟨1205905

1205909

12059010

⟩ = ⟨1205905

12059010

12059011

⟩ = ⟨1205905

12059010

12059012

⟩ = ⟨1205905

12059011

12059012

⟩ = 1205905

1205906

12059013

(29) ⟨120590

5

1205907

⟩ = 1205905

1205907

12059013

(30) ⟨120590

5

1205908

⟩ = 1205905

1205908

12059013

(31) ⟨120590

5

12059010

⟩ = ⟨1205905

1205907

1205908

⟩ = 1205905

1205907

1205908

12059010

12059013

(32) ⟨120590

5

12059011

⟩ = ⟨1205905

1205907

1205909

⟩ = 1205905

1205907

1205909

12059011

12059013

(33) ⟨120590

5

12059012

⟩ = ⟨1205905

1205908

1205909

⟩ = 1205905

1205908

1205909

12059012

12059013

(34) ⟨120590

6

1205907

⟩ = ⟨1205906

12059011

⟩ = ⟨12059010

12059011

⟩ = 1205906

1205907

12059010

12059011

(35) ⟨120590

6

1205908

⟩ = ⟨1205906

12059012

⟩ = ⟨12059010

12059012

⟩ = 1205906

1205908

12059010

12059012

(36) ⟨120590

6

1205909

⟩ = ⟨1205906

12059013

⟩ = ⟨1205907

12059012

⟩ = ⟨1205908

12059011

⟩ = ⟨1205909

12059010

⟩ = ⟨12059011

12059012

⟩ = ⟨1205906

1205907

1205908

⟩ = ⟨1205907

1205908

1205909

⟩ =⟨12059010

12059011

12059013

⟩ = ⟨12059010

12059012

12059013

⟩ = 1205906

1205907

12059013

(37) ⟨120590

7

1205908

⟩ = 1205907

1205908

12059010

12059013

(38) ⟨120590

7

1205909

⟩ = 1205907

1205909

12059011

12059013

(39) ⟨120590

7

12059010

⟩ = 1205907

12059010

(40) ⟨120590

7

12059013

⟩ = 1205907

12059013

(41) ⟨120590

8

1205909

⟩ = 1205908

1205909

12059012

12059013

(42) ⟨120590

8

12059010

⟩ = 1205908

12059010

(43) ⟨120590

8

12059013

⟩ = 1205908

12059013

93 Isomorphism Types of Semigroups Contained in M Sys-tematize the list of all isomorphism types of semigroupscontained in themonoidM Isomorphisms which are restric-tions of the isomorphism A will be regarded as self-evidentand therefore they will not be commented on

(i) The monoid M contains 12 groups with 2 isomor-phism types These are 7 one-element groups and 5two-element groups

(ii) ThemonoidM contains 8 two-element monoids with2 isomorphism types These are 120590

0

added to 6 one-element groups and ⟨120590

2

12059010

⟩ cong ⟨1205905

12059013

⟩ Also itcontains 4 two-element semigroups not monoidswith 2 isomorphism types These are ⟨120590

7

12059010

⟩ cong

⟨1205908

12059013

⟩ and ⟨1205908

12059010

⟩ cong ⟨1205907

12059013

⟩(iii) The monoid M contains 12 three-element monoids

with 5 isomorphism types These are 1205900

addedto 4 two-element groups and also ⟨120590

0

1205902

12059010

⟩ cong

⟨1205900

1205905

12059013

⟩ ⟨1205900

1205907

12059010

⟩ cong ⟨1205900

1205908

12059013

⟩⟨1205900

1205908

12059010

⟩ cong ⟨1205900

1205907

12059013

⟩ and ⟨1205902

1205906

⟩ cong ⟨1205905

1205909

⟩(iv) The monoidM contains 6 three-element semigroups

not monoids with 3 isomorphism types These are⟨1205902

1205907

⟩ cong ⟨1205905

1205908

⟩ ⟨1205902

1205908

⟩ cong ⟨1205905

1205907

⟩ and ⟨1205903

⟩ cong

⟨1205904

⟩

Journal of Mathematics 7

(v) The monoid M contains 8 four-element monoidseach contains 120590

0

with 4 isomorphism types Theseare semigroups from two preceding items that can besubstantially extended by 120590

0

(vi) The monoid M contains 5 four-element semigroups

not monoids with 3 isomorphism types These are⟨1205906

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong ⟨1205907

1205909

⟩ and ⟨1205907

1205908

⟩These semigroups extended by 120590

0

yield 5 monoids allwhich consist of five elements with 3 isomorphismtypes

(vii) ThemonoidM contains 10 five-element semigroupsmdashnot monoids with 5 isomorphism types These are⟨1205902

12059011

⟩ cong ⟨1205905

12059012

⟩ ⟨1205902

12059012

⟩ cong ⟨1205905

12059011

⟩ ⟨1205902

12059013

⟩ cong

⟨1205905

12059010

⟩ ⟨1205903

1205906

⟩ cong ⟨1205904

1205909

⟩ and ⟨1205903

1205909

⟩ cong ⟨1205904

1205906

⟩These semigroups extended by120590

0

yield 10monoids allof which consist of six elements with 5 isomorphismtypes We get 10 new isomorphism types since thesemigroups are distinguished by semigroups ⟨120590

3

⟩

and ⟨1205904

⟩ and by nonisomorphic semigroups ⟨1205906

1205907

⟩⟨1205906

1205908

⟩ and ⟨1205907

1205908

⟩(viii) The monoid M contains 5 six-element semigroups

not monoids with 3 isomorphism types These are⟨1205902

1205903

⟩ cong ⟨1205904

1205905

⟩ ⟨1205902

1205904

⟩ cong ⟨1205903

1205905

⟩ and ⟨1205902

1205905

⟩These semigroups extended by 120590

0

yield 5 monoidsall of which consist of seven elements with 3 isomor-phism types We get 6 new isomorphism types sincethe semigroups are distinguished by non isomorphicsemigroups ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ and ⟨1205907

1205908

⟩(ix) ThemonoidM contains no seven-element semigroup

not a monoid no eight-element monoid and theonly semigroup ⟨120590

6

1205909

⟩ with exactly eight elementsand the only monoid ⟨120590

0

1205906

1205909

⟩ with exactly nineelements

(x) The monoid M contains 4 nine-element semigroupsnot monoids with 2 isomorphism types These are⟨1205902

1205909

⟩ cong ⟨1205905

1205906

⟩ and ⟨1205903

1205908

⟩ cong ⟨1205904

1205907

⟩ Thesemigroup ⟨120590

2

1205909

⟩ does not contain a semigroupisomorphic to ⟨120590

3

⟩ hence it is not isomorphic to⟨1205903

1205908

⟩ These semigroups extended by 1205900

yield 4monoids all of which consist of ten elements with 2isomorphism types

(xi) The monoid M contains 6 ten-element semigroupsnot monoids with 4 isomorphism typesThese are ⟨120590

1

1205906

⟩ ⟨1205902

1205903

1205908

⟩ cong ⟨1205904

1205905

1205906

⟩⟨1205902

1205904

1205907

⟩ cong ⟨1205903

1205905

1205906

⟩ and ⟨1205902

1205905

1205906

⟩ Thesesemigroups (except ⟨120590

1

1205906

⟩) extended by 1205900

yield 5monoids all of which consist of ten elements with3 isomorphism types We get 6 new isomorphismtypes since the semigroups are distinguished bynot isomorphic semigroups ⟨120590

2

1205903

⟩ ⟨1205902

1205904

⟩ and⟨1205902

1205905

⟩(xii) The monoid M contains 2 isomorphic semigroups

not monoids which consist of 11 elements that is⟨1205902

1205903

1205905

⟩ cong ⟨1205902

1205904

1205905

⟩These semigroups extendedby 1205900

yield 2 isomorphic monoids which consistof 12 elements The monoid M contains no largersemigroup with the exception of itself

10 Cancellation Rules Motivated by SomeTopological Properties

101 Some Consequences of the Axiom 0 = 0minus So far we usedonly the following relations (above named cancellation rules)1205902

∘ 1205902

= 1205902

1205901

∘ 1205901

= 1205900

1205902

∘ 12059012

= 1205906

and 1205902

∘ 12059013

= 1205907

When one assumes119883 = 0 = 0

minus then

119883 = 1205900

(119883) = 1205902

(119883) = 1205905

(119883) = 1205907

(119883)

= 1205908

(119883) = 12059010

(119883) = 12059013

(119883)

0 = 1205901

(119883) = 1205903

(119883) = 1205904

(119883) = 1205906

(119883)

= 1205909

(119883) = 12059011

(119883) = 12059012

(119883)

(18)

Using the substitution 119860 997891rarr 119860119888 one obtains equiv-

alent relations between operations from the set 1205901

1205903

1205904

1205906

1205909

12059011

12059012

and conversely Therefore cancellationrules are topologically reasonable only between the opera-tions from the monoid as follows

⟨1205900

1205902

1205905

⟩ = 1205900

1205902

1205905

1205907

1205908

12059010

12059013

(19)

Chapman see [3] considered properties of subsets withrespect to such relations Below we are going to identifyrelations that are determined by some topological spacescompare [10 11]

102 The Relation 1205900 = 1205902 If a topological space 119883 isdiscrete then there exist two Kuratowski operation onlyThese are 120590

0

and 1205901

So themonoid of Kuratowski operationsreduced to the group ⟨120590

1

⟩The relation 120590

0

= 1205902

is equivalent to any relation 1205900

=

120590119894

where 119894 isin 5 7 8 10 13 Any such relation implies thatevery subset of 119883 has to be closed and open that is 119883 hasto be discrete However one can check these using (only) thefacts that 120590

1

is an involution and 1205902

is an idempotent and thecancellation rules that is the Cayley table forM So 120590

0

= 1205902

follows

1205900

= 1205902

= 1205905

= 1205907

= 1205908

= 12059010

= 12059013

(20)

103 The Relation 1205902 = 1205905 Topologically 1205902 = 1205905

meansthat 119883 must be discrete This is so because 119860119888minus119888 sube 119860 sube 119860

minus

for any 119860 sube 119883

104 The Relation 1205902 = 1205907 Topologically the relation 1205902 =1205907

implies 1205900

= 1205902

But it requires the use of topology axioms0 = 0minus and 119862minus cup 119861minus = (119862 cup 119861)minus for each 119862 and 119861

Lemma 5 For any topological space 1205902

= 1205907

implies 1205902

= 1205908

Proof If 1205902

= 1205907

then 119860 = 0 rArr 119860119888minus119888

= 0 for any 119860 sube 119883Indeed if 119860119888minus119888 = 0 then 120590

7

(119860) = 0minus

= 0 Since 119860 = 0 then1205902

(119860) = 0 Hence 1205902

(119860) = 1205907

(119860) a contradictionThe axiom 119862

minus

cup 119861minus

= (119862 cup 119861)minus implies that always

(119860minus

cap 119860minus119888minus

)119888minus119888

= 0 (21)

Thus the additional assumption 1205902

= 1205907

follows that always119860minus

cap 119860minus119888minus

= 0 Therefore always 119860minus119888minus = 119860minus119888 but this means

8 Journal of Mathematics

that any closed set has to be open in other words 1205902

= 1205908

Proposition 6 For any topological space 1205902

= 1205907

implies 1205900

=

1205902

Proof But the relation 1205902

= 1205907

is equivalent with 1205905

= 1205908

ByLemma 5 we get 120590

2

= 1205905

Finally 1205900

= 1205902

The relation 1205902

= 1205907

has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put

1205902

(0) = 119886 = 1205902

(119886) 119883 = 1205902

(119883) = 1205902

(119887) (22)

Then check that 1205902

= 1205907

and

1205908

(0) = 1205905

(119886) = 1205904

(119887) = 1205901

(119883) = 0 (23)

in other words 1205902

= 1205907

and 1205902

= 1205908

However 1205902

= 1205907

isequivalent to 120590

5

= 1205908

This relation implies 120590

2

= 1205907

= 12059010

1205905

= 1205908

= 12059013

1205903

= 1205906

= 12059011

and 1205904

= 1205909

= 12059012

For this interpretationthe monoidM(120590

2

= 1205907

) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590

2

= 1205907

) thereare relations covered by the following proposition only

Proposition 7 For any monoid with the Cayley table as forM the relation 120590

2

= 1205907

implies

(i) 1205902

= 1205907

= 12059010

= 1205907

∘ 1205902

(ii) 1205905

= 1205901

∘1205902

∘1205901

= 1205901

∘1205907

∘1205901

= 1205908

= 1205905

∘1205902

= 1205905

∘1205907

=

12059013

(iii) 120590

3

= 1205902

∘ 1205901

= 1205907

∘ 1205901

= 1205906

= 12059010

∘ 1205901

= 12059011

(iv) 1205904

= 1205901

∘ 1205902

= 1205901

∘ 1205907

= 1205909

= 1205901

∘ 12059010

= 12059012

Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology

105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908

means that any closed set is open tooThus if119883 = 119886 119887 119888

is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590

2

= 1205908

) is the monoid of all Kuratowski operationsover 119883 Relations 120590

2

= 1205908

and 1205905

= 1205907

are equivalent Theyimply relations 120590

2

= 1205908

= 12059010

1205905

= 1205907

= 12059013

1205903

= 1205909

= 12059011

and 120590

4

= 1205906

= 12059012

The permutation

(1205900

1205901

1205902

1205903

1205904

1205905

1205900

1205901

1205905

1205904

1205903

1205902

) (24)

determines the isomorphism between monoidsM(1205902

= 1205907

)

andM(1205902

= 1205908

)

106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590

2

= 1205908

means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590

2

= 12059010

implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902

= 1205908

The relation 1205902

= 12059013

follows that each closed sethas to be open so it implies 120590

2

= 1205908

too

107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590

7

= 1205908

and 12059010

= 12059013

areequivalent Each of them gives 120590

7

= 1205908

= 12059010

= 12059013

and1205906

= 1205909

= 12059011

= 12059012

If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590

7

= 1205908

) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements

108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590

7

= 12059010

1205908

= 12059013

1205906

= 12059011

and 120590

9

= 12059012

are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590

7

= 12059010

) is the monoid of allKuratowski operations over119883 and consists of 10 elements

109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590

7

= 12059013

and 1205908

= 12059010

areequivalent Also 120590

6

= 12059012

and 1205909

= 12059011

These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted

1205901

1205902

1205903

1205904

1205905

1205906

1205907

1205908

1205909

1205901

1205900

1205904

1205905

1205902

1205903

1205908

1205909

1205906

1205907

1205902

1205903

1205902

1205903

1205906

1205907

1205906

1205907

1205908

1205909

1205903

1205902

1205906

1205907

1205902

1205903

1205908

1205909

1205906

1205907

1205904

1205905

1205904

1205905

1205908

1205909

1205908

1205909

1205906

1205907

1205905

1205904

1205908

1205909

1205904

1205905

1205906

1205907

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

1205907

1205906

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205909

1205908

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

(25)

If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906

= 12059012

The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590

7

= 1205908

Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590

7

= 1205908

and 1205907

= 12059013

To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that

(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860

119888minus119888Then check that

(i) 1205900

(119860) = 119860 and 1205901

(119860) = 119883 (119860119888minus119888

cup 119861 cup 119862)(ii) 1205902

(119860) = 119880 cup 119861minus

cup119882 and 1205903

(119860) = 119883 119860119888minus119888

(iii) 1205904

(119860) = 119881 119861minus and 120590

5

(119860) = 119860119888minus119888

(iv) 1205906

(119860) = 12059012

(119860) = 119881 and 1205907

(119860) = 12059013

(119860) = 119880(v) 1205908

(119860) = 12059010

(119860) = 119880 cup 119882 and 1205909

(119860) = 12059011

(119860) =

119881 cup 119882Hence we have that 120590

7

= 1205908

Note that if119882 = 0 then 1205907

(119860) =

1205908

(119860) This is the case of subsets of 120573119873

Journal of Mathematics 9

References

[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922

[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977

[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962

[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010

[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008

[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966

[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996

[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956

[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889

[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967

[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968

[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Mathematics

cancellation rules (relations) motivated by some topologicalspaces

2 Cancellation Rules

A map 119891 119875(119883) rarr 119875(119883) is called

(i) increasing if 119860 sube 119861 implies 119891(119860) sube 119891(119861)(ii) decreasing if 119860 sube 119861 implies 119891(119861) sube 119891(119860)(iii) an involution if the composition 119891 ∘ 119891 is the identity(iv) an idempotent if 119891 ∘ 119891 = 119891

Assume that 119860 997891rarr 1205900

(119860) is the identity 119860 997891rarr 1205901

(119860) isa decreasing involution and 119860 997891rarr 120590

2

(119860) is an increasingidempotent map Other operations 120590

119894

let be compositions of1205901

and 1205902

as it was with the kuratowski operationsWe get thefollowing cancellation rules

Lemma 1 If 119861 sube 1205902

(119861) then

1205902

∘ 12059012

= 1205906

1205902

∘ 12059013

= 1205907

(2)

Proof For clarity of this proof use designations 1205901

(119860) = 119860119888

and 1205902

(119860) = 119860minus Thus we shall prove

119860minus119888minus119888minus119888minus

= 119860minus119888minus

119860119888minus119888minus119888minus119888minus

= 119860119888minus119888minus

(3)

We start with 119860minus119888minus119888 sube 119860minus119888minus119888minus substituting 119861 = 119860minus119888minus119888 in 119861 sube119861minus This corresponds to 119860minus119888minus119888minus119888 sube 119860minus119888minus119888119888 = 119860minus119888minus since 120590

1

isa decreasing involution Hence 119860minus119888minus119888minus119888minus sube 119860minus119888minus since 120590

2

anincreasing idempotent

Since 1205901

is decreasing 1205902

is increasing and 119861 sube 119861minus wehave

119861minus119888

sube 119861119888

sube 119861119888minus

(4)

Thus119860minus119888minus119888 sube 119860minus119888119888minus = 119860minus if we put119860minus119888 = 119861 Again using that1205902

is increasing and 1205901

is decreasing we obtain 119860minus119888minus119888minus sube 119860minusand then 119860minus119888 sube 119860minus119888minus119888minus119888 Finally we get 119860minus119888minus sube 119860minus119888minus119888minus119888minus

With 119860119888 in the place of 119860 in the rule 119860minus119888minus = 119860minus119888minus119888minus119888minus weget the second rule

In academic textbooks of general topology for example[2 Problem 171] one can find a hint suggested to prove theabove cancellation rules Students go like this steps 119860 sube 119860

minus

and 119860119888 sube 119860119888minus lead to 119860119888minus119888 sube 119860 the special case 119860minus119888minus119888minus119888 sube

119860minus119888minus (of119860119888minus119888 sube 119860) leads to119860minus119888minus119888minus119888minus sube 119860minus119888minus steps119860minus119888minus119888 sube 119860minus

and 119860minus119888minus119888minus sube 119860minus lead to 119860minus119888minus sube 119860minus119888minus119888minus119888minus at the end use thelast step of the proof of Lemma 1 Note that the above proofsdo not use the axioms of topology as follows

(i) 0 = 0minus(ii) (119860 cup 119861)minus = 119860minus cup 119861minus

In the literature there are articles in which Kuratowski oper-ations are replaced by some other mappings For exampleKoenen [6] considered linear spaces and put 120590

2

(119860) to bethe convex hull of 119860 In fact Shum [7] considered 120590

2

as theclosure due to the algebraic operations Add to this that theseoperations can be applied to so-called Frechet (V) spaceswhich were considered in the book [8 pages 3ndash37]

3 The Monoid M

LetM be the monoid consisting of all Kuratowski operationsthat is there are assumed cancellation rules 120590

6

= 1205902

∘12059012

and1205907

= 1205902

∘ 12059013

Fill in the Cayley table for M where the rowand column marked by the identity are omitted Similarly asin [9] the factor that labels the row comes first and that thefactor that labels the column is second For example 120590

119894

∘ 120590119896

isin the row marked by 120590

119894

and the column marked by 120590119896

1205901

1205902

1205903

1205904

1205905

1205906

1205907

1205908

1205909

12059010

12059011

12059012

12059013

1205901

1205900

1205904

1205905

1205902

1205903

1205908

1205909

1205906

1205907

12059012

12059013

12059010

12059011

1205902

1205903

1205902

1205903

1205906

1205907

1205906

1205907

12059010

12059011

12059010

12059011

1205906

1205907

1205903

1205902

1205906

1205907

1205902

1205903

12059010

12059011

1205906

1205907

1205906

1205907

12059010

12059011

1205904

1205905

1205904

1205905

1205908

1205909

1205908

1205909

12059012

12059013

12059012

12059013

1205908

1205909

1205905

1205904

1205908

1205909

1205904

1205905

12059012

12059013

1205908

1205909

1205908

1205909

12059012

12059013

1205906

1205907

1205906

1205907

12059010

12059011

12059010

12059011

1205906

1205907

1205906

1205907

12059010

12059011

1205907

1205906

12059010

12059011

1205906

1205907

1205906

1205907

12059010

12059011

12059010

12059011

1205906

1205907

1205908

1205909

1205908

1205909

12059012

12059013

12059012

12059013

1205908

1205909

1205908

1205909

12059012

12059013

1205909

1205908

12059012

12059013

1205908

1205909

1205908

1205909

12059012

12059013

12059012

12059013

1205908

1205909

12059010

12059011

12059010

12059011

1205906

1205907

1205906

1205907

12059010

12059011

12059010

12059011

1205906

1205907

12059011

12059010

1205906

1205907

12059010

12059011

12059010

12059011

1205906

1205907

1205906

1205907

12059010

12059011

12059012

12059013

12059012

12059013

1205908

1205909

1205908

1205909

12059012

12059013

12059012

12059013

1205908

1205909

12059013

12059012

1205908

1205909

12059012

12059013

12059012

12059013

1205908

1205909

1205908

1205909

12059012

12059013

(5)

It turns out that the above table allows us to describeall semigroups contained inM using pencil-and-paper tech-niques only The argument will be by a systematic study ofpossibilities Preparing the list of all semigroups consisting ofKuratowski operations we used following principles

(i) minimal collection of generators is written using⟨119860 119861 119885⟩ where letters denote generators

(ii) when a semigroup has a few minimal collections ofgenerators then its name is the first collection in thedictionary order

(iii) all minimal collections of generators are written withthe exception of some containing 120590

0

(iv) we leave to the readers verification that our list is

complete sometimes we add hints

4 Semigroups with 1205901

Observe that each semigroup which contains 1205900

is a monoidSince120590

1

∘ 1205901

= 1205900

a semigroupwhich contains1205901

is amonoidtoo

Theorem 2 There are three monoids containing 1205901

(1) ⟨1205901

⟩ = 1205900

1205901

(2) M = ⟨120590

1

120590119894

⟩ = 1205900

1205901

12059013

where 119894 isin 2 3 4 5(3) letM

1

= ⟨1205901

1205906

⟩ If 119895 isin 6 7 13 then

M1

= ⟨1205901

120590119895

⟩ = 1205900

1205901

cup 1205906

1205907

12059013

(6)

Proof The equality ⟨1205901

⟩ = 1205900

1205901

is obviousSince 120590

2

= 1205903

∘ 1205901

= 1205901

∘ 1205904

= 1205901

∘ 1205905

∘ 1205901

we have⟨1205901

1205902

⟩ = ⟨1205901

1205903

⟩ = ⟨1205901

1205904

⟩ = ⟨1205901

1205905

⟩ = MIf 119895 isin 6 7 13 then any composition 120590

119895

∘ 120590119894

or 120590119896

∘ 120590119895

belongs to M1

and so ⟨1205901

120590119895

⟩ sube M1

We have 1205907

= 1205906

∘ 1205901

Journal of Mathematics 3

1205908

= 1205901

∘ 1205906

1205909

= 1205901

∘ 1205907

12059010

= 1205906

∘ 1205906

12059011

= 1205906

∘ 1205907

12059012

= 1205908

∘ 1205906

and 12059013

= 1205908

∘ 1205907

and so M1

= ⟨1205901

1205906

⟩ =

1205900

1205901

cup 1205906

1205907

12059013

Since 120590

6

= 1205907

∘ 1205901

= 1205901

∘ 1205908

= 1205901

∘ 1205909

∘ 1205901

= 12059010

∘ 1205901

∘ 12059010

=

12059011

∘ 12059011

∘ 1205901

= 1205901

∘ 12059012

∘ 12059012

and 1205906

= 1205901

∘ 12059013

∘ 1205901

∘ 12059013

∘ 1205901

we have ⟨120590

1

120590119895

⟩ = M1

for 119895 isin 6 7 13

Consider the permutation

(1205900

1205901

1205902

1205903

1205904

1205905

1205906

1205907

1205908

1205909

12059010

12059011

12059012

12059013

1205900

1205901

1205905

1205904

1205903

1205902

1205909

1205908

1205907

1205906

12059013

12059012

12059011

12059010

)

(7)

It determines an automorphism A M rarr M

Theorem 3 The identity and A are the only automorphismsofM

Proof Delete rows and columns marked by 1205901

in the Cayleytable for M Then check that the operation 120590

3

is in the rowor the column marked by 120590

3

only Also the operation 1205904

isin the row or the column marked by 120590

4

only Therefore thesemigroup

⟨1205903

1205904

⟩ = 1205902

1205903

12059013

(8)

has a unique minimal set of generators 1205903

1205904

The reader isleft to check this with the Cayley table forM

Suppose G is an automorphism of M By Theorem 2 Gtransforms the set 120590

2

1205903

1205904

1205905

onto itself However 1205902

and1205905

are idempotents but 1205903

and 1205904

are not idempotents Sothere are two possibilities G(120590

3

) = 1205903

and G(1205904

) = 1205904

whichimplies that G is the identity G(120590

3

) = 1205904

and G(1205904

) = 1205903

which implies G = A The reader is left to check this with theCayley table forM We offer hints 120590

2

= 1205903

∘ 1205904

1205905

= 1205904

∘ 1205903

1205906

= 1205903

∘ 1205902

1205907

= 1205903

∘ 1205903

1205908

= 1205904

∘ 1205904

1205909

= 1205904

∘ 1205905

12059010

= 1205906

∘ 1205906

12059011

= 1205906

∘ 1205907

12059012

= 1205908

∘ 1205906

and 12059013

= 1205908

∘ 1205907

to verify the details of this proof

5 The Monoid of All Idempotents

The set 1205900

1205902

1205905

1205907

1205908

12059010

12059013

consists of all squares inMThese squares are idempotents and lie on the main diagonalin the Cayley table forM They constitute a monoid and

1205900

1205902

1205905

1205907

1205908

12059010

12059013

= ⟨1205900

1205902

1205905

⟩ (9)

The permutation

(1205900

1205902

1205905

1205907

1205908

12059010

12059013

1205900

1205902

1205905

1205908

1205907

12059010

12059013

) (10)

determines the bijection I ⟨1205900

1205902

1205905

⟩ rarr ⟨1205900

1205902

1205905

⟩ suchthat

I (120572 ∘ 120573) = I (120573) ∘ I (120572) (11)

for any 120572 120573 isin ⟨1205900

1205902

1205905

⟩ To verify this apply equalities1205902

∘ 1205905

= 1205907

1205905

∘ 1205902

= 1205908

1205902

∘ 1205905

∘ 1205902

= 12059010

and1205905

∘ 1205902

∘ 1205905

= 12059013

Any bijection I 119866 rarr 119867 having propertyI(120572∘120573) = I(120573)∘I(120572) transposesCayley tables for semigroups119866

and119867 Several semigroups contained inM have this propertyWe leave the reader to verify this

We shall classify all semigroups contained in the semi-group ⟨120590

2

1205905

⟩ Every such semigroup can be extended toa monoid by attaching 120590

0

to it This gives a completeclassification of all semigroups in ⟨120590

0

1205902

1205905

⟩The semigroup ⟨120590

2

1205905

⟩ contains six groups with exactlyone element

Semigroups ⟨1205902

12059010

⟩ = 1205902

12059010

and ⟨1205905

12059013

⟩ = 1205905

12059013

aremonoids Both consist of exactly two elements and are notgroups so they are isomorphic

Semigroups ⟨1205907

12059010

⟩ and ⟨1205908

12059013

⟩ are isomorphic in par-ticular A[⟨120590

7

12059010

⟩] = ⟨1205908

12059013

⟩ Also semigroups ⟨1205907

12059013

⟩

and ⟨1205908

12059010

⟩ are isomorphic by A Every of these four semi-groups has exactly two elements None of them is a monoidThey form two types of nonisomorphic semigroups becausethe bijections (120590

7

12059010

) (12059010

1205908

) and (1205907

1205908

) (12059010

12059010

) arenot isomorphisms

Semigroups ⟨1205902

1205907

⟩ = 1205902

1205907

12059010

and ⟨1205905

1205908

⟩ =

1205905

1205908

12059013

are isomorphic Also semigroups ⟨1205902

1205908

⟩ =

1205902

1205908

12059010

and ⟨1205905

1205907

⟩ = 1205905

1205907

12059013

are isomorphic Infact A[⟨120590

2

1205907

⟩] = ⟨1205905

1205908

⟩ and A[⟨1205902

1205908

⟩] = ⟨1205905

1205907

⟩None of these semigroups is a monoid They form two typesof nonisomorphic semigroups Indeed any isomorphismbetween ⟨120590

2

1205907

⟩ and ⟨1205902

1205908

⟩ must be the identity on themonoid ⟨120590

2

12059010

⟩ Therefore would have to be the restrictionof I But I restricted to ⟨120590

7

12059010

⟩ is not an isomorphismSemigroups ⟨120590

2

12059013

⟩ = ⟨1205902

1205907

1205908

⟩ = 1205902

1205907

1205908

12059010

12059013

and ⟨1205905

12059010

⟩ = ⟨1205905

1205907

1205908

⟩ = 1205905

1205907

1205908

12059010

12059013

are notmonoids They are isomorphic by A

The semigroup ⟨1205902

1205905

⟩ contains exactly one semigroupwith four elements ⟨120590

7

1205908

⟩ = ⟨12059010

12059013

⟩ = 1205907

1205908

12059010

12059013

which is not a monoidNote that ⟨120590

2

1205905

⟩ contains twenty different semigroupswith nine non-isomorphism types These are six isomorphicgroups with exactly one element ⟨120590

2

⟩ cong ⟨1205905

⟩ cong ⟨1205907

⟩ cong

⟨1205908

⟩ cong ⟨12059010

⟩ cong ⟨12059013

⟩ two isomorphic monoids with exactlytwo elements ⟨120590

2

12059010

⟩ cong ⟨1205905

12059013

⟩ two pairs of isomorphicsemigroups with exactly two elements ⟨120590

7

12059010

⟩ cong ⟨1205908

12059013

⟩

and ⟨1205907

12059013

⟩ cong ⟨1205908

12059010

⟩ two pairs of isomorphic semigroupswith exactly three elements ⟨120590

2

1205907

⟩ cong ⟨1205905

1205908

⟩ and ⟨1205902

1205908

⟩ cong

⟨1205905

1205907

⟩ a semigroup with exactly four elements ⟨1205907

1205908

⟩and two isomorphic semigroups with exactly five elements⟨1205902

12059013

⟩ cong ⟨1205905

12059010

⟩ and also ⟨1205902

1205905

⟩ Thus ⟨1205902

1205905

⟩ containstwenty different semigroups with nine isomorphism typesBut ⟨120590

0

1205902

1205905

⟩ contains forty-one different semigroups withseventeen isomorphism types Indeed adding 120590

0

to semi-groups contained in ⟨120590

2

1205905

⟩ which are not monoids weget twenty monoids with eight non-isomorphism typesAdding 120590

0

to a group contained in ⟨1205902

1205905

⟩ we get a monoidisomorphic to ⟨120590

2

12059010

⟩

6 The Semigroup Consisting of 12059061205907 120590

13

Using the Cayley table forM check thatA [1205906

1205907

12059013

] = 1205906

1205907

12059013

(12)Similarly check that the semigroup ⟨120590

6

1205909

⟩ = 1205906

1205907

12059013

can be represented as ⟨1205906

12059013

⟩ ⟨1205907

12059012

⟩ ⟨1205908

12059011

⟩

4 Journal of Mathematics

⟨1205909

12059010

⟩ or ⟨12059011

12059012

⟩ Also ⟨1205906

1205909

⟩ = ⟨1205906

1205907

1205908

⟩ =

⟨1205907

1205908

1205909

⟩ = ⟨12059010

12059011

12059013

⟩ = ⟨12059010

12059012

12059013

⟩ These rep-resentations exhaust all minimal collections of the Kura-towski operations which generate ⟨120590

6

1205909

⟩ Other semigroupsincluded in ⟨120590

6

1205909

⟩ have one or two minimal collectionof generators One generator has groups ⟨120590

6

⟩ = 1205906

12059010

⟨1205909

⟩ = 1205909

12059013

⟨12059011

⟩ = 1205907

12059011

and ⟨12059012

⟩ = 1205908

12059012

Each of them has exactly two elements so they are isomor-phic Semigroups ⟨120590

7

12059010

⟩ ⟨1205907

12059013

⟩ ⟨1205908

12059010

⟩ ⟨1205908

12059013

⟩ and⟨1205907

1205908

⟩ are discussed in the previous section Contained in⟨1205906

1205909

⟩ and not previously discussed semigroups are ⟨1205906

1205907

⟩⟨1205906

1205908

⟩⟨1205907

1205909

⟩ and ⟨1205908

1205909

⟩ We leave the reader to verifythat the following are all possible pairs of Kuratowski oper-ations which constitute a minimal collection of generatorsfor semigroups contained in ⟨120590

6

1205909

⟩ but different from thewhole One has

(i) ⟨1205906

1205907

⟩ = ⟨1205906

12059011

⟩ = ⟨12059010

12059011

⟩ = 1205906

1205907

12059010

12059011

(ii) ⟨120590

6

1205908

⟩ = ⟨1205906

12059012

⟩ = ⟨12059010

12059012

⟩ = 1205906

1205908

12059010

12059012

(iii) ⟨120590

7

1205909

⟩ = ⟨1205909

12059011

⟩ = ⟨12059011

12059013

⟩ = 1205907

1205909

12059011

12059013

(iv) ⟨120590

8

1205909

⟩ = ⟨1205909

12059012

⟩ = ⟨12059012

12059013

⟩ = 1205908

1205909

12059012

12059013

Proposition 4 Semigroups ⟨1205906

1205907

⟩ and ⟨1205908

1205909

⟩ are isomor-phic and also semigroups ⟨120590

6

1205908

⟩ and ⟨1205907

1205909

⟩ are isomorphicbut semigroups ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ are not isomorphic

Proof Isomorphisms are defined by A Suppose 119869 ⟨1205906

1205907

⟩ rarr ⟨1205906

1205908

⟩ is an isomorphism Thus 119869[⟨1205907

12059010

⟩] =

⟨1205908

12059010

⟩ Given 119869(1205907

) = 1205908

we get 12059010

= 119869(12059010

) = 119869(1205907

∘

12059010

) = 1205908

∘ 12059010

= 1205908

But 119869(1205907

) = 12059010

implies 12059010

= 119869(1205907

) =

119869(12059010

∘ 1205907

) = 1205908

∘ 12059010

= 1205908

Both possibilities lead to acontradiction

So ⟨1205906

1205909

⟩ contains eighteen different semigroups witheight non-isomorphism types Indeed these are four iso-morphic groups with exactly one element ⟨120590

7

⟩ cong ⟨1205908

⟩ cong

⟨12059010

⟩ cong ⟨12059013

⟩ four isomorphic groups with exactly twoelements ⟨120590

6

⟩ cong ⟨1205909

⟩ cong ⟨12059011

⟩ cong ⟨12059012

⟩ two pairs ofisomorphic semigroups with exactly two elements ⟨120590

7

12059010

⟩ cong

⟨1205908

12059013

⟩ and ⟨1205907

12059013

⟩ cong ⟨1205908

12059010

⟩ and five semigroupswith exactly four elements ⟨120590

6

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong

⟨1205907

1205909

⟩ ⟨1205907

1205908

⟩ and also ⟨1205906

1205909

⟩

7 Remaining Semigroups in ⟨1205903 1205904⟩

We have yet to discuss semigroups included in ⟨1205903

1205904

⟩ notincluded in ⟨120590

2

1205905

⟩ and containing at least one of Kuratowskioperation 120590

2

1205903

1205904

or 1205905

It will be discussed up to theisomorphism A Obviously ⟨120590

2

⟩ = 1205902

and ⟨1205905

⟩ = 1205905

aregroups

71 Extensions of ⟨1205902⟩ and ⟨1205905⟩ with Elements of ⟨12059061205909⟩Monoids ⟨120590

2

1205906

⟩ = 1205902

1205906

12059010

and ⟨1205902

12059010

⟩ = 1205902

12059010

havedifferent numbers of elements Also ⟨120590

2

1205906

⟩ is isomorphic to⟨1205900

1205906

⟩ Nonisomorphic semigroups ⟨1205902

1205907

⟩ and ⟨1205902

1205908

⟩ arediscussed above The following three semigroups

(i) ⟨1205902

12059011

⟩ = ⟨1205902

1205906

1205907

⟩ = 1205902

1205906

1205907

12059010

12059011

(ii) ⟨1205902

12059012

⟩ = ⟨1205902

1205906

1205908

⟩ = 1205902

1205906

1205908

12059010

12059012

(iii) ⟨120590

2

12059013

⟩ = ⟨1205902

1205907

1205908

⟩ = 1205902

1205907

1205908

12059010

12059013

are not monoids They are not isomorphic Indeed anyisomorphism between these semigroups would lead an iso-morphism between ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ or ⟨1205907

1205908

⟩ This isimpossible by Proposition 4 and because ⟨120590

7

1205908

⟩ consists ofidempotents but 120590

6

is not an idempotent The nine-elementsemigroup on the set 120590

2

1205906

1205907

12059013

is represented as⟨1205902

1205909

⟩ So we have added four new semigroups which arenot isomorphic with the semigroups previously discussedThese are ⟨120590

2

12059011

⟩ ⟨1205902

12059012

⟩ ⟨1205902

12059013

⟩ and ⟨1205902

1205909

⟩Using A we have described eight semigroupsmdasheach

one isomorphic to a semigroup previously discussedmdashwhichcontains 120590

5

and elements (at least one) of ⟨1205906

1205909

⟩ Collec-tions of generators ⟨120590

2

1205906

12059013

⟩ ⟨1205902

1205907

12059012

⟩ ⟨1205902

1205908

12059011

⟩⟨1205902

12059011

12059012

⟩ ⟨1205902

12059011

12059013

⟩ and ⟨1205902

12059012

12059013

⟩ are minimalin ⟨120590

2

1205909

⟩ Also collections of generators ⟨1205905

1205907

12059012

⟩⟨1205905

1205908

12059011

⟩ ⟨1205905

1205909

12059010

⟩ ⟨1205905

12059010

12059011

⟩ ⟨1205905

12059010

12059012

⟩ and⟨1205905

12059011

12059012

⟩ are minimal in ⟨1205905

1205906

⟩

72 Extensions of ⟨1205903⟩ and ⟨1205904⟩ by Elements from the Semi-group ⟨12059061205909⟩ Semigroups ⟨120590

3

⟩ = 1205903

1205907

12059011

and ⟨1205904

⟩ =

1205904

1205908

12059012

are isomorphic by A They are not monoids Thesemigroup ⟨120590

3

⟩ can be extended using elements of ⟨1205906

1205909

⟩in three following ways

(i) ⟨1205903

1205906

⟩ = ⟨1205903

12059010

⟩ = 1205903

1205906

1205907

12059010

12059011

(ii) ⟨120590

3

1205908

⟩ = ⟨1205903

12059012

⟩ = 1205903

1205906

1205907

12059013

(iii) ⟨120590

3

1205909

⟩ = ⟨1205903

12059013

⟩ = 1205903

1205907

1205909

12059011

12059013

Semigroups ⟨1205903

1205906

⟩ and ⟨1205903

1205909

⟩ are not isomorphic Indeedsuppose 119869 ⟨120590

3

1205906

⟩ rarr ⟨1205903

1205909

⟩ is an isomorphismThus 119869 isthe identity on ⟨120590

3

⟩ 119869(1205906

) = 1205909

and 119869(12059010

) = 12059013

This givesa contradiction since 120590

3

∘ 1205906

= 12059010

and 1205903

∘ 1205909

= 1205907

Neither ⟨120590

3

1205906

⟩ nor ⟨1205903

1205909

⟩ has a minimal collection ofgenerators with three elements so they give new isomor-phism types Also ⟨120590

2

1205909

⟩ is not isomorphic to ⟨1205903

1205908

⟩since ⟨120590

2

1205909

⟩ has a unique pair of generators and ⟨1205903

1205908

⟩ =

⟨1205903

12059012

⟩Using A we getmdashisomorphic to previously discussed

onesmdashsemigroups ⟨1205904

1205909

⟩ = ⟨1205904

12059013

⟩ = 1205904

1205908

1205909

12059012

12059013

⟨1205904

1205906

⟩ = ⟨1205904

12059010

⟩ = 1205904

1205906

1205908

12059010

12059012

and ⟨1205904

1205907

⟩ =

⟨1205904

12059011

⟩ = 1205904

1205906

1205907

12059013

There exist minimal collec-tions of generators such as follows

⟨1205903

1205908

⟩ = ⟨1205903

1205906

1205909

⟩ = ⟨1205903

1205906

12059013

⟩

= ⟨1205903

1205909

12059010

⟩ = ⟨1205903

12059010

12059013

⟩

⟨1205904

1205907

⟩ = ⟨1205904

1205906

1205909

⟩ = ⟨1205904

1205909

12059010

⟩

= ⟨1205904

1205906

12059013

⟩ = ⟨1205904

12059010

12059013

⟩

(13)

73 More Generators from the Set 1205902120590312059041205905 Nowwe check that ⟨120590

2

1205903

⟩ = 1205902

1205903

1205906

1205907

12059010

12059011

and⟨1205904

1205905

⟩ = 1205904

1205905

1205908

1205909

12059012

12059013

= A[⟨1205902

1205903

⟩] andalso ⟨120590

2

1205904

⟩ = 1205902

1205904

1205906

1205908

12059010

12059012

and ⟨1205903

1205905

⟩ =

1205903

1205905

1205907

1205909

12059011

12059013

= A[⟨1205902

1205904

⟩] are two pairs of iso-morphic semigroupswhich give two new isomorphism types

Journal of Mathematics 5

Each of these semigroups has six element so in M thereare five six-element semigroups of three isomorphism typessince ⟨120590

2

1205905

⟩ has 6 elements which are idempotentsIn ⟨1205903

1205904

⟩ there are seven semigroups which have threegenerators and have not two generators These are

(1) ⟨1205902

1205903

1205908

⟩ = ⟨1205902

1205903

1205909

⟩ = ⟨1205902

1205903

12059012

⟩ = ⟨1205902

1205903

12059013

⟩

(2) ⟨1205904

1205905

1205906

⟩ = ⟨1205904

1205905

1205907

⟩ = ⟨1205904

1205905

12059010

⟩ = ⟨1205904

1205905

12059011

⟩

(3) ⟨1205902

1205904

1205907

⟩ = ⟨1205902

1205904

1205909

⟩ = ⟨1205902

1205904

12059011

⟩ = ⟨1205902

1205904

12059013

⟩

(4) ⟨1205903

1205905

1205906

⟩ = ⟨1205903

1205905

1205908

⟩ = ⟨1205903

1205905

12059010

⟩ = ⟨1205903

1205905

12059012

⟩

(5) ⟨1205902

1205905

1205906

⟩ = ⟨1205902

1205905

1205909

⟩ = ⟨1205902

1205905

12059011

⟩ = ⟨1205902

1205905

12059012

⟩

(6) ⟨1205902

1205903

1205905

⟩

(7) ⟨1205902

1205904

1205905

⟩

8 Semigroups which Are Contained in ⟨12059031205904⟩

Groups ⟨1205902

⟩ cong ⟨1205905

⟩ cong ⟨1205907

⟩ cong ⟨1205908

⟩ cong ⟨12059010

⟩ cong ⟨12059013

⟩ have oneelement and are isomorphic

Groups ⟨1205906

⟩ cong ⟨1205909

⟩ cong ⟨12059011

⟩ cong ⟨12059012

⟩ monoids ⟨1205902

12059010

⟩ cong

⟨1205905

12059013

⟩ and also semigroups ⟨1205907

12059010

⟩ cong ⟨1205908

12059013

⟩ and⟨1205907

12059013

⟩ cong ⟨1205908

12059010

⟩ have two elements and Cayley tables asfollows

119860 119861

119860 119861 119860

119861 119860 119861

119860 119861

119860 119860 119861

119861 119861 119860

119860 119861

119860 119860 119861

119861 119860 119861

119860 119861

119860 119860 119860

119861 119861 119861

(14)

Monoids ⟨1205902

1205906

⟩ cong ⟨1205905

1205909

⟩ and also semigroups⟨1205902

1205907

⟩ cong ⟨1205905

1205908

⟩ ⟨1205902

1205908

⟩ cong ⟨1205905

1205907

⟩ and ⟨1205903

⟩ cong ⟨1205904

⟩ havethree elements and Cayley tables as follows

119860 119861 119862

119860 119860 119861 119862

119861 119861 119862 119861

119862 119862 119861 119862

119860 119861 119862

119860 119860 119861 119862

119861 119862 119861 119862

119862 119862 119861 119862

119860 119861 119862

119860 119860 119862 119862

119861 119861 119861 119861

119862 119862 119862 119862

119860 119861 119862

119860 119861 119862 119861

119861 119862 119861 119862

119862 119861 119862 119861

(15)

Semigroups ⟨1205906

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong ⟨1205907

1205909

⟩ and⟨1205907

1205908

⟩ have four elements and Cayley tables as follows

119860 119861 119862 119863

119860 119862 119863 119860 119861

119861 119860 119861 119862 119863

119862 119860 119861 119862 119863

119863 119862 119863 119860 119861

119860 119861 119862 119863

119860 119862 119860 119860 119862

119861 119863 119861 119861 119863

119862 119860 119862 119862 119860

119863 119861 119863 119863 119861

119860 119861 119862 119863

119860 119860 119862 119862 119860

119861 119863 119861 119861 119863

119862 119860 119862 119862 119860

119863 119863 119861 119861 119863

(16)

Semigroups ⟨1205902

12059011

⟩ cong ⟨1205905

12059012

⟩ ⟨1205902

12059012

⟩ cong ⟨1205905

12059011

⟩⟨1205902

12059013

⟩ cong ⟨1205905

12059010

⟩ ⟨1205903

1205906

⟩ cong ⟨1205904

1205909

⟩ and ⟨1205903

1205909

⟩ cong

⟨1205904

1205906

⟩ have five elements Semigroups ⟨1205902

1205903

⟩ cong ⟨1205904

1205905

⟩⟨1205902

1205904

⟩ cong ⟨1205903

1205905

⟩ and ⟨1205902

1205905

⟩ have six elements Construc-tion of Cayley tables for these semigroups as well as othersemigroups we leave it to the readers One can prepare suchtables removing from the Cayley table for M some columnsand rows Then change Kuratowski operations onto lettersof the alphabet For example ⟨120590

0

1205902

1205905

⟩ has the followingCayley table

0 119860 119861 119862 119863 119864 119865

0 0 119860 119861 119862 119863 119864 119865

119860 119860 119860 119862 119862 119864 119864 119862

119861 119861 119863 119861 119865 119863 119863 119865

119862 119862 119864 119862 119862 119864 119864 119862

119863 119863 119863 119865 119865 119863 119863 119865

119864 119864 119864 119862 119862 119864 119864 119862

119865 119865 119863 119865 119865 119863 119863 119865

(17)

Preparing theCayley table for ⟨1205900

1205902

1205905

⟩weput1205900

= 0 1205902

=

119860 1205905

= 119861 1205907

= 119862 1205908

= 119863 12059010

= 119864 and 12059013

= 119865 This tableimmediately shows that semigroups ⟨120590

2

1205905

⟩ and ⟨1205900

1205902

1205905

⟩

have exactly two automorphisms These are identities andrestrictions of A

Semigroup ⟨1205906

1205909

⟩ has eight elements Semigroups⟨1205902

1205909

⟩ cong ⟨1205905

1205906

⟩ and ⟨1205903

1205908

⟩ cong ⟨1205904

1205907

⟩ have nine ele-ments Semigroups ⟨120590

2

1205903

1205908

⟩ cong ⟨1205904

1205905

1205907

⟩ ⟨1205902

1205904

1205907

⟩ cong

⟨1205903

1205905

1205908

⟩ and ⟨1205902

1205905

1205906

⟩ have ten elements Semigroups⟨1205902

1205903

1205905

⟩ cong ⟨1205902

1205904

1205905

⟩ have eleven elements In the endthe semigroup ⟨120590

3

1205904

⟩ has twelve elementsThus the semigroup ⟨120590

3

1205904

⟩ includes fifty-seven semi-groups amongwhich there are ten groups fourteenmonoidsand also forty-three semigroups which are not monoidsThese semigroups consist of twenty-eight types of noniso-morphic semigroups two non-isomorphism types of groupstwo non-isomorphism types of monoids which are notgroups and twenty four non-isomorphism types of semi-groups which are not monoids

9 Viewing Semigroups Contained in M

91 Descriptive Data on Semigroups which Are Contained inM There are one hundred eighteen that is 118 = 2 sdot 57 +

6 Journal of Mathematics

4 semigroups which are contained in M These are fifty-seven semigroups contained in ⟨120590

3

1205904

⟩ fifty-seven monoidsformed by adding 120590

0

to a semigroup contained in ⟨1205903

1205904

⟩groups ⟨120590

0

⟩ ⟨1205901

⟩ and monoids ⟨1205901

1205906

⟩ = M1

and ⟨1205901

1205902

⟩ =

MThere are fifty-six types of nonisomorphic semigroups

in M These are twenty-eight non-isomorphism types ofsemigroups contained in ⟨120590

3

1205904

⟩ twenty-six types of non-isomorphic monoids formed by adding 120590

0

to a semigroupcontained in ⟨120590

3

1205904

⟩ and alsoM1

andM Indeed adding 1205900

to a semigroup which is not a monoid we obtain a monoidIn this way we get twenty-four types of nonisomorphicmonoids Adding 120590

0

to a monoid contained in ⟨1205903

1205904

⟩ weget two new non-isomorphism types ofmonoids But adding1205900

to a group contained in ⟨1205903

1205904

⟩ we get no new type ofmonoid since we get a monoid isomorphic to ⟨120590

2

12059010

⟩ or⟨1205902

1205906

⟩ The other two types areM1

andM

92 Semigroups which Are Not Monoids Below we havereproduced using the smallest number of generators andthe dictionary order a list of all 43 semigroups which areincluded in theM as follows

(1) ⟨1205902

1205903

⟩ = 1205902

1205903

1205906

1205907

12059010

12059011

(2) ⟨120590

2

1205903

1205905

⟩ = 1205902

1205903

1205905

1205906

12059013

(3) ⟨120590

2

1205903

1205908

⟩ = ⟨1205902

1205903

1205909

⟩ = ⟨1205902

1205903

12059012

⟩ = ⟨1205902

1205903

12059013

⟩ = 1205902

1205903

1205906

1205907

12059013

(4) ⟨120590

2

1205904

⟩ = 1205902

1205904

1205906

1205908

12059010

12059012

(5) ⟨120590

2

1205904

1205905

⟩ = 1205902

1205904

1205905

12059013

(6) ⟨120590

2

1205904

1205907

⟩ = ⟨1205902

1205904

1205909

⟩ = ⟨1205902

1205904

12059011

⟩ = ⟨1205902

1205904

12059013

⟩ = 1205902

1205904

1205906

1205907

12059013

(7) ⟨120590

2

1205905

⟩ = 1205902

1205905

1205907

1205908

12059010

12059013

(8) ⟨120590

2

1205905

1205906

⟩ = ⟨1205902

1205905

1205909

⟩ = ⟨1205902

1205905

12059011

⟩ = ⟨1205902

1205905

12059012

⟩ = 1205902

1205905

1205906

12059013

(9) ⟨120590

2

1205907

⟩ = 1205902

1205907

12059010

(10) ⟨120590

2

1205908

⟩ = 1205902

1205908

12059010

(11) ⟨120590

2

1205909

⟩ = ⟨1205902

1205906

12059013

⟩ = ⟨1205902

1205907

12059012

⟩ = ⟨1205902

1205908

12059011

⟩ = ⟨1205902

12059011

12059012

⟩ = ⟨1205902

12059011

12059013

⟩ = ⟨1205902

12059012

12059013

⟩ = 1205902

1205906

1205907

12059013

(12) ⟨120590

2

12059011

⟩ = ⟨1205902

1205906

1205907

⟩ = 1205902

1205906

1205907

12059010

12059011

(13) ⟨120590

2

12059012

⟩ = ⟨1205902

1205906

1205908

⟩ = 1205902

1205906

1205908

12059010

12059012

(14) ⟨120590

2

12059013

⟩ = ⟨1205902

1205907

1205908

⟩ = 1205902

1205907

1205908

12059010

12059013

(15) ⟨120590

3

⟩ = 1205903

1205907

12059011

(16) ⟨120590

3

1205904

⟩ = 1205902

1205903

12059013

(17) ⟨120590

3

1205905

⟩ = 1205903

1205905

1205907

1205909

12059011

12059013

(18) ⟨120590

3

1205905

1205906

⟩ = ⟨1205903

1205905

1205908

⟩ = ⟨1205903

1205905

12059010

⟩ = ⟨1205903

1205905

12059012

⟩ = 1205903

1205905

1205906

12059013

(19) ⟨120590

3

1205906

⟩ = ⟨1205903

12059010

⟩ = 1205903

1205906

1205907

12059010

12059011

(20) ⟨120590

3

1205908

⟩ = ⟨1205903

12059012

⟩ = 1205903

1205906

1205907

12059013

(21) ⟨120590

3

1205909

⟩ = ⟨1205903

12059013

⟩ = 1205903

1205907

1205909

12059011

12059013

(22) ⟨120590

4

⟩ = 1205904

1205908

12059012

(23) ⟨1205904

1205905

⟩ = 1205904

1205905

1205908

1205909

12059012

12059013

(24) ⟨120590

4

1205905

1205906

⟩ = ⟨1205904

1205905

1205907

⟩ = ⟨1205904

1205905

12059010

⟩ = ⟨1205904

1205905

12059011

⟩ = 1205904

1205905

12059013

(25) ⟨120590

4

1205906

⟩ = ⟨1205904

12059010

⟩ = 1205904

1205906

1205908

12059010

12059012

(26) ⟨120590

4

1205907

⟩ = ⟨1205904

12059011

⟩ = 1205904

1205906

1205907

12059013

(27) ⟨120590

4

1205909

⟩ = ⟨1205904

12059013

⟩ = 1205904

1205908

1205909

12059012

12059013

(28) ⟨120590

5

1205906

⟩ = ⟨1205905

1205907

12059012

⟩ = ⟨1205905

1205908

12059011

⟩ = ⟨1205905

1205909

12059010

⟩ = ⟨1205905

12059010

12059011

⟩ = ⟨1205905

12059010

12059012

⟩ = ⟨1205905

12059011

12059012

⟩ = 1205905

1205906

12059013

(29) ⟨120590

5

1205907

⟩ = 1205905

1205907

12059013

(30) ⟨120590

5

1205908

⟩ = 1205905

1205908

12059013

(31) ⟨120590

5

12059010

⟩ = ⟨1205905

1205907

1205908

⟩ = 1205905

1205907

1205908

12059010

12059013

(32) ⟨120590

5

12059011

⟩ = ⟨1205905

1205907

1205909

⟩ = 1205905

1205907

1205909

12059011

12059013

(33) ⟨120590

5

12059012

⟩ = ⟨1205905

1205908

1205909

⟩ = 1205905

1205908

1205909

12059012

12059013

(34) ⟨120590

6

1205907

⟩ = ⟨1205906

12059011

⟩ = ⟨12059010

12059011

⟩ = 1205906

1205907

12059010

12059011

(35) ⟨120590

6

1205908

⟩ = ⟨1205906

12059012

⟩ = ⟨12059010

12059012

⟩ = 1205906

1205908

12059010

12059012

(36) ⟨120590

6

1205909

⟩ = ⟨1205906

12059013

⟩ = ⟨1205907

12059012

⟩ = ⟨1205908

12059011

⟩ = ⟨1205909

12059010

⟩ = ⟨12059011

12059012

⟩ = ⟨1205906

1205907

1205908

⟩ = ⟨1205907

1205908

1205909

⟩ =⟨12059010

12059011

12059013

⟩ = ⟨12059010

12059012

12059013

⟩ = 1205906

1205907

12059013

(37) ⟨120590

7

1205908

⟩ = 1205907

1205908

12059010

12059013

(38) ⟨120590

7

1205909

⟩ = 1205907

1205909

12059011

12059013

(39) ⟨120590

7

12059010

⟩ = 1205907

12059010

(40) ⟨120590

7

12059013

⟩ = 1205907

12059013

(41) ⟨120590

8

1205909

⟩ = 1205908

1205909

12059012

12059013

(42) ⟨120590

8

12059010

⟩ = 1205908

12059010

(43) ⟨120590

8

12059013

⟩ = 1205908

12059013

93 Isomorphism Types of Semigroups Contained in M Sys-tematize the list of all isomorphism types of semigroupscontained in themonoidM Isomorphisms which are restric-tions of the isomorphism A will be regarded as self-evidentand therefore they will not be commented on

(i) The monoid M contains 12 groups with 2 isomor-phism types These are 7 one-element groups and 5two-element groups

(ii) ThemonoidM contains 8 two-element monoids with2 isomorphism types These are 120590

0

added to 6 one-element groups and ⟨120590

2

12059010

⟩ cong ⟨1205905

12059013

⟩ Also itcontains 4 two-element semigroups not monoidswith 2 isomorphism types These are ⟨120590

7

12059010

⟩ cong

⟨1205908

12059013

⟩ and ⟨1205908

12059010

⟩ cong ⟨1205907

12059013

⟩(iii) The monoid M contains 12 three-element monoids

with 5 isomorphism types These are 1205900

addedto 4 two-element groups and also ⟨120590

0

1205902

12059010

⟩ cong

⟨1205900

1205905

12059013

⟩ ⟨1205900

1205907

12059010

⟩ cong ⟨1205900

1205908

12059013

⟩⟨1205900

1205908

12059010

⟩ cong ⟨1205900

1205907

12059013

⟩ and ⟨1205902

1205906

⟩ cong ⟨1205905

1205909

⟩(iv) The monoidM contains 6 three-element semigroups

not monoids with 3 isomorphism types These are⟨1205902

1205907

⟩ cong ⟨1205905

1205908

⟩ ⟨1205902

1205908

⟩ cong ⟨1205905

1205907

⟩ and ⟨1205903

⟩ cong

⟨1205904

⟩

Journal of Mathematics 7

(v) The monoid M contains 8 four-element monoidseach contains 120590

0

with 4 isomorphism types Theseare semigroups from two preceding items that can besubstantially extended by 120590

0

(vi) The monoid M contains 5 four-element semigroups

not monoids with 3 isomorphism types These are⟨1205906

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong ⟨1205907

1205909

⟩ and ⟨1205907

1205908

⟩These semigroups extended by 120590

0

yield 5 monoids allwhich consist of five elements with 3 isomorphismtypes

(vii) ThemonoidM contains 10 five-element semigroupsmdashnot monoids with 5 isomorphism types These are⟨1205902

12059011

⟩ cong ⟨1205905

12059012

⟩ ⟨1205902

12059012

⟩ cong ⟨1205905

12059011

⟩ ⟨1205902

12059013

⟩ cong

⟨1205905

12059010

⟩ ⟨1205903

1205906

⟩ cong ⟨1205904

1205909

⟩ and ⟨1205903

1205909

⟩ cong ⟨1205904

1205906

⟩These semigroups extended by120590

0

yield 10monoids allof which consist of six elements with 5 isomorphismtypes We get 10 new isomorphism types since thesemigroups are distinguished by semigroups ⟨120590

3

⟩

and ⟨1205904

⟩ and by nonisomorphic semigroups ⟨1205906

1205907

⟩⟨1205906

1205908

⟩ and ⟨1205907

1205908

⟩(viii) The monoid M contains 5 six-element semigroups

not monoids with 3 isomorphism types These are⟨1205902

1205903

⟩ cong ⟨1205904

1205905

⟩ ⟨1205902

1205904

⟩ cong ⟨1205903

1205905

⟩ and ⟨1205902

1205905

⟩These semigroups extended by 120590

0

yield 5 monoidsall of which consist of seven elements with 3 isomor-phism types We get 6 new isomorphism types sincethe semigroups are distinguished by non isomorphicsemigroups ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ and ⟨1205907

1205908

⟩(ix) ThemonoidM contains no seven-element semigroup

not a monoid no eight-element monoid and theonly semigroup ⟨120590

6

1205909

⟩ with exactly eight elementsand the only monoid ⟨120590

0

1205906

1205909

⟩ with exactly nineelements

(x) The monoid M contains 4 nine-element semigroupsnot monoids with 2 isomorphism types These are⟨1205902

1205909

⟩ cong ⟨1205905

1205906

⟩ and ⟨1205903

1205908

⟩ cong ⟨1205904

1205907

⟩ Thesemigroup ⟨120590

2

1205909

⟩ does not contain a semigroupisomorphic to ⟨120590

3

⟩ hence it is not isomorphic to⟨1205903

1205908

⟩ These semigroups extended by 1205900

yield 4monoids all of which consist of ten elements with 2isomorphism types

(xi) The monoid M contains 6 ten-element semigroupsnot monoids with 4 isomorphism typesThese are ⟨120590

1

1205906

⟩ ⟨1205902

1205903

1205908

⟩ cong ⟨1205904

1205905

1205906

⟩⟨1205902

1205904

1205907

⟩ cong ⟨1205903

1205905

1205906

⟩ and ⟨1205902

1205905

1205906

⟩ Thesesemigroups (except ⟨120590

1

1205906

⟩) extended by 1205900

yield 5monoids all of which consist of ten elements with3 isomorphism types We get 6 new isomorphismtypes since the semigroups are distinguished bynot isomorphic semigroups ⟨120590

2

1205903

⟩ ⟨1205902

1205904

⟩ and⟨1205902

1205905

⟩(xii) The monoid M contains 2 isomorphic semigroups

not monoids which consist of 11 elements that is⟨1205902

1205903

1205905

⟩ cong ⟨1205902

1205904

1205905

⟩These semigroups extendedby 1205900

yield 2 isomorphic monoids which consistof 12 elements The monoid M contains no largersemigroup with the exception of itself

10 Cancellation Rules Motivated by SomeTopological Properties

101 Some Consequences of the Axiom 0 = 0minus So far we usedonly the following relations (above named cancellation rules)1205902

∘ 1205902

= 1205902

1205901

∘ 1205901

= 1205900

1205902

∘ 12059012

= 1205906

and 1205902

∘ 12059013

= 1205907

When one assumes119883 = 0 = 0

minus then

119883 = 1205900

(119883) = 1205902

(119883) = 1205905

(119883) = 1205907

(119883)

= 1205908

(119883) = 12059010

(119883) = 12059013

(119883)

0 = 1205901

(119883) = 1205903

(119883) = 1205904

(119883) = 1205906

(119883)

= 1205909

(119883) = 12059011

(119883) = 12059012

(119883)

(18)

Using the substitution 119860 997891rarr 119860119888 one obtains equiv-

alent relations between operations from the set 1205901

1205903

1205904

1205906

1205909

12059011

12059012

and conversely Therefore cancellationrules are topologically reasonable only between the opera-tions from the monoid as follows

⟨1205900

1205902

1205905

⟩ = 1205900

1205902

1205905

1205907

1205908

12059010

12059013

(19)

Chapman see [3] considered properties of subsets withrespect to such relations Below we are going to identifyrelations that are determined by some topological spacescompare [10 11]

102 The Relation 1205900 = 1205902 If a topological space 119883 isdiscrete then there exist two Kuratowski operation onlyThese are 120590

0

and 1205901

So themonoid of Kuratowski operationsreduced to the group ⟨120590

1

⟩The relation 120590

0

= 1205902

is equivalent to any relation 1205900

=

120590119894

where 119894 isin 5 7 8 10 13 Any such relation implies thatevery subset of 119883 has to be closed and open that is 119883 hasto be discrete However one can check these using (only) thefacts that 120590

1

is an involution and 1205902

is an idempotent and thecancellation rules that is the Cayley table forM So 120590

0

= 1205902

follows

1205900

= 1205902

= 1205905

= 1205907

= 1205908

= 12059010

= 12059013

(20)

103 The Relation 1205902 = 1205905 Topologically 1205902 = 1205905

meansthat 119883 must be discrete This is so because 119860119888minus119888 sube 119860 sube 119860

minus

for any 119860 sube 119883

104 The Relation 1205902 = 1205907 Topologically the relation 1205902 =1205907

implies 1205900

= 1205902

But it requires the use of topology axioms0 = 0minus and 119862minus cup 119861minus = (119862 cup 119861)minus for each 119862 and 119861

Lemma 5 For any topological space 1205902

= 1205907

implies 1205902

= 1205908

Proof If 1205902

= 1205907

then 119860 = 0 rArr 119860119888minus119888

= 0 for any 119860 sube 119883Indeed if 119860119888minus119888 = 0 then 120590

7

(119860) = 0minus

= 0 Since 119860 = 0 then1205902

(119860) = 0 Hence 1205902

(119860) = 1205907

(119860) a contradictionThe axiom 119862

minus

cup 119861minus

= (119862 cup 119861)minus implies that always

(119860minus

cap 119860minus119888minus

)119888minus119888

= 0 (21)

Thus the additional assumption 1205902

= 1205907

follows that always119860minus

cap 119860minus119888minus

= 0 Therefore always 119860minus119888minus = 119860minus119888 but this means

8 Journal of Mathematics

that any closed set has to be open in other words 1205902

= 1205908

Proposition 6 For any topological space 1205902

= 1205907

implies 1205900

=

1205902

Proof But the relation 1205902

= 1205907

is equivalent with 1205905

= 1205908

ByLemma 5 we get 120590

2

= 1205905

Finally 1205900

= 1205902

The relation 1205902

= 1205907

has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put

1205902

(0) = 119886 = 1205902

(119886) 119883 = 1205902

(119883) = 1205902

(119887) (22)

Then check that 1205902

= 1205907

and

1205908

(0) = 1205905

(119886) = 1205904

(119887) = 1205901

(119883) = 0 (23)

in other words 1205902

= 1205907

and 1205902

= 1205908

However 1205902

= 1205907

isequivalent to 120590

5

= 1205908

This relation implies 120590

2

= 1205907

= 12059010

1205905

= 1205908

= 12059013

1205903

= 1205906

= 12059011

and 1205904

= 1205909

= 12059012

For this interpretationthe monoidM(120590

2

= 1205907

) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590

2

= 1205907

) thereare relations covered by the following proposition only

Proposition 7 For any monoid with the Cayley table as forM the relation 120590

2

= 1205907

implies

(i) 1205902

= 1205907

= 12059010

= 1205907

∘ 1205902

(ii) 1205905

= 1205901

∘1205902

∘1205901

= 1205901

∘1205907

∘1205901

= 1205908

= 1205905

∘1205902

= 1205905

∘1205907

=

12059013

(iii) 120590

3

= 1205902

∘ 1205901

= 1205907

∘ 1205901

= 1205906

= 12059010

∘ 1205901

= 12059011

(iv) 1205904

= 1205901

∘ 1205902

= 1205901

∘ 1205907

= 1205909

= 1205901

∘ 12059010

= 12059012

Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology

105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908

means that any closed set is open tooThus if119883 = 119886 119887 119888

is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590

2

= 1205908

) is the monoid of all Kuratowski operationsover 119883 Relations 120590

2

= 1205908

and 1205905

= 1205907

are equivalent Theyimply relations 120590

2

= 1205908

= 12059010

1205905

= 1205907

= 12059013

1205903

= 1205909

= 12059011

and 120590

4

= 1205906

= 12059012

The permutation

(1205900

1205901

1205902

1205903

1205904

1205905

1205900

1205901

1205905

1205904

1205903

1205902

) (24)

determines the isomorphism between monoidsM(1205902

= 1205907

)

andM(1205902

= 1205908

)

106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590

2

= 1205908

means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590

2

= 12059010

implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902

= 1205908

The relation 1205902

= 12059013

follows that each closed sethas to be open so it implies 120590

2

= 1205908

too

107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590

7

= 1205908

and 12059010

= 12059013

areequivalent Each of them gives 120590

7

= 1205908

= 12059010

= 12059013

and1205906

= 1205909

= 12059011

= 12059012

If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590

7

= 1205908

) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements

108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590

7

= 12059010

1205908

= 12059013

1205906

= 12059011

and 120590

9

= 12059012

are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590

7

= 12059010

) is the monoid of allKuratowski operations over119883 and consists of 10 elements

109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590

7

= 12059013

and 1205908

= 12059010

areequivalent Also 120590

6

= 12059012

and 1205909

= 12059011

These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted

1205901

1205902

1205903

1205904

1205905

1205906

1205907

1205908

1205909

1205901

1205900

1205904

1205905

1205902

1205903

1205908

1205909

1205906

1205907

1205902

1205903

1205902

1205903

1205906

1205907

1205906

1205907

1205908

1205909

1205903

1205902

1205906

1205907

1205902

1205903

1205908

1205909

1205906

1205907

1205904

1205905

1205904

1205905

1205908

1205909

1205908

1205909

1205906

1205907

1205905

1205904

1205908

1205909

1205904

1205905

1205906

1205907

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

1205907

1205906

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205909

1205908

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

(25)

If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906

= 12059012

The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590

7

= 1205908

Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590

7

= 1205908

and 1205907

= 12059013

To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that

(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860

119888minus119888Then check that

(i) 1205900

(119860) = 119860 and 1205901

(119860) = 119883 (119860119888minus119888

cup 119861 cup 119862)(ii) 1205902

(119860) = 119880 cup 119861minus

cup119882 and 1205903

(119860) = 119883 119860119888minus119888

(iii) 1205904

(119860) = 119881 119861minus and 120590

5

(119860) = 119860119888minus119888

(iv) 1205906

(119860) = 12059012

(119860) = 119881 and 1205907

(119860) = 12059013

(119860) = 119880(v) 1205908

(119860) = 12059010

(119860) = 119880 cup 119882 and 1205909

(119860) = 12059011

(119860) =

119881 cup 119882Hence we have that 120590

7

= 1205908

Note that if119882 = 0 then 1205907

(119860) =

1205908

(119860) This is the case of subsets of 120573119873

Journal of Mathematics 9

References

[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922

[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977

[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962

[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010

[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008

[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966

[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996

[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956

[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889

[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967

[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968

[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 3

1205908

= 1205901

∘ 1205906

1205909

= 1205901

∘ 1205907

12059010

= 1205906

∘ 1205906

12059011

= 1205906

∘ 1205907

12059012

= 1205908

∘ 1205906

and 12059013

= 1205908

∘ 1205907

and so M1

= ⟨1205901

1205906

⟩ =

1205900

1205901

cup 1205906

1205907

12059013

Since 120590

6

= 1205907

∘ 1205901

= 1205901

∘ 1205908

= 1205901

∘ 1205909

∘ 1205901

= 12059010

∘ 1205901

∘ 12059010

=

12059011

∘ 12059011

∘ 1205901

= 1205901

∘ 12059012

∘ 12059012

and 1205906

= 1205901

∘ 12059013

∘ 1205901

∘ 12059013

∘ 1205901

we have ⟨120590

1

120590119895

⟩ = M1

for 119895 isin 6 7 13

Consider the permutation

(1205900

1205901

1205902

1205903

1205904

1205905

1205906

1205907

1205908

1205909

12059010

12059011

12059012

12059013

1205900

1205901

1205905

1205904

1205903

1205902

1205909

1205908

1205907

1205906

12059013

12059012

12059011

12059010

)

(7)

It determines an automorphism A M rarr M

Theorem 3 The identity and A are the only automorphismsofM

Proof Delete rows and columns marked by 1205901

in the Cayleytable for M Then check that the operation 120590

3

is in the rowor the column marked by 120590

3

only Also the operation 1205904

isin the row or the column marked by 120590

4

only Therefore thesemigroup

⟨1205903

1205904

⟩ = 1205902

1205903

12059013

(8)

has a unique minimal set of generators 1205903

1205904

The reader isleft to check this with the Cayley table forM

Suppose G is an automorphism of M By Theorem 2 Gtransforms the set 120590

2

1205903

1205904

1205905

onto itself However 1205902

and1205905

are idempotents but 1205903

and 1205904

are not idempotents Sothere are two possibilities G(120590

3

) = 1205903

and G(1205904

) = 1205904

whichimplies that G is the identity G(120590

3

) = 1205904

and G(1205904

) = 1205903

which implies G = A The reader is left to check this with theCayley table forM We offer hints 120590

2

= 1205903

∘ 1205904

1205905

= 1205904

∘ 1205903

1205906

= 1205903

∘ 1205902

1205907

= 1205903

∘ 1205903

1205908

= 1205904

∘ 1205904

1205909

= 1205904

∘ 1205905

12059010

= 1205906

∘ 1205906

12059011

= 1205906

∘ 1205907

12059012

= 1205908

∘ 1205906

and 12059013

= 1205908

∘ 1205907

to verify the details of this proof

5 The Monoid of All Idempotents

The set 1205900

1205902

1205905

1205907

1205908

12059010

12059013

consists of all squares inMThese squares are idempotents and lie on the main diagonalin the Cayley table forM They constitute a monoid and

1205900

1205902

1205905

1205907

1205908

12059010

12059013

= ⟨1205900

1205902

1205905

⟩ (9)

The permutation

(1205900

1205902

1205905

1205907

1205908

12059010

12059013

1205900

1205902

1205905

1205908

1205907

12059010

12059013

) (10)

determines the bijection I ⟨1205900

1205902

1205905

⟩ rarr ⟨1205900

1205902

1205905

⟩ suchthat

I (120572 ∘ 120573) = I (120573) ∘ I (120572) (11)

for any 120572 120573 isin ⟨1205900

1205902

1205905

⟩ To verify this apply equalities1205902

∘ 1205905

= 1205907

1205905

∘ 1205902

= 1205908

1205902

∘ 1205905

∘ 1205902

= 12059010

and1205905

∘ 1205902

∘ 1205905

= 12059013

Any bijection I 119866 rarr 119867 having propertyI(120572∘120573) = I(120573)∘I(120572) transposesCayley tables for semigroups119866

and119867 Several semigroups contained inM have this propertyWe leave the reader to verify this

We shall classify all semigroups contained in the semi-group ⟨120590

2

1205905

⟩ Every such semigroup can be extended toa monoid by attaching 120590

0

to it This gives a completeclassification of all semigroups in ⟨120590

0

1205902

1205905

⟩The semigroup ⟨120590

2

1205905

⟩ contains six groups with exactlyone element

Semigroups ⟨1205902

12059010

⟩ = 1205902

12059010

and ⟨1205905

12059013

⟩ = 1205905

12059013

aremonoids Both consist of exactly two elements and are notgroups so they are isomorphic

Semigroups ⟨1205907

12059010

⟩ and ⟨1205908

12059013

⟩ are isomorphic in par-ticular A[⟨120590

7

12059010

⟩] = ⟨1205908

12059013

⟩ Also semigroups ⟨1205907

12059013

⟩

and ⟨1205908

12059010

⟩ are isomorphic by A Every of these four semi-groups has exactly two elements None of them is a monoidThey form two types of nonisomorphic semigroups becausethe bijections (120590

7

12059010

) (12059010

1205908

) and (1205907

1205908

) (12059010

12059010

) arenot isomorphisms

Semigroups ⟨1205902

1205907

⟩ = 1205902

1205907

12059010

and ⟨1205905

1205908

⟩ =

1205905

1205908

12059013

are isomorphic Also semigroups ⟨1205902

1205908

⟩ =

1205902

1205908

12059010

and ⟨1205905

1205907

⟩ = 1205905

1205907

12059013

are isomorphic Infact A[⟨120590

2

1205907

⟩] = ⟨1205905

1205908

⟩ and A[⟨1205902

1205908

⟩] = ⟨1205905

1205907

⟩None of these semigroups is a monoid They form two typesof nonisomorphic semigroups Indeed any isomorphismbetween ⟨120590

2

1205907

⟩ and ⟨1205902

1205908

⟩ must be the identity on themonoid ⟨120590

2

12059010

⟩ Therefore would have to be the restrictionof I But I restricted to ⟨120590

7

12059010

⟩ is not an isomorphismSemigroups ⟨120590

2

12059013

⟩ = ⟨1205902

1205907

1205908

⟩ = 1205902

1205907

1205908

12059010

12059013

and ⟨1205905

12059010

⟩ = ⟨1205905

1205907

1205908

⟩ = 1205905

1205907

1205908

12059010

12059013

are notmonoids They are isomorphic by A

The semigroup ⟨1205902

1205905

⟩ contains exactly one semigroupwith four elements ⟨120590

7

1205908

⟩ = ⟨12059010

12059013

⟩ = 1205907

1205908

12059010

12059013

which is not a monoidNote that ⟨120590

2

1205905

⟩ contains twenty different semigroupswith nine non-isomorphism types These are six isomorphicgroups with exactly one element ⟨120590

2

⟩ cong ⟨1205905

⟩ cong ⟨1205907

⟩ cong

⟨1205908

⟩ cong ⟨12059010

⟩ cong ⟨12059013

⟩ two isomorphic monoids with exactlytwo elements ⟨120590

2

12059010

⟩ cong ⟨1205905

12059013

⟩ two pairs of isomorphicsemigroups with exactly two elements ⟨120590

7

12059010

⟩ cong ⟨1205908

12059013

⟩

and ⟨1205907

12059013

⟩ cong ⟨1205908

12059010

⟩ two pairs of isomorphic semigroupswith exactly three elements ⟨120590

2

1205907

⟩ cong ⟨1205905

1205908

⟩ and ⟨1205902

1205908

⟩ cong

⟨1205905

1205907

⟩ a semigroup with exactly four elements ⟨1205907

1205908

⟩and two isomorphic semigroups with exactly five elements⟨1205902

12059013

⟩ cong ⟨1205905

12059010

⟩ and also ⟨1205902

1205905

⟩ Thus ⟨1205902

1205905

⟩ containstwenty different semigroups with nine isomorphism typesBut ⟨120590

0

1205902

1205905

⟩ contains forty-one different semigroups withseventeen isomorphism types Indeed adding 120590

0

to semi-groups contained in ⟨120590

2

1205905

⟩ which are not monoids weget twenty monoids with eight non-isomorphism typesAdding 120590

0

to a group contained in ⟨1205902

1205905

⟩ we get a monoidisomorphic to ⟨120590

2

12059010

⟩

6 The Semigroup Consisting of 12059061205907 120590

13

Using the Cayley table forM check thatA [1205906

1205907

12059013

] = 1205906

1205907

12059013

(12)Similarly check that the semigroup ⟨120590

6

1205909

⟩ = 1205906

1205907

12059013

can be represented as ⟨1205906

12059013

⟩ ⟨1205907

12059012

⟩ ⟨1205908

12059011

⟩

4 Journal of Mathematics

⟨1205909

12059010

⟩ or ⟨12059011

12059012

⟩ Also ⟨1205906

1205909

⟩ = ⟨1205906

1205907

1205908

⟩ =

⟨1205907

1205908

1205909

⟩ = ⟨12059010

12059011

12059013

⟩ = ⟨12059010

12059012

12059013

⟩ These rep-resentations exhaust all minimal collections of the Kura-towski operations which generate ⟨120590

6

1205909

⟩ Other semigroupsincluded in ⟨120590

6

1205909

⟩ have one or two minimal collectionof generators One generator has groups ⟨120590

6

⟩ = 1205906

12059010

⟨1205909

⟩ = 1205909

12059013

⟨12059011

⟩ = 1205907

12059011

and ⟨12059012

⟩ = 1205908

12059012

Each of them has exactly two elements so they are isomor-phic Semigroups ⟨120590

7

12059010

⟩ ⟨1205907

12059013

⟩ ⟨1205908

12059010

⟩ ⟨1205908

12059013

⟩ and⟨1205907

1205908

⟩ are discussed in the previous section Contained in⟨1205906

1205909

⟩ and not previously discussed semigroups are ⟨1205906

1205907

⟩⟨1205906

1205908

⟩⟨1205907

1205909

⟩ and ⟨1205908

1205909

⟩ We leave the reader to verifythat the following are all possible pairs of Kuratowski oper-ations which constitute a minimal collection of generatorsfor semigroups contained in ⟨120590

6

1205909

⟩ but different from thewhole One has

(i) ⟨1205906

1205907

⟩ = ⟨1205906

12059011

⟩ = ⟨12059010

12059011

⟩ = 1205906

1205907

12059010

12059011

(ii) ⟨120590

6

1205908

⟩ = ⟨1205906

12059012

⟩ = ⟨12059010

12059012

⟩ = 1205906

1205908

12059010

12059012

(iii) ⟨120590

7

1205909

⟩ = ⟨1205909

12059011

⟩ = ⟨12059011

12059013

⟩ = 1205907

1205909

12059011

12059013

(iv) ⟨120590

8

1205909

⟩ = ⟨1205909

12059012

⟩ = ⟨12059012

12059013

⟩ = 1205908

1205909

12059012

12059013

Proposition 4 Semigroups ⟨1205906

1205907

⟩ and ⟨1205908

1205909

⟩ are isomor-phic and also semigroups ⟨120590

6

1205908

⟩ and ⟨1205907

1205909

⟩ are isomorphicbut semigroups ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ are not isomorphic

Proof Isomorphisms are defined by A Suppose 119869 ⟨1205906

1205907

⟩ rarr ⟨1205906

1205908

⟩ is an isomorphism Thus 119869[⟨1205907

12059010

⟩] =

⟨1205908

12059010

⟩ Given 119869(1205907

) = 1205908

we get 12059010

= 119869(12059010

) = 119869(1205907

∘

12059010

) = 1205908

∘ 12059010

= 1205908

But 119869(1205907

) = 12059010

implies 12059010

= 119869(1205907

) =

119869(12059010

∘ 1205907

) = 1205908

∘ 12059010

= 1205908

Both possibilities lead to acontradiction

So ⟨1205906

1205909

⟩ contains eighteen different semigroups witheight non-isomorphism types Indeed these are four iso-morphic groups with exactly one element ⟨120590

7

⟩ cong ⟨1205908

⟩ cong

⟨12059010

⟩ cong ⟨12059013

⟩ four isomorphic groups with exactly twoelements ⟨120590

6

⟩ cong ⟨1205909

⟩ cong ⟨12059011

⟩ cong ⟨12059012

⟩ two pairs ofisomorphic semigroups with exactly two elements ⟨120590

7

12059010

⟩ cong

⟨1205908

12059013

⟩ and ⟨1205907

12059013

⟩ cong ⟨1205908

12059010

⟩ and five semigroupswith exactly four elements ⟨120590

6

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong

⟨1205907

1205909

⟩ ⟨1205907

1205908

⟩ and also ⟨1205906

1205909

⟩

7 Remaining Semigroups in ⟨1205903 1205904⟩

We have yet to discuss semigroups included in ⟨1205903

1205904

⟩ notincluded in ⟨120590

2

1205905

⟩ and containing at least one of Kuratowskioperation 120590

2

1205903

1205904

or 1205905

It will be discussed up to theisomorphism A Obviously ⟨120590

2

⟩ = 1205902

and ⟨1205905

⟩ = 1205905

aregroups

71 Extensions of ⟨1205902⟩ and ⟨1205905⟩ with Elements of ⟨12059061205909⟩Monoids ⟨120590

2

1205906

⟩ = 1205902

1205906

12059010

and ⟨1205902

12059010

⟩ = 1205902

12059010

havedifferent numbers of elements Also ⟨120590

2

1205906

⟩ is isomorphic to⟨1205900

1205906

⟩ Nonisomorphic semigroups ⟨1205902

1205907

⟩ and ⟨1205902

1205908

⟩ arediscussed above The following three semigroups

(i) ⟨1205902

12059011

⟩ = ⟨1205902

1205906

1205907

⟩ = 1205902

1205906

1205907

12059010

12059011

(ii) ⟨1205902

12059012

⟩ = ⟨1205902

1205906

1205908

⟩ = 1205902

1205906

1205908

12059010

12059012

(iii) ⟨120590

2

12059013

⟩ = ⟨1205902

1205907

1205908

⟩ = 1205902

1205907

1205908

12059010

12059013

are not monoids They are not isomorphic Indeed anyisomorphism between these semigroups would lead an iso-morphism between ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ or ⟨1205907

1205908

⟩ This isimpossible by Proposition 4 and because ⟨120590

7

1205908

⟩ consists ofidempotents but 120590

6

is not an idempotent The nine-elementsemigroup on the set 120590

2

1205906

1205907

12059013

is represented as⟨1205902

1205909

⟩ So we have added four new semigroups which arenot isomorphic with the semigroups previously discussedThese are ⟨120590

2

12059011

⟩ ⟨1205902

12059012

⟩ ⟨1205902

12059013

⟩ and ⟨1205902

1205909

⟩Using A we have described eight semigroupsmdasheach

one isomorphic to a semigroup previously discussedmdashwhichcontains 120590

5

and elements (at least one) of ⟨1205906

1205909

⟩ Collec-tions of generators ⟨120590

2

1205906

12059013

⟩ ⟨1205902

1205907

12059012

⟩ ⟨1205902

1205908

12059011

⟩⟨1205902

12059011

12059012

⟩ ⟨1205902

12059011

12059013

⟩ and ⟨1205902

12059012

12059013

⟩ are minimalin ⟨120590

2

1205909

⟩ Also collections of generators ⟨1205905

1205907

12059012

⟩⟨1205905

1205908

12059011

⟩ ⟨1205905

1205909

12059010

⟩ ⟨1205905

12059010

12059011

⟩ ⟨1205905

12059010

12059012

⟩ and⟨1205905

12059011

12059012

⟩ are minimal in ⟨1205905

1205906

⟩

72 Extensions of ⟨1205903⟩ and ⟨1205904⟩ by Elements from the Semi-group ⟨12059061205909⟩ Semigroups ⟨120590

3

⟩ = 1205903

1205907

12059011

and ⟨1205904

⟩ =

1205904

1205908

12059012

are isomorphic by A They are not monoids Thesemigroup ⟨120590

3

⟩ can be extended using elements of ⟨1205906

1205909

⟩in three following ways

(i) ⟨1205903

1205906

⟩ = ⟨1205903

12059010

⟩ = 1205903

1205906

1205907

12059010

12059011

(ii) ⟨120590

3

1205908

⟩ = ⟨1205903

12059012

⟩ = 1205903

1205906

1205907

12059013

(iii) ⟨120590

3

1205909

⟩ = ⟨1205903

12059013

⟩ = 1205903

1205907

1205909

12059011

12059013

Semigroups ⟨1205903

1205906

⟩ and ⟨1205903

1205909

⟩ are not isomorphic Indeedsuppose 119869 ⟨120590

3

1205906

⟩ rarr ⟨1205903

1205909

⟩ is an isomorphismThus 119869 isthe identity on ⟨120590

3

⟩ 119869(1205906

) = 1205909

and 119869(12059010

) = 12059013

This givesa contradiction since 120590

3

∘ 1205906

= 12059010

and 1205903

∘ 1205909

= 1205907

Neither ⟨120590

3

1205906

⟩ nor ⟨1205903

1205909

⟩ has a minimal collection ofgenerators with three elements so they give new isomor-phism types Also ⟨120590

2

1205909

⟩ is not isomorphic to ⟨1205903

1205908

⟩since ⟨120590

2

1205909

⟩ has a unique pair of generators and ⟨1205903

1205908

⟩ =

⟨1205903

12059012

⟩Using A we getmdashisomorphic to previously discussed

onesmdashsemigroups ⟨1205904

1205909

⟩ = ⟨1205904

12059013

⟩ = 1205904

1205908

1205909

12059012

12059013

⟨1205904

1205906

⟩ = ⟨1205904

12059010

⟩ = 1205904

1205906

1205908

12059010

12059012

and ⟨1205904

1205907

⟩ =

⟨1205904

12059011

⟩ = 1205904

1205906

1205907

12059013

There exist minimal collec-tions of generators such as follows

⟨1205903

1205908

⟩ = ⟨1205903

1205906

1205909

⟩ = ⟨1205903

1205906

12059013

⟩

= ⟨1205903

1205909

12059010

⟩ = ⟨1205903

12059010

12059013

⟩

⟨1205904

1205907

⟩ = ⟨1205904

1205906

1205909

⟩ = ⟨1205904

1205909

12059010

⟩

= ⟨1205904

1205906

12059013

⟩ = ⟨1205904

12059010

12059013

⟩

(13)

73 More Generators from the Set 1205902120590312059041205905 Nowwe check that ⟨120590

2

1205903

⟩ = 1205902

1205903

1205906

1205907

12059010

12059011

and⟨1205904

1205905

⟩ = 1205904

1205905

1205908

1205909

12059012

12059013

= A[⟨1205902

1205903

⟩] andalso ⟨120590

2

1205904

⟩ = 1205902

1205904

1205906

1205908

12059010

12059012

and ⟨1205903

1205905

⟩ =

1205903

1205905

1205907

1205909

12059011

12059013

= A[⟨1205902

1205904

⟩] are two pairs of iso-morphic semigroupswhich give two new isomorphism types

Journal of Mathematics 5

Each of these semigroups has six element so in M thereare five six-element semigroups of three isomorphism typessince ⟨120590

2

1205905

⟩ has 6 elements which are idempotentsIn ⟨1205903

1205904

⟩ there are seven semigroups which have threegenerators and have not two generators These are

(1) ⟨1205902

1205903

1205908

⟩ = ⟨1205902

1205903

1205909

⟩ = ⟨1205902

1205903

12059012

⟩ = ⟨1205902

1205903

12059013

⟩

(2) ⟨1205904

1205905

1205906

⟩ = ⟨1205904

1205905

1205907

⟩ = ⟨1205904

1205905

12059010

⟩ = ⟨1205904

1205905

12059011

⟩

(3) ⟨1205902

1205904

1205907

⟩ = ⟨1205902

1205904

1205909

⟩ = ⟨1205902

1205904

12059011

⟩ = ⟨1205902

1205904

12059013

⟩

(4) ⟨1205903

1205905

1205906

⟩ = ⟨1205903

1205905

1205908

⟩ = ⟨1205903

1205905

12059010

⟩ = ⟨1205903

1205905

12059012

⟩

(5) ⟨1205902

1205905

1205906

⟩ = ⟨1205902

1205905

1205909

⟩ = ⟨1205902

1205905

12059011

⟩ = ⟨1205902

1205905

12059012

⟩

(6) ⟨1205902

1205903

1205905

⟩

(7) ⟨1205902

1205904

1205905

⟩

8 Semigroups which Are Contained in ⟨12059031205904⟩

Groups ⟨1205902

⟩ cong ⟨1205905

⟩ cong ⟨1205907

⟩ cong ⟨1205908

⟩ cong ⟨12059010

⟩ cong ⟨12059013

⟩ have oneelement and are isomorphic

Groups ⟨1205906

⟩ cong ⟨1205909

⟩ cong ⟨12059011

⟩ cong ⟨12059012

⟩ monoids ⟨1205902

12059010

⟩ cong

⟨1205905

12059013

⟩ and also semigroups ⟨1205907

12059010

⟩ cong ⟨1205908

12059013

⟩ and⟨1205907

12059013

⟩ cong ⟨1205908

12059010

⟩ have two elements and Cayley tables asfollows

119860 119861

119860 119861 119860

119861 119860 119861

119860 119861

119860 119860 119861

119861 119861 119860

119860 119861

119860 119860 119861

119861 119860 119861

119860 119861

119860 119860 119860

119861 119861 119861

(14)

Monoids ⟨1205902

1205906

⟩ cong ⟨1205905

1205909

⟩ and also semigroups⟨1205902

1205907

⟩ cong ⟨1205905

1205908

⟩ ⟨1205902

1205908

⟩ cong ⟨1205905

1205907

⟩ and ⟨1205903

⟩ cong ⟨1205904

⟩ havethree elements and Cayley tables as follows

119860 119861 119862

119860 119860 119861 119862

119861 119861 119862 119861

119862 119862 119861 119862

119860 119861 119862

119860 119860 119861 119862

119861 119862 119861 119862

119862 119862 119861 119862

119860 119861 119862

119860 119860 119862 119862

119861 119861 119861 119861

119862 119862 119862 119862

119860 119861 119862

119860 119861 119862 119861

119861 119862 119861 119862

119862 119861 119862 119861

(15)

Semigroups ⟨1205906

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong ⟨1205907

1205909

⟩ and⟨1205907

1205908

⟩ have four elements and Cayley tables as follows

119860 119861 119862 119863

119860 119862 119863 119860 119861

119861 119860 119861 119862 119863

119862 119860 119861 119862 119863

119863 119862 119863 119860 119861

119860 119861 119862 119863

119860 119862 119860 119860 119862

119861 119863 119861 119861 119863

119862 119860 119862 119862 119860

119863 119861 119863 119863 119861

119860 119861 119862 119863

119860 119860 119862 119862 119860

119861 119863 119861 119861 119863

119862 119860 119862 119862 119860

119863 119863 119861 119861 119863

(16)

Semigroups ⟨1205902

12059011

⟩ cong ⟨1205905

12059012

⟩ ⟨1205902

12059012

⟩ cong ⟨1205905

12059011

⟩⟨1205902

12059013

⟩ cong ⟨1205905

12059010

⟩ ⟨1205903

1205906

⟩ cong ⟨1205904

1205909

⟩ and ⟨1205903

1205909

⟩ cong

⟨1205904

1205906

⟩ have five elements Semigroups ⟨1205902

1205903

⟩ cong ⟨1205904

1205905

⟩⟨1205902

1205904

⟩ cong ⟨1205903

1205905

⟩ and ⟨1205902

1205905

⟩ have six elements Construc-tion of Cayley tables for these semigroups as well as othersemigroups we leave it to the readers One can prepare suchtables removing from the Cayley table for M some columnsand rows Then change Kuratowski operations onto lettersof the alphabet For example ⟨120590

0

1205902

1205905

⟩ has the followingCayley table

0 119860 119861 119862 119863 119864 119865

0 0 119860 119861 119862 119863 119864 119865

119860 119860 119860 119862 119862 119864 119864 119862

119861 119861 119863 119861 119865 119863 119863 119865

119862 119862 119864 119862 119862 119864 119864 119862

119863 119863 119863 119865 119865 119863 119863 119865

119864 119864 119864 119862 119862 119864 119864 119862

119865 119865 119863 119865 119865 119863 119863 119865

(17)

Preparing theCayley table for ⟨1205900

1205902

1205905

⟩weput1205900

= 0 1205902

=

119860 1205905

= 119861 1205907

= 119862 1205908

= 119863 12059010

= 119864 and 12059013

= 119865 This tableimmediately shows that semigroups ⟨120590

2

1205905

⟩ and ⟨1205900

1205902

1205905

⟩

have exactly two automorphisms These are identities andrestrictions of A

Semigroup ⟨1205906

1205909

⟩ has eight elements Semigroups⟨1205902

1205909

⟩ cong ⟨1205905

1205906

⟩ and ⟨1205903

1205908

⟩ cong ⟨1205904

1205907

⟩ have nine ele-ments Semigroups ⟨120590

2

1205903

1205908

⟩ cong ⟨1205904

1205905

1205907

⟩ ⟨1205902

1205904

1205907

⟩ cong

⟨1205903

1205905

1205908

⟩ and ⟨1205902

1205905

1205906

⟩ have ten elements Semigroups⟨1205902

1205903

1205905

⟩ cong ⟨1205902

1205904

1205905

⟩ have eleven elements In the endthe semigroup ⟨120590

3

1205904

⟩ has twelve elementsThus the semigroup ⟨120590

3

1205904

⟩ includes fifty-seven semi-groups amongwhich there are ten groups fourteenmonoidsand also forty-three semigroups which are not monoidsThese semigroups consist of twenty-eight types of noniso-morphic semigroups two non-isomorphism types of groupstwo non-isomorphism types of monoids which are notgroups and twenty four non-isomorphism types of semi-groups which are not monoids

9 Viewing Semigroups Contained in M

91 Descriptive Data on Semigroups which Are Contained inM There are one hundred eighteen that is 118 = 2 sdot 57 +

6 Journal of Mathematics

4 semigroups which are contained in M These are fifty-seven semigroups contained in ⟨120590

3

1205904

⟩ fifty-seven monoidsformed by adding 120590

0

to a semigroup contained in ⟨1205903

1205904

⟩groups ⟨120590

0

⟩ ⟨1205901

⟩ and monoids ⟨1205901

1205906

⟩ = M1

and ⟨1205901

1205902

⟩ =

MThere are fifty-six types of nonisomorphic semigroups

in M These are twenty-eight non-isomorphism types ofsemigroups contained in ⟨120590

3

1205904

⟩ twenty-six types of non-isomorphic monoids formed by adding 120590

0

to a semigroupcontained in ⟨120590

3

1205904

⟩ and alsoM1

andM Indeed adding 1205900

to a semigroup which is not a monoid we obtain a monoidIn this way we get twenty-four types of nonisomorphicmonoids Adding 120590

0

to a monoid contained in ⟨1205903

1205904

⟩ weget two new non-isomorphism types ofmonoids But adding1205900

to a group contained in ⟨1205903

1205904

⟩ we get no new type ofmonoid since we get a monoid isomorphic to ⟨120590

2

12059010

⟩ or⟨1205902

1205906

⟩ The other two types areM1

andM

92 Semigroups which Are Not Monoids Below we havereproduced using the smallest number of generators andthe dictionary order a list of all 43 semigroups which areincluded in theM as follows

(1) ⟨1205902

1205903

⟩ = 1205902

1205903

1205906

1205907

12059010

12059011

(2) ⟨120590

2

1205903

1205905

⟩ = 1205902

1205903

1205905

1205906

12059013

(3) ⟨120590

2

1205903

1205908

⟩ = ⟨1205902

1205903

1205909

⟩ = ⟨1205902

1205903

12059012

⟩ = ⟨1205902

1205903

12059013

⟩ = 1205902

1205903

1205906

1205907

12059013

(4) ⟨120590

2

1205904

⟩ = 1205902

1205904

1205906

1205908

12059010

12059012

(5) ⟨120590

2

1205904

1205905

⟩ = 1205902

1205904

1205905

12059013

(6) ⟨120590

2

1205904

1205907

⟩ = ⟨1205902

1205904

1205909

⟩ = ⟨1205902

1205904

12059011

⟩ = ⟨1205902

1205904

12059013

⟩ = 1205902

1205904

1205906

1205907

12059013

(7) ⟨120590

2

1205905

⟩ = 1205902

1205905

1205907

1205908

12059010

12059013

(8) ⟨120590

2

1205905

1205906

⟩ = ⟨1205902

1205905

1205909

⟩ = ⟨1205902

1205905

12059011

⟩ = ⟨1205902

1205905

12059012

⟩ = 1205902

1205905

1205906

12059013

(9) ⟨120590

2

1205907

⟩ = 1205902

1205907

12059010

(10) ⟨120590

2

1205908

⟩ = 1205902

1205908

12059010

(11) ⟨120590

2

1205909

⟩ = ⟨1205902

1205906

12059013

⟩ = ⟨1205902

1205907

12059012

⟩ = ⟨1205902

1205908

12059011

⟩ = ⟨1205902

12059011

12059012

⟩ = ⟨1205902

12059011

12059013

⟩ = ⟨1205902

12059012

12059013

⟩ = 1205902

1205906

1205907

12059013

(12) ⟨120590

2

12059011

⟩ = ⟨1205902

1205906

1205907

⟩ = 1205902

1205906

1205907

12059010

12059011

(13) ⟨120590

2

12059012

⟩ = ⟨1205902

1205906

1205908

⟩ = 1205902

1205906

1205908

12059010

12059012

(14) ⟨120590

2

12059013

⟩ = ⟨1205902

1205907

1205908

⟩ = 1205902

1205907

1205908

12059010

12059013

(15) ⟨120590

3

⟩ = 1205903

1205907

12059011

(16) ⟨120590

3

1205904

⟩ = 1205902

1205903

12059013

(17) ⟨120590

3

1205905

⟩ = 1205903

1205905

1205907

1205909

12059011

12059013

(18) ⟨120590

3

1205905

1205906

⟩ = ⟨1205903

1205905

1205908

⟩ = ⟨1205903

1205905

12059010

⟩ = ⟨1205903

1205905

12059012

⟩ = 1205903

1205905

1205906

12059013

(19) ⟨120590

3

1205906

⟩ = ⟨1205903

12059010

⟩ = 1205903

1205906

1205907

12059010

12059011

(20) ⟨120590

3

1205908

⟩ = ⟨1205903

12059012

⟩ = 1205903

1205906

1205907

12059013

(21) ⟨120590

3

1205909

⟩ = ⟨1205903

12059013

⟩ = 1205903

1205907

1205909

12059011

12059013

(22) ⟨120590

4

⟩ = 1205904

1205908

12059012

(23) ⟨1205904

1205905

⟩ = 1205904

1205905

1205908

1205909

12059012

12059013

(24) ⟨120590

4

1205905

1205906

⟩ = ⟨1205904

1205905

1205907

⟩ = ⟨1205904

1205905

12059010

⟩ = ⟨1205904

1205905

12059011

⟩ = 1205904

1205905

12059013

(25) ⟨120590

4

1205906

⟩ = ⟨1205904

12059010

⟩ = 1205904

1205906

1205908

12059010

12059012

(26) ⟨120590

4

1205907

⟩ = ⟨1205904

12059011

⟩ = 1205904

1205906

1205907

12059013

(27) ⟨120590

4

1205909

⟩ = ⟨1205904

12059013

⟩ = 1205904

1205908

1205909

12059012

12059013

(28) ⟨120590

5

1205906

⟩ = ⟨1205905

1205907

12059012

⟩ = ⟨1205905

1205908

12059011

⟩ = ⟨1205905

1205909

12059010

⟩ = ⟨1205905

12059010

12059011

⟩ = ⟨1205905

12059010

12059012

⟩ = ⟨1205905

12059011

12059012

⟩ = 1205905

1205906

12059013

(29) ⟨120590

5

1205907

⟩ = 1205905

1205907

12059013

(30) ⟨120590

5

1205908

⟩ = 1205905

1205908

12059013

(31) ⟨120590

5

12059010

⟩ = ⟨1205905

1205907

1205908

⟩ = 1205905

1205907

1205908

12059010

12059013

(32) ⟨120590

5

12059011

⟩ = ⟨1205905

1205907

1205909

⟩ = 1205905

1205907

1205909

12059011

12059013

(33) ⟨120590

5

12059012

⟩ = ⟨1205905

1205908

1205909

⟩ = 1205905

1205908

1205909

12059012

12059013

(34) ⟨120590

6

1205907

⟩ = ⟨1205906

12059011

⟩ = ⟨12059010

12059011

⟩ = 1205906

1205907

12059010

12059011

(35) ⟨120590

6

1205908

⟩ = ⟨1205906

12059012

⟩ = ⟨12059010

12059012

⟩ = 1205906

1205908

12059010

12059012

(36) ⟨120590

6

1205909

⟩ = ⟨1205906

12059013

⟩ = ⟨1205907

12059012

⟩ = ⟨1205908

12059011

⟩ = ⟨1205909

12059010

⟩ = ⟨12059011

12059012

⟩ = ⟨1205906

1205907

1205908

⟩ = ⟨1205907

1205908

1205909

⟩ =⟨12059010

12059011

12059013

⟩ = ⟨12059010

12059012

12059013

⟩ = 1205906

1205907

12059013

(37) ⟨120590

7

1205908

⟩ = 1205907

1205908

12059010

12059013

(38) ⟨120590

7

1205909

⟩ = 1205907

1205909

12059011

12059013

(39) ⟨120590

7

12059010

⟩ = 1205907

12059010

(40) ⟨120590

7

12059013

⟩ = 1205907

12059013

(41) ⟨120590

8

1205909

⟩ = 1205908

1205909

12059012

12059013

(42) ⟨120590

8

12059010

⟩ = 1205908

12059010

(43) ⟨120590

8

12059013

⟩ = 1205908

12059013

93 Isomorphism Types of Semigroups Contained in M Sys-tematize the list of all isomorphism types of semigroupscontained in themonoidM Isomorphisms which are restric-tions of the isomorphism A will be regarded as self-evidentand therefore they will not be commented on

(i) The monoid M contains 12 groups with 2 isomor-phism types These are 7 one-element groups and 5two-element groups

(ii) ThemonoidM contains 8 two-element monoids with2 isomorphism types These are 120590

0

added to 6 one-element groups and ⟨120590

2

12059010

⟩ cong ⟨1205905

12059013

⟩ Also itcontains 4 two-element semigroups not monoidswith 2 isomorphism types These are ⟨120590

7

12059010

⟩ cong

⟨1205908

12059013

⟩ and ⟨1205908

12059010

⟩ cong ⟨1205907

12059013

⟩(iii) The monoid M contains 12 three-element monoids

with 5 isomorphism types These are 1205900

addedto 4 two-element groups and also ⟨120590

0

1205902

12059010

⟩ cong

⟨1205900

1205905

12059013

⟩ ⟨1205900

1205907

12059010

⟩ cong ⟨1205900

1205908

12059013

⟩⟨1205900

1205908

12059010

⟩ cong ⟨1205900

1205907

12059013

⟩ and ⟨1205902

1205906

⟩ cong ⟨1205905

1205909

⟩(iv) The monoidM contains 6 three-element semigroups

not monoids with 3 isomorphism types These are⟨1205902

1205907

⟩ cong ⟨1205905

1205908

⟩ ⟨1205902

1205908

⟩ cong ⟨1205905

1205907

⟩ and ⟨1205903

⟩ cong

⟨1205904

⟩

Journal of Mathematics 7

(v) The monoid M contains 8 four-element monoidseach contains 120590

0

with 4 isomorphism types Theseare semigroups from two preceding items that can besubstantially extended by 120590

0

(vi) The monoid M contains 5 four-element semigroups

not monoids with 3 isomorphism types These are⟨1205906

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong ⟨1205907

1205909

⟩ and ⟨1205907

1205908

⟩These semigroups extended by 120590

0

yield 5 monoids allwhich consist of five elements with 3 isomorphismtypes

(vii) ThemonoidM contains 10 five-element semigroupsmdashnot monoids with 5 isomorphism types These are⟨1205902

12059011

⟩ cong ⟨1205905

12059012

⟩ ⟨1205902

12059012

⟩ cong ⟨1205905

12059011

⟩ ⟨1205902

12059013

⟩ cong

⟨1205905

12059010

⟩ ⟨1205903

1205906

⟩ cong ⟨1205904

1205909

⟩ and ⟨1205903

1205909

⟩ cong ⟨1205904

1205906

⟩These semigroups extended by120590

0

yield 10monoids allof which consist of six elements with 5 isomorphismtypes We get 10 new isomorphism types since thesemigroups are distinguished by semigroups ⟨120590

3

⟩

and ⟨1205904

⟩ and by nonisomorphic semigroups ⟨1205906

1205907

⟩⟨1205906

1205908

⟩ and ⟨1205907

1205908

⟩(viii) The monoid M contains 5 six-element semigroups

not monoids with 3 isomorphism types These are⟨1205902

1205903

⟩ cong ⟨1205904

1205905

⟩ ⟨1205902

1205904

⟩ cong ⟨1205903

1205905

⟩ and ⟨1205902

1205905

⟩These semigroups extended by 120590

0

yield 5 monoidsall of which consist of seven elements with 3 isomor-phism types We get 6 new isomorphism types sincethe semigroups are distinguished by non isomorphicsemigroups ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ and ⟨1205907

1205908

⟩(ix) ThemonoidM contains no seven-element semigroup

not a monoid no eight-element monoid and theonly semigroup ⟨120590

6

1205909

⟩ with exactly eight elementsand the only monoid ⟨120590

0

1205906

1205909

⟩ with exactly nineelements

(x) The monoid M contains 4 nine-element semigroupsnot monoids with 2 isomorphism types These are⟨1205902

1205909

⟩ cong ⟨1205905

1205906

⟩ and ⟨1205903

1205908

⟩ cong ⟨1205904

1205907

⟩ Thesemigroup ⟨120590

2

1205909

⟩ does not contain a semigroupisomorphic to ⟨120590

3

⟩ hence it is not isomorphic to⟨1205903

1205908

⟩ These semigroups extended by 1205900

yield 4monoids all of which consist of ten elements with 2isomorphism types

(xi) The monoid M contains 6 ten-element semigroupsnot monoids with 4 isomorphism typesThese are ⟨120590

1

1205906

⟩ ⟨1205902

1205903

1205908

⟩ cong ⟨1205904

1205905

1205906

⟩⟨1205902

1205904

1205907

⟩ cong ⟨1205903

1205905

1205906

⟩ and ⟨1205902

1205905

1205906

⟩ Thesesemigroups (except ⟨120590

1

1205906

⟩) extended by 1205900

yield 5monoids all of which consist of ten elements with3 isomorphism types We get 6 new isomorphismtypes since the semigroups are distinguished bynot isomorphic semigroups ⟨120590

2

1205903

⟩ ⟨1205902

1205904

⟩ and⟨1205902

1205905

⟩(xii) The monoid M contains 2 isomorphic semigroups

not monoids which consist of 11 elements that is⟨1205902

1205903

1205905

⟩ cong ⟨1205902

1205904

1205905

⟩These semigroups extendedby 1205900

yield 2 isomorphic monoids which consistof 12 elements The monoid M contains no largersemigroup with the exception of itself

10 Cancellation Rules Motivated by SomeTopological Properties

101 Some Consequences of the Axiom 0 = 0minus So far we usedonly the following relations (above named cancellation rules)1205902

∘ 1205902

= 1205902

1205901

∘ 1205901

= 1205900

1205902

∘ 12059012

= 1205906

and 1205902

∘ 12059013

= 1205907

When one assumes119883 = 0 = 0

minus then

119883 = 1205900

(119883) = 1205902

(119883) = 1205905

(119883) = 1205907

(119883)

= 1205908

(119883) = 12059010

(119883) = 12059013

(119883)

0 = 1205901

(119883) = 1205903

(119883) = 1205904

(119883) = 1205906

(119883)

= 1205909

(119883) = 12059011

(119883) = 12059012

(119883)

(18)

Using the substitution 119860 997891rarr 119860119888 one obtains equiv-

alent relations between operations from the set 1205901

1205903

1205904

1205906

1205909

12059011

12059012

and conversely Therefore cancellationrules are topologically reasonable only between the opera-tions from the monoid as follows

⟨1205900

1205902

1205905

⟩ = 1205900

1205902

1205905

1205907

1205908

12059010

12059013

(19)

Chapman see [3] considered properties of subsets withrespect to such relations Below we are going to identifyrelations that are determined by some topological spacescompare [10 11]

102 The Relation 1205900 = 1205902 If a topological space 119883 isdiscrete then there exist two Kuratowski operation onlyThese are 120590

0

and 1205901

So themonoid of Kuratowski operationsreduced to the group ⟨120590

1

⟩The relation 120590

0

= 1205902

is equivalent to any relation 1205900

=

120590119894

where 119894 isin 5 7 8 10 13 Any such relation implies thatevery subset of 119883 has to be closed and open that is 119883 hasto be discrete However one can check these using (only) thefacts that 120590

1

is an involution and 1205902

is an idempotent and thecancellation rules that is the Cayley table forM So 120590

0

= 1205902

follows

1205900

= 1205902

= 1205905

= 1205907

= 1205908

= 12059010

= 12059013

(20)

103 The Relation 1205902 = 1205905 Topologically 1205902 = 1205905

meansthat 119883 must be discrete This is so because 119860119888minus119888 sube 119860 sube 119860

minus

for any 119860 sube 119883

104 The Relation 1205902 = 1205907 Topologically the relation 1205902 =1205907

implies 1205900

= 1205902

But it requires the use of topology axioms0 = 0minus and 119862minus cup 119861minus = (119862 cup 119861)minus for each 119862 and 119861

Lemma 5 For any topological space 1205902

= 1205907

implies 1205902

= 1205908

Proof If 1205902

= 1205907

then 119860 = 0 rArr 119860119888minus119888

= 0 for any 119860 sube 119883Indeed if 119860119888minus119888 = 0 then 120590

7

(119860) = 0minus

= 0 Since 119860 = 0 then1205902

(119860) = 0 Hence 1205902

(119860) = 1205907

(119860) a contradictionThe axiom 119862

minus

cup 119861minus

= (119862 cup 119861)minus implies that always

(119860minus

cap 119860minus119888minus

)119888minus119888

= 0 (21)

Thus the additional assumption 1205902

= 1205907

follows that always119860minus

cap 119860minus119888minus

= 0 Therefore always 119860minus119888minus = 119860minus119888 but this means

8 Journal of Mathematics

that any closed set has to be open in other words 1205902

= 1205908

Proposition 6 For any topological space 1205902

= 1205907

implies 1205900

=

1205902

Proof But the relation 1205902

= 1205907

is equivalent with 1205905

= 1205908

ByLemma 5 we get 120590

2

= 1205905

Finally 1205900

= 1205902

The relation 1205902

= 1205907

has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put

1205902

(0) = 119886 = 1205902

(119886) 119883 = 1205902

(119883) = 1205902

(119887) (22)

Then check that 1205902

= 1205907

and

1205908

(0) = 1205905

(119886) = 1205904

(119887) = 1205901

(119883) = 0 (23)

in other words 1205902

= 1205907

and 1205902

= 1205908

However 1205902

= 1205907

isequivalent to 120590

5

= 1205908

This relation implies 120590

2

= 1205907

= 12059010

1205905

= 1205908

= 12059013

1205903

= 1205906

= 12059011

and 1205904

= 1205909

= 12059012

For this interpretationthe monoidM(120590

2

= 1205907

) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590

2

= 1205907

) thereare relations covered by the following proposition only

Proposition 7 For any monoid with the Cayley table as forM the relation 120590

2

= 1205907

implies

(i) 1205902

= 1205907

= 12059010

= 1205907

∘ 1205902

(ii) 1205905

= 1205901

∘1205902

∘1205901

= 1205901

∘1205907

∘1205901

= 1205908

= 1205905

∘1205902

= 1205905

∘1205907

=

12059013

(iii) 120590

3

= 1205902

∘ 1205901

= 1205907

∘ 1205901

= 1205906

= 12059010

∘ 1205901

= 12059011

(iv) 1205904

= 1205901

∘ 1205902

= 1205901

∘ 1205907

= 1205909

= 1205901

∘ 12059010

= 12059012

Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology

105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908

means that any closed set is open tooThus if119883 = 119886 119887 119888

is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590

2

= 1205908

) is the monoid of all Kuratowski operationsover 119883 Relations 120590

2

= 1205908

and 1205905

= 1205907

are equivalent Theyimply relations 120590

2

= 1205908

= 12059010

1205905

= 1205907

= 12059013

1205903

= 1205909

= 12059011

and 120590

4

= 1205906

= 12059012

The permutation

(1205900

1205901

1205902

1205903

1205904

1205905

1205900

1205901

1205905

1205904

1205903

1205902

) (24)

determines the isomorphism between monoidsM(1205902

= 1205907

)

andM(1205902

= 1205908

)

106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590

2

= 1205908

means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590

2

= 12059010

implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902

= 1205908

The relation 1205902

= 12059013

follows that each closed sethas to be open so it implies 120590

2

= 1205908

too

107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590

7

= 1205908

and 12059010

= 12059013

areequivalent Each of them gives 120590

7

= 1205908

= 12059010

= 12059013

and1205906

= 1205909

= 12059011

= 12059012

If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590

7

= 1205908

) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements

108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590

7

= 12059010

1205908

= 12059013

1205906

= 12059011

and 120590

9

= 12059012

are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590

7

= 12059010

) is the monoid of allKuratowski operations over119883 and consists of 10 elements

109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590

7

= 12059013

and 1205908

= 12059010

areequivalent Also 120590

6

= 12059012

and 1205909

= 12059011

These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted

1205901

1205902

1205903

1205904

1205905

1205906

1205907

1205908

1205909

1205901

1205900

1205904

1205905

1205902

1205903

1205908

1205909

1205906

1205907

1205902

1205903

1205902

1205903

1205906

1205907

1205906

1205907

1205908

1205909

1205903

1205902

1205906

1205907

1205902

1205903

1205908

1205909

1205906

1205907

1205904

1205905

1205904

1205905

1205908

1205909

1205908

1205909

1205906

1205907

1205905

1205904

1205908

1205909

1205904

1205905

1205906

1205907

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

1205907

1205906

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205909

1205908

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

(25)

If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906

= 12059012

The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590

7

= 1205908

Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590

7

= 1205908

and 1205907

= 12059013

To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that

(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860

119888minus119888Then check that

(i) 1205900

(119860) = 119860 and 1205901

(119860) = 119883 (119860119888minus119888

cup 119861 cup 119862)(ii) 1205902

(119860) = 119880 cup 119861minus

cup119882 and 1205903

(119860) = 119883 119860119888minus119888

(iii) 1205904

(119860) = 119881 119861minus and 120590

5

(119860) = 119860119888minus119888

(iv) 1205906

(119860) = 12059012

(119860) = 119881 and 1205907

(119860) = 12059013

(119860) = 119880(v) 1205908

(119860) = 12059010

(119860) = 119880 cup 119882 and 1205909

(119860) = 12059011

(119860) =

119881 cup 119882Hence we have that 120590

7

= 1205908

Note that if119882 = 0 then 1205907

(119860) =

1205908

(119860) This is the case of subsets of 120573119873

Journal of Mathematics 9

References

[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922

[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977

[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962

[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010

[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008

[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966

[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996

[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956

[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889

[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967

[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968

[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Mathematics

⟨1205909

12059010

⟩ or ⟨12059011

12059012

⟩ Also ⟨1205906

1205909

⟩ = ⟨1205906

1205907

1205908

⟩ =

⟨1205907

1205908

1205909

⟩ = ⟨12059010

12059011

12059013

⟩ = ⟨12059010

12059012

12059013

⟩ These rep-resentations exhaust all minimal collections of the Kura-towski operations which generate ⟨120590

6

1205909

⟩ Other semigroupsincluded in ⟨120590

6

1205909

⟩ have one or two minimal collectionof generators One generator has groups ⟨120590

6

⟩ = 1205906

12059010

⟨1205909

⟩ = 1205909

12059013

⟨12059011

⟩ = 1205907

12059011

and ⟨12059012

⟩ = 1205908

12059012

Each of them has exactly two elements so they are isomor-phic Semigroups ⟨120590

7

12059010

⟩ ⟨1205907

12059013

⟩ ⟨1205908

12059010

⟩ ⟨1205908

12059013

⟩ and⟨1205907

1205908

⟩ are discussed in the previous section Contained in⟨1205906

1205909

⟩ and not previously discussed semigroups are ⟨1205906

1205907

⟩⟨1205906

1205908

⟩⟨1205907

1205909

⟩ and ⟨1205908

1205909

⟩ We leave the reader to verifythat the following are all possible pairs of Kuratowski oper-ations which constitute a minimal collection of generatorsfor semigroups contained in ⟨120590

6

1205909

⟩ but different from thewhole One has

(i) ⟨1205906

1205907

⟩ = ⟨1205906

12059011

⟩ = ⟨12059010

12059011

⟩ = 1205906

1205907

12059010

12059011

(ii) ⟨120590

6

1205908

⟩ = ⟨1205906

12059012

⟩ = ⟨12059010

12059012

⟩ = 1205906

1205908

12059010

12059012

(iii) ⟨120590

7

1205909

⟩ = ⟨1205909

12059011

⟩ = ⟨12059011

12059013

⟩ = 1205907

1205909

12059011

12059013

(iv) ⟨120590

8

1205909

⟩ = ⟨1205909

12059012

⟩ = ⟨12059012

12059013

⟩ = 1205908

1205909

12059012

12059013

Proposition 4 Semigroups ⟨1205906

1205907

⟩ and ⟨1205908

1205909

⟩ are isomor-phic and also semigroups ⟨120590

6

1205908

⟩ and ⟨1205907

1205909

⟩ are isomorphicbut semigroups ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ are not isomorphic

Proof Isomorphisms are defined by A Suppose 119869 ⟨1205906

1205907

⟩ rarr ⟨1205906

1205908

⟩ is an isomorphism Thus 119869[⟨1205907

12059010

⟩] =

⟨1205908

12059010

⟩ Given 119869(1205907

) = 1205908

we get 12059010

= 119869(12059010

) = 119869(1205907

∘

12059010

) = 1205908

∘ 12059010

= 1205908

But 119869(1205907

) = 12059010

implies 12059010

= 119869(1205907

) =

119869(12059010

∘ 1205907

) = 1205908

∘ 12059010

= 1205908

Both possibilities lead to acontradiction

So ⟨1205906

1205909

⟩ contains eighteen different semigroups witheight non-isomorphism types Indeed these are four iso-morphic groups with exactly one element ⟨120590

7

⟩ cong ⟨1205908

⟩ cong

⟨12059010

⟩ cong ⟨12059013

⟩ four isomorphic groups with exactly twoelements ⟨120590

6

⟩ cong ⟨1205909

⟩ cong ⟨12059011

⟩ cong ⟨12059012

⟩ two pairs ofisomorphic semigroups with exactly two elements ⟨120590

7

12059010

⟩ cong

⟨1205908

12059013

⟩ and ⟨1205907

12059013

⟩ cong ⟨1205908

12059010

⟩ and five semigroupswith exactly four elements ⟨120590

6

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong

⟨1205907

1205909

⟩ ⟨1205907

1205908

⟩ and also ⟨1205906

1205909

⟩

7 Remaining Semigroups in ⟨1205903 1205904⟩

We have yet to discuss semigroups included in ⟨1205903

1205904

⟩ notincluded in ⟨120590

2

1205905

⟩ and containing at least one of Kuratowskioperation 120590

2

1205903

1205904

or 1205905

It will be discussed up to theisomorphism A Obviously ⟨120590

2

⟩ = 1205902

and ⟨1205905

⟩ = 1205905

aregroups

71 Extensions of ⟨1205902⟩ and ⟨1205905⟩ with Elements of ⟨12059061205909⟩Monoids ⟨120590

2

1205906

⟩ = 1205902

1205906

12059010

and ⟨1205902

12059010

⟩ = 1205902

12059010

havedifferent numbers of elements Also ⟨120590

2

1205906

⟩ is isomorphic to⟨1205900

1205906

⟩ Nonisomorphic semigroups ⟨1205902

1205907

⟩ and ⟨1205902

1205908

⟩ arediscussed above The following three semigroups

(i) ⟨1205902

12059011

⟩ = ⟨1205902

1205906

1205907

⟩ = 1205902

1205906

1205907

12059010

12059011

(ii) ⟨1205902

12059012

⟩ = ⟨1205902

1205906

1205908

⟩ = 1205902

1205906

1205908

12059010

12059012

(iii) ⟨120590

2

12059013

⟩ = ⟨1205902

1205907

1205908

⟩ = 1205902

1205907

1205908

12059010

12059013

are not monoids They are not isomorphic Indeed anyisomorphism between these semigroups would lead an iso-morphism between ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ or ⟨1205907

1205908

⟩ This isimpossible by Proposition 4 and because ⟨120590

7

1205908

⟩ consists ofidempotents but 120590

6

is not an idempotent The nine-elementsemigroup on the set 120590

2

1205906

1205907

12059013

is represented as⟨1205902

1205909

⟩ So we have added four new semigroups which arenot isomorphic with the semigroups previously discussedThese are ⟨120590

2

12059011

⟩ ⟨1205902

12059012

⟩ ⟨1205902

12059013

⟩ and ⟨1205902

1205909

⟩Using A we have described eight semigroupsmdasheach

one isomorphic to a semigroup previously discussedmdashwhichcontains 120590

5

and elements (at least one) of ⟨1205906

1205909

⟩ Collec-tions of generators ⟨120590

2

1205906

12059013

⟩ ⟨1205902

1205907

12059012

⟩ ⟨1205902

1205908

12059011

⟩⟨1205902

12059011

12059012

⟩ ⟨1205902

12059011

12059013

⟩ and ⟨1205902

12059012

12059013

⟩ are minimalin ⟨120590

2

1205909

⟩ Also collections of generators ⟨1205905

1205907

12059012

⟩⟨1205905

1205908

12059011

⟩ ⟨1205905

1205909

12059010

⟩ ⟨1205905

12059010

12059011

⟩ ⟨1205905

12059010

12059012

⟩ and⟨1205905

12059011

12059012

⟩ are minimal in ⟨1205905

1205906

⟩

72 Extensions of ⟨1205903⟩ and ⟨1205904⟩ by Elements from the Semi-group ⟨12059061205909⟩ Semigroups ⟨120590

3

⟩ = 1205903

1205907

12059011

and ⟨1205904

⟩ =

1205904

1205908

12059012

are isomorphic by A They are not monoids Thesemigroup ⟨120590

3

⟩ can be extended using elements of ⟨1205906

1205909

⟩in three following ways

(i) ⟨1205903

1205906

⟩ = ⟨1205903

12059010

⟩ = 1205903

1205906

1205907

12059010

12059011

(ii) ⟨120590

3

1205908

⟩ = ⟨1205903

12059012

⟩ = 1205903

1205906

1205907

12059013

(iii) ⟨120590

3

1205909

⟩ = ⟨1205903

12059013

⟩ = 1205903

1205907

1205909

12059011

12059013

Semigroups ⟨1205903

1205906

⟩ and ⟨1205903

1205909

⟩ are not isomorphic Indeedsuppose 119869 ⟨120590

3

1205906

⟩ rarr ⟨1205903

1205909

⟩ is an isomorphismThus 119869 isthe identity on ⟨120590

3

⟩ 119869(1205906

) = 1205909

and 119869(12059010

) = 12059013

This givesa contradiction since 120590

3

∘ 1205906

= 12059010

and 1205903

∘ 1205909

= 1205907

Neither ⟨120590

3

1205906

⟩ nor ⟨1205903

1205909

⟩ has a minimal collection ofgenerators with three elements so they give new isomor-phism types Also ⟨120590

2

1205909

⟩ is not isomorphic to ⟨1205903

1205908

⟩since ⟨120590

2

1205909

⟩ has a unique pair of generators and ⟨1205903

1205908

⟩ =

⟨1205903

12059012

⟩Using A we getmdashisomorphic to previously discussed

onesmdashsemigroups ⟨1205904

1205909

⟩ = ⟨1205904

12059013

⟩ = 1205904

1205908

1205909

12059012

12059013

⟨1205904

1205906

⟩ = ⟨1205904

12059010

⟩ = 1205904

1205906

1205908

12059010

12059012

and ⟨1205904

1205907

⟩ =

⟨1205904

12059011

⟩ = 1205904

1205906

1205907

12059013

There exist minimal collec-tions of generators such as follows

⟨1205903

1205908

⟩ = ⟨1205903

1205906

1205909

⟩ = ⟨1205903

1205906

12059013

⟩

= ⟨1205903

1205909

12059010

⟩ = ⟨1205903

12059010

12059013

⟩

⟨1205904

1205907

⟩ = ⟨1205904

1205906

1205909

⟩ = ⟨1205904

1205909

12059010

⟩

= ⟨1205904

1205906

12059013

⟩ = ⟨1205904

12059010

12059013

⟩

(13)

73 More Generators from the Set 1205902120590312059041205905 Nowwe check that ⟨120590

2

1205903

⟩ = 1205902

1205903

1205906

1205907

12059010

12059011

and⟨1205904

1205905

⟩ = 1205904

1205905

1205908

1205909

12059012

12059013

= A[⟨1205902

1205903

⟩] andalso ⟨120590

2

1205904

⟩ = 1205902

1205904

1205906

1205908

12059010

12059012

and ⟨1205903

1205905

⟩ =

1205903

1205905

1205907

1205909

12059011

12059013

= A[⟨1205902

1205904

⟩] are two pairs of iso-morphic semigroupswhich give two new isomorphism types

Journal of Mathematics 5

Each of these semigroups has six element so in M thereare five six-element semigroups of three isomorphism typessince ⟨120590

2

1205905

⟩ has 6 elements which are idempotentsIn ⟨1205903

1205904

⟩ there are seven semigroups which have threegenerators and have not two generators These are

(1) ⟨1205902

1205903

1205908

⟩ = ⟨1205902

1205903

1205909

⟩ = ⟨1205902

1205903

12059012

⟩ = ⟨1205902

1205903

12059013

⟩

(2) ⟨1205904

1205905

1205906

⟩ = ⟨1205904

1205905

1205907

⟩ = ⟨1205904

1205905

12059010

⟩ = ⟨1205904

1205905

12059011

⟩

(3) ⟨1205902

1205904

1205907

⟩ = ⟨1205902

1205904

1205909

⟩ = ⟨1205902

1205904

12059011

⟩ = ⟨1205902

1205904

12059013

⟩

(4) ⟨1205903

1205905

1205906

⟩ = ⟨1205903

1205905

1205908

⟩ = ⟨1205903

1205905

12059010

⟩ = ⟨1205903

1205905

12059012

⟩

(5) ⟨1205902

1205905

1205906

⟩ = ⟨1205902

1205905

1205909

⟩ = ⟨1205902

1205905

12059011

⟩ = ⟨1205902

1205905

12059012

⟩

(6) ⟨1205902

1205903

1205905

⟩

(7) ⟨1205902

1205904

1205905

⟩

8 Semigroups which Are Contained in ⟨12059031205904⟩

Groups ⟨1205902

⟩ cong ⟨1205905

⟩ cong ⟨1205907

⟩ cong ⟨1205908

⟩ cong ⟨12059010

⟩ cong ⟨12059013

⟩ have oneelement and are isomorphic

Groups ⟨1205906

⟩ cong ⟨1205909

⟩ cong ⟨12059011

⟩ cong ⟨12059012

⟩ monoids ⟨1205902

12059010

⟩ cong

⟨1205905

12059013

⟩ and also semigroups ⟨1205907

12059010

⟩ cong ⟨1205908

12059013

⟩ and⟨1205907

12059013

⟩ cong ⟨1205908

12059010

⟩ have two elements and Cayley tables asfollows

119860 119861

119860 119861 119860

119861 119860 119861

119860 119861

119860 119860 119861

119861 119861 119860

119860 119861

119860 119860 119861

119861 119860 119861

119860 119861

119860 119860 119860

119861 119861 119861

(14)

Monoids ⟨1205902

1205906

⟩ cong ⟨1205905

1205909

⟩ and also semigroups⟨1205902

1205907

⟩ cong ⟨1205905

1205908

⟩ ⟨1205902

1205908

⟩ cong ⟨1205905

1205907

⟩ and ⟨1205903

⟩ cong ⟨1205904

⟩ havethree elements and Cayley tables as follows

119860 119861 119862

119860 119860 119861 119862

119861 119861 119862 119861

119862 119862 119861 119862

119860 119861 119862

119860 119860 119861 119862

119861 119862 119861 119862

119862 119862 119861 119862

119860 119861 119862

119860 119860 119862 119862

119861 119861 119861 119861

119862 119862 119862 119862

119860 119861 119862

119860 119861 119862 119861

119861 119862 119861 119862

119862 119861 119862 119861

(15)

Semigroups ⟨1205906

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong ⟨1205907

1205909

⟩ and⟨1205907

1205908

⟩ have four elements and Cayley tables as follows

119860 119861 119862 119863

119860 119862 119863 119860 119861

119861 119860 119861 119862 119863

119862 119860 119861 119862 119863

119863 119862 119863 119860 119861

119860 119861 119862 119863

119860 119862 119860 119860 119862

119861 119863 119861 119861 119863

119862 119860 119862 119862 119860

119863 119861 119863 119863 119861

119860 119861 119862 119863

119860 119860 119862 119862 119860

119861 119863 119861 119861 119863

119862 119860 119862 119862 119860

119863 119863 119861 119861 119863

(16)

Semigroups ⟨1205902

12059011

⟩ cong ⟨1205905

12059012

⟩ ⟨1205902

12059012

⟩ cong ⟨1205905

12059011

⟩⟨1205902

12059013

⟩ cong ⟨1205905

12059010

⟩ ⟨1205903

1205906

⟩ cong ⟨1205904

1205909

⟩ and ⟨1205903

1205909

⟩ cong

⟨1205904

1205906

⟩ have five elements Semigroups ⟨1205902

1205903

⟩ cong ⟨1205904

1205905

⟩⟨1205902

1205904

⟩ cong ⟨1205903

1205905

⟩ and ⟨1205902

1205905

⟩ have six elements Construc-tion of Cayley tables for these semigroups as well as othersemigroups we leave it to the readers One can prepare suchtables removing from the Cayley table for M some columnsand rows Then change Kuratowski operations onto lettersof the alphabet For example ⟨120590

0

1205902

1205905

⟩ has the followingCayley table

0 119860 119861 119862 119863 119864 119865

0 0 119860 119861 119862 119863 119864 119865

119860 119860 119860 119862 119862 119864 119864 119862

119861 119861 119863 119861 119865 119863 119863 119865

119862 119862 119864 119862 119862 119864 119864 119862

119863 119863 119863 119865 119865 119863 119863 119865

119864 119864 119864 119862 119862 119864 119864 119862

119865 119865 119863 119865 119865 119863 119863 119865

(17)

Preparing theCayley table for ⟨1205900

1205902

1205905

⟩weput1205900

= 0 1205902

=

119860 1205905

= 119861 1205907

= 119862 1205908

= 119863 12059010

= 119864 and 12059013

= 119865 This tableimmediately shows that semigroups ⟨120590

2

1205905

⟩ and ⟨1205900

1205902

1205905

⟩

have exactly two automorphisms These are identities andrestrictions of A

Semigroup ⟨1205906

1205909

⟩ has eight elements Semigroups⟨1205902

1205909

⟩ cong ⟨1205905

1205906

⟩ and ⟨1205903

1205908

⟩ cong ⟨1205904

1205907

⟩ have nine ele-ments Semigroups ⟨120590

2

1205903

1205908

⟩ cong ⟨1205904

1205905

1205907

⟩ ⟨1205902

1205904

1205907

⟩ cong

⟨1205903

1205905

1205908

⟩ and ⟨1205902

1205905

1205906

⟩ have ten elements Semigroups⟨1205902

1205903

1205905

⟩ cong ⟨1205902

1205904

1205905

⟩ have eleven elements In the endthe semigroup ⟨120590

3

1205904

⟩ has twelve elementsThus the semigroup ⟨120590

3

1205904

⟩ includes fifty-seven semi-groups amongwhich there are ten groups fourteenmonoidsand also forty-three semigroups which are not monoidsThese semigroups consist of twenty-eight types of noniso-morphic semigroups two non-isomorphism types of groupstwo non-isomorphism types of monoids which are notgroups and twenty four non-isomorphism types of semi-groups which are not monoids

9 Viewing Semigroups Contained in M

91 Descriptive Data on Semigroups which Are Contained inM There are one hundred eighteen that is 118 = 2 sdot 57 +

6 Journal of Mathematics

4 semigroups which are contained in M These are fifty-seven semigroups contained in ⟨120590

3

1205904

⟩ fifty-seven monoidsformed by adding 120590

0

to a semigroup contained in ⟨1205903

1205904

⟩groups ⟨120590

0

⟩ ⟨1205901

⟩ and monoids ⟨1205901

1205906

⟩ = M1

and ⟨1205901

1205902

⟩ =

MThere are fifty-six types of nonisomorphic semigroups

in M These are twenty-eight non-isomorphism types ofsemigroups contained in ⟨120590

3

1205904

⟩ twenty-six types of non-isomorphic monoids formed by adding 120590

0

to a semigroupcontained in ⟨120590

3

1205904

⟩ and alsoM1

andM Indeed adding 1205900

to a semigroup which is not a monoid we obtain a monoidIn this way we get twenty-four types of nonisomorphicmonoids Adding 120590

0

to a monoid contained in ⟨1205903

1205904

⟩ weget two new non-isomorphism types ofmonoids But adding1205900

to a group contained in ⟨1205903

1205904

⟩ we get no new type ofmonoid since we get a monoid isomorphic to ⟨120590

2

12059010

⟩ or⟨1205902

1205906

⟩ The other two types areM1

andM

92 Semigroups which Are Not Monoids Below we havereproduced using the smallest number of generators andthe dictionary order a list of all 43 semigroups which areincluded in theM as follows

(1) ⟨1205902

1205903

⟩ = 1205902

1205903

1205906

1205907

12059010

12059011

(2) ⟨120590

2

1205903

1205905

⟩ = 1205902

1205903

1205905

1205906

12059013

(3) ⟨120590

2

1205903

1205908

⟩ = ⟨1205902

1205903

1205909

⟩ = ⟨1205902

1205903

12059012

⟩ = ⟨1205902

1205903

12059013

⟩ = 1205902

1205903

1205906

1205907

12059013

(4) ⟨120590

2

1205904

⟩ = 1205902

1205904

1205906

1205908

12059010

12059012

(5) ⟨120590

2

1205904

1205905

⟩ = 1205902

1205904

1205905

12059013

(6) ⟨120590

2

1205904

1205907

⟩ = ⟨1205902

1205904

1205909

⟩ = ⟨1205902

1205904

12059011

⟩ = ⟨1205902

1205904

12059013

⟩ = 1205902

1205904

1205906

1205907

12059013

(7) ⟨120590

2

1205905

⟩ = 1205902

1205905

1205907

1205908

12059010

12059013

(8) ⟨120590

2

1205905

1205906

⟩ = ⟨1205902

1205905

1205909

⟩ = ⟨1205902

1205905

12059011

⟩ = ⟨1205902

1205905

12059012

⟩ = 1205902

1205905

1205906

12059013

(9) ⟨120590

2

1205907

⟩ = 1205902

1205907

12059010

(10) ⟨120590

2

1205908

⟩ = 1205902

1205908

12059010

(11) ⟨120590

2

1205909

⟩ = ⟨1205902

1205906

12059013

⟩ = ⟨1205902

1205907

12059012

⟩ = ⟨1205902

1205908

12059011

⟩ = ⟨1205902

12059011

12059012

⟩ = ⟨1205902

12059011

12059013

⟩ = ⟨1205902

12059012

12059013

⟩ = 1205902

1205906

1205907

12059013

(12) ⟨120590

2

12059011

⟩ = ⟨1205902

1205906

1205907

⟩ = 1205902

1205906

1205907

12059010

12059011

(13) ⟨120590

2

12059012

⟩ = ⟨1205902

1205906

1205908

⟩ = 1205902

1205906

1205908

12059010

12059012

(14) ⟨120590

2

12059013

⟩ = ⟨1205902

1205907

1205908

⟩ = 1205902

1205907

1205908

12059010

12059013

(15) ⟨120590

3

⟩ = 1205903

1205907

12059011

(16) ⟨120590

3

1205904

⟩ = 1205902

1205903

12059013

(17) ⟨120590

3

1205905

⟩ = 1205903

1205905

1205907

1205909

12059011

12059013

(18) ⟨120590

3

1205905

1205906

⟩ = ⟨1205903

1205905

1205908

⟩ = ⟨1205903

1205905

12059010

⟩ = ⟨1205903

1205905

12059012

⟩ = 1205903

1205905

1205906

12059013

(19) ⟨120590

3

1205906

⟩ = ⟨1205903

12059010

⟩ = 1205903

1205906

1205907

12059010

12059011

(20) ⟨120590

3

1205908

⟩ = ⟨1205903

12059012

⟩ = 1205903

1205906

1205907

12059013

(21) ⟨120590

3

1205909

⟩ = ⟨1205903

12059013

⟩ = 1205903

1205907

1205909

12059011

12059013

(22) ⟨120590

4

⟩ = 1205904

1205908

12059012

(23) ⟨1205904

1205905

⟩ = 1205904

1205905

1205908

1205909

12059012

12059013

(24) ⟨120590

4

1205905

1205906

⟩ = ⟨1205904

1205905

1205907

⟩ = ⟨1205904

1205905

12059010

⟩ = ⟨1205904

1205905

12059011

⟩ = 1205904

1205905

12059013

(25) ⟨120590

4

1205906

⟩ = ⟨1205904

12059010

⟩ = 1205904

1205906

1205908

12059010

12059012

(26) ⟨120590

4

1205907

⟩ = ⟨1205904

12059011

⟩ = 1205904

1205906

1205907

12059013

(27) ⟨120590

4

1205909

⟩ = ⟨1205904

12059013

⟩ = 1205904

1205908

1205909

12059012

12059013

(28) ⟨120590

5

1205906

⟩ = ⟨1205905

1205907

12059012

⟩ = ⟨1205905

1205908

12059011

⟩ = ⟨1205905

1205909

12059010

⟩ = ⟨1205905

12059010

12059011

⟩ = ⟨1205905

12059010

12059012

⟩ = ⟨1205905

12059011

12059012

⟩ = 1205905

1205906

12059013

(29) ⟨120590

5

1205907

⟩ = 1205905

1205907

12059013

(30) ⟨120590

5

1205908

⟩ = 1205905

1205908

12059013

(31) ⟨120590

5

12059010

⟩ = ⟨1205905

1205907

1205908

⟩ = 1205905

1205907

1205908

12059010

12059013

(32) ⟨120590

5

12059011

⟩ = ⟨1205905

1205907

1205909

⟩ = 1205905

1205907

1205909

12059011

12059013

(33) ⟨120590

5

12059012

⟩ = ⟨1205905

1205908

1205909

⟩ = 1205905

1205908

1205909

12059012

12059013

(34) ⟨120590

6

1205907

⟩ = ⟨1205906

12059011

⟩ = ⟨12059010

12059011

⟩ = 1205906

1205907

12059010

12059011

(35) ⟨120590

6

1205908

⟩ = ⟨1205906

12059012

⟩ = ⟨12059010

12059012

⟩ = 1205906

1205908

12059010

12059012

(36) ⟨120590

6

1205909

⟩ = ⟨1205906

12059013

⟩ = ⟨1205907

12059012

⟩ = ⟨1205908

12059011

⟩ = ⟨1205909

12059010

⟩ = ⟨12059011

12059012

⟩ = ⟨1205906

1205907

1205908

⟩ = ⟨1205907

1205908

1205909

⟩ =⟨12059010

12059011

12059013

⟩ = ⟨12059010

12059012

12059013

⟩ = 1205906

1205907

12059013

(37) ⟨120590

7

1205908

⟩ = 1205907

1205908

12059010

12059013

(38) ⟨120590

7

1205909

⟩ = 1205907

1205909

12059011

12059013

(39) ⟨120590

7

12059010

⟩ = 1205907

12059010

(40) ⟨120590

7

12059013

⟩ = 1205907

12059013

(41) ⟨120590

8

1205909

⟩ = 1205908

1205909

12059012

12059013

(42) ⟨120590

8

12059010

⟩ = 1205908

12059010

(43) ⟨120590

8

12059013

⟩ = 1205908

12059013

93 Isomorphism Types of Semigroups Contained in M Sys-tematize the list of all isomorphism types of semigroupscontained in themonoidM Isomorphisms which are restric-tions of the isomorphism A will be regarded as self-evidentand therefore they will not be commented on

(i) The monoid M contains 12 groups with 2 isomor-phism types These are 7 one-element groups and 5two-element groups

(ii) ThemonoidM contains 8 two-element monoids with2 isomorphism types These are 120590

0

added to 6 one-element groups and ⟨120590

2

12059010

⟩ cong ⟨1205905

12059013

⟩ Also itcontains 4 two-element semigroups not monoidswith 2 isomorphism types These are ⟨120590

7

12059010

⟩ cong

⟨1205908

12059013

⟩ and ⟨1205908

12059010

⟩ cong ⟨1205907

12059013

⟩(iii) The monoid M contains 12 three-element monoids

with 5 isomorphism types These are 1205900

addedto 4 two-element groups and also ⟨120590

0

1205902

12059010

⟩ cong

⟨1205900

1205905

12059013

⟩ ⟨1205900

1205907

12059010

⟩ cong ⟨1205900

1205908

12059013

⟩⟨1205900

1205908

12059010

⟩ cong ⟨1205900

1205907

12059013

⟩ and ⟨1205902

1205906

⟩ cong ⟨1205905

1205909

⟩(iv) The monoidM contains 6 three-element semigroups

not monoids with 3 isomorphism types These are⟨1205902

1205907

⟩ cong ⟨1205905

1205908

⟩ ⟨1205902

1205908

⟩ cong ⟨1205905

1205907

⟩ and ⟨1205903

⟩ cong

⟨1205904

⟩

Journal of Mathematics 7

(v) The monoid M contains 8 four-element monoidseach contains 120590

0

with 4 isomorphism types Theseare semigroups from two preceding items that can besubstantially extended by 120590

0

(vi) The monoid M contains 5 four-element semigroups

not monoids with 3 isomorphism types These are⟨1205906

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong ⟨1205907

1205909

⟩ and ⟨1205907

1205908

⟩These semigroups extended by 120590

0

yield 5 monoids allwhich consist of five elements with 3 isomorphismtypes

(vii) ThemonoidM contains 10 five-element semigroupsmdashnot monoids with 5 isomorphism types These are⟨1205902

12059011

⟩ cong ⟨1205905

12059012

⟩ ⟨1205902

12059012

⟩ cong ⟨1205905

12059011

⟩ ⟨1205902

12059013

⟩ cong

⟨1205905

12059010

⟩ ⟨1205903

1205906

⟩ cong ⟨1205904

1205909

⟩ and ⟨1205903

1205909

⟩ cong ⟨1205904

1205906

⟩These semigroups extended by120590

0

yield 10monoids allof which consist of six elements with 5 isomorphismtypes We get 10 new isomorphism types since thesemigroups are distinguished by semigroups ⟨120590

3

⟩

and ⟨1205904

⟩ and by nonisomorphic semigroups ⟨1205906

1205907

⟩⟨1205906

1205908

⟩ and ⟨1205907

1205908

⟩(viii) The monoid M contains 5 six-element semigroups

not monoids with 3 isomorphism types These are⟨1205902

1205903

⟩ cong ⟨1205904

1205905

⟩ ⟨1205902

1205904

⟩ cong ⟨1205903

1205905

⟩ and ⟨1205902

1205905

⟩These semigroups extended by 120590

0

yield 5 monoidsall of which consist of seven elements with 3 isomor-phism types We get 6 new isomorphism types sincethe semigroups are distinguished by non isomorphicsemigroups ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ and ⟨1205907

1205908

⟩(ix) ThemonoidM contains no seven-element semigroup

not a monoid no eight-element monoid and theonly semigroup ⟨120590

6

1205909

⟩ with exactly eight elementsand the only monoid ⟨120590

0

1205906

1205909

⟩ with exactly nineelements

(x) The monoid M contains 4 nine-element semigroupsnot monoids with 2 isomorphism types These are⟨1205902

1205909

⟩ cong ⟨1205905

1205906

⟩ and ⟨1205903

1205908

⟩ cong ⟨1205904

1205907

⟩ Thesemigroup ⟨120590

2

1205909

⟩ does not contain a semigroupisomorphic to ⟨120590

3

⟩ hence it is not isomorphic to⟨1205903

1205908

⟩ These semigroups extended by 1205900

yield 4monoids all of which consist of ten elements with 2isomorphism types

(xi) The monoid M contains 6 ten-element semigroupsnot monoids with 4 isomorphism typesThese are ⟨120590

1

1205906

⟩ ⟨1205902

1205903

1205908

⟩ cong ⟨1205904

1205905

1205906

⟩⟨1205902

1205904

1205907

⟩ cong ⟨1205903

1205905

1205906

⟩ and ⟨1205902

1205905

1205906

⟩ Thesesemigroups (except ⟨120590

1

1205906

⟩) extended by 1205900

yield 5monoids all of which consist of ten elements with3 isomorphism types We get 6 new isomorphismtypes since the semigroups are distinguished bynot isomorphic semigroups ⟨120590

2

1205903

⟩ ⟨1205902

1205904

⟩ and⟨1205902

1205905

⟩(xii) The monoid M contains 2 isomorphic semigroups

not monoids which consist of 11 elements that is⟨1205902

1205903

1205905

⟩ cong ⟨1205902

1205904

1205905

⟩These semigroups extendedby 1205900

yield 2 isomorphic monoids which consistof 12 elements The monoid M contains no largersemigroup with the exception of itself

10 Cancellation Rules Motivated by SomeTopological Properties

101 Some Consequences of the Axiom 0 = 0minus So far we usedonly the following relations (above named cancellation rules)1205902

∘ 1205902

= 1205902

1205901

∘ 1205901

= 1205900

1205902

∘ 12059012

= 1205906

and 1205902

∘ 12059013

= 1205907

When one assumes119883 = 0 = 0

minus then

119883 = 1205900

(119883) = 1205902

(119883) = 1205905

(119883) = 1205907

(119883)

= 1205908

(119883) = 12059010

(119883) = 12059013

(119883)

0 = 1205901

(119883) = 1205903

(119883) = 1205904

(119883) = 1205906

(119883)

= 1205909

(119883) = 12059011

(119883) = 12059012

(119883)

(18)

Using the substitution 119860 997891rarr 119860119888 one obtains equiv-

alent relations between operations from the set 1205901

1205903

1205904

1205906

1205909

12059011

12059012

and conversely Therefore cancellationrules are topologically reasonable only between the opera-tions from the monoid as follows

⟨1205900

1205902

1205905

⟩ = 1205900

1205902

1205905

1205907

1205908

12059010

12059013

(19)

Chapman see [3] considered properties of subsets withrespect to such relations Below we are going to identifyrelations that are determined by some topological spacescompare [10 11]

102 The Relation 1205900 = 1205902 If a topological space 119883 isdiscrete then there exist two Kuratowski operation onlyThese are 120590

0

and 1205901

So themonoid of Kuratowski operationsreduced to the group ⟨120590

1

⟩The relation 120590

0

= 1205902

is equivalent to any relation 1205900

=

120590119894

where 119894 isin 5 7 8 10 13 Any such relation implies thatevery subset of 119883 has to be closed and open that is 119883 hasto be discrete However one can check these using (only) thefacts that 120590

1

is an involution and 1205902

is an idempotent and thecancellation rules that is the Cayley table forM So 120590

0

= 1205902

follows

1205900

= 1205902

= 1205905

= 1205907

= 1205908

= 12059010

= 12059013

(20)

103 The Relation 1205902 = 1205905 Topologically 1205902 = 1205905

meansthat 119883 must be discrete This is so because 119860119888minus119888 sube 119860 sube 119860

minus

for any 119860 sube 119883

104 The Relation 1205902 = 1205907 Topologically the relation 1205902 =1205907

implies 1205900

= 1205902

But it requires the use of topology axioms0 = 0minus and 119862minus cup 119861minus = (119862 cup 119861)minus for each 119862 and 119861

Lemma 5 For any topological space 1205902

= 1205907

implies 1205902

= 1205908

Proof If 1205902

= 1205907

then 119860 = 0 rArr 119860119888minus119888

= 0 for any 119860 sube 119883Indeed if 119860119888minus119888 = 0 then 120590

7

(119860) = 0minus

= 0 Since 119860 = 0 then1205902

(119860) = 0 Hence 1205902

(119860) = 1205907

(119860) a contradictionThe axiom 119862

minus

cup 119861minus

= (119862 cup 119861)minus implies that always

(119860minus

cap 119860minus119888minus

)119888minus119888

= 0 (21)

Thus the additional assumption 1205902

= 1205907

follows that always119860minus

cap 119860minus119888minus

= 0 Therefore always 119860minus119888minus = 119860minus119888 but this means

8 Journal of Mathematics

that any closed set has to be open in other words 1205902

= 1205908

Proposition 6 For any topological space 1205902

= 1205907

implies 1205900

=

1205902

Proof But the relation 1205902

= 1205907

is equivalent with 1205905

= 1205908

ByLemma 5 we get 120590

2

= 1205905

Finally 1205900

= 1205902

The relation 1205902

= 1205907

has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put

1205902

(0) = 119886 = 1205902

(119886) 119883 = 1205902

(119883) = 1205902

(119887) (22)

Then check that 1205902

= 1205907

and

1205908

(0) = 1205905

(119886) = 1205904

(119887) = 1205901

(119883) = 0 (23)

in other words 1205902

= 1205907

and 1205902

= 1205908

However 1205902

= 1205907

isequivalent to 120590

5

= 1205908

This relation implies 120590

2

= 1205907

= 12059010

1205905

= 1205908

= 12059013

1205903

= 1205906

= 12059011

and 1205904

= 1205909

= 12059012

For this interpretationthe monoidM(120590

2

= 1205907

) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590

2

= 1205907

) thereare relations covered by the following proposition only

Proposition 7 For any monoid with the Cayley table as forM the relation 120590

2

= 1205907

implies

(i) 1205902

= 1205907

= 12059010

= 1205907

∘ 1205902

(ii) 1205905

= 1205901

∘1205902

∘1205901

= 1205901

∘1205907

∘1205901

= 1205908

= 1205905

∘1205902

= 1205905

∘1205907

=

12059013

(iii) 120590

3

= 1205902

∘ 1205901

= 1205907

∘ 1205901

= 1205906

= 12059010

∘ 1205901

= 12059011

(iv) 1205904

= 1205901

∘ 1205902

= 1205901

∘ 1205907

= 1205909

= 1205901

∘ 12059010

= 12059012

Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology

105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908

means that any closed set is open tooThus if119883 = 119886 119887 119888

is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590

2

= 1205908

) is the monoid of all Kuratowski operationsover 119883 Relations 120590

2

= 1205908

and 1205905

= 1205907

are equivalent Theyimply relations 120590

2

= 1205908

= 12059010

1205905

= 1205907

= 12059013

1205903

= 1205909

= 12059011

and 120590

4

= 1205906

= 12059012

The permutation

(1205900

1205901

1205902

1205903

1205904

1205905

1205900

1205901

1205905

1205904

1205903

1205902

) (24)

determines the isomorphism between monoidsM(1205902

= 1205907

)

andM(1205902

= 1205908

)

106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590

2

= 1205908

means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590

2

= 12059010

implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902

= 1205908

The relation 1205902

= 12059013

follows that each closed sethas to be open so it implies 120590

2

= 1205908

too

107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590

7

= 1205908

and 12059010

= 12059013

areequivalent Each of them gives 120590

7

= 1205908

= 12059010

= 12059013

and1205906

= 1205909

= 12059011

= 12059012

If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590

7

= 1205908

) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements

108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590

7

= 12059010

1205908

= 12059013

1205906

= 12059011

and 120590

9

= 12059012

are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590

7

= 12059010

) is the monoid of allKuratowski operations over119883 and consists of 10 elements

109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590

7

= 12059013

and 1205908

= 12059010

areequivalent Also 120590

6

= 12059012

and 1205909

= 12059011

These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted

1205901

1205902

1205903

1205904

1205905

1205906

1205907

1205908

1205909

1205901

1205900

1205904

1205905

1205902

1205903

1205908

1205909

1205906

1205907

1205902

1205903

1205902

1205903

1205906

1205907

1205906

1205907

1205908

1205909

1205903

1205902

1205906

1205907

1205902

1205903

1205908

1205909

1205906

1205907

1205904

1205905

1205904

1205905

1205908

1205909

1205908

1205909

1205906

1205907

1205905

1205904

1205908

1205909

1205904

1205905

1205906

1205907

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

1205907

1205906

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205909

1205908

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

(25)

If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906

= 12059012

The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590

7

= 1205908

Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590

7

= 1205908

and 1205907

= 12059013

To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that

(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860

119888minus119888Then check that

(i) 1205900

(119860) = 119860 and 1205901

(119860) = 119883 (119860119888minus119888

cup 119861 cup 119862)(ii) 1205902

(119860) = 119880 cup 119861minus

cup119882 and 1205903

(119860) = 119883 119860119888minus119888

(iii) 1205904

(119860) = 119881 119861minus and 120590

5

(119860) = 119860119888minus119888

(iv) 1205906

(119860) = 12059012

(119860) = 119881 and 1205907

(119860) = 12059013

(119860) = 119880(v) 1205908

(119860) = 12059010

(119860) = 119880 cup 119882 and 1205909

(119860) = 12059011

(119860) =

119881 cup 119882Hence we have that 120590

7

= 1205908

Note that if119882 = 0 then 1205907

(119860) =

1205908

(119860) This is the case of subsets of 120573119873

Journal of Mathematics 9

References

[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922

[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977

[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962

[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010

[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008

[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966

[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996

[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956

[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889

[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967

[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968

[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 5

Each of these semigroups has six element so in M thereare five six-element semigroups of three isomorphism typessince ⟨120590

2

1205905

⟩ has 6 elements which are idempotentsIn ⟨1205903

1205904

⟩ there are seven semigroups which have threegenerators and have not two generators These are

(1) ⟨1205902

1205903

1205908

⟩ = ⟨1205902

1205903

1205909

⟩ = ⟨1205902

1205903

12059012

⟩ = ⟨1205902

1205903

12059013

⟩

(2) ⟨1205904

1205905

1205906

⟩ = ⟨1205904

1205905

1205907

⟩ = ⟨1205904

1205905

12059010

⟩ = ⟨1205904

1205905

12059011

⟩

(3) ⟨1205902

1205904

1205907

⟩ = ⟨1205902

1205904

1205909

⟩ = ⟨1205902

1205904

12059011

⟩ = ⟨1205902

1205904

12059013

⟩

(4) ⟨1205903

1205905

1205906

⟩ = ⟨1205903

1205905

1205908

⟩ = ⟨1205903

1205905

12059010

⟩ = ⟨1205903

1205905

12059012

⟩

(5) ⟨1205902

1205905

1205906

⟩ = ⟨1205902

1205905

1205909

⟩ = ⟨1205902

1205905

12059011

⟩ = ⟨1205902

1205905

12059012

⟩

(6) ⟨1205902

1205903

1205905

⟩

(7) ⟨1205902

1205904

1205905

⟩

8 Semigroups which Are Contained in ⟨12059031205904⟩

Groups ⟨1205902

⟩ cong ⟨1205905

⟩ cong ⟨1205907

⟩ cong ⟨1205908

⟩ cong ⟨12059010

⟩ cong ⟨12059013

⟩ have oneelement and are isomorphic

Groups ⟨1205906

⟩ cong ⟨1205909

⟩ cong ⟨12059011

⟩ cong ⟨12059012

⟩ monoids ⟨1205902

12059010

⟩ cong

⟨1205905

12059013

⟩ and also semigroups ⟨1205907

12059010

⟩ cong ⟨1205908

12059013

⟩ and⟨1205907

12059013

⟩ cong ⟨1205908

12059010

⟩ have two elements and Cayley tables asfollows

119860 119861

119860 119861 119860

119861 119860 119861

119860 119861

119860 119860 119861

119861 119861 119860

119860 119861

119860 119860 119861

119861 119860 119861

119860 119861

119860 119860 119860

119861 119861 119861

(14)

Monoids ⟨1205902

1205906

⟩ cong ⟨1205905

1205909

⟩ and also semigroups⟨1205902

1205907

⟩ cong ⟨1205905

1205908

⟩ ⟨1205902

1205908

⟩ cong ⟨1205905

1205907

⟩ and ⟨1205903

⟩ cong ⟨1205904

⟩ havethree elements and Cayley tables as follows

119860 119861 119862

119860 119860 119861 119862

119861 119861 119862 119861

119862 119862 119861 119862

119860 119861 119862

119860 119860 119861 119862

119861 119862 119861 119862

119862 119862 119861 119862

119860 119861 119862

119860 119860 119862 119862

119861 119861 119861 119861

119862 119862 119862 119862

119860 119861 119862

119860 119861 119862 119861

119861 119862 119861 119862

119862 119861 119862 119861

(15)

Semigroups ⟨1205906

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong ⟨1205907

1205909

⟩ and⟨1205907

1205908

⟩ have four elements and Cayley tables as follows

119860 119861 119862 119863

119860 119862 119863 119860 119861

119861 119860 119861 119862 119863

119862 119860 119861 119862 119863

119863 119862 119863 119860 119861

119860 119861 119862 119863

119860 119862 119860 119860 119862

119861 119863 119861 119861 119863

119862 119860 119862 119862 119860

119863 119861 119863 119863 119861

119860 119861 119862 119863

119860 119860 119862 119862 119860

119861 119863 119861 119861 119863

119862 119860 119862 119862 119860

119863 119863 119861 119861 119863

(16)

Semigroups ⟨1205902

12059011

⟩ cong ⟨1205905

12059012

⟩ ⟨1205902

12059012

⟩ cong ⟨1205905

12059011

⟩⟨1205902

12059013

⟩ cong ⟨1205905

12059010

⟩ ⟨1205903

1205906

⟩ cong ⟨1205904

1205909

⟩ and ⟨1205903

1205909

⟩ cong

⟨1205904

1205906

⟩ have five elements Semigroups ⟨1205902

1205903

⟩ cong ⟨1205904

1205905

⟩⟨1205902

1205904

⟩ cong ⟨1205903

1205905

⟩ and ⟨1205902

1205905

⟩ have six elements Construc-tion of Cayley tables for these semigroups as well as othersemigroups we leave it to the readers One can prepare suchtables removing from the Cayley table for M some columnsand rows Then change Kuratowski operations onto lettersof the alphabet For example ⟨120590

0

1205902

1205905

⟩ has the followingCayley table

0 119860 119861 119862 119863 119864 119865

0 0 119860 119861 119862 119863 119864 119865

119860 119860 119860 119862 119862 119864 119864 119862

119861 119861 119863 119861 119865 119863 119863 119865

119862 119862 119864 119862 119862 119864 119864 119862

119863 119863 119863 119865 119865 119863 119863 119865

119864 119864 119864 119862 119862 119864 119864 119862

119865 119865 119863 119865 119865 119863 119863 119865

(17)

Preparing theCayley table for ⟨1205900

1205902

1205905

⟩weput1205900

= 0 1205902

=

119860 1205905

= 119861 1205907

= 119862 1205908

= 119863 12059010

= 119864 and 12059013

= 119865 This tableimmediately shows that semigroups ⟨120590

2

1205905

⟩ and ⟨1205900

1205902

1205905

⟩

have exactly two automorphisms These are identities andrestrictions of A

Semigroup ⟨1205906

1205909

⟩ has eight elements Semigroups⟨1205902

1205909

⟩ cong ⟨1205905

1205906

⟩ and ⟨1205903

1205908

⟩ cong ⟨1205904

1205907

⟩ have nine ele-ments Semigroups ⟨120590

2

1205903

1205908

⟩ cong ⟨1205904

1205905

1205907

⟩ ⟨1205902

1205904

1205907

⟩ cong

⟨1205903

1205905

1205908

⟩ and ⟨1205902

1205905

1205906

⟩ have ten elements Semigroups⟨1205902

1205903

1205905

⟩ cong ⟨1205902

1205904

1205905

⟩ have eleven elements In the endthe semigroup ⟨120590

3

1205904

⟩ has twelve elementsThus the semigroup ⟨120590

3

1205904

⟩ includes fifty-seven semi-groups amongwhich there are ten groups fourteenmonoidsand also forty-three semigroups which are not monoidsThese semigroups consist of twenty-eight types of noniso-morphic semigroups two non-isomorphism types of groupstwo non-isomorphism types of monoids which are notgroups and twenty four non-isomorphism types of semi-groups which are not monoids

9 Viewing Semigroups Contained in M

91 Descriptive Data on Semigroups which Are Contained inM There are one hundred eighteen that is 118 = 2 sdot 57 +

6 Journal of Mathematics

4 semigroups which are contained in M These are fifty-seven semigroups contained in ⟨120590

3

1205904

⟩ fifty-seven monoidsformed by adding 120590

0

to a semigroup contained in ⟨1205903

1205904

⟩groups ⟨120590

0

⟩ ⟨1205901

⟩ and monoids ⟨1205901

1205906

⟩ = M1

and ⟨1205901

1205902

⟩ =

MThere are fifty-six types of nonisomorphic semigroups

in M These are twenty-eight non-isomorphism types ofsemigroups contained in ⟨120590

3

1205904

⟩ twenty-six types of non-isomorphic monoids formed by adding 120590

0

to a semigroupcontained in ⟨120590

3

1205904

⟩ and alsoM1

andM Indeed adding 1205900

to a semigroup which is not a monoid we obtain a monoidIn this way we get twenty-four types of nonisomorphicmonoids Adding 120590

0

to a monoid contained in ⟨1205903

1205904

⟩ weget two new non-isomorphism types ofmonoids But adding1205900

to a group contained in ⟨1205903

1205904

⟩ we get no new type ofmonoid since we get a monoid isomorphic to ⟨120590

2

12059010

⟩ or⟨1205902

1205906

⟩ The other two types areM1

andM

92 Semigroups which Are Not Monoids Below we havereproduced using the smallest number of generators andthe dictionary order a list of all 43 semigroups which areincluded in theM as follows

(1) ⟨1205902

1205903

⟩ = 1205902

1205903

1205906

1205907

12059010

12059011

(2) ⟨120590

2

1205903

1205905

⟩ = 1205902

1205903

1205905

1205906

12059013

(3) ⟨120590

2

1205903

1205908

⟩ = ⟨1205902

1205903

1205909

⟩ = ⟨1205902

1205903

12059012

⟩ = ⟨1205902

1205903

12059013

⟩ = 1205902

1205903

1205906

1205907

12059013

(4) ⟨120590

2

1205904

⟩ = 1205902

1205904

1205906

1205908

12059010

12059012

(5) ⟨120590

2

1205904

1205905

⟩ = 1205902

1205904

1205905

12059013

(6) ⟨120590

2

1205904

1205907

⟩ = ⟨1205902

1205904

1205909

⟩ = ⟨1205902

1205904

12059011

⟩ = ⟨1205902

1205904

12059013

⟩ = 1205902

1205904

1205906

1205907

12059013

(7) ⟨120590

2

1205905

⟩ = 1205902

1205905

1205907

1205908

12059010

12059013

(8) ⟨120590

2

1205905

1205906

⟩ = ⟨1205902

1205905

1205909

⟩ = ⟨1205902

1205905

12059011

⟩ = ⟨1205902

1205905

12059012

⟩ = 1205902

1205905

1205906

12059013

(9) ⟨120590

2

1205907

⟩ = 1205902

1205907

12059010

(10) ⟨120590

2

1205908

⟩ = 1205902

1205908

12059010

(11) ⟨120590

2

1205909

⟩ = ⟨1205902

1205906

12059013

⟩ = ⟨1205902

1205907

12059012

⟩ = ⟨1205902

1205908

12059011

⟩ = ⟨1205902

12059011

12059012

⟩ = ⟨1205902

12059011

12059013

⟩ = ⟨1205902

12059012

12059013

⟩ = 1205902

1205906

1205907

12059013

(12) ⟨120590

2

12059011

⟩ = ⟨1205902

1205906

1205907

⟩ = 1205902

1205906

1205907

12059010

12059011

(13) ⟨120590

2

12059012

⟩ = ⟨1205902

1205906

1205908

⟩ = 1205902

1205906

1205908

12059010

12059012

(14) ⟨120590

2

12059013

⟩ = ⟨1205902

1205907

1205908

⟩ = 1205902

1205907

1205908

12059010

12059013

(15) ⟨120590

3

⟩ = 1205903

1205907

12059011

(16) ⟨120590

3

1205904

⟩ = 1205902

1205903

12059013

(17) ⟨120590

3

1205905

⟩ = 1205903

1205905

1205907

1205909

12059011

12059013

(18) ⟨120590

3

1205905

1205906

⟩ = ⟨1205903

1205905

1205908

⟩ = ⟨1205903

1205905

12059010

⟩ = ⟨1205903

1205905

12059012

⟩ = 1205903

1205905

1205906

12059013

(19) ⟨120590

3

1205906

⟩ = ⟨1205903

12059010

⟩ = 1205903

1205906

1205907

12059010

12059011

(20) ⟨120590

3

1205908

⟩ = ⟨1205903

12059012

⟩ = 1205903

1205906

1205907

12059013

(21) ⟨120590

3

1205909

⟩ = ⟨1205903

12059013

⟩ = 1205903

1205907

1205909

12059011

12059013

(22) ⟨120590

4

⟩ = 1205904

1205908

12059012

(23) ⟨1205904

1205905

⟩ = 1205904

1205905

1205908

1205909

12059012

12059013

(24) ⟨120590

4

1205905

1205906

⟩ = ⟨1205904

1205905

1205907

⟩ = ⟨1205904

1205905

12059010

⟩ = ⟨1205904

1205905

12059011

⟩ = 1205904

1205905

12059013

(25) ⟨120590

4

1205906

⟩ = ⟨1205904

12059010

⟩ = 1205904

1205906

1205908

12059010

12059012

(26) ⟨120590

4

1205907

⟩ = ⟨1205904

12059011

⟩ = 1205904

1205906

1205907

12059013

(27) ⟨120590

4

1205909

⟩ = ⟨1205904

12059013

⟩ = 1205904

1205908

1205909

12059012

12059013

(28) ⟨120590

5

1205906

⟩ = ⟨1205905

1205907

12059012

⟩ = ⟨1205905

1205908

12059011

⟩ = ⟨1205905

1205909

12059010

⟩ = ⟨1205905

12059010

12059011

⟩ = ⟨1205905

12059010

12059012

⟩ = ⟨1205905

12059011

12059012

⟩ = 1205905

1205906

12059013

(29) ⟨120590

5

1205907

⟩ = 1205905

1205907

12059013

(30) ⟨120590

5

1205908

⟩ = 1205905

1205908

12059013

(31) ⟨120590

5

12059010

⟩ = ⟨1205905

1205907

1205908

⟩ = 1205905

1205907

1205908

12059010

12059013

(32) ⟨120590

5

12059011

⟩ = ⟨1205905

1205907

1205909

⟩ = 1205905

1205907

1205909

12059011

12059013

(33) ⟨120590

5

12059012

⟩ = ⟨1205905

1205908

1205909

⟩ = 1205905

1205908

1205909

12059012

12059013

(34) ⟨120590

6

1205907

⟩ = ⟨1205906

12059011

⟩ = ⟨12059010

12059011

⟩ = 1205906

1205907

12059010

12059011

(35) ⟨120590

6

1205908

⟩ = ⟨1205906

12059012

⟩ = ⟨12059010

12059012

⟩ = 1205906

1205908

12059010

12059012

(36) ⟨120590

6

1205909

⟩ = ⟨1205906

12059013

⟩ = ⟨1205907

12059012

⟩ = ⟨1205908

12059011

⟩ = ⟨1205909

12059010

⟩ = ⟨12059011

12059012

⟩ = ⟨1205906

1205907

1205908

⟩ = ⟨1205907

1205908

1205909

⟩ =⟨12059010

12059011

12059013

⟩ = ⟨12059010

12059012

12059013

⟩ = 1205906

1205907

12059013

(37) ⟨120590

7

1205908

⟩ = 1205907

1205908

12059010

12059013

(38) ⟨120590

7

1205909

⟩ = 1205907

1205909

12059011

12059013

(39) ⟨120590

7

12059010

⟩ = 1205907

12059010

(40) ⟨120590

7

12059013

⟩ = 1205907

12059013

(41) ⟨120590

8

1205909

⟩ = 1205908

1205909

12059012

12059013

(42) ⟨120590

8

12059010

⟩ = 1205908

12059010

(43) ⟨120590

8

12059013

⟩ = 1205908

12059013

93 Isomorphism Types of Semigroups Contained in M Sys-tematize the list of all isomorphism types of semigroupscontained in themonoidM Isomorphisms which are restric-tions of the isomorphism A will be regarded as self-evidentand therefore they will not be commented on

(i) The monoid M contains 12 groups with 2 isomor-phism types These are 7 one-element groups and 5two-element groups

(ii) ThemonoidM contains 8 two-element monoids with2 isomorphism types These are 120590

0

added to 6 one-element groups and ⟨120590

2

12059010

⟩ cong ⟨1205905

12059013

⟩ Also itcontains 4 two-element semigroups not monoidswith 2 isomorphism types These are ⟨120590

7

12059010

⟩ cong

⟨1205908

12059013

⟩ and ⟨1205908

12059010

⟩ cong ⟨1205907

12059013

⟩(iii) The monoid M contains 12 three-element monoids

with 5 isomorphism types These are 1205900

addedto 4 two-element groups and also ⟨120590

0

1205902

12059010

⟩ cong

⟨1205900

1205905

12059013

⟩ ⟨1205900

1205907

12059010

⟩ cong ⟨1205900

1205908

12059013

⟩⟨1205900

1205908

12059010

⟩ cong ⟨1205900

1205907

12059013

⟩ and ⟨1205902

1205906

⟩ cong ⟨1205905

1205909

⟩(iv) The monoidM contains 6 three-element semigroups

not monoids with 3 isomorphism types These are⟨1205902

1205907

⟩ cong ⟨1205905

1205908

⟩ ⟨1205902

1205908

⟩ cong ⟨1205905

1205907

⟩ and ⟨1205903

⟩ cong

⟨1205904

⟩

Journal of Mathematics 7

(v) The monoid M contains 8 four-element monoidseach contains 120590

0

with 4 isomorphism types Theseare semigroups from two preceding items that can besubstantially extended by 120590

0

(vi) The monoid M contains 5 four-element semigroups

not monoids with 3 isomorphism types These are⟨1205906

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong ⟨1205907

1205909

⟩ and ⟨1205907

1205908

⟩These semigroups extended by 120590

0

yield 5 monoids allwhich consist of five elements with 3 isomorphismtypes

(vii) ThemonoidM contains 10 five-element semigroupsmdashnot monoids with 5 isomorphism types These are⟨1205902

12059011

⟩ cong ⟨1205905

12059012

⟩ ⟨1205902

12059012

⟩ cong ⟨1205905

12059011

⟩ ⟨1205902

12059013

⟩ cong

⟨1205905

12059010

⟩ ⟨1205903

1205906

⟩ cong ⟨1205904

1205909

⟩ and ⟨1205903

1205909

⟩ cong ⟨1205904

1205906

⟩These semigroups extended by120590

0

yield 10monoids allof which consist of six elements with 5 isomorphismtypes We get 10 new isomorphism types since thesemigroups are distinguished by semigroups ⟨120590

3

⟩

and ⟨1205904

⟩ and by nonisomorphic semigroups ⟨1205906

1205907

⟩⟨1205906

1205908

⟩ and ⟨1205907

1205908

⟩(viii) The monoid M contains 5 six-element semigroups

not monoids with 3 isomorphism types These are⟨1205902

1205903

⟩ cong ⟨1205904

1205905

⟩ ⟨1205902

1205904

⟩ cong ⟨1205903

1205905

⟩ and ⟨1205902

1205905

⟩These semigroups extended by 120590

0

yield 5 monoidsall of which consist of seven elements with 3 isomor-phism types We get 6 new isomorphism types sincethe semigroups are distinguished by non isomorphicsemigroups ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ and ⟨1205907

1205908

⟩(ix) ThemonoidM contains no seven-element semigroup

not a monoid no eight-element monoid and theonly semigroup ⟨120590

6

1205909

⟩ with exactly eight elementsand the only monoid ⟨120590

0

1205906

1205909

⟩ with exactly nineelements

(x) The monoid M contains 4 nine-element semigroupsnot monoids with 2 isomorphism types These are⟨1205902

1205909

⟩ cong ⟨1205905

1205906

⟩ and ⟨1205903

1205908

⟩ cong ⟨1205904

1205907

⟩ Thesemigroup ⟨120590

2

1205909

⟩ does not contain a semigroupisomorphic to ⟨120590

3

⟩ hence it is not isomorphic to⟨1205903

1205908

⟩ These semigroups extended by 1205900

yield 4monoids all of which consist of ten elements with 2isomorphism types

(xi) The monoid M contains 6 ten-element semigroupsnot monoids with 4 isomorphism typesThese are ⟨120590

1

1205906

⟩ ⟨1205902

1205903

1205908

⟩ cong ⟨1205904

1205905

1205906

⟩⟨1205902

1205904

1205907

⟩ cong ⟨1205903

1205905

1205906

⟩ and ⟨1205902

1205905

1205906

⟩ Thesesemigroups (except ⟨120590

1

1205906

⟩) extended by 1205900

yield 5monoids all of which consist of ten elements with3 isomorphism types We get 6 new isomorphismtypes since the semigroups are distinguished bynot isomorphic semigroups ⟨120590

2

1205903

⟩ ⟨1205902

1205904

⟩ and⟨1205902

1205905

⟩(xii) The monoid M contains 2 isomorphic semigroups

not monoids which consist of 11 elements that is⟨1205902

1205903

1205905

⟩ cong ⟨1205902

1205904

1205905

⟩These semigroups extendedby 1205900

yield 2 isomorphic monoids which consistof 12 elements The monoid M contains no largersemigroup with the exception of itself

10 Cancellation Rules Motivated by SomeTopological Properties

101 Some Consequences of the Axiom 0 = 0minus So far we usedonly the following relations (above named cancellation rules)1205902

∘ 1205902

= 1205902

1205901

∘ 1205901

= 1205900

1205902

∘ 12059012

= 1205906

and 1205902

∘ 12059013

= 1205907

When one assumes119883 = 0 = 0

minus then

119883 = 1205900

(119883) = 1205902

(119883) = 1205905

(119883) = 1205907

(119883)

= 1205908

(119883) = 12059010

(119883) = 12059013

(119883)

0 = 1205901

(119883) = 1205903

(119883) = 1205904

(119883) = 1205906

(119883)

= 1205909

(119883) = 12059011

(119883) = 12059012

(119883)

(18)

Using the substitution 119860 997891rarr 119860119888 one obtains equiv-

alent relations between operations from the set 1205901

1205903

1205904

1205906

1205909

12059011

12059012

and conversely Therefore cancellationrules are topologically reasonable only between the opera-tions from the monoid as follows

⟨1205900

1205902

1205905

⟩ = 1205900

1205902

1205905

1205907

1205908

12059010

12059013

(19)

Chapman see [3] considered properties of subsets withrespect to such relations Below we are going to identifyrelations that are determined by some topological spacescompare [10 11]

102 The Relation 1205900 = 1205902 If a topological space 119883 isdiscrete then there exist two Kuratowski operation onlyThese are 120590

0

and 1205901

So themonoid of Kuratowski operationsreduced to the group ⟨120590

1

⟩The relation 120590

0

= 1205902

is equivalent to any relation 1205900

=

120590119894

where 119894 isin 5 7 8 10 13 Any such relation implies thatevery subset of 119883 has to be closed and open that is 119883 hasto be discrete However one can check these using (only) thefacts that 120590

1

is an involution and 1205902

is an idempotent and thecancellation rules that is the Cayley table forM So 120590

0

= 1205902

follows

1205900

= 1205902

= 1205905

= 1205907

= 1205908

= 12059010

= 12059013

(20)

103 The Relation 1205902 = 1205905 Topologically 1205902 = 1205905

meansthat 119883 must be discrete This is so because 119860119888minus119888 sube 119860 sube 119860

minus

for any 119860 sube 119883

104 The Relation 1205902 = 1205907 Topologically the relation 1205902 =1205907

implies 1205900

= 1205902

But it requires the use of topology axioms0 = 0minus and 119862minus cup 119861minus = (119862 cup 119861)minus for each 119862 and 119861

Lemma 5 For any topological space 1205902

= 1205907

implies 1205902

= 1205908

Proof If 1205902

= 1205907

then 119860 = 0 rArr 119860119888minus119888

= 0 for any 119860 sube 119883Indeed if 119860119888minus119888 = 0 then 120590

7

(119860) = 0minus

= 0 Since 119860 = 0 then1205902

(119860) = 0 Hence 1205902

(119860) = 1205907

(119860) a contradictionThe axiom 119862

minus

cup 119861minus

= (119862 cup 119861)minus implies that always

(119860minus

cap 119860minus119888minus

)119888minus119888

= 0 (21)

Thus the additional assumption 1205902

= 1205907

follows that always119860minus

cap 119860minus119888minus

= 0 Therefore always 119860minus119888minus = 119860minus119888 but this means

8 Journal of Mathematics

that any closed set has to be open in other words 1205902

= 1205908

Proposition 6 For any topological space 1205902

= 1205907

implies 1205900

=

1205902

Proof But the relation 1205902

= 1205907

is equivalent with 1205905

= 1205908

ByLemma 5 we get 120590

2

= 1205905

Finally 1205900

= 1205902

The relation 1205902

= 1205907

has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put

1205902

(0) = 119886 = 1205902

(119886) 119883 = 1205902

(119883) = 1205902

(119887) (22)

Then check that 1205902

= 1205907

and

1205908

(0) = 1205905

(119886) = 1205904

(119887) = 1205901

(119883) = 0 (23)

in other words 1205902

= 1205907

and 1205902

= 1205908

However 1205902

= 1205907

isequivalent to 120590

5

= 1205908

This relation implies 120590

2

= 1205907

= 12059010

1205905

= 1205908

= 12059013

1205903

= 1205906

= 12059011

and 1205904

= 1205909

= 12059012

For this interpretationthe monoidM(120590

2

= 1205907

) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590

2

= 1205907

) thereare relations covered by the following proposition only

Proposition 7 For any monoid with the Cayley table as forM the relation 120590

2

= 1205907

implies

(i) 1205902

= 1205907

= 12059010

= 1205907

∘ 1205902

(ii) 1205905

= 1205901

∘1205902

∘1205901

= 1205901

∘1205907

∘1205901

= 1205908

= 1205905

∘1205902

= 1205905

∘1205907

=

12059013

(iii) 120590

3

= 1205902

∘ 1205901

= 1205907

∘ 1205901

= 1205906

= 12059010

∘ 1205901

= 12059011

(iv) 1205904

= 1205901

∘ 1205902

= 1205901

∘ 1205907

= 1205909

= 1205901

∘ 12059010

= 12059012

Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology

105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908

means that any closed set is open tooThus if119883 = 119886 119887 119888

is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590

2

= 1205908

) is the monoid of all Kuratowski operationsover 119883 Relations 120590

2

= 1205908

and 1205905

= 1205907

are equivalent Theyimply relations 120590

2

= 1205908

= 12059010

1205905

= 1205907

= 12059013

1205903

= 1205909

= 12059011

and 120590

4

= 1205906

= 12059012

The permutation

(1205900

1205901

1205902

1205903

1205904

1205905

1205900

1205901

1205905

1205904

1205903

1205902

) (24)

determines the isomorphism between monoidsM(1205902

= 1205907

)

andM(1205902

= 1205908

)

106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590

2

= 1205908

means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590

2

= 12059010

implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902

= 1205908

The relation 1205902

= 12059013

follows that each closed sethas to be open so it implies 120590

2

= 1205908

too

107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590

7

= 1205908

and 12059010

= 12059013

areequivalent Each of them gives 120590

7

= 1205908

= 12059010

= 12059013

and1205906

= 1205909

= 12059011

= 12059012

If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590

7

= 1205908

) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements

108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590

7

= 12059010

1205908

= 12059013

1205906

= 12059011

and 120590

9

= 12059012

are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590

7

= 12059010

) is the monoid of allKuratowski operations over119883 and consists of 10 elements

109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590

7

= 12059013

and 1205908

= 12059010

areequivalent Also 120590

6

= 12059012

and 1205909

= 12059011

These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted

1205901

1205902

1205903

1205904

1205905

1205906

1205907

1205908

1205909

1205901

1205900

1205904

1205905

1205902

1205903

1205908

1205909

1205906

1205907

1205902

1205903

1205902

1205903

1205906

1205907

1205906

1205907

1205908

1205909

1205903

1205902

1205906

1205907

1205902

1205903

1205908

1205909

1205906

1205907

1205904

1205905

1205904

1205905

1205908

1205909

1205908

1205909

1205906

1205907

1205905

1205904

1205908

1205909

1205904

1205905

1205906

1205907

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

1205907

1205906

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205909

1205908

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

(25)

If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906

= 12059012

The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590

7

= 1205908

Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590

7

= 1205908

and 1205907

= 12059013

To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that

(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860

119888minus119888Then check that

(i) 1205900

(119860) = 119860 and 1205901

(119860) = 119883 (119860119888minus119888

cup 119861 cup 119862)(ii) 1205902

(119860) = 119880 cup 119861minus

cup119882 and 1205903

(119860) = 119883 119860119888minus119888

(iii) 1205904

(119860) = 119881 119861minus and 120590

5

(119860) = 119860119888minus119888

(iv) 1205906

(119860) = 12059012

(119860) = 119881 and 1205907

(119860) = 12059013

(119860) = 119880(v) 1205908

(119860) = 12059010

(119860) = 119880 cup 119882 and 1205909

(119860) = 12059011

(119860) =

119881 cup 119882Hence we have that 120590

7

= 1205908

Note that if119882 = 0 then 1205907

(119860) =

1205908

(119860) This is the case of subsets of 120573119873

Journal of Mathematics 9

References

[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922

[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977

[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962

[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010

[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008

[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966

[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996

[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956

[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889

[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967

[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968

[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Mathematics

4 semigroups which are contained in M These are fifty-seven semigroups contained in ⟨120590

3

1205904

⟩ fifty-seven monoidsformed by adding 120590

0

to a semigroup contained in ⟨1205903

1205904

⟩groups ⟨120590

0

⟩ ⟨1205901

⟩ and monoids ⟨1205901

1205906

⟩ = M1

and ⟨1205901

1205902

⟩ =

MThere are fifty-six types of nonisomorphic semigroups

in M These are twenty-eight non-isomorphism types ofsemigroups contained in ⟨120590

3

1205904

⟩ twenty-six types of non-isomorphic monoids formed by adding 120590

0

to a semigroupcontained in ⟨120590

3

1205904

⟩ and alsoM1

andM Indeed adding 1205900

to a semigroup which is not a monoid we obtain a monoidIn this way we get twenty-four types of nonisomorphicmonoids Adding 120590

0

to a monoid contained in ⟨1205903

1205904

⟩ weget two new non-isomorphism types ofmonoids But adding1205900

to a group contained in ⟨1205903

1205904

⟩ we get no new type ofmonoid since we get a monoid isomorphic to ⟨120590

2

12059010

⟩ or⟨1205902

1205906

⟩ The other two types areM1

andM

92 Semigroups which Are Not Monoids Below we havereproduced using the smallest number of generators andthe dictionary order a list of all 43 semigroups which areincluded in theM as follows

(1) ⟨1205902

1205903

⟩ = 1205902

1205903

1205906

1205907

12059010

12059011

(2) ⟨120590

2

1205903

1205905

⟩ = 1205902

1205903

1205905

1205906

12059013

(3) ⟨120590

2

1205903

1205908

⟩ = ⟨1205902

1205903

1205909

⟩ = ⟨1205902

1205903

12059012

⟩ = ⟨1205902

1205903

12059013

⟩ = 1205902

1205903

1205906

1205907

12059013

(4) ⟨120590

2

1205904

⟩ = 1205902

1205904

1205906

1205908

12059010

12059012

(5) ⟨120590

2

1205904

1205905

⟩ = 1205902

1205904

1205905

12059013

(6) ⟨120590

2

1205904

1205907

⟩ = ⟨1205902

1205904

1205909

⟩ = ⟨1205902

1205904

12059011

⟩ = ⟨1205902

1205904

12059013

⟩ = 1205902

1205904

1205906

1205907

12059013

(7) ⟨120590

2

1205905

⟩ = 1205902

1205905

1205907

1205908

12059010

12059013

(8) ⟨120590

2

1205905

1205906

⟩ = ⟨1205902

1205905

1205909

⟩ = ⟨1205902

1205905

12059011

⟩ = ⟨1205902

1205905

12059012

⟩ = 1205902

1205905

1205906

12059013

(9) ⟨120590

2

1205907

⟩ = 1205902

1205907

12059010

(10) ⟨120590

2

1205908

⟩ = 1205902

1205908

12059010

(11) ⟨120590

2

1205909

⟩ = ⟨1205902

1205906

12059013

⟩ = ⟨1205902

1205907

12059012

⟩ = ⟨1205902

1205908

12059011

⟩ = ⟨1205902

12059011

12059012

⟩ = ⟨1205902

12059011

12059013

⟩ = ⟨1205902

12059012

12059013

⟩ = 1205902

1205906

1205907

12059013

(12) ⟨120590

2

12059011

⟩ = ⟨1205902

1205906

1205907

⟩ = 1205902

1205906

1205907

12059010

12059011

(13) ⟨120590

2

12059012

⟩ = ⟨1205902

1205906

1205908

⟩ = 1205902

1205906

1205908

12059010

12059012

(14) ⟨120590

2

12059013

⟩ = ⟨1205902

1205907

1205908

⟩ = 1205902

1205907

1205908

12059010

12059013

(15) ⟨120590

3

⟩ = 1205903

1205907

12059011

(16) ⟨120590

3

1205904

⟩ = 1205902

1205903

12059013

(17) ⟨120590

3

1205905

⟩ = 1205903

1205905

1205907

1205909

12059011

12059013

(18) ⟨120590

3

1205905

1205906

⟩ = ⟨1205903

1205905

1205908

⟩ = ⟨1205903

1205905

12059010

⟩ = ⟨1205903

1205905

12059012

⟩ = 1205903

1205905

1205906

12059013

(19) ⟨120590

3

1205906

⟩ = ⟨1205903

12059010

⟩ = 1205903

1205906

1205907

12059010

12059011

(20) ⟨120590

3

1205908

⟩ = ⟨1205903

12059012

⟩ = 1205903

1205906

1205907

12059013

(21) ⟨120590

3

1205909

⟩ = ⟨1205903

12059013

⟩ = 1205903

1205907

1205909

12059011

12059013

(22) ⟨120590

4

⟩ = 1205904

1205908

12059012

(23) ⟨1205904

1205905

⟩ = 1205904

1205905

1205908

1205909

12059012

12059013

(24) ⟨120590

4

1205905

1205906

⟩ = ⟨1205904

1205905

1205907

⟩ = ⟨1205904

1205905

12059010

⟩ = ⟨1205904

1205905

12059011

⟩ = 1205904

1205905

12059013

(25) ⟨120590

4

1205906

⟩ = ⟨1205904

12059010

⟩ = 1205904

1205906

1205908

12059010

12059012

(26) ⟨120590

4

1205907

⟩ = ⟨1205904

12059011

⟩ = 1205904

1205906

1205907

12059013

(27) ⟨120590

4

1205909

⟩ = ⟨1205904

12059013

⟩ = 1205904

1205908

1205909

12059012

12059013

(28) ⟨120590

5

1205906

⟩ = ⟨1205905

1205907

12059012

⟩ = ⟨1205905

1205908

12059011

⟩ = ⟨1205905

1205909

12059010

⟩ = ⟨1205905

12059010

12059011

⟩ = ⟨1205905

12059010

12059012

⟩ = ⟨1205905

12059011

12059012

⟩ = 1205905

1205906

12059013

(29) ⟨120590

5

1205907

⟩ = 1205905

1205907

12059013

(30) ⟨120590

5

1205908

⟩ = 1205905

1205908

12059013

(31) ⟨120590

5

12059010

⟩ = ⟨1205905

1205907

1205908

⟩ = 1205905

1205907

1205908

12059010

12059013

(32) ⟨120590

5

12059011

⟩ = ⟨1205905

1205907

1205909

⟩ = 1205905

1205907

1205909

12059011

12059013

(33) ⟨120590

5

12059012

⟩ = ⟨1205905

1205908

1205909

⟩ = 1205905

1205908

1205909

12059012

12059013

(34) ⟨120590

6

1205907

⟩ = ⟨1205906

12059011

⟩ = ⟨12059010

12059011

⟩ = 1205906

1205907

12059010

12059011

(35) ⟨120590

6

1205908

⟩ = ⟨1205906

12059012

⟩ = ⟨12059010

12059012

⟩ = 1205906

1205908

12059010

12059012

(36) ⟨120590

6

1205909

⟩ = ⟨1205906

12059013

⟩ = ⟨1205907

12059012

⟩ = ⟨1205908

12059011

⟩ = ⟨1205909

12059010

⟩ = ⟨12059011

12059012

⟩ = ⟨1205906

1205907

1205908

⟩ = ⟨1205907

1205908

1205909

⟩ =⟨12059010

12059011

12059013

⟩ = ⟨12059010

12059012

12059013

⟩ = 1205906

1205907

12059013

(37) ⟨120590

7

1205908

⟩ = 1205907

1205908

12059010

12059013

(38) ⟨120590

7

1205909

⟩ = 1205907

1205909

12059011

12059013

(39) ⟨120590

7

12059010

⟩ = 1205907

12059010

(40) ⟨120590

7

12059013

⟩ = 1205907

12059013

(41) ⟨120590

8

1205909

⟩ = 1205908

1205909

12059012

12059013

(42) ⟨120590

8

12059010

⟩ = 1205908

12059010

(43) ⟨120590

8

12059013

⟩ = 1205908

12059013

93 Isomorphism Types of Semigroups Contained in M Sys-tematize the list of all isomorphism types of semigroupscontained in themonoidM Isomorphisms which are restric-tions of the isomorphism A will be regarded as self-evidentand therefore they will not be commented on

(i) The monoid M contains 12 groups with 2 isomor-phism types These are 7 one-element groups and 5two-element groups

(ii) ThemonoidM contains 8 two-element monoids with2 isomorphism types These are 120590

0

added to 6 one-element groups and ⟨120590

2

12059010

⟩ cong ⟨1205905

12059013

⟩ Also itcontains 4 two-element semigroups not monoidswith 2 isomorphism types These are ⟨120590

7

12059010

⟩ cong

⟨1205908

12059013

⟩ and ⟨1205908

12059010

⟩ cong ⟨1205907

12059013

⟩(iii) The monoid M contains 12 three-element monoids

with 5 isomorphism types These are 1205900

addedto 4 two-element groups and also ⟨120590

0

1205902

12059010

⟩ cong

⟨1205900

1205905

12059013

⟩ ⟨1205900

1205907

12059010

⟩ cong ⟨1205900

1205908

12059013

⟩⟨1205900

1205908

12059010

⟩ cong ⟨1205900

1205907

12059013

⟩ and ⟨1205902

1205906

⟩ cong ⟨1205905

1205909

⟩(iv) The monoidM contains 6 three-element semigroups

not monoids with 3 isomorphism types These are⟨1205902

1205907

⟩ cong ⟨1205905

1205908

⟩ ⟨1205902

1205908

⟩ cong ⟨1205905

1205907

⟩ and ⟨1205903

⟩ cong

⟨1205904

⟩

Journal of Mathematics 7

(v) The monoid M contains 8 four-element monoidseach contains 120590

0

with 4 isomorphism types Theseare semigroups from two preceding items that can besubstantially extended by 120590

0

(vi) The monoid M contains 5 four-element semigroups

not monoids with 3 isomorphism types These are⟨1205906

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong ⟨1205907

1205909

⟩ and ⟨1205907

1205908

⟩These semigroups extended by 120590

0

yield 5 monoids allwhich consist of five elements with 3 isomorphismtypes

(vii) ThemonoidM contains 10 five-element semigroupsmdashnot monoids with 5 isomorphism types These are⟨1205902

12059011

⟩ cong ⟨1205905

12059012

⟩ ⟨1205902

12059012

⟩ cong ⟨1205905

12059011

⟩ ⟨1205902

12059013

⟩ cong

⟨1205905

12059010

⟩ ⟨1205903

1205906

⟩ cong ⟨1205904

1205909

⟩ and ⟨1205903

1205909

⟩ cong ⟨1205904

1205906

⟩These semigroups extended by120590

0

yield 10monoids allof which consist of six elements with 5 isomorphismtypes We get 10 new isomorphism types since thesemigroups are distinguished by semigroups ⟨120590

3

⟩

and ⟨1205904

⟩ and by nonisomorphic semigroups ⟨1205906

1205907

⟩⟨1205906

1205908

⟩ and ⟨1205907

1205908

⟩(viii) The monoid M contains 5 six-element semigroups

not monoids with 3 isomorphism types These are⟨1205902

1205903

⟩ cong ⟨1205904

1205905

⟩ ⟨1205902

1205904

⟩ cong ⟨1205903

1205905

⟩ and ⟨1205902

1205905

⟩These semigroups extended by 120590

0

yield 5 monoidsall of which consist of seven elements with 3 isomor-phism types We get 6 new isomorphism types sincethe semigroups are distinguished by non isomorphicsemigroups ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ and ⟨1205907

1205908

⟩(ix) ThemonoidM contains no seven-element semigroup

not a monoid no eight-element monoid and theonly semigroup ⟨120590

6

1205909

⟩ with exactly eight elementsand the only monoid ⟨120590

0

1205906

1205909

⟩ with exactly nineelements

(x) The monoid M contains 4 nine-element semigroupsnot monoids with 2 isomorphism types These are⟨1205902

1205909

⟩ cong ⟨1205905

1205906

⟩ and ⟨1205903

1205908

⟩ cong ⟨1205904

1205907

⟩ Thesemigroup ⟨120590

2

1205909

⟩ does not contain a semigroupisomorphic to ⟨120590

3

⟩ hence it is not isomorphic to⟨1205903

1205908

⟩ These semigroups extended by 1205900

yield 4monoids all of which consist of ten elements with 2isomorphism types

(xi) The monoid M contains 6 ten-element semigroupsnot monoids with 4 isomorphism typesThese are ⟨120590

1

1205906

⟩ ⟨1205902

1205903

1205908

⟩ cong ⟨1205904

1205905

1205906

⟩⟨1205902

1205904

1205907

⟩ cong ⟨1205903

1205905

1205906

⟩ and ⟨1205902

1205905

1205906

⟩ Thesesemigroups (except ⟨120590

1

1205906

⟩) extended by 1205900

yield 5monoids all of which consist of ten elements with3 isomorphism types We get 6 new isomorphismtypes since the semigroups are distinguished bynot isomorphic semigroups ⟨120590

2

1205903

⟩ ⟨1205902

1205904

⟩ and⟨1205902

1205905

⟩(xii) The monoid M contains 2 isomorphic semigroups

not monoids which consist of 11 elements that is⟨1205902

1205903

1205905

⟩ cong ⟨1205902

1205904

1205905

⟩These semigroups extendedby 1205900

yield 2 isomorphic monoids which consistof 12 elements The monoid M contains no largersemigroup with the exception of itself

10 Cancellation Rules Motivated by SomeTopological Properties

101 Some Consequences of the Axiom 0 = 0minus So far we usedonly the following relations (above named cancellation rules)1205902

∘ 1205902

= 1205902

1205901

∘ 1205901

= 1205900

1205902

∘ 12059012

= 1205906

and 1205902

∘ 12059013

= 1205907

When one assumes119883 = 0 = 0

minus then

119883 = 1205900

(119883) = 1205902

(119883) = 1205905

(119883) = 1205907

(119883)

= 1205908

(119883) = 12059010

(119883) = 12059013

(119883)

0 = 1205901

(119883) = 1205903

(119883) = 1205904

(119883) = 1205906

(119883)

= 1205909

(119883) = 12059011

(119883) = 12059012

(119883)

(18)

Using the substitution 119860 997891rarr 119860119888 one obtains equiv-

alent relations between operations from the set 1205901

1205903

1205904

1205906

1205909

12059011

12059012

and conversely Therefore cancellationrules are topologically reasonable only between the opera-tions from the monoid as follows

⟨1205900

1205902

1205905

⟩ = 1205900

1205902

1205905

1205907

1205908

12059010

12059013

(19)

Chapman see [3] considered properties of subsets withrespect to such relations Below we are going to identifyrelations that are determined by some topological spacescompare [10 11]

102 The Relation 1205900 = 1205902 If a topological space 119883 isdiscrete then there exist two Kuratowski operation onlyThese are 120590

0

and 1205901

So themonoid of Kuratowski operationsreduced to the group ⟨120590

1

⟩The relation 120590

0

= 1205902

is equivalent to any relation 1205900

=

120590119894

where 119894 isin 5 7 8 10 13 Any such relation implies thatevery subset of 119883 has to be closed and open that is 119883 hasto be discrete However one can check these using (only) thefacts that 120590

1

is an involution and 1205902

is an idempotent and thecancellation rules that is the Cayley table forM So 120590

0

= 1205902

follows

1205900

= 1205902

= 1205905

= 1205907

= 1205908

= 12059010

= 12059013

(20)

103 The Relation 1205902 = 1205905 Topologically 1205902 = 1205905

meansthat 119883 must be discrete This is so because 119860119888minus119888 sube 119860 sube 119860

minus

for any 119860 sube 119883

104 The Relation 1205902 = 1205907 Topologically the relation 1205902 =1205907

implies 1205900

= 1205902

But it requires the use of topology axioms0 = 0minus and 119862minus cup 119861minus = (119862 cup 119861)minus for each 119862 and 119861

Lemma 5 For any topological space 1205902

= 1205907

implies 1205902

= 1205908

Proof If 1205902

= 1205907

then 119860 = 0 rArr 119860119888minus119888

= 0 for any 119860 sube 119883Indeed if 119860119888minus119888 = 0 then 120590

7

(119860) = 0minus

= 0 Since 119860 = 0 then1205902

(119860) = 0 Hence 1205902

(119860) = 1205907

(119860) a contradictionThe axiom 119862

minus

cup 119861minus

= (119862 cup 119861)minus implies that always

(119860minus

cap 119860minus119888minus

)119888minus119888

= 0 (21)

Thus the additional assumption 1205902

= 1205907

follows that always119860minus

cap 119860minus119888minus

= 0 Therefore always 119860minus119888minus = 119860minus119888 but this means

8 Journal of Mathematics

that any closed set has to be open in other words 1205902

= 1205908

Proposition 6 For any topological space 1205902

= 1205907

implies 1205900

=

1205902

Proof But the relation 1205902

= 1205907

is equivalent with 1205905

= 1205908

ByLemma 5 we get 120590

2

= 1205905

Finally 1205900

= 1205902

The relation 1205902

= 1205907

has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put

1205902

(0) = 119886 = 1205902

(119886) 119883 = 1205902

(119883) = 1205902

(119887) (22)

Then check that 1205902

= 1205907

and

1205908

(0) = 1205905

(119886) = 1205904

(119887) = 1205901

(119883) = 0 (23)

in other words 1205902

= 1205907

and 1205902

= 1205908

However 1205902

= 1205907

isequivalent to 120590

5

= 1205908

This relation implies 120590

2

= 1205907

= 12059010

1205905

= 1205908

= 12059013

1205903

= 1205906

= 12059011

and 1205904

= 1205909

= 12059012

For this interpretationthe monoidM(120590

2

= 1205907

) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590

2

= 1205907

) thereare relations covered by the following proposition only

Proposition 7 For any monoid with the Cayley table as forM the relation 120590

2

= 1205907

implies

(i) 1205902

= 1205907

= 12059010

= 1205907

∘ 1205902

(ii) 1205905

= 1205901

∘1205902

∘1205901

= 1205901

∘1205907

∘1205901

= 1205908

= 1205905

∘1205902

= 1205905

∘1205907

=

12059013

(iii) 120590

3

= 1205902

∘ 1205901

= 1205907

∘ 1205901

= 1205906

= 12059010

∘ 1205901

= 12059011

(iv) 1205904

= 1205901

∘ 1205902

= 1205901

∘ 1205907

= 1205909

= 1205901

∘ 12059010

= 12059012

Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology

105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908

means that any closed set is open tooThus if119883 = 119886 119887 119888

is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590

2

= 1205908

) is the monoid of all Kuratowski operationsover 119883 Relations 120590

2

= 1205908

and 1205905

= 1205907

are equivalent Theyimply relations 120590

2

= 1205908

= 12059010

1205905

= 1205907

= 12059013

1205903

= 1205909

= 12059011

and 120590

4

= 1205906

= 12059012

The permutation

(1205900

1205901

1205902

1205903

1205904

1205905

1205900

1205901

1205905

1205904

1205903

1205902

) (24)

determines the isomorphism between monoidsM(1205902

= 1205907

)

andM(1205902

= 1205908

)

106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590

2

= 1205908

means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590

2

= 12059010

implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902

= 1205908

The relation 1205902

= 12059013

follows that each closed sethas to be open so it implies 120590

2

= 1205908

too

107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590

7

= 1205908

and 12059010

= 12059013

areequivalent Each of them gives 120590

7

= 1205908

= 12059010

= 12059013

and1205906

= 1205909

= 12059011

= 12059012

If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590

7

= 1205908

) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements

108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590

7

= 12059010

1205908

= 12059013

1205906

= 12059011

and 120590

9

= 12059012

are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590

7

= 12059010

) is the monoid of allKuratowski operations over119883 and consists of 10 elements

109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590

7

= 12059013

and 1205908

= 12059010

areequivalent Also 120590

6

= 12059012

and 1205909

= 12059011

These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted

1205901

1205902

1205903

1205904

1205905

1205906

1205907

1205908

1205909

1205901

1205900

1205904

1205905

1205902

1205903

1205908

1205909

1205906

1205907

1205902

1205903

1205902

1205903

1205906

1205907

1205906

1205907

1205908

1205909

1205903

1205902

1205906

1205907

1205902

1205903

1205908

1205909

1205906

1205907

1205904

1205905

1205904

1205905

1205908

1205909

1205908

1205909

1205906

1205907

1205905

1205904

1205908

1205909

1205904

1205905

1205906

1205907

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

1205907

1205906

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205909

1205908

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

(25)

If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906

= 12059012

The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590

7

= 1205908

Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590

7

= 1205908

and 1205907

= 12059013

To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that

(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860

119888minus119888Then check that

(i) 1205900

(119860) = 119860 and 1205901

(119860) = 119883 (119860119888minus119888

cup 119861 cup 119862)(ii) 1205902

(119860) = 119880 cup 119861minus

cup119882 and 1205903

(119860) = 119883 119860119888minus119888

(iii) 1205904

(119860) = 119881 119861minus and 120590

5

(119860) = 119860119888minus119888

(iv) 1205906

(119860) = 12059012

(119860) = 119881 and 1205907

(119860) = 12059013

(119860) = 119880(v) 1205908

(119860) = 12059010

(119860) = 119880 cup 119882 and 1205909

(119860) = 12059011

(119860) =

119881 cup 119882Hence we have that 120590

7

= 1205908

Note that if119882 = 0 then 1205907

(119860) =

1205908

(119860) This is the case of subsets of 120573119873

Journal of Mathematics 9

References

[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922

[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977

[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962

[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010

[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008

[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966

[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996

[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956

[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889

[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967

[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968

[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 7

(v) The monoid M contains 8 four-element monoidseach contains 120590

0

with 4 isomorphism types Theseare semigroups from two preceding items that can besubstantially extended by 120590

0

(vi) The monoid M contains 5 four-element semigroups

not monoids with 3 isomorphism types These are⟨1205906

1205907

⟩ cong ⟨1205908

1205909

⟩ ⟨1205906

1205908

⟩ cong ⟨1205907

1205909

⟩ and ⟨1205907

1205908

⟩These semigroups extended by 120590

0

yield 5 monoids allwhich consist of five elements with 3 isomorphismtypes

(vii) ThemonoidM contains 10 five-element semigroupsmdashnot monoids with 5 isomorphism types These are⟨1205902

12059011

⟩ cong ⟨1205905

12059012

⟩ ⟨1205902

12059012

⟩ cong ⟨1205905

12059011

⟩ ⟨1205902

12059013

⟩ cong

⟨1205905

12059010

⟩ ⟨1205903

1205906

⟩ cong ⟨1205904

1205909

⟩ and ⟨1205903

1205909

⟩ cong ⟨1205904

1205906

⟩These semigroups extended by120590

0

yield 10monoids allof which consist of six elements with 5 isomorphismtypes We get 10 new isomorphism types since thesemigroups are distinguished by semigroups ⟨120590

3

⟩

and ⟨1205904

⟩ and by nonisomorphic semigroups ⟨1205906

1205907

⟩⟨1205906

1205908

⟩ and ⟨1205907

1205908

⟩(viii) The monoid M contains 5 six-element semigroups

not monoids with 3 isomorphism types These are⟨1205902

1205903

⟩ cong ⟨1205904

1205905

⟩ ⟨1205902

1205904

⟩ cong ⟨1205903

1205905

⟩ and ⟨1205902

1205905

⟩These semigroups extended by 120590

0

yield 5 monoidsall of which consist of seven elements with 3 isomor-phism types We get 6 new isomorphism types sincethe semigroups are distinguished by non isomorphicsemigroups ⟨120590

6

1205907

⟩ ⟨1205906

1205908

⟩ and ⟨1205907

1205908

⟩(ix) ThemonoidM contains no seven-element semigroup

not a monoid no eight-element monoid and theonly semigroup ⟨120590

6

1205909

⟩ with exactly eight elementsand the only monoid ⟨120590

0

1205906

1205909

⟩ with exactly nineelements

(x) The monoid M contains 4 nine-element semigroupsnot monoids with 2 isomorphism types These are⟨1205902

1205909

⟩ cong ⟨1205905

1205906

⟩ and ⟨1205903

1205908

⟩ cong ⟨1205904

1205907

⟩ Thesemigroup ⟨120590

2

1205909

⟩ does not contain a semigroupisomorphic to ⟨120590

3

⟩ hence it is not isomorphic to⟨1205903

1205908

⟩ These semigroups extended by 1205900

yield 4monoids all of which consist of ten elements with 2isomorphism types

(xi) The monoid M contains 6 ten-element semigroupsnot monoids with 4 isomorphism typesThese are ⟨120590

1

1205906

⟩ ⟨1205902

1205903

1205908

⟩ cong ⟨1205904

1205905

1205906

⟩⟨1205902

1205904

1205907

⟩ cong ⟨1205903

1205905

1205906

⟩ and ⟨1205902

1205905

1205906

⟩ Thesesemigroups (except ⟨120590

1

1205906

⟩) extended by 1205900

yield 5monoids all of which consist of ten elements with3 isomorphism types We get 6 new isomorphismtypes since the semigroups are distinguished bynot isomorphic semigroups ⟨120590

2

1205903

⟩ ⟨1205902

1205904

⟩ and⟨1205902

1205905

⟩(xii) The monoid M contains 2 isomorphic semigroups

not monoids which consist of 11 elements that is⟨1205902

1205903

1205905

⟩ cong ⟨1205902

1205904

1205905

⟩These semigroups extendedby 1205900

yield 2 isomorphic monoids which consistof 12 elements The monoid M contains no largersemigroup with the exception of itself

10 Cancellation Rules Motivated by SomeTopological Properties

101 Some Consequences of the Axiom 0 = 0minus So far we usedonly the following relations (above named cancellation rules)1205902

∘ 1205902

= 1205902

1205901

∘ 1205901

= 1205900

1205902

∘ 12059012

= 1205906

and 1205902

∘ 12059013

= 1205907

When one assumes119883 = 0 = 0

minus then

119883 = 1205900

(119883) = 1205902

(119883) = 1205905

(119883) = 1205907

(119883)

= 1205908

(119883) = 12059010

(119883) = 12059013

(119883)

0 = 1205901

(119883) = 1205903

(119883) = 1205904

(119883) = 1205906

(119883)

= 1205909

(119883) = 12059011

(119883) = 12059012

(119883)

(18)

Using the substitution 119860 997891rarr 119860119888 one obtains equiv-

alent relations between operations from the set 1205901

1205903

1205904

1205906

1205909

12059011

12059012

and conversely Therefore cancellationrules are topologically reasonable only between the opera-tions from the monoid as follows

⟨1205900

1205902

1205905

⟩ = 1205900

1205902

1205905

1205907

1205908

12059010

12059013

(19)

Chapman see [3] considered properties of subsets withrespect to such relations Below we are going to identifyrelations that are determined by some topological spacescompare [10 11]

102 The Relation 1205900 = 1205902 If a topological space 119883 isdiscrete then there exist two Kuratowski operation onlyThese are 120590

0

and 1205901

So themonoid of Kuratowski operationsreduced to the group ⟨120590

1

⟩The relation 120590

0

= 1205902

is equivalent to any relation 1205900

=

120590119894

where 119894 isin 5 7 8 10 13 Any such relation implies thatevery subset of 119883 has to be closed and open that is 119883 hasto be discrete However one can check these using (only) thefacts that 120590

1

is an involution and 1205902

is an idempotent and thecancellation rules that is the Cayley table forM So 120590

0

= 1205902

follows

1205900

= 1205902

= 1205905

= 1205907

= 1205908

= 12059010

= 12059013

(20)

103 The Relation 1205902 = 1205905 Topologically 1205902 = 1205905

meansthat 119883 must be discrete This is so because 119860119888minus119888 sube 119860 sube 119860

minus

for any 119860 sube 119883

104 The Relation 1205902 = 1205907 Topologically the relation 1205902 =1205907

implies 1205900

= 1205902

But it requires the use of topology axioms0 = 0minus and 119862minus cup 119861minus = (119862 cup 119861)minus for each 119862 and 119861

Lemma 5 For any topological space 1205902

= 1205907

implies 1205902

= 1205908

Proof If 1205902

= 1205907

then 119860 = 0 rArr 119860119888minus119888

= 0 for any 119860 sube 119883Indeed if 119860119888minus119888 = 0 then 120590

7

(119860) = 0minus

= 0 Since 119860 = 0 then1205902

(119860) = 0 Hence 1205902

(119860) = 1205907

(119860) a contradictionThe axiom 119862

minus

cup 119861minus

= (119862 cup 119861)minus implies that always

(119860minus

cap 119860minus119888minus

)119888minus119888

= 0 (21)

Thus the additional assumption 1205902

= 1205907

follows that always119860minus

cap 119860minus119888minus

= 0 Therefore always 119860minus119888minus = 119860minus119888 but this means

8 Journal of Mathematics

that any closed set has to be open in other words 1205902

= 1205908

Proposition 6 For any topological space 1205902

= 1205907

implies 1205900

=

1205902

Proof But the relation 1205902

= 1205907

is equivalent with 1205905

= 1205908

ByLemma 5 we get 120590

2

= 1205905

Finally 1205900

= 1205902

The relation 1205902

= 1205907

has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put

1205902

(0) = 119886 = 1205902

(119886) 119883 = 1205902

(119883) = 1205902

(119887) (22)

Then check that 1205902

= 1205907

and

1205908

(0) = 1205905

(119886) = 1205904

(119887) = 1205901

(119883) = 0 (23)

in other words 1205902

= 1205907

and 1205902

= 1205908

However 1205902

= 1205907

isequivalent to 120590

5

= 1205908

This relation implies 120590

2

= 1205907

= 12059010

1205905

= 1205908

= 12059013

1205903

= 1205906

= 12059011

and 1205904

= 1205909

= 12059012

For this interpretationthe monoidM(120590

2

= 1205907

) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590

2

= 1205907

) thereare relations covered by the following proposition only

Proposition 7 For any monoid with the Cayley table as forM the relation 120590

2

= 1205907

implies

(i) 1205902

= 1205907

= 12059010

= 1205907

∘ 1205902

(ii) 1205905

= 1205901

∘1205902

∘1205901

= 1205901

∘1205907

∘1205901

= 1205908

= 1205905

∘1205902

= 1205905

∘1205907

=

12059013

(iii) 120590

3

= 1205902

∘ 1205901

= 1205907

∘ 1205901

= 1205906

= 12059010

∘ 1205901

= 12059011

(iv) 1205904

= 1205901

∘ 1205902

= 1205901

∘ 1205907

= 1205909

= 1205901

∘ 12059010

= 12059012

Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology

105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908

means that any closed set is open tooThus if119883 = 119886 119887 119888

is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590

2

= 1205908

) is the monoid of all Kuratowski operationsover 119883 Relations 120590

2

= 1205908

and 1205905

= 1205907

are equivalent Theyimply relations 120590

2

= 1205908

= 12059010

1205905

= 1205907

= 12059013

1205903

= 1205909

= 12059011

and 120590

4

= 1205906

= 12059012

The permutation

(1205900

1205901

1205902

1205903

1205904

1205905

1205900

1205901

1205905

1205904

1205903

1205902

) (24)

determines the isomorphism between monoidsM(1205902

= 1205907

)

andM(1205902

= 1205908

)

106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590

2

= 1205908

means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590

2

= 12059010

implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902

= 1205908

The relation 1205902

= 12059013

follows that each closed sethas to be open so it implies 120590

2

= 1205908

too

107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590

7

= 1205908

and 12059010

= 12059013

areequivalent Each of them gives 120590

7

= 1205908

= 12059010

= 12059013

and1205906

= 1205909

= 12059011

= 12059012

If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590

7

= 1205908

) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements

108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590

7

= 12059010

1205908

= 12059013

1205906

= 12059011

and 120590

9

= 12059012

are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590

7

= 12059010

) is the monoid of allKuratowski operations over119883 and consists of 10 elements

109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590

7

= 12059013

and 1205908

= 12059010

areequivalent Also 120590

6

= 12059012

and 1205909

= 12059011

These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted

1205901

1205902

1205903

1205904

1205905

1205906

1205907

1205908

1205909

1205901

1205900

1205904

1205905

1205902

1205903

1205908

1205909

1205906

1205907

1205902

1205903

1205902

1205903

1205906

1205907

1205906

1205907

1205908

1205909

1205903

1205902

1205906

1205907

1205902

1205903

1205908

1205909

1205906

1205907

1205904

1205905

1205904

1205905

1205908

1205909

1205908

1205909

1205906

1205907

1205905

1205904

1205908

1205909

1205904

1205905

1205906

1205907

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

1205907

1205906

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205909

1205908

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

(25)

If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906

= 12059012

The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590

7

= 1205908

Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590

7

= 1205908

and 1205907

= 12059013

To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that

(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860

119888minus119888Then check that

(i) 1205900

(119860) = 119860 and 1205901

(119860) = 119883 (119860119888minus119888

cup 119861 cup 119862)(ii) 1205902

(119860) = 119880 cup 119861minus

cup119882 and 1205903

(119860) = 119883 119860119888minus119888

(iii) 1205904

(119860) = 119881 119861minus and 120590

5

(119860) = 119860119888minus119888

(iv) 1205906

(119860) = 12059012

(119860) = 119881 and 1205907

(119860) = 12059013

(119860) = 119880(v) 1205908

(119860) = 12059010

(119860) = 119880 cup 119882 and 1205909

(119860) = 12059011

(119860) =

119881 cup 119882Hence we have that 120590

7

= 1205908

Note that if119882 = 0 then 1205907

(119860) =

1205908

(119860) This is the case of subsets of 120573119873

Journal of Mathematics 9

References

[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922

[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977

[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962

[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010

[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008

[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966

[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996

[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956

[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889

[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967

[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968

[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Mathematics

that any closed set has to be open in other words 1205902

= 1205908

Proposition 6 For any topological space 1205902

= 1205907

implies 1205900

=

1205902

Proof But the relation 1205902

= 1205907

is equivalent with 1205905

= 1205908

ByLemma 5 we get 120590

2

= 1205905

Finally 1205900

= 1205902

The relation 1205902

= 1205907

has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put

1205902

(0) = 119886 = 1205902

(119886) 119883 = 1205902

(119883) = 1205902

(119887) (22)

Then check that 1205902

= 1205907

and

1205908

(0) = 1205905

(119886) = 1205904

(119887) = 1205901

(119883) = 0 (23)

in other words 1205902

= 1205907

and 1205902

= 1205908

However 1205902

= 1205907

isequivalent to 120590

5

= 1205908

This relation implies 120590

2

= 1205907

= 12059010

1205905

= 1205908

= 12059013

1205903

= 1205906

= 12059011

and 1205904

= 1205909

= 12059012

For this interpretationthe monoidM(120590

2

= 1205907

) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590

2

= 1205907

) thereare relations covered by the following proposition only

Proposition 7 For any monoid with the Cayley table as forM the relation 120590

2

= 1205907

implies

(i) 1205902

= 1205907

= 12059010

= 1205907

∘ 1205902

(ii) 1205905

= 1205901

∘1205902

∘1205901

= 1205901

∘1205907

∘1205901

= 1205908

= 1205905

∘1205902

= 1205905

∘1205907

=

12059013

(iii) 120590

3

= 1205902

∘ 1205901

= 1205907

∘ 1205901

= 1205906

= 12059010

∘ 1205901

= 12059011

(iv) 1205904

= 1205901

∘ 1205902

= 1205901

∘ 1205907

= 1205909

= 1205901

∘ 12059010

= 12059012

Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology

105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908

means that any closed set is open tooThus if119883 = 119886 119887 119888

is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590

2

= 1205908

) is the monoid of all Kuratowski operationsover 119883 Relations 120590

2

= 1205908

and 1205905

= 1205907

are equivalent Theyimply relations 120590

2

= 1205908

= 12059010

1205905

= 1205907

= 12059013

1205903

= 1205909

= 12059011

and 120590

4

= 1205906

= 12059012

The permutation

(1205900

1205901

1205902

1205903

1205904

1205905

1205900

1205901

1205905

1205904

1205903

1205902

) (24)

determines the isomorphism between monoidsM(1205902

= 1205907

)

andM(1205902

= 1205908

)

106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590

2

= 1205908

means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590

2

= 12059010

implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902

= 1205908

The relation 1205902

= 12059013

follows that each closed sethas to be open so it implies 120590

2

= 1205908

too

107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590

7

= 1205908

and 12059010

= 12059013

areequivalent Each of them gives 120590

7

= 1205908

= 12059010

= 12059013

and1205906

= 1205909

= 12059011

= 12059012

If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590

7

= 1205908

) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements

108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590

7

= 12059010

1205908

= 12059013

1205906

= 12059011

and 120590

9

= 12059012

are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590

7

= 12059010

) is the monoid of allKuratowski operations over119883 and consists of 10 elements

109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590

7

= 12059013

and 1205908

= 12059010

areequivalent Also 120590

6

= 12059012

and 1205909

= 12059011

These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted

1205901

1205902

1205903

1205904

1205905

1205906

1205907

1205908

1205909

1205901

1205900

1205904

1205905

1205902

1205903

1205908

1205909

1205906

1205907

1205902

1205903

1205902

1205903

1205906

1205907

1205906

1205907

1205908

1205909

1205903

1205902

1205906

1205907

1205902

1205903

1205908

1205909

1205906

1205907

1205904

1205905

1205904

1205905

1205908

1205909

1205908

1205909

1205906

1205907

1205905

1205904

1205908

1205909

1205904

1205905

1205906

1205907

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

1205907

1205906

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205908

1205909

1205908

1205909

1205906

1205907

1205906

1205907

1205908

1205909

1205909

1205908

1205906

1205907

1205908

1205909

1205908

1205909

1205906

1205907

(25)

If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906

= 12059012

The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590

7

= 1205908

Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590

7

= 1205908

and 1205907

= 12059013

To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that

(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860

119888minus119888Then check that

(i) 1205900

(119860) = 119860 and 1205901

(119860) = 119883 (119860119888minus119888

cup 119861 cup 119862)(ii) 1205902

(119860) = 119880 cup 119861minus

cup119882 and 1205903

(119860) = 119883 119860119888minus119888

(iii) 1205904

(119860) = 119881 119861minus and 120590

5

(119860) = 119860119888minus119888

(iv) 1205906

(119860) = 12059012

(119860) = 119881 and 1205907

(119860) = 12059013

(119860) = 119880(v) 1205908

(119860) = 12059010

(119860) = 119880 cup 119882 and 1205909

(119860) = 12059011

(119860) =

119881 cup 119882Hence we have that 120590

7

= 1205908

Note that if119882 = 0 then 1205907

(119860) =

1205908

(119860) This is the case of subsets of 120573119873

Journal of Mathematics 9

References

[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922

[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977

[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962

[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010

[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008

[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966

[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996

[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956

[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889

[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967

[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968

[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 9

References

[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922

[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977

[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962

[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010

[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008

[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966

[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996

[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956

[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889

[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967

[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968

[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of