TOMUS LX.web.cs.elte.hu/~annalesm/ANNALES-LX-2018-03-20-.pdfANNALES Universitatis Scientiarum Budapestinensis de Rolando Eötvös nominatae SECTIO MATHEMATICA TOMUS LX. REDIGIT Á

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REDIGIT

Á. CSÁSZÁR

ADIUVANTIBUS

L. BABAI, A. BENCZÚR, K. BEZDEK., K. BÖRÖCZKY, Z. BUCZOLICH,I. CSISZÁR, J. DEMETROVICS, I. FARAGÓ, A. FRANK, J. FRITZ,

V. GROLMUSZ, A. HAJNAL , G. HALÁSZ, A. IVÁNYI, A. JÁRAI, P. KACSUK,

GY. KÁROLYI, I. KÁTAI, T. KELETI, E. KISS, P. KOMJÁTH, M. LACZKOVICH,

L. LOVÁSZ, GY. MICHALETZKY, J. MOLNÁR, P. P. PÁLFY, A. PRÉKOPA ,

A. RECSKI, A. SÁRKÖZY, CS. SZABÓ, F. SCHIPP, Z. SEBESTYÉN, L. SIMON,P. SIMON, P. SIMON, L. SZEIDL, T. SZNYI, G. STOYAN, J. SZENTHE,

G. SZÉKELY, A. SZCS, L. VARGA, F. WEISZ

2017

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ANNALES UNIV. SCI. BUDAPEST., 60 (2017), 311

NANO IDEAL GENERALIZED CLOSED SETS IN NANO IDEAL

TOPOLOGICAL SPACES

By

M. PARIMALA, S. JAFARI, AND S. MURALI

(Received February 22, 2017)

Abstract. The purpose of this paper is to introduce a new type of generalized

closed and open sets called nIg-closed set and nIg-open set in nano ideal topological

spaces and investigate the relation between this set with other sets in nano topological

spaces and nano ideal topological spaces. Characterizations and properties of nIg-closedsets and nIg-open sets are given.

1. Introduction

An ideal [7] I on a nonempty set X is a nonempty collection of subsets

of X which satises (i) A ∈ I and B ⊂ A implies B ∈ I and (ii) A ∈ I and

B ∈ I implies A ∪ B ∈ I. Given a topological space (X, τ) with an ideal I onX and if P(X ) is the set of all subsets of X , a set operator (.)∗ : P(X ) → P(X ),called a local function [6] of A with respect to τ and I is dened as follows:

for A ⊂ X , A∗(I, τ) = x ∈ X : U ∩ A < I for every U ∈ τ(x) where τ(x) == U ∈ τ : x ∈ U . A Kuratowski closure operator cl∗(.) for a topology τ∗(I, τ),called the ∗-topology, ner than τ is dened by cl∗(A) = A ∪ A∗(I, τ) [13].

When there is no chance for confusion, we will simply write A∗ for A∗(I, τ) andτ∗ for τ∗(I, τ). If I is an ideal on X , then the space (X, τ, I) is called an ideal

topological space. A subset A of an ideal topological space is said to be ∗-dense

in itself [4] (resp. ∗-closed [6]) if A ⊂ A∗ (resp. A∗ ⊂ A). By a space (X, τ), wealways mean a topological space (X, τ) with no separation properties assumed.If A ⊂ X , then cl(A) and int(A) denote the closure and interior of A in (X, τ)respectively, and int∗(A) will denote the interior of A in (X, τ∗). The notion of

2000 Mathematics Subject Classication 54A05, 54A10, 54B05

4 M. PARIMALA, S. JAFARI, S. MURALI

I-open sets was introduced by Jankovic et al. [5] and it was investigated by AbdEl-Monsef [1].

The notion of nano topology was introduced by Lellis Thivagar [9] which

was dened in terms of approximations and boundary region of a subset of

an universe using an equivalence relation on it and also dened nano closed

sets, nano-interior and nano-closure. K. Bhuvaneswari et al. [2] introduced and

studied the concept of nano generalised closed sets in nano topological spaces.

M. Parimala et al. [10, 11, 12] introduced the concept of nano ideal

topological spaces and investvestigated some of its basic properties. They also

introduced the notion of nI-open sets, nI-closed sets, qnI-open sets and qnI-closed sets in nano ideal topological spaces. In this paper, we introduce a new

type of generalized closed and open sets called nIg-closed set and nIg-openset in nano ideal topological spaces and investigate the relationships between

this set with other sets in nano topological spaces and nano ideal topological

spaces. Characterizations and properties of nIg-closed sets and nIg-open sets

are studied.

2. Preliminaries

We recall the following denitions, which will be used in the sequel.

Denition 2.1 ([8]). LetU be a non-empty nite set of objects called the universe

and R be an equivalence relation on U named as indiscernibility relation. Then

U is divided into disjoint equivalence classes. Elements belonging to the same

equivalence class are said to be indiscernible with one another. The pair (U, R)is said to be the approximation space.

Let X ⊆ U . Then,

(i) The lower approximation of X with respect to R is the set of all objects

which can be for certain classied as X with respect to R and is denoted

by LR (X ). That is, LR (X ) = ∪ R(X ) : R(X ) ⊆ X, x ∈ U where R(X )denotes the equivalence class determined by x ∈ U.

(ii) The upper approximation of X with respect to R is the set of all objects

which can be possibly classied as X with respect to R and is denoted

byUR (X ). That is, UR (X ) = ∪ R(X ) : R(X ) ∩ X , φ, x ∈ U .(iii) The boundary region of X with respect to R is the set of all objects which

can be classied neither as X nor as not -X with respect to R and is denoted

by BR (X ). BR (X ) = UR (X ) − LR (X ).

NANO IDEAL GENERALIZED CLOSED SETS IN NANO IDEAL TOPOLOGICAL SPACES 5

Property 2.2 ([9]). If (U, R) is an approximation space and X,Y ⊆ U, then

(i) LR (X ) ⊆ X ⊆ UR (X )(ii) LR (φ) = UR (φ) = φ

(iii) LR (U) = UR (U) = U(iv) UR(X∪Y) =UR (X ) ∪UR (Y )(v) UR(X∩ Y)⊆UR(X)∩UR(Y)

(vi) LR(X∪Y)⊇LR(X)∪ LR(Y)

(vii) LR(X∩Y) = LR(X)∩ LR(Y)

(viii) LR(X)⊆ LR(Y) and UR(X)⊆UR(Y) whenever X⊆Y

(ix) UR (Xc ) = [LR (X )]c and LR (Xc ) = [UR (X )]c

(x) UR[UR (X )] = LR[UR (X )] = UR (X )(xi) LR[LR (X )] = UR[LR (X )] = LR (X )

Denition 2.3 ([9]). LetU be the universe, R be an equivalence relation onU and

τR (X )= U, φ, LR (X ),UR (X ), BR (X ) where X ⊆ U and by the Property 2.2,

τR (X ) satises the following axioms:(i) U and φ ∈ τR (X ).

(ii) The union of the elements of any sub-collection of τR (X ) is in τR (X ).(iii) The intersection of the elements of any nite subcollection of τR (X ) is in

τR (X ).

Therefore, τR (X ) is a topology on U called the nano topology on U with

respect to X . We call (U, τR (X )) as the nano topological space. The elements

of τR (X ) are called nano open sets (briey n-open sets). The complement of anano open set is called a nano closed set (briey n-closed set).

Denition 2.4 ([2]). Let (U, τR (X )) be a nano topological space and A ⊆ U.

Then A is said to be ng-closed set if ncl (A) ⊆ B whenever A ⊆ B ⊆ τR (X ).The complement of a ng-closed set is called a ng-open set.

Denition 2.5 ([11]). Let (U,N , I) be a nano ideal topological space with an

ideal I on U , where N = τR (X ) and (.)∗nbe a set operator from P(U) to P(U)

(P(U) is the set of all subsets of U). For a subset A ⊂ U , A∗n

(I,N ) = x ∈∈ U : Gn ∩ A < I, for every Gn ∈ Gn (x), where Gn = Gn | x ∈ Gn,Gn ∈ N

is called the nano local function (briey, n-local function) of A with respect to Iand N . We will simply write A∗

nfor A∗

n(I,N ).

Theorem 2.6 ([11]). Let (U,N ) be a nano topological space with ideals I, I ′

on U and A, B be subsets of U. Then

(i) A ⊆ B ⇒ A∗n⊆ B∗

n,

(ii) I ⊆ I ′ ⇒ A∗n

(I ′) ⊆ A∗n

(I),


(iii) A∗n= n-cl(A∗

n) ⊆ n-cl(A) (A∗

nis a nano closed subset of n-cl(A)),

(iv) (A∗n

)∗n⊆ A∗

n,

(v) A∗n∪ B∗

n= (A ∪ B)∗

n,

(vi) A∗n− B∗

n= (A − B)∗

n− B∗

n⊆ (A − B)∗

n,

(vii) V ∈ N ⇒ V ∩ A∗n= V ∩ (V ∩ A)∗

n⊆ (V ∩ A)∗

nand

(viii) J ∈ I ⇒ (A ∪ J)∗n= A∗

n= (A − J)∗

n.

Theorem 2.7 ([11]). If (U,N , I) is a nano topological space with an ideal Iand A ⊆ A∗

n, then A∗

n= n-cl(A∗

n) = n-cl(A).

Denition 2.8 ([11]). Let (U,N ) be a nano topological space with an ideal

I on U. The set operator n-cl∗ is called a nano∗-closure and is dened as

n-cl∗(A) = A ∪ A∗nfor A ⊆ X .

Theorem 2.9 ([11]). The set operator n-cl∗ satises the following conditions:

(i) A ⊆ n-cl∗(A),(ii) n-cl∗(φ) = φ and n-cl∗(U) = U,

(iii) If A ⊂ B, then n-cl∗(A) ⊆ n-cl∗(B),(iv) n-cl∗(A) ∪ n-cl∗(B) = n-cl∗(A ∪ B).(v) n-cl∗(n-cl∗(A)) = n-cl∗(A).

Denition 2.10 ([11]). An ideal I in a space (U,N , I) is calledN -codense ideal

if N ∩ I = φ.

Denition 2.11 ([11]). A subset A of a nano ideal topological space (U,N , I)is n∗-dense in itself (resp. n∗-perfect and n∗-closed) if A ⊆ A∗

n(resp. A = A∗

n,

A∗n⊆ A).

Lemma 2.12 ([11]). Let (U,N , I) be a nano ideal topological space and A ⊆ U.

If A is n∗-dense in itself, then A∗n= n-cl(A∗

n) = n-cl(A) = n-cl∗(A).

3. On nano-Ig-closed sets and nano-Ig-open sets

Denition 3.1. A subset A of a nano ideal topological space (U,N , I) is said tobe nano-I-generalized closed (briey, nIg-closed) if A∗

n⊆ V whenever A ⊆ V

and V is n-open.A subset A of a nano ideal topological space (U,N , I) is saidto be nano-I-generalized open (briey, nIg-open) if X − A is nIg-closed.

Theorem 3.2. Let (U,N , I) be a nano ideal topological space. Every ng-closedset is nIg-closed.


Proof. Let V be any n-open set containing A. Since A is ng-closed, thenn-cl(A) ⊆ V . By Theorem 2.6(iii), we have A∗

n⊆ V .

Example 3.3. Let U = a, b, c, d be the universe, X = a, d ⊂ U , U/R == b, d, a, c and N = U, φ, d, a, c, d, a, c and the ideal I = φ, d.The set A = d is nIg-closed but not ng-closed.

Theorem 3.4. If (U,N , I) is a nano ideal topological space and A ⊆ X , then

A is nIg-closed if and only if n-cl∗n

(A) ⊆ V whenever A ⊆ V and V is n-openin U .

Proof. Necessity: Since A is nIg-closed, we have A∗n⊆ V whenever A ⊆ V

and V is n-open inU . n-cl∗n

(A) = A∪ A∗n⊆ V whenever A ⊆ V and V is n-open

in U .

Suciency: Let A ⊆ V and V be n-open inU . By hypothesis n-cl∗n

(A) ⊆ V .

Since n-cl∗n

(A) = A ∪ A∗n, we have A∗

n⊆ V .

Theorem 3.5. If (U,N , I) is a nano ideal topological space and A ⊆ X , then

the following are equivalent:

(i) A is nIg-closed.(ii) n-cl∗(A) ⊆ V whenever A ⊆ V and V is n-open in U .

(iii) For all x ∈ n-cl∗(A), n-cl(x) ∩ A , φ.(iv) n-cl∗(A) − A contains no nonempty n-closed set.

(v) A∗n− A contains no nonempty n-closed set.

Proof. (i) ⇒ (ii) If A is nIg-closed, then A∗n⊆ V whenever A ⊆ V and V is

open in U and so cl∗n

(A) = A ∪ A∗n⊆ V whenever A ⊆ V and V is open in U .

This proves (ii).(ii) ⇒ (iii) Suppose x ∈ n-cl∗

n(A) and n-cl(x) ∩ A = φ. In this case,

A ⊆ X −n-cl(x). We have n-cl∗((x) ⊆ U − (n-cl(x)) and hence n-cl∗(A)∩∩ x = φ. This is a contradiction, since x ∈ n-cl∗(A).

(iii) ⇒ (iv) Suppose F ⊆ n-cl∗(A) − A, F is n-closed ans x ∈ F. SinceF ⊆ U − A, F ∩ A = φ. We have n-cl(x) ∩ A = φ because F is n-closed andx ∈ F. From (iii), this is a contradiction.

(iv) ⇒ (v) This is obvious from the denition of n-cl∗(A).(v) ⇒ (i) Let V be an n-open subset containing A. Since A∗

nis n-closed

by means of Theorem 2.6(iii) we obtain A∗n∩ (U − V ) ⊆ A∗

n− A is an n-closed

set contained in A∗n− A. By assumption, A∗

n∩ (U − V ) = φ. Hence, we have

A∗n⊆ V .

Corollary 3.6. Let (U,N , I) be a nano ideal topological space and A ⊆ U is

an nIg-closed set, then the following are equivalent:


(i) A is an n∗-closed set.

(ii) n-cl∗(A) − A is an n-closed set.

(iii) A∗n− A is an n-closed set.

Proof. (i) ⇒ (ii) If A is n∗-closed, then n-cl∗(A) − A = φ and n-cl∗(A) − A is

n-closed.(ii) ⇒ (iii) Since n-cl∗(A) − A = A∗

n− A, it is clear.

(iii) ⇒ (i) If A∗n− A is n-closed and A is nIg-closed, from Theorem 3.5(v),

A∗n− A = φ and so A is n∗-closed.

Theorem 3.7. If (U,N , I) is a nano ideal topological space and A is a n∗-densein itself, nIg-closed subset of U , then A is ng-closed.

Proof. Suppose A is a n∗-dense in itself, nIg-closed subset of U. If V is

any open set containing A, then A∗n⊆ V . Since A is n∗-dense in itself, by

Lemma 2.12, n-cl(A) ⊆ V and so A is ng-closed.

Corollary 3.8. If (U,N , I) is any nano ideal topological space where I = φ,then A is nIg-closed if and only if A is ng-closed.

Proof. The proof follows from the fact that for I = φ, A∗n= n-cl(A) ⊃ A and

so every subset of U is n∗-dense in itself.

Theorem 3.9. Let (U,N , I) be a nano ideal topological space. Then every

subset of U is nIg-closed if and only if every n-open set is n∗-closed.

Proof. Suppose every subset of U is nIg-closed. If V is n-open, then V is

nIg-closed and so V ∗n⊆ V . Hence V is n∗-closed. Conversely, suppose that

every n-open set is n∗-closed. If A ⊆ U and V is an n-open set such that A ⊆ V ,

then A∗n⊂ V ∗

nand so A is nIg-closed.

Theorem 3.10. Let (U,N , I) be a nano ideal topological space. Every n∗-closedset is nIg-closed.

Proof. Let A be a subset of X and A be n∗-closed. Assume that A ⊆ V and Vis n-open. Since A is n∗-closed, we have A∗

n⊆ A and so A is nIg-closed.

Example 3.11. Let U = a, b, c, d be the universe, X = a, d ⊂ U , U/R == b, d, a, c and N = U, φ, d, a, c, d, a, c and the ideal I = φ, d.The set A = a, b is nIg-closed but not n∗-closed. Since, A∗

n= a, b, c *

* a, b = A.

For the relationship related to several sets dened in the paper, we have the

following diagram:


n-closed =⇒ ng-closed =⇒ nIg-closed~www

n∗-dense in itself⇐= n∗-perfect =⇒ n∗-closed

Theorem 3.12. Let (U,N , I) be a nano ideal topological space and A ⊆ U . If

A is n∗-dense in itself and nIg-closed, then A is ng-closed.

Proof. Assume A is n∗-dense in itself and nIg-closed onU . IfV is an n-open setcontaining A, then we have A∗

m⊆ V . Since A is n∗-dense in itself, Lemma 2.12

implies n-cl(A) ⊆ V and so A is ng-closed.

Theorem 3.13. Let (U,N , I) be a nano ideal topological space and A ⊆ U . If

A is nIg-closed and n-open then A is n∗-closed.

Proof. Let A be n-open. Since A is nIg-closed, we have A∗n⊆ A. Hence A is

n∗-closed.

Theorem 3.14. If A and B are nIg-closed, then A ∪ B is nIg-closed.

Proof. Let A and B are nIg-closed sets. Then A∗n⊆ V where A ⊆ V and V is

n-open and B∗n⊆ V where B ⊆ V and V is n-open. Since A and B are subsets

of V , (A∗n∪ B∗

n) = (A∪ B)∗

nis a subset of V and V is n-open which implies that

(A ∪ B) is nIg-closed.

Remark 3.15. The Intersection of two nIg-closed sets need not be nIg-closedset which is shown in the following example.

Example 3.16. Let U = a, b, c, d be the universe, X = b, d ⊂ U, U/R == c, d, a, b and N = U, φ, c, d, a, b, d and the ideal I = φ, a.The sets A = b, d and B = c, d are nIg-closed sets, A ∩ B = d is not anIg-closed set.

Theorem 3.17. If A is nIg-closed and A ⊆ B ⊆ A∗n, then B is nIg-closed.

Proof. Let B ⊆ V where V is n-open inN . Then A ⊆ B implies A ⊆ V . Since

A is nIg-closed, A∗n⊆ V . Also B ⊆ A∗

nimplies B∗

n⊆ A∗

n. Thus B∗

n⊆ V and so

B is nIg-closed.

Theorem 3.18. Let (U,N , I) be a nano ideal topological space and A ⊆ U .

Then A is nIg-open if and only if F ⊆ n-int∗(A) whenever F is closed and

F ⊆ A.

Proof. Suppose A is nIg-open. If F is closed and F ⊆ A, then U − A ⊆ U − Fand so n-cl∗(U − A) ⊆ U − F. Therefore, F ⊆ n-int∗(A). Conversely, suppose


the condition holds. Let V be an open set such thatU − A ⊆ V . ThenU −V ⊆ Aand so U − V ⊆ n-int∗(A) which implies that n-cl∗(U − A) ⊆ V . Therefore,

U − A is nIg-closed and so A is nIg-open.

Theorem 3.19. Let (U,N , I) be a nano ideal topological space and A ⊆ U.

Then the following are equivalent.

(i) A is nIg-closed.(ii) A ∪ (U − A∗

n) is nIg-closed.

(iii) A∗n− A is nIg-open.

Proof. (i) ⇒ (ii). Suppose A is nIg-closed. If V is any open set such that

(A ∪ (U − A∗n

)) ⊆ V , then U − V ⊆ U − (A ∪ (U − A∗n

)) = A∗n− A. Since A is

nIg-closed, by Theorem 3.5(v), it follows thatU −V = φ and soU = V . SinceUis the only open set containing A∪ (U− A∗

n), clearly, A∪ (U− A∗

n) is nIg-closed.

(ii) ⇒ (i). Suppose A∪ (U − A∗n

) is nIg-closed. If F is any closed set such

that F ⊆ A∗n− A, then A ∪ (U − A∗

n) ⊆ U − F and U − F is open. Therefore,

(A ∪ (U − A∗n

))∗n⊆ U − F which implies that A∗

n∪ (U − A∗

n)∗n⊆ U − F and so

F ⊆ U − A∗n. Since F ⊆ A∗

n, it follows that F = φ. Hence A is nIg-closed.

The equivalence of (ii) and (iii) follows from the fact that U − (A∗n− A) =

= A ∪ (U − A∗n

).

References

[1] M. E. Abd El-Monsef, E. F. Lashien and A. A. Nasef, On I-open sets and

I-continuous functions, Kyungpook Math. J., 32 (1992), 2130.

[2] K. Bhuvaneswari and K. Mythili Gnanapriya, Nano generalized closed sets

in nano topological spaces, International Journal of Scientic and Research

Publications, Volume 4, Issue 5, May 2014, 13.

[3] T. R. Hamlett andD. Jankovic, Ideals in topological spaces and the set operator

ψ, Boll. Un. Mat. Ital., (7), 4-B (1990), 863874.

[4] E. Hayashi, Topologies dened by local properties. Math. Ann., 156 1964,

205215.

[5] D. Jankovic and T. R. Hamlett,Compatible extensions of ideals,Boll. Un.Mat.

Ital. (7), 6-B (1992), 453465.

[6] D. Jankovic andT. R. Hamlett,New topologies from old via ideals,Amer. Math.

Monthly, 97(4) (1990), 295310.

[7] K. Kuratowski, Topology, Vol. I, Academic Press (New York, 1966).

[8] M. Lellis Thivagar and Carmel Richard, On nano continuity, International

Journal of Mathematics and statistics invention, 1 2013, 3137.


[9] M. Lellis Thivagar and Carmel Richard,On nano forms of weakly open sets,

Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522

(Online), Vol. 3, No. 7, 2013, 3237.

[10] M. Parimala and S. Jafari, On some new notions in nano ideal topological

spaces (communicated).

[11] M. Parimala, T. Noiri and S. Jafari,New types of nano topological spacess via

nano ideals (communicated).

[12] M.Parimala andR.Perumal,Weaker formof open sets in nano ideal topological

spaces, Global Journal of Pure and Applied Mathematics (GJPAM), 12, (2016),

302305.

[13] R. Vaidyanathaswamy, The localization theory in set topology, Proc. Indian

Acad. Sci., 20 (1945), 5161.

M. Parimala

Department of Mathematics

Bannari Amman Institute of Technology

Sathyamangalam-638401

Tamil Nadu, India

[email protected]

S. Jafari

College of Vestsjaelland South

Herrestraede 11,4200 Slagelse

Denmark

[email protected]

S. Murali


Jansons Institute of Technology

Karumathampatti-641659

Tamil Nadu, India

[email protected]


MIXED CONNECTED SPACES

By

V. VIJAYA BHARATHI, T. NOIRI AND N. RAJESH

(Received February 28, 2017)

Abstract. We introduce and investigate the notion of mixed connected on two

generalized topologies µ1 and µ2.

1. Introduction

In the past fewyears, dierent forms of open sets have been studied.Recently,a signicant contribution to the theory of generalized open sets, was extendedby Á. Császár. Especially, he dened some basic operators on generalizedtopological spaces. In [3], he introduced and investigated the notions of mixedgeneralized open ((ν1, ν2)-semiopen, (ν1, ν2)-preopen, (ν1, ν2)-β-open) sets ontwo generalized topologies. Császár and Makai Jr. [4] modied the notions ofδ and θ by mixing two generalized topologies. In this paper, we introduce andinvestigate the notion of mixed connected on two generalized topologies µ1and µ2.

2. Preliminaries

A subfamily µ of the power set P(X ) of a nonempty set X is called ageneralized topology [1] on X if ∅ ∈ µ and Gi ∈ µ for i ∈ I , ∅ implies∪i∈I

Gi ∈ µ. We call the pair (X, µ) a generalized topological space (briey

GTS) on X . The members of µ are called µ-open sets [1] and the complementof a µ-open set is called a µ-closed set. For A ⊂ X , we denote by c (A) the

2000 Mathematics Subject Classication 54A40

14 V. VIJAYA BHARATHI, T. NOIRI, N. RAJESH

intersection of all µ-closed sets containing A, that is, the smallest µ-closed setcontaining A; and by i (A) the union of all µ-open sets contained in A, that is,the largest µ-open set contained in A. Obviously in a topological space (X, µ),if one takes µ as the generalized topology, then c becomes the usual closureoperator.

3. Mixed connected sets

Definition 3.1. Two nonempty subsets A and B of (X, µ1, µ2) are said to be(µ1, µ2)-separated if (A ∩ c1

(B)) ∪ (c2(A) ∩ B) = ∅.

If X = A∪ B such that A and B are (µ1, µ2)-separated sets, then A, B forma (µ1, µ2)-separation of X and we write it as X = A|B.

Remark 3.2. Let µ1 and µ2 be two generalized topologies on a nonempty setX . If µ1 = µ2, then the (µ1, µ2)-separated is just a µ1-separated [2].

Definition 3.3. A subset A of (X, µ1, µ2) is said to be (µ1, µ2)-connected if itcannot be expressed as the union of two (µ1, µ2)-separated sets. Otherwise, theset A is said to be (µ1, µ2)-disconnected.

Theorem 3.4. For (X, µ1, µ2), the following statements are equivalent:(1) X is (µ1, µ2)-connected.(2) X can not be expressed as the union of two nonempty disjoint sets A and B

such that A is µ1-open and B is µ2-open.(3) X contains no nonempty proper subset which is both µ1-open and µ2-closed.

Proof. (1) ⇒ (2): Suppose that X is (µ1, µ2)-connected and if X can beexpressed as the union of two nonempty disjoint sets A and B such that A isµ1-open and B is µ2-open. Then A ∩ B = ∅. Consequently A ⊂ X \ B. Thenc2

(A) ⊂ c2(X \B) = X \B. Therefore, c2

(A)∩B = ∅. Similarly we can proveA∩ c1

(B) = ∅. Hence (A∩ c1(B))∪ (c2

(A)∩ B) = ∅. This is a contradictionto the fact that X is (µ1, µ2)-connected. Therefore, X cannot be expressed as theunion of two nonempty disjoint sets A and B such that A is µ1-open and B isµ2-open.

(2) ⇒ (3): Suppose that X cannot be expressed as the union of twononempty disjoint sets A and B such that A is µ1-open and B is µ2-open. IfX contains a nonempty proper subset A which is both µ1-open and µ2-closed.Then X = A ∪ (X \ A), where A is µ1-open, X \ A is µ2-open and A, X \ A are

MIXED CONNECTED SPACES 15

disjoint sets. This is the contradiction to our assumption. Therefore, X containsno nonempty proper subset which is both µ1-open and µ2-closed.

(3) ⇒ (1): Suppose that X contains no nonempty proper subset which isboth µ1-open and µ2-closed. Suppose that X is not (µ1, µ2)-connected. Then Xcan be expressed as the union of two nonempty disjoint sets A and B such that(A ∩ c1

(B)) ∪ (c2(A) ∩ B) = ∅. Since A ∩ B = ∅, A = X \ B and B = X \ A.

Since c2(A) ∩ B = ∅, c2

(A) ⊂ X \ B. Hence c2(A) ⊂ A. Therefore, A is

µ2-closed. Similarly, B is µ1-closed. Since A = X \ B, A is µ1-open. Therefore,there exists a nonempty proper set A which is both µ1-open and µ2-closed. Thisis a contradiction to our assumption. Therefore, X is (µ1, µ2)-connected.

Theorem 3.5. Let A be a (µ1, µ2)-connected subset of (X, µ1, µ2). If A ⊂ C∪D,where C and D are (µ1, µ2)-separated sets in (X, µ1, µ2), then either A ⊂ C orA ⊂ D.

Proof. Let A ⊂ C∪D. Since A = (A∩C)∪ (A∩D), then (A∩D)∩ (c2(C∩

∩ A)) ⊂ D∩ (c2(C)) = ∅. By similar way we have (A∩C)∩ (c1

(D∩ A)) = ∅.Suppose that A∩C and A∩ D are nonempty. Then A is not (µ1, µ2)-connected.This is a contradiction. Thus, either A ∩ C = ∅ or A ∩ D = ∅. This implies thatA ⊂ C or A ⊂ D.

Theorem 3.6. If A is a (µ1, µ2)-connected subset (X, µ1, µ2) and A ⊂ B ⊂⊂ c1

(A) ∩ c2(A), then B is also a (µ1, µ2)-connected subset of X .

Proof. Suppose that B is not a (µ1, µ2)-connected subset of X . Then B = C∪D,whereC and D are two nonempty disjoint sets such that (C∩c1

(D))∪(c2(C)∩

∩ D) = ∅. Since A is (µ1, µ2)-connected, by Theorem 3.5 A ⊂ C or A ⊂ D. IfA ⊂ C, then D = D∩B ⊂ D∩c2

(A) ⊂ D∩c2(C) = ∅. This is a contradiction.

The case A ⊂ D is proved in the same way.

Theorem 3.7. If Ai : i ∈ ∆ is a nonempty family of (µ1, µ2)-connected setsof (X, µ1, µ2) with ∩

i∈∆Ai , ∅, then ∪

i∈∆Ai is (µ1, µ2)-connected.

Proof. Suppose that ∪i∈∆

Ai is not (µ1, µ2)-connected. Then we have ∪i∈∆

Ai =

= H ∪G, where H and G are (µ1, µ2)-separated sets in X . Since ∩i∈∆

Ai , ∅, we

have a point x in ∩i∈∆

Ai . Since x ∈ ∪i∈∆

Ai , either x ∈ G or x ∈ H . Suppose that

x ∈ H . Since x ∈ Ai for each i ∈ ∆, then Ai and H intersect for each i ∈ ∆.By Theorem 3.5, Ai ⊂ H or Ai ⊂ G. Since H and G are disjoint, Ai ⊂ Hfor all i ∈ ∆ and hence ∪

i∈∆Ai ⊂ H . This implies that G is empty. This is a


contradiction. Suppose that x ∈ G. By similar way, we have that H is empty.This is a contradiction. Thus ∪

i∈∆Ai is (µ1, µ2)-connected.

Proposition 3.8. Let A and B be subsets of (X, µ1, µ2). If A and B are (µ1, µ2)-separated, ∅ , C ⊂ A and ∅ , D ⊂ B, then C and D are (µ1, µ2)-separated.

Proof. Since A and B are (µ1, µ2)-separated sets of (X, µ1, µ2), A∩c1(B) = ∅

and c2(A) ∩ B = ∅. By hypothesis C ⊂ A, we have c2

(C) ∩ D = ∅. Similarly,we have C ∩ c1

(D) = ∅. Therefore, C and D are (µ1, µ2)-separated sets.

Theorem 3.9. Let Fi : i ∈ ∆ be a family of (µ1, µ2)-connected sets of(X, µ1, µ2). If a pair (Fi, Fj ) is not an (µ1, µ2)-separation for any i, j ∈ ∆,then ∪Fi : i ∈ ∆ is (µ1, µ2)-connected.

Proof. Suppose that ∪Fi : i ∈ ∆ is not (µ1, µ2)-connected. Then there exist(µ1, µ2)-separated sets A and B such that ∪Fi : i ∈ ∆ = A ∪ B. Since for eachi ∈ ∆ Fi is (µ1, µ2)-connected and Fi ⊂ A ∪ B, by Theorem 3.5 Fi ⊂ A orFi ⊂ B. Now, put ∆a = i ∈ ∆ : Fi ⊂ A and ∆b = i ∈ ∆ : Fi ⊂ B. Then∆a , ∅, ∆b , ∅ and ∆a ∪ ∆b = ∆. Let ia ∈ ∆a and ib ∈ ∆b , then Fia ⊂ A andFib ⊂ B. By Proposition 3.8, Fia and Fib are (µ1, µ2)-separated sets. This iscontrary to our hypothesis.

Proposition 3.10. If A and B are (µ1, µ2)-separation of (X, µ1, µ2) itself, thenA is µ2-closed and B is µ1-closed.

Proof. Since A and B are (µ1, µ2)-separated, A ∩ c1(B) = c2

(A) ∩ B = ∅.Then A ∩ c1

(B) = ∅ if and only if B is µ1-closed in A ∪ B = X . Similarly, wecan show that A is µ2-closed in X .

Theorem 3.11. If a subset A of (X, µ1, µ2) is (µ1, µ2)-connected, then(A, µ1A, µ2A ) is (µ1, µ2)-connected.

Proof. In fact if U |V is an (µ1, µ2)-separation of A in (A, µ1A, µ2A ), then it isalso an (µ1, µ2)-separation of A in X , since c1

(B) ∩ A ⊂ c1A(B) for every

B ⊂ A.

Definition 3.12. The (µ1, µ2)-component of x in (X, µ1, µ2), denoted by(µ1, µ2)-C(x), is the union of all (µ1, µ2)-connected subsets of X containingx.

Further, if E ⊂ X and x ∈ E, then the union of all (µ1, µ2)-connected setscontaining x and contained in E is called the (µ1, µ2)-component of E.

Theorem 3.13. For (X, µ1, µ2), the following properties hold:

MIXED CONNECTED SPACES 17

(1) Each (µ1, µ2)-component (µ1, µ2)-C(x) is a maximal (µ1, µ2)-connectedset in X .

(2) The set of all distinct (µ1, µ2)-components of points of X froms a partitionof X .

(3) Each (µ1, µ2)-C(x) satifies the equation (µ1, µ2)-C(x) = c1((µ1, µ2)-

C(x)) ∩ c2((µ1, µ2)-C(x)).

(4) Any subset S of X is the union of all its (µ1, µ2)-components.

Proof. (1) Let C be a (µ1, µ2)-connected subset of X containing x. Now byTheorem 3.7 (µ1, µ2)-C(x) is (µ1, µ2)-connected. Since C ⊂ (µ1, µ2)-C(x),then (µ1, µ2)-C(x) is a maximal (µ1, µ2)-connected subset of X containing x.

(2) Let x and y be two distinct points of X and (µ1, µ2)-C(x), (µ1, µ2)-C(y) be their (µ1, µ2)-components. If (µ1, µ2)-C(x) ∩ (µ1, µ2)-C(y) , ∅,then by Theorem 3.7, (µ1, µ2)-C(x) ∪ (µ1, µ2)-C(y) is a (µ1, µ2)-connectedset. But (µ1, µ2)-C(x) ⊂ (µ1, µ2)-C(x) ∪ (µ1, µ2)-C(y), which contradicts themaximality of (µ1, µ2)-C(x).

(3) Let x be any point of X and (µ1, µ2)-C(x) its (µ1, µ2)-component.Suppose that a point p in X does not belong to (µ1, µ2)-C(x). Then (µ1, µ2)-C(x)∪p is not (µ1, µ2)-connected and hence there exists a (µ1, µ2)-separationA|B in X such that (µ1, µ2)-C(x)∪ p = A|B. By Theorem 3.5, either (µ1, µ2)-C(x) ⊂ A and p ⊂ B or (µ1, µ2)-C(x) ⊂ B and p ⊂ A. Thus (µ1, µ2)-C(x)∪∪ p = (µ1, µ2)-C(x) |p or (µ1, µ2)-C(x) ∪ p = p|(µ1, µ2)-C(x). Hencep < c1

((µ1, µ2)-C(x)) or p < c2((µ1, µ2)-C(x)). Hence p < c1

((µ1, µ2)-C(x)) ∩ c2

((µ1, µ2)-C(x)) and so c1((µ1, µ2)-C(x)) ∩ c2

((µ1, µ2)-C(x)) ⊂⊂ (µ1, µ2)-C(x).

(4) For x ∈ S, x is contained in the (µ1, µ2)-component (µ1, µ2)-C(x) ofS containing x. Hence x ⊂ (µ1, µ2)-C(x). Since S = ∪

x∈S(µ1, µ2)-C(x), then

the proof follows.

References

[1] Á. Császár, Generalized topology, generalized continuity, Acta Math. Hungar.,

96 (2002), 351357.

[2] Á. Császár, γ-connected sets, Acta Math. Hungar., 101 (2003), 273279.

[3] Á. Császár, Mixed constructions for generalized topologies, Acta Math. Hungar.,

122 (2009), 153159.

[4] Á. Császár and E. Makai Jr., Further remarks on δ- and θ-modications, Acta

Math. Hungar., 123 (2009), 223228.


V. Vijaya Bharathi


National Institute of Technology

Tiruchirappalli, Tamilnadu, India

[email protected]

T. Noiri

2949-1 Shiokita-cho,

Hinagu, Yatsushiro-shi

Kumamoto-ken, 869-5142 Japan

[email protected]

N. Rajesh


Rajah Serfoji Govt. College

Thanjavur-613005

Tamilnadu, India

[email protected]


GEOMETRIC INEQUALITIES CONCERNING MEDIANS OF ACUTE

TRIANGLES

ByBÉLA FINTA

(Received March 10, 2017)

Abstract. The goal of the paper is to study the validity of the following statement:

if a, b, c denote the sides of an arbitrary acute triangle ABC such that a < b < c, andma,mb,mc are the corresponding medians, then a +m

a < b +m

b< c +m

c , where

α is a given positive real number.

1. Introduction

Let us consider an acute triangle ABC with sides a = BC, b = AC and

c = AB. In [1] the following conjecture due to Erd®s is formulated: if ABC isan acute triangle such that a < b < c then a + la < b + lb < c + lc , where la, lband lc denote the lengths of the interior bisectrices corresponding to the sidesBC, AC and AB, respectively. The origin of this Erd®s' problem is unknown for

me.

Later Bencze generalized the problem of Erd®s and has formulated the

following conjecture [2]: determine all interior points M in the acute triangleABC for which BC < AC < AB implies that BC+AA′ < AC+BB′ < AB+CC ′,where A′ = AM ∩ BC, B′ = BM ∩ AC and C ′ = CM ∩ AB.

In [3] an acute triangle ABC was constructed for which the Erd®s' inequality

is false: let ABC be a triangle such that c = 10+ε, b = 10 and a = 1, where ε > 0

is suciently small. Combining trigonometric techniques with some elementary

results of algebra and mathematical analysis, we established that c + lc < b+ lb ,if a < b < c.

2000 Mathematics Subject Classication 97H30

20 BÉLA FINTA

Furthermore, in [4] Sándor proved some new geometric inequalities replac-

ing in Erd®s' conjecture the bisectrices with altitudes and medians, respectively.

In [5] Dáné and Ignát studied the open question of Bencze with the aid of

computational methods.

In connection with Erd®s' problem, we formulated and proved a related

inequality in [6]: if ABC is an acute triangle such that a < b < c thena2+l2a < b2+l2

b< c2+l2c .Besides this another questionwas also formulated: in an

arbitrary acute triangle ABC does it follow from the inequalities a 0 rather small) the condition a 0 rather small) we have a4+ l4a > b4+ l4

b,

if a < b < c.As usual, we denote by ha, hb and hc the lengths of the altitudes corre-

sponding to the sides BC, AC and AB of the triangle ABC. Replacing the

bisectrices with altitudes in Erd®s' conjecture, we may formulate the following

more general statement: if ABC is an acute triangle such that a < b < c thena + ha < b + h

b< c + hc , where α is a real number. In [4] and [8] it was

shown that this property is true for every α , 0.

Analogously to the above mentioned results, we may study the validity of

the following statement:

(1.1)if ABC is an acute triangle and α is a real number, then a < b < cimplies that a + m

a < b + mb< c + m

c .

Here ma,mb and mc denote the lengths of the medians corresponding to the

sides BC, AC and AB.We mention, that the statement (1.1) is not obvious, because a < b is

equivalent to mb < ma due to the well-known formulae from elementary

geometry:

(1.2) m2

a =2(b2 + c2) − a2

4and m2

b =2(a2 + c2) − b2

2.

Analogously, b < c is equivalent to mc < mb . It is clear that (1.1) is false for

α = 0.

Finally, we summarize some earlier results in connection with (1.1). The

problem (1.1) was solved in [8] for α ∈ 1, 2, 4, obtaining that (1.1) is false

for α = 1 and (1.1) is true for α ∈ 2, 4. The case α = 8 was studied in [9],

proving that (1.1) is false, while the case α = −2 was studied in [10], proving

that (1.1) is also false. In [11] it was obtained that for every even natural number

GEOMETRIC INEQUALITIES CONCERNING MEDIANS OF ACUTE TRIANGLES 21

α = 2n, n = 3, 4, . . ., the statement (1.1) is false. Analogously, in [12] it was

proved for the odd natural numbers α = 2n + 1, n = 2, 3, . . ., that (1.1) is false.If α = −2n, n = 1, 2, . . ., then the statement (1.1) is not valid, as it was shown

in [13]. For α = −2n − 1, n = 0, 1, . . . we proved in [14] that the problem (1.1)

is also false. The case of rational numbers α =p

q> 2 log 5

4

5

3was solved in [15],

giving negative answer to (1.1). In [16] it was proved that the statement (1.1) is

not true, if α is an arbitrary negative rational number.

We mention in connection with the above problems that in [17] we stud-

ied the characterization of the isosceles and equilateral triangles by algebraic

relations.

The goal of the paper is to study the validity of the geometric inequalities

formulated in (1.1) for an arbitrary positive number α.

2. Main results

Taking into account [8], we have the following results concerning (1.1):

Proposition 1, Proposition 2 and Proposition 3 (see below). For completeness

we give their proofs.

Proposition 1 (case α = 1). In every acute triangle ABC the inequality

a + ma c + mc .

Proof. First we prove that in every acute triangle ABC with a < b < cwe have a + ma < b + mb . Indeed, in view of the Cosine Rule, we get

c2 = a2 + b2 − 2ab cosC. Therefore,

(2.1) if a < b < c, then the triangle ABC is acute if and only if c2 < a2 + b2.

Hence a < b < c <√

a2 + b2. Dividing these inequalities by a, and using the

notations x = ba, y = c

a, we obtain

(2.2) 1 < x < y <√1 + x2.

On the other hand, a + ma < b + mb means in view of (1.2) that

a +

√2(b2 + c2) − a2

2< b +

√2(a2 + c2) − b2

2

22 BÉLA FINTA

which takes the form

1 +

√2(x2 + y2) − 1

2< x +

√2(1 + y2) − x2

2.

After simple algebraic manipulations this is equivalent to

(2.3)3

2(x + 1) <

√(x2 − 1) + (x2 + 2y2) +

√(y2 − x2) + (y2 + 2).

Because of (2.2), we obtain x2 − 1 > 0, x2 + 2y2 > 3x2, y2 − x2 > 0 and

y2 + 2 > 3. Thus in order to prove (2.3), it is enough to verify the inequality

3

2(x + 1) <

√3x +

√3, which obviously holds.

Next we show the existence of an acute triangle ABC such that b + mb << c + mc . Indeed, let A"B"C" be the triangle with sides a = 1, b = 1 + ε and

c =√2, which is an acute triangle for suciently small ε > 0 due to (2.1). Then

b + mb < c + mc takes the form

(1 + ε) +

√2(12 +

√22

) − (1 + ε)2

2<√2 +

√2(12 + (1 + ε)2) −

√22

2

which is equivalent to

(2.4) ε +

√5 − 2ε − ε2

2−

√2 + 4ε + 2ε2

2<√2 − 1.

If ε → 0, then the left-hand side of (2.4) tends to√5

2−√2

2which is clearly less

than√2 − 1, so by the denition of the limit we obtain that there exists ε0 > 0

rather small such that (2.4) holds for all ε ≤ ε0. This means that the inequality

(2.4) is true for the acute triangle A"0 B"0C"0 .

In the next step we prove that there exists an acute triangle ABC such that

b+mb > c+mc . Indeed, let A"B"C" be the acute triangle with sides a = 3, b = 4

and c = 5− ε (see (2.1)), where ε > 0 is a suciently small quantity. As before,

we have that the inequality b + mb > c + mc is equivalent to

(2.5) ε +

√52 − 20ε + 2ε2

2−

√25 + 10ε − ε2

2> 1.

The left-hand side of (2.5) tends to√13 − 5

2> 1. Then, in view of denition of

the limit, there exists ε1 > 0 rather small such that (2.5) holds for all ε ≤ ε1.This means that the inequality (2.5) is true for the acute triangle A"1 B"1C"1 .

Consequently, the conditions a c + mc .


Proposition 2 (case α = 2). In every triangle ABC the inequalities a < b < cimply a2 + m2

a < b2 + m2

b< c2 + m2

c .

Proof. Due to (1.2), the inequality a2 + m2a < b2 + m2

bhas the form

a2 +2(b2 + c2) − a2

4< b2 +

2(a2 + c2) − b2

4

which is clearly equivalent to a2 < b2 and hence to a < b.Analogously b2 + m2

b< c2 + m2

c is equivalent to b < c.

Proposition 3 (case α = 4). If the sides of the arbitrary acute triangle ABCsatisfy a < b < c then a4 + m4

a < b4 + m4

b< c4 + m4

c .

Proof. Taking into account (1.2), the inequality a4 + m4a < b4 + m4

btakes the

form

a4 +

"2(b2 + c2) − a2

4

#2< b4 +

"2(a2 + c2) − b2

4

#2,

and after some simple algebraic manipulations it is equivalent to

(2.6) (a2 − b2)[13(a2 + b2) − 12c2] < 0.

In view of (2.1), we have a2 + b2 > c2 or 13(a2 + b2) > 12c2. In conclusion,

(2.6) is true.

Analogously to the above procedure, we get b4 + m4

b< c4 + m4

c . This

completes the proof of our proposition.

We have the following new result in connection with (1.1).

Proposition 4. Let α ∈(2, 2 log 5

4

5

3

)and ABC be an arbitrary acute triangle

such that a < b < c. Then a + ma < b + m

b< c + m

c .

Proof. With the notations used in the proof of Proposition 1, we have that the

inequality

a + ma 1 we consider the function f : [1, y]→ R,

f (x) = x − 1 +[2(1 + y

2) − x2]

2

2−

[2(x2 + y2) − 1]

2

2.

24 BÉLA FINTA

We will show that f ′(x) > 0 for all x ∈ (1, y), which means that f is strictly

increasing, therefore f (x) > f (1) = 0. Hence (2.7) follows.

Since by simple calculation we have

f ′(x) = αx−1 +α

2·[2(1 + y

2) − x2]

2−1

2(−2x) −

α

2·[2(x2 + y

2) − 1]

2−1

24x,

thus f ′(x) > 0 is equivalent to

(2.8) 2 x−2 > [2(1 + y2) − x2]

2−1 + 2[2(x2 + y

2) − 1]

2−1.

But using (2.2) we have

(2.9) 2(1 + y2) − x2 < 2(1 + 1 + x2) − x2 = 4 + x2 < 4x2 + x2 = 5x2

and

2(x2 + y2) − 1 < 2(x2 + 1 + x2) − 1 = 4x2 + 1 < 4x2 + x2 = 5x2,

therefore (2.8) follows from the inequality 2 x−2 > (5x2)

2−1 + (5x2)

2−1 · 2,

which is true for α < 2 log 5

4

5

3.

For the inequality b + mb< c + m

c we use the same ideas as above. It is

equivalent to

x +[2(1 + y

2) − x2]

2

2< y

+[2(1 + x2) − y

2]

2

2.

This inequality follows from the strictly decreasing property of the function

f : [1, y]→ R,

f (x) = y − x +

[2(1 + x2) − y2]

2

2−

[2(1 + y2) − x2]

2

2,

where y > 1 is xed. If f ′(x) < 0 for all x ∈ (1, y), then f (x) > f (y) = 0, and

the inequality follows.

Since by simple calculation we have

f ′(x) = −αx−1 +α

2·[2(1 + x2) − y

2]

2−1

24x −

α

2·[2(1 + y

2) − x2]

2−1

2(−2x),

therefore f ′(x) < 0 is equivalent to

2 x−2 > [2(1 + y2) − x2]

2−1 + 2[2(x2 + y

2) − 1]

2−1.

But using (2.2) we have

2(1 + x2) − y2 < 2(1 + x2) − x2 = 2 + x2 < 2x2 + x2 = 3x2,

and (2.9) is also valid, thuswe have to prove that 2 x−2 > 2(3x2)

2−1+(5x2)

2−1

or 2 > 2(√3)−2 + (

√5)−2. Since (

√3)−2 < (

√5)−2 for α > 2, it suces


to verify that 2 > 2(√5)−2 + (

√5)−2, which is true for α < 2 log 5

4

5

3.

Consequently, if α ∈ (2, 2 log 5

4

5

3), then we get b + m

b< c + m

c .

Let α =p

qbe a positive rational number. If q = 1 and p ≥ 5, then (1.1) is

false, taking into account [11] and [12]. For q = 2, 3, . . ., using [15], we have thefollowing result.

Proposition 5. There exist acute triangles ABC with the properties a < b < cand a

p

q +(ma )p

q < bp

q +(mb )p

q , where p = 1, 2, . . . and q = 2, 3, . . . are arbitrary.

Proof. The inequality ap

q + (ma )p

q < bp

q + (mb )p

q is equivalent to (ma )p

q −

− (mb )p

q < bp

q − ap

q . Using the formulae un − vn = (u− v)(un−1 +un−2v + . . . +

+ uvn−2 + vn−1) four times, the latter inequality is equivalent to

(ma − mb ) ·

p−1∑i=0

( q√

ma )p−1−i ( q√

mb )i

q−1∑i=0

( q√

ma )q−1−i ( q√

mb )i< (b − a) ·

p−1∑i=0

(q√

b)p−1−i ( q√a)i

q−1∑i=0

(q√

b)q−1−i ( q√a)i

or

(2.10)m2a − m2

b

ma + mb

·

p−1∑i=0

( q√

ma )p−1−i ( q√

mb )i

q−1∑i=0

( q√

ma )q−1−i ( q√

mb )i<

b2 − a2

b + a·

p−1∑i=0

(q√

b)p−1−i ( q√a)i

q−1∑i=0

(q√

b)q−1−i ( q√a)i.

By (1.2), we have m2a − m2

b= 3

4(b2 − a2) > 0. Hence, in view of (2.10) and the

notation

E(

pq

)=

3

4

ma + mb

·

p−1∑i=0

( q√

ma )p−1−i ( q√

mb )i

q−1∑i=0

( q√

ma )q−1−i ( q√

mb )i

/ *.....,1

a + b

p−1∑i=0

( q√a)p−1−i (q√

b)i

q−1∑i=0

( q√a)q−1−i (q√

b)i

+/////-,

wherep

q(p = 1, 2, . . . ; q = 2, 3, . . .) are arbitrary positive rational numbers, we

get

(2.11) E(

pq

)< 1.

26 BÉLA FINTA

Nowwe choose the acute triangle A"B"C" such that b = a+ε and c = a+2ε,where ε > 0 is a small quantity (see also (2.1)). Then, by (1.2),

lim"0

m2

a = lim"0

2[(a + ε)2 + (a + 2ε)2] − a2

4=

3a2

4,

lim"0

m2

b = lim"0

2[a2 + (a + 2ε)2] − (a + ε)2

4=

3a2

4

and lim"0

b = lim"0(a + ε) = a. This means that

lim"0

E(

pq

)=

3

4·

1

2√3

2a·

(√3

2a) p−1

q

p(√3

2a) q−1

q

q

/ *.,1

2aap−1

q p

aq−1

q q

+/- = *,√3

2+-p

q

.

But (√3

2)p

q < 1 for p = 1, 2, . . . and q = 2, 3, . . ., therefore the denition of

the limit implies that there exists ε0 > 0 suciently small such that (2.11) is

satised for the triangle A"0 B"0C"0 . This completes the proof.


Proposition 6. Let α be a positive irrational number. Then there exist acute

triangles ABC such that a + ma 0 is suciently small. Then

a < b < c and the triangle A"B"C" is acute due to (2.1).

Now let ( pnqn

) be a sequence of positive rational numbers such that limn→∞

pnqn=

= α. In this case limn→∞

qn = +∞. We show that for all suciently small xed

ε > 0 the inequality

(2.12) apn

qn + (ma )pn

qn < bpn

qn + (mb )pn

qn

holds if n is large enough. From this we will obtain the desired inequality by

passing to the limit as n → ∞.Recall from the proof of Proposition 5 that (2.12) is equivalent to

E(pn/qn ) < 1. In order to guarantee this, we use a mb and

we obtain

(2.13) E(

pnqn

)≤

3

4 · 2mb

·(ma )

pn−1

qn pn

(mb )qn−1

qn qn

/ *.,1

2bapn−1

qn pn

bqn−1

qn qn

+/- .


We claim that the right-hand side of (2.13) is less than 1 for n large enough.

Indeed,

lim"0

ma

a=

√3

2, lim

"0

mb

a=

√3

2and lim

"0

mb

b= lim

"0

mb

aab=

√3

2.

Furthermore

lim"0

3

4

(b

mb

)2 (ma

a

)=

3

4

(2√3

)2 *,√3

2+-

= *,√3

2+-

< 1,

therefore the denition of the limit implies that there exists ε0 > 0 such that

(2.14)3

4

(b

mb

)2 (ma

a

)< 1

for every ε ∈ (0, ε0). By (2.13) and (2.14)

limn→∞

3

4 · 2mb

·(ma )

pn−1

qn pn

(mb )qn−1

qn qn

/ *.,1

2bapn−1

qn pn

bqn−1

qn qn

+/- == lim

n→∞

3

4·

bmb

·

(ma

a

) pn

qn− 1

qn

(b

mb

)1− 1

qn

=3

4

(b

mb

)2 (ma

a

)< 1

for every acute triangle A"B"C" with ε ∈ (0, ε0). Now let A"B"C" be a xed

acute triangle, where ε ∈ (0, ε0). According to the denition of the limit, there

exists a natural number n0 such that the right-hand side of the inequality (2.13)

is less than 1 for each n ≥ n0. Consequently E( pnqn

) < 1 for n ≥ n0.

It remains to show that there exists ε1 ∈ (0, ε0) such that a +ma < b +m

b

in the triangle A"1 B"1C"1 . In order to prove this assertion we use the method of

reductio ad absurdum. Let us suppose that a + ma = b + m

bis satised in the

acute triangle A"B"C" for every ε ∈ (0, ε0). Then

a +

"2[(a + ε)2 + (a + 2ε)2] − a2

4

# 2

=

= (a + ε) +

"2[a2 + (a + 2ε)2] − (a + ε)2

4

# 2

.

28 BÉLA FINTA

Dierentiating both sides of the equality with respect to ε, we obtain

α

2

"2[(a + ε)2 + (a + 2ε)2] − a2

4

# 2−1

2[2(a + ε) + 2(a + 2ε) · 2]4

=

=α(a+ε)−1+α

2

"2[a2+(a+2ε)2]−(a+ε)2

4

# 2−1

2[2(a+2ε) · 2]−2(a + ε)4

,

where 0 < ε < ε0. Taking the limit as ε approaches zero, we nd that

α

2

(3a2

4

) 2−1

3a = αa−1 +α

2

(3a2

4

) 2−1

3a2

which simplies to (3/4)=2 = 1 and thus we obtain α = 0 as a contradiction.

The next result was proved in [15]. For completeness we present its proof.

Proposition 7. For everyp

q> 2 log 5

4

5

3, where p = 1, 2, . . . and q = 2, 3, . . .,

there exist acute triangles ABC with the properties a 

> bp

q + (mb )p

q .

Proof. Proceeding as above, we have that ap

q + (ma )p

q > bp

q + (mb )p

q is

equivalent to E( pq

) > 1.

We choose the acute triangle A"B"C" with sides b = a + ε and c = a√2,

where ε > 0 is rather small (see (2.1)). Then, by (1.2)

lim"0

m2

a = lim"0

2[(a + ε)2 + 2a2] − a2

4=

5a2

4,

lim"0

m2

b = lim"0

2[a2 + 2a2] − (a + ε)2

4=

5a2

4

and lim"0

b = lim"0(a + ε) = a. Hence

lim"0

E(

pq

)=

3

4·

1

2√5

2a·

(√5

2a) p−1

q

p(√5

2a) q−1

q

q

/ *.,1

2aap−1

q p

aq−1

q q

+/- =3

5*,√5

2+-p

q

.

Since 3

5

(√5

2

) p

q

> 1 forp

q> 2 log 5

4

5

3, we obtain in view of denition of the limit

that there exists ε1 > 0 small enough such that the triangle A"1 B"1C"1 satises

ap

q + (ma )p

q > bp

q + (mb )p

q .


Remark 1. Let α =p

q> 0 be a rational number. If q = 1 and α = p ≥ 5, then

(1.1) is false. If q ≥ 2 and α =p

q> 2 log 5

4

5

3then (1.1) is also false because of

Proposition 5 and Proposition 7. But 5 > 2 log 5

4

5

3> 4, therefore the problem

(1.1) will be false for every rational number α =p

q> 2 log 5

4

5

3.


Proposition 8. Let α be a positive irrational number such that α > 2 log 5

4

5

3.

Then there exist acute triangles ABC forwhich a b+m

b.

Proof. Let us consider the triangle A"B"C" with B"C" = a, A"C" = b = a+ ε

and A"B" = c = a√2, where ε > 0 is a small quantity. Then a < b < c and the

triangle A"B"C" is acute due to (2.1).

Now we proceed similarly as in the proof of Proposition 6. We choose a

sequence ( pnqn

) of positive rationals such that limn→∞

pnqn= α and we show that the

inequality apn

qn + (ma )pn

qn > bpn

qn + (mb )pn

qn or equivalently E(pn/qn ) > 1 holds

for suciently large n.Indeed, by using a mb , we obtain

(2.15) E(

pnqn

)≥

3

4 · 2ma

(mb )pn−1

qn pn

(ma )qn−1

qn qn

/ *.,1

2abpn−1

qn pn

aqn−1

qn qn

+/- .But

lim"0

ma

a=

√5

2, lim

"0

mb

a=

√5

2and lim

"0

mb

b= lim

"0

mb

aab=

√5

2.

Since

lim"0

3

4

(a

ma

)2 (mb

b

)=

3

4

(2√5

)2 *,√5

2+-

=3

5*,√5

2+-

> 1,

we obtain that there exists ε1 > 0 such that

(2.16)3

4

(a

ma

)2 (mb

b

)> 1

30 BÉLA FINTA

for every ε ∈ (0, ε1). By the right-hand side of (2.15) and (2.16)

limn→∞

3

4 · 2ma

·(mb )

pn−1

qn pn

(ma )qn−1

qn qn

/ *.,1

2abpn−1

qn pn

aqn−1

qn qn

+/- == lim

n→∞

3

4·

ama

(mb

b

) pn

qn− 1

qn

(a

ma

)1− 1

qn

=3

4

(a

ma

)2 (mb

b

)> 1

for every acute triangle A"B"C" with ε ∈ (0, ε1). Now let A"B"C" be a xed

acute triangle, ε ∈ (0, ε1). According to the denition of the limit, there exists a

natural number n1 such that the right-hand side of the inequality (2.15) is largerthan 1 for each n ≥ n1. Consequently E( pn

qn) > 1 for n ≥ n1.

It remains to show that there exists ε2 ∈ (0, ε1) such that a +ma > b +m

b

in the triangle A"2 B"2C"2 . In order to prove this assertion we use the method of

reductio ad absurdum as before in the proof of Proposition 6. Let us suppose

that a + ma = b + m

bis satised in the acute triangle A"B"C" for every

ε ∈ (0, ε1). Then

a +

"2[(a + ε)2 + 2a2] − a2

4

# 2

= (a + ε) +

"2[a2 + 2a2] − (a + ε)2

4

# 2

.

Dierentiating both sides of the equality with respect to ε, we obtain

α

2

"2[(a + ε)2 + 2a2] − a2

4

# 2−1

2 · 2(a + ε)4

=

= α(a + ε)−1 +α

2

"2[a2 + 2a2] − (a + ε)2

4

# 2−1−2(a + ε)

4,

where 0 < ε < ε1. Taking the limit as ε approaches zero, we nd that

α

2

(5a2

4

) 2−1

a = αa−1 +α

2

(5a2

4

) 2−1−a2

which simplies to(5

4

)−2=

(4

3

)2and we obtain α = 2 log 5

4

5

3, as a contradic-

tion.

Remark 2. As we have shown in Proposition 6 and Proposition 8, the problem

(1.1) is false for every irrational number α > 2 log 5

4

5

3. In conclusion, (1.1) is

false for all real numbers α > 2 log 5

4

5

3.


3. Discussion and conclusion

It is worth mentioning that (1.1) is also false for α ∈ (−∞, 1), whichhas been proved with similar technique used in the present paper (unpublished

result). In conclusion, (1.1) is true for α ∈2, 2 log 5

4

5

3

)and (1.1) is false for

α ∈ (−∞, 1] ∪(2 log 5

4

5

3, +∞

).

The case α ∈ (1, 2) ∪2 log 5

4

5

3

is unsolved.

Acknowledgements. The author would like to thank the unknown referee for

the suggestions made in order to improve the presentation.

References

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Didactica Matematicii, Babe³-Bolyai University, 25 (1) (2007), 7577.

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Medians III, in: International Conference Mathematical Education in the Current

32 BÉLA FINTA

European Context, 3rd edition, November 23, 2012, Bra³ov, Romania, 151156,

ISBN 978606-624475-6, StudIS Publishing House, Ia³i, 2013.

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Medians IV, in: International Conference Mathematical Education in the Current

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ISSN 23605324, ISSN-L 23605324, StudIS Publishing House, Ia³i, 2014.

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Scientic Bulletin of the Petru Maior University of Tîrgu Mure³, Vol. 11(XXVIII)

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Scientic Bulletin of the PetruMaior University of TîrguMure³, Vol. 12(XXIX) no.

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Béla Finta

Petru Maior University of Tîrgu Mure³

Department of Informatics

540088 Tîrgu Mure³, Romania

[email protected]


SOME GENERALIZATIONS OF ULTRA WEAKLY CONTINUOUS

MULTIFUNCTIONS

By

TAKASHI NOIRI AND VALERIU POPA


Abstract. We introduce the notion of upper/lower weakly (τ,m)-continuousmultifunctions and obtain many charactrizations of such multifunctions. The notion

of upper/lower weakly (τ,m)-continuous multifunctions is a generalization of weakly

(τ,m)-continuous functions [32] and upper/lower ultra weakly continuous multifunc-

tions due to Rajeswari [34].

1. Introduction

Semi-open sets, preopen sets, α-open sets and β-open sets play an important

role in the researching of generalizations of continuity of functions and multi-

functions in topological spaces and bitopological spaces. By using these sets,

many authors introduced and studied various types ofmodications of continuity

in bitopological spaces.

The notions of (i, j)-semi-open sets [17], (i, j)-preopen sets [9], (i, j)-α-open sets [10], (i, j)-semi-preopen sets [11]; (i, j)-semi-continuity, (i, j)-precontinuity, (i, j)-α-continuity and (i, j)-semipre-continuity are introduced

and investigated in bitopological spaces.

On the other hand, the notions of quasi-open sets or τ1τ2-open sets [7], [18],[36], [37], quasi-semi-open sets [12], quasi preopen sets [28], quasi-α-open sets[39], quasi-semipreopen sets [39]; quasi-continuity, quasi-semi-continuity, quasi

precontinuity, quasi α-continuity and quasi semiprecontinuity are introduced and

studied in bitopological spaces.

2000 Mathematics Subject Classication 54C08, 54C60, 54E55

34 TAKASHI NOIRI, VALERIU POPA

As variations of quasi-open sets and quasi-continuity, the notions of (1, 2)-semi-open sets, (1, 2)-preopen sets, (1, 2)-α-open sets; (1, 2)-semi-continuity,

(1, 2)-precontinuity, and (1, 2)-α-continuity are introduced in [13]. The notionsof (1, 2)-semi-preopen sets and (1, 2)-semi-precontinuity are introduced and

studied in [14].

Similarly, the notions of (1, 2)∗-semiopen sets, (1, 2)∗-preopen sets,

(1, 2)∗-α-open sets, (1, 2)∗-semi-preopen sets; (1, 2)∗-semi-continuity, (1, 2)∗-precontinuity, (1, 2)∗-α-continuity, and (1, 2)∗-semi-precontinuity are intro-

duced in [35].

In 1961, Levine [15] introduced the notion of weakly continuous func-

tions. Popa [27] and Smithson [38] introduced the notion of weakly continuous

multifunctions. The present authors introduced and studied another weak forms

of continuous multifunctions: weakly quasi-continuous multifunctions, almost

weakly continuous multifunctions, weakly α-continuous multifunctions, weakly

β-continuous multifunctions. Recently, Rajeswari [34] introduced and studied

the notion of ultra weakly continuous multifunctions.

The present authors introduced and investigated the notions of minimal

structures, m-spaces [29] and [31], m-continuity [31], M-continuity [29] and

(τ,m)-continuity [32] for functions. The notion of weakly M-continuous multi-

functions is introduced and investigated in [24]. Moreover, in [25] some forms

of generalizations of ultra continuous multifunctions are studied.

In this paper, we introduce the notion of upper/lower weakly (τ,m)-continuous multifunctions as multifunctions from a topological space (X, τ)into an m-space (Y,m). The notion of weakly (τ,m)-continuous multifunctions

is a generalization of the notions of weakly (τ,m)-continuous functions [32]

and ultra weakly continuous multifunctions due to Rajeswari [34]. In the last

section, we transfer the study of a multifunction F from a topological space

(X, τ) into a bitopological space (Y, σ1, σ2) to the study of a (τ,m)-continuousmultifunction F : (X, τ) → (Y,m(σ1, σ2)) which enable us to obtain the uniedtheory of many generalizations of ultra continuous multifunctions.

2. Preliminaries

Let (X, τ) be a topological space and A a subset of X . The closure of A and

the interior of A are denoted by Cl(A) and Int(A), respectively. Let (X, τ1, τ2) bea bitopological space and A be a subset of X . The closure and the interior of Awith respect to τi are denoted by i Cl(A) and i Int(A), respectively, for i = 1, 2.

SOME GENERALIZATIONS OF ULTRA WEAKLY CONTINUOUS MULTIFUNCTIONS 35

Definition 2.1. Let (X, τ) be a topological space. A subset A of X is said to be

(1) semi-open [16] if A ⊂ Cl(Int(A)),(2) preopen [20] if A ⊂ Int(Cl(A)),(3) α-open [22] if A ⊂ Int(Cl(Int(A))),(4) β-open [1] or semi-preopen [3] if A ⊂ Cl(Int(Cl(A))).

The family of all semi-open (resp. preopen, α-open, β-open) sets in (X, τ)is denoted by SO(X ) (resp. PO(X ), α(X ), β(X ) or SPO(X )).

Definition 2.2. The complement of a semi-open (resp. preopen, α-open, β-open) set is said to be semi-closed [4] (resp. preclosed [20], α-closed [21],

β-closed [1] or semi-preclosed [3]).

Definition 2.3. The intersection of all semi-closed (resp. preclosed, α-closed,β-closed) sets of X containing A is called the semi-closure [4] (resp. preclosure

[8], α-closure [21], β-closure [2] or semi-preclosure [3]) of A and is denoted by

sCl(A) (resp. pCl(A), αCl(A), Cl(A) or spCl(A)).

Definition 2.4. The union of all semi-open (resp. preopen, α-open, β-open)sets of X contained in A is called the semi-interior (resp. preinterior, α-interior,β-interior or semi-preinterior) of A and is denoted by sInt(A) (resp. pInt(A),α Int(A), Int(A) or spInt(A)).

Throughout the present paper, (X, τ) and (Y, σ) (or simply X andY ) denotetopological spaces and F : X → Y (resp. f : X → Y ) presents a multivalued

(resp. singlevalued) function. For a multifunction F : X → Y , we shall denotethe upper and lower inverse of a subset B ofY by F+(B) and F−(B), respectively,that is,

F+(B) = x ∈ X : F (x) ⊂ B and F−(B) = x ∈ X : F (x) ∩ B , ∅.

3. Minimal structures

Definition 3.1. A subfamily mX of the power set P (X ) of a nonempty set X is

called a minimal structure (briey m-structure) on X [29], [30] if ∅ ∈ mX and

X ∈ mX .

By (X,mX ), we denote a nonempty subset X with a minimal structure mX

on X and call it an m-space. Each member of mX is said to be mX -open (briey


m-open) and the complement of an mX -open set is said to be mX -closed (briey

m-closed).

Remark 3.1. Let (X, τ) be a topological space. Then the families τ, SO(X),

PO(X ), α(X ), β(X ), SPO(X ) are all m-structures on X .

Definition 3.2. Let X be a nonempty set and mX an m-structure on X . For a

subset A of X , the mX -closure of A and the mX -interior of A are dened in [19]

as follows:

(1) mCl(A) =⋂F : A ⊂ F, X \ F ∈ mX ,

(2) mInt(A) =⋃U : U ⊂ A,U ∈ mX .

Remark 3.2. Let (X, τ) be a topological space and A a subset of X . If mX = τ(resp. SO(X ), PO(X ), α(X ), β(X ), SPO(X )), then we have

(1) mCl(A) = Cl(A) (resp. sCl(A), pCl(A), αCl(A), Cl(A), spCl(A)),(2) mInt(A) = Int(A) (resp. sInt(A), pInt(A), α Int(A), Int(A), spInt(A)).

Lemma 3.1 (Maki et al. [19]). Let X be a nonempty set and mX an m-structureon X . For subsets A and B of X , the following properties hold:

(1) mCl(X \ A) = X \mInt(A) and mInt(X \ A) = X \mCl(A),(2) If (X \ A) ∈ mX , then mCl(A) = A and if A ∈ mX , then mInt(A) = A,(3) mCl(∅) = ∅, mCl(X ) = X , mInt(∅) = ∅ and mInt(X ) = X ,(4) If A ⊂ B, then mCl(A) ⊂ mCl(B) and mInt(A) ⊂ mInt(B),(5) A ⊂ mCl(A) and mInt(A) ⊂ A,(6) mCl(mCl(A)) = mCl(A) and mInt(mInt(A)) = mInt(A).

Lemma 3.2 (Popa and Noiri [30]). Let (X,mX ) be an m-space and A a subsetof X . Then x ∈ mCl(A) if and only if U ∩ A , ∅ for every U ∈ mX containing x.

Definition 3.3. Anm-structure mX on a nonempty set X is said to have property

B [19] if the union of any family of subsets belonging to mX belongs to mX .

Remark 3.3. Let (X, τ) be a topological space. Then the families τ, SO(X ),PO(X ), α(X ), β(X ), and SPO(X ) have property B.

Lemma 3.3 (Popa and Noiri [33]). Let X be a nonempty set and mX anm-structure on X satisfying property B. For a subset A of X , the followingproperties hold:

(1) A ∈ mX if and only if mInt(A) = A,(2) A is mX -closed if and only if mCl(A) = A,(3) mInt(A) ∈ mX and mCl(A) is mX -closed.


Definition 3.4. A function f : (X, τ) → (Y,mY ) is said to be weakly (τ,m)-continuous (resp. (τ,m)-continuous) [32] at x ∈ X if for eachV ∈ mY containing

f (x), there exists U ∈ τ containing x such that f (U) ⊂ mCl(V ) (resp. f (U) ⊂⊂ V ). The function f : (X, τ) → (Y,mY ) is said to be weakly (τ,m)-continuous(resp. (τ,m)-continuous) if it has the property at each point x ∈ X .

Theorem3.1 (PopaandNoiri [32]). A function f : (X, τ) → (Y,mY ) is weakly(τ,m)-continuous if and only if f −1(V ) ⊂ Int( f −1(mCl (V )) for every m-openset V of Y .

Definition 3.5. Let S be a subset of an m-space (Y,mY ). A point y ∈ Y is

called an m-adherent point of S [32] if mCl(V ) ∩ S , ∅ for every mY -open set

V containing y.

The set of all m-adherent points of S is called the m-closure of S and

is denoted by mCl (S). If S = mCl (S), then S is said to be m-closed. The

complement of an m-closed set is said to be m-open.

Remark 3.4. Let S be a subset of a topological space (X, τ) and mX = τ (resp.SO(X ), PO(X )), thenmCl (S) = Cl (S) [41] (resp. sCl (S) [5], pCl (S) [26]).

Definition 3.6. An m-space (Y,mY ) is said to be m-regular [32] if for each

mY -closed set F and each y < F, there exist disjoint mY -open sets U and V such

that y ∈ U and F ⊂ V .

Remark 3.5. Let (X, τ) be a topological space and mX = τ (resp. SO(X ),PO(X ), β(X )). Then m-regularity coincides with regularity (resp. semi-

regularity [6], pre-regularity [26], semi-preregularity [23]).

Lemma 3.4 (Popa and Noiri [32]). Let (X,mX ) be an m-space and A a subsetof X . Then the following hold:

(1) If A is mX -open or (X,mX ) is m-regular, then mCl (A) = mCl(A),(2) If mX satisfies property B, then mCl (A) is mX -closed.

4. Weakly (τ,m)-continuous multifunctions

Definition 4.1. A multifunction F : (X, τ) → (Y,mY ) is said to be

(1) upper weakly (τ,m)-continuous (resp. upper (τ,m)-continuous [25]) at

x ∈ X if for each V ∈ mY containing F (x), there exists U ∈ τ containing

x such that F (U) ⊂ mCl(V ) (resp. F (U) ⊂ V ),


(2) lower weakly (τ,m)-continuous (resp. lower (τ,m)-continuous [25]) at

x ∈ X if for each V ∈ mY such that F (x) ∩ V , ∅, there exists U ∈ τcontaining x such that F (u) ∩mCl(V ) , ∅ (resp. F (u) ∩ V , ∅) for everyu ∈ U ,

(3) upper/lower weakly (τ,m)-continuous if F has this property at each x ∈ X .

Theorem4.1. For a multifunction F : (X, τ) → (Y,mY ), where mY has propertyB, the following properties are equivalent:

(1) F is upper weakly (τ,m)-continuous at x ∈ X ;(2) x ∈ Int(F+(mCl(V ))) for each m-open set V of Y containing F (x);(3) x ∈ F−(K ) for every m-closed set K of Y such that x ∈ Cl(F−(mInt(K )));(4) x ∈ F−(mCl(B)) for every subset B of Y such that x ∈

∈ Cl(F−(mInt(mCl(B))));(5) x ∈ Int(F+(mCl(mInt(B)))) for every subset B of Y such that x ∈∈ F+(mInt(B)).

Proof. (1) ⇒ (2): Let V be any m-open set V of Y containing F (x).There exists an open set U containing x such that F (U) ⊂ mCl(V ). Thenx ∈ U ⊂ F+(mCl(V )). Since U is open, we have x ∈ Int(F+(mCl(V ))).

(2) ⇒ (3): Suppose that K is an m-closed set of Y such that x ∈∈ Cl(F−(mInt(K ))) and x < F−1(K ), then x ∈ X − F−(K ) = F+(Y − K ).Since Y − K is m-open, by (2) we have x ∈ Int(F+(mCl(Y − K ))) = Int(X −− F−(mInt(K ))) = X − Cl(F−(mInt(K ))). Hence x < Cl(F−(mInt(K ))). Thisis a contradiction.

(3) ⇒ (4): Let B be any subset of Y . Since mY has property B,mCl(B) ism-closed. By (3), we have x ∈ F−(mCl(B)) if x ∈ Cl(F−(mInt(mCl(B)))).

(4) ⇒ (5): Suppose that x ∈F+(mInt(B)) and x< Int(F+(mCl(mInt(B))))for a subset B of Y . Then

x ∈ X − Int(F+(mCl(mInt(B)))) = Cl(X − F+(mCl(mInt(B)))) =

= Cl(F−(Y −mCl(mInt(B)))) = Cl(F−(mInt(mCl(Y − B)))).

By (4) x ∈ F−(mCl(Y − B)) = F−(Y −mInt(B)) = X − F+(mInt(B)). This is acontradiction.

(5) ⇒ (1): Let V be any m-open set of Y containing F (x). Since mY

has property B, V = mInt(V ) and x ∈ F+(V ) = F+(mInt(V )). Then, by (5),

x ∈ Int(F+(mCl(V ))). Therefore, there exists an open set U of X containing xsuch thatU ⊂ F+(mCl(V )). Hence F (U) ⊂ mCl(V ). This shows that F is upper

weakly (τ,m)-continuous at x ∈ X .



(1) F is lower weakly (τ,m)-continuous at x ∈ X ;(2) x ∈ Int(F−(mCl(V ))) for each m-open set V of Y meeting F (x);(3) x ∈ F+(K ) for every m-closed set K of Y such that x ∈ Cl(F+(mInt(K )));(4) x ∈ F+(mCl(B)) for every subset B of Y such that x ∈

∈ Cl(F+(mInt(mCl(B))));(5) x ∈ Int(F−(mCl(mInt(B)))) for every subset B of Y such that x ∈∈ F−(mInt(B)).

Proof. The proof is similar to the proof of Theorem 4.1.

For a multifunction F : (X, τ) → (Y,mY ), we dene D+wm (F) and

D−wm (F) as follows:

D+

wm (F) = x ∈ X : F is not upper weakly (τ,m)-continuous at x ∈ X ,

D−wm (F) = x ∈ X : F is not lower weakly (τ,m)-continuous at x ∈ X .

Theorem4.3. For a multifunction F : (X, τ) → (Y,mY ), where mY has propertyB, the following equalities hold:

D+

wm (F) =⋃

G∈mY

F+(G) − Int(F+(mCl(G)))

=⋃

B∈P(Y )

F+(mInt(B)) − Int(F+(mCl(mInt(B))))

=⋃

B∈P(Y )

Cl(F−(mInt(mCl(B)))) − F−(mCl(B))

=⋃K ∈F

Cl(F−(mInt(K ))) − F−(K ),

where P(Y ) is the family of all subsets of Y and F is the family of all m-closedsets of (Y,mY ).

Proof. We shall show only the rst equality since the proofs of other are similarto the rst.

Let x ∈ D+wm (F). Then, by Theorem 4.1, there exists V ∈ mY containing

F (x) such that x < Int(F+(mCl(V ))). Therefore, we havex ∈ F+(V ) − Int(F+(mCl(V ))) ⊂

⋃G∈mY

F+(G) − Int(F+(mCl(G))).


Conversely, let x ∈⋃G∈mY

F+(G) − Int(F+(mCl(G))). Then there existsV ∈ mY such that x ∈ F+(V ) − Int(F+(mCl(V ))). By Theorem 4.1, x ∈∈ D+

wm (F).

Theorem4.4. For a multifunction F : (X, τ) → (Y,mY ), where mY has propertyB, the following equalities hold:

D−wm (F) =⋃

G∈mY

F−(G) − Int(F−(mCl(G)))

=⋃

B∈P(Y )

F−(mInt(B)) − Int(F−(mCl(mInt(B))))

=⋃

B∈P(Y )

Cl(F+(mInt(mCl(B)))) − F+mCl(B)

=⋃K ∈F

Cl(F+(mInt(K ))) − F+(K ),



Let (X, τ) be a topological space and (Y,mY ) an m-space. For a functionf : (X, τ) → (Y,mY ), we dene Dwm ( f ) as follows:

Dwm ( f ) = x ∈ X : f is not weakly (τ,m)-continuous at x.

Corollary 4.1. For a function f : (X, τ) → (Y,mY ), where mY has propertyB, the following equalities hold:

Dwm ( f ) =⋃

G∈mY

f −1(G) − Int( f −1(mCl(G)))

=⋃

B∈P(Y )

f −1(mInt(B)) − Int( f −1(mCl(mInt(B))))

=⋃

B∈P(Y )

Cl( f −1(mInt(mCl(B)))) − f −1(mCl(B))

=⋃K ∈F

Cl( f −1(mInt(K ))) − f −1(K ),


Theorem 4.5. For a multifunction F : (X, τ) → (Y,mY ), D+wm (F) (resp.

D−wm (F)) is identical with the union of the frontiers of the upper (resp. lower)inverse images of the m-closures of m-open sets containing (resp. meeting) F (x).


Proof. We shall prove the rst case since the proof of the second is similar.

Let x ∈ D+wm (F). Then, there exists an m-open set V of Y containing

F (x) such that U ∩ (X − F+(mCl(V ))) , ∅ for every open set U containing x.Then, we have x ∈ Cl(X − F+(mCl(V ))). On the other hand, since x ∈ F+(V ),x ∈ Cl(F+(mCl(V ))) and hence x ∈ Fr(F+(mCl(V ))).

Conversely, suppose that F is upper weakly (τ,m)-continuous at x ∈ X .

Then, for any m-open setV ofY containing F (x), there existsU ∈ τ containing xsuch that F (U) ⊂ mCl(V ); hence x ∈ U ⊂ F+(mCl(V )). Therefore, we have x ∈U ⊂ Int(F+(mCl(V ))). This is contrary to the fact that x ∈Fr(F+(mCl(V ))).

Corollary 4.2. For a function f : (X, τ) → (Y,mY ), Dwm ( f ) is identicalwith the union of the frontiers of the inverse images of the m-closure of m-opensets containing f (x).


(1) F is upper weakly (τ,m)-continuous;(2) F+(V ) ⊂ Int(F+(mCl(V ))) for every V ∈ mY ;(3) Cl(F−(mInt(K ))) ⊂ F−(K ) for every mY -closed set K ;(4) Cl(F−(mInt(mCl(B)))) ⊂ F−(mCl(B)) for every subset B of Y ;(5) F+(mInt(B)) ⊂ Int(F+(mCl(mInt(B)))) for every subset B of Y .

Proof. (1) ⇒ (2): Let V be any m-open set of Y and x ∈ F+(V ).Then F (x) ⊂ V . By Theorem 4.1, x ∈ Int(F+(mCl(V ))). Hence F+(V ) ⊂⊂ Int(F+(mCl(V ))).

(2) ⇒ (3): Let K be an m-closed set of Y . Then Y − K is m-open in Y . By(2) we obtain F+(Y − K ) ⊂ Int(F+(mCl(Y − K ))) = Int(F+(Y − mInt(K ))) == X − Cl(F−(mInt(K ))). This implies that Cl(F−(mInt(K ))) ⊂ F−(K ).

(3) ⇒ (4): Let B be any subset of Y . Since mY has property B,mCl(B) ism-closed. By (3), we obtain Cl(F−(mInt(mCl(B)))) ⊂ F−(mCl(B)).

(4) ⇒ (5): By (4), we have

Cl(F−(mInt(mCl(Y − B)))) ⊂ F−(mCl(Y − B)) =

= F−(Y −mInt(B)) = X − F+(mInt(B)).

On the other hand,

Cl(F−(mInt(mCl(Y − B)))) = Cl(F−(Y −mCl(mInt(B)))) =

= Cl(X − F+(mCl(mInt(B)))) = X − Int(F+(mCl(mInt(B)))).


Hence F+(mInt(B)) ⊂ Int(F+(mCl(mInt(B)))).(5) ⇒ (2): Let V be any m-open set of Y . By Lemma 3.1, V = mInt(V )

and by (5) we obtain F+(V ) ⊂ Int(F+(mCl(V ))).(2) ⇒ (1): let x be any point of X and V any m-open set of Y containing

F (x). Then x ∈ F+(V ). By (2) x ∈ Int(F+(mCl(V ))). Therefore, there exists anopen set U such that x ∈ U ⊂ F+(mCl(V )); hence F (U) ⊂ mCl(V ). Hence Fis upper weakly (τ,m)-continuous.


(1) F is lower weakly (τ,m)-continuous;(2) F−(V ) ⊂ Int(F−(mCl(V ))) for every V ∈ mY ;(3) Cl(F+(mInt(K ))) ⊂ F+(K ) for every mY -closed set K ;(4) Cl(F+(mInt(mCl(B)))) ⊂ F+(mCl(B)) for every subset B of Y ;(5) F−(mInt(B)) ⊂ Int(F−(mCl(mInt(B)))) for every subset B of Y .


Remark 4.1. For a function f : (X, τ) → (Y,mY ), by Theorems 4.6 and 4.7 we

obtain Theorem 3.1.

Theorem 4.8. Let (Y,mY ) be m-regular and mY have property B. For a multi-function F : (X, τ) → (Y,mY ), the following properties are equivalent:

(1) F is upper weakly (τ,m)-continuous;(2) Cl(F−(mInt(mCl (B)))) ⊂ F−(mCl (B)) for every subset B of Y ;(3) Cl(F−(mInt(mCl(B)))) ⊂ F−(mCl (B)) for every subset B of Y .

Proof. (1) ⇒ (2): Let B be any subset of Y . By Lemma 3.4, mCl (B) is

m-closed. Then, by Theorem 4.6, Cl(F−(mInt(mCl (B)))) ⊂ F−(mCl (B)).(2) ⇒ (3): This is obvious by mCl(B) ⊂ mCl (B).(3) ⇒ (1): Let B be any subset of Y . By (3) and Lemma 3.4, we obtain

Cl(F−(mInt(mCl(B)))) ⊂ F−(mCl(B)).

Then, by Theorem 4.6 (4), F is upper weakly (τ,m)-continuous.

Theorem 4.9. Let (Y,mY ) be m-regular and mY have property B. For a multi-function F : (X, τ) → (Y,mY ), the following properties are equivalent:

(1) F is lower weakly (τ,m)-continuous;(2) Cl(F+(mInt(mCl (B)))) ⊂ F+(mCl (B)) for every subset B of Y ;(3) Cl(F+(mInt(mCl(B)))) ⊂ F+(mCl (B)) for every subset B of Y .


Corollary 4.3. Let (Y,mY ) be m-regular and mY have property B. Then for afunction f : (X, τ) → (Y,mY ), the following properties are equivalent:

(1) f is weakly (τ,m)-continuous;(2) Cl( f −1(mInt(mCl (B)))) ⊂ f −1(mCl (B)) for every subset B of Y ;(3) Cl( f −1(mInt(mCl(B)))) ⊂ f −1(mCl (B)) for every subset B of Y .

Lemma 4.1. Let (X,mX ) be an m-regular space, where mX has property B, andA be a subset of X . Then, for every m-open set D which intersects with A, thereexists an m-open set DA such that A ∩ DA , ∅ and mCl(DA) ⊂ D.

Proof. Since A ∩ D , ∅, let x ∈ A ∩ D. Then x < X − D. Since D ∈ mX ,

X − D is m-closed. By Denition 3.6, there exist disjoint m-open sets U and Vsuch that x ∈ U and X − D ⊂ V . Hence, by Lemma 3.2 U ∩ mCl(V ) = ∅ andx < mCl(V ). Let DA = X − mCl(V ). Then, since mX has property B, mCl(V )is m-closed and DA is m-open, x ∈ DA and A ∩ DA , ∅. On the other hand,

mCl(DA) = mCl(X − mCl(V )) ⊂ mCl(X − V ) = X − V ⊂ D because X − Vis m-closed and by Lemma 3.3 mCl(X − V ) = X − V . Hence x ∈ DA and

mCl(DA) ⊂ D.

Theorem 4.10. Let (Y,mY ) be an m-regular space, where mY has property B.Then, for a multifunction F : (X, τ) → (Y,mY ), the following properties areequivalent:

(1) F is upper (τ,m)-continuous;(2) F−(mCl (B)) is closed in X for every subset B of Y ;(3) F−(K ) is closed in X for every m-closed set K of Y ;(4) F+(V ) is open in X for every m-open set V of Y .

Proof. (1) ⇒ (2): Let B be any subset of Y . By Lemma 3.4, mCl (B) is

m-closed and by Theorem 4.3 of [25], F−(mCl (B)) is closed in X .

(2) ⇒ (3): Let K be an m-closed set of Y . Then K = mCl (K ) and by

(2) F−(K ) is closed in X .

(3) ⇒ (4): Let V be an m-open set of Y . Then Y − V is m-closed and

hence F−(Y − V ) = X − F+(V ) is closed in X . Hence F+(V ) is open in X .

(4) ⇒ (1): Let V be any m-open set of Y . Since (Y,mY ) is m-regular,

by Lemma 3.4 V is m-open. By (4), F+(V ) is open in X and by Theorem 4.3

of [25] F is upper (τ,m)-continuous.

Theorem 4.11. Let (Y,mY ) be an m-regular space, where mY has property B.Then, for a multifunction F : (X, τ) → (Y,mY ), the following properties areequivalent:


(1) F is lower (τ,m)-continuous;(2) F+(mCl (B)) is closed in X for every subset B of Y ;(3) F+(K ) is closed in X for every m-closed set K of Y ;(4) F−(V ) is open in X for every m-open set V of Y ;(5) F is lower weakly (τ,m)-continuous.

Proof. The proofs of implications (1) ⇒ (2) ⇒ (3) ⇒ (4) are similar as in

Theorem 4.10.

(4) ⇒ (5): Let G be any m-open set of Y . Since (Y,mY ) is m-regular, G is

m-open and by (4) F−(G) = Int(F−(G)) ⊂ Int(F−(mCl(G))). ByTheorem4.7,

F is lower (τ,m)-continuous.(5) ⇒ (1): Let x ∈ X andV be anym-open sets ofY such that F (x)∩V , ∅.

Since (Y,mY ) is m-regular, by Lemma 4.1 there exists an m-open setW such that

F (x) ∩W , ∅ and mCl(W ) ⊂ V . Since F is lower weakly (τ,m)-continuous,there exists an open setU of X containing x such that F (u)∩mCl(W ) , ∅; henceF (u) ∩V , ∅ for every u ∈ U. This shows that F is lower (τ,m)-continuous.

Corollary 4.4 (Popa and Noiri [32]). Let (Y,mY ) be an m-regular space,where mY has property B. Then, for a function f : (X, τ) → (Y,mY ), thefollowing properties are equivalent:

(1) f is (τ,m)-continuous;(2) f −1(mCl (B)) is closed in X for every subset B of Y ;(3) f −1(K ) is closed in X for every m-closed set K of Y ;(4) f −1(V ) is open in X for every m-open set V of Y ;(5) f is weakly (τ,m)-continuous.

5. Minimal structures in bitopological spaces

In this section, we recall four types of generalizations of open sets in

bitopological spaces. Every family belonging to these types is an m-space having

property B.

A. (i, j)mX -open sets

Definition 5.1. A subset A of a bitopological space (X, τ1, τ2) is said to be

(1) (i, j)-semi-open [17] if A ⊂ j Cl(i Int(A)), where i , j, i, j = 1, 2,(2) (i, j)-preopen [9] if A ⊂ i Int( j Cl(A)), where i , j, i, j = 1, 2,(3) (i, j)-α-open [10] if A ⊂ i Int( j Cl(i Int(A))), where i , j, i, j = 1, 2,


(4) (i, j)-semi-preopen (briey (i, j)-sp-open) [11] if there exists an (i, j)-preopen set U such that U ⊂ A ⊂ j Cl(U), where i , j, i, j = 1, 2.

The family of all (i, j)-semi-open (resp. (i, j)-preopen, (i, j)-α-open, (i, j)-sp-open) sets of (X, τ1, τ2) is denoted by (i, j)SO(X ) (resp. (i, j) PO(X ),(i, j)α(X ), (i, j) SPO(X )).

Remark 5.1. Let (X, τ1, τ2) be a bitoppological space and A a subset of X .

Then (i, j) SO(X ), (i, j) PO(X ), (i, j)α(X ), and (i, j) SPO(X ) are all minimal

structures on X . If (i, j)mX = (i, j) SO(X ) (resp. (i, j) PO(X ), (i, j)α(X ),(i, j) SPO(X )), then

(1) (i, j) mCl(A) = (i, j)-sCl(A) (resp. (i, j)-pCl(A), (i, j)-αCl(A), (i, j)-spCl(A)),

(2) (i, j) mInt(A) = (i, j)-sInt(A) (resp. (i, j)-pInt(A), (i, j)-α Int(A), (i, j)-spInt(A)).

B. Quasi m-open sets


(1) quasi open [7], [18] or τ1τ2-open [36], [37] if A = B ∪ C, where B ∈ τ1and C ∈ τ2,

(2) quasi semi-open [12] if A = B ∪ C, where B ∈ SO(X, τ1) and C ∈

∈ SO(X, τ2),(3) quasi preopen [28] if A = B∪C, where B ∈ PO(X, τ1) andC ∈ PO(X, τ2),(4) quasi semipreopen [40] if A = B ∪ C, where B ∈ SPO(X, τ1) and

C ∈ SPO(X, τ2),(5) quasi α-open [39] if A = B ∪ C, where B ∈ α(X, τ1) and C ∈ α(X, τ2).

The family of all quasi open (resp. quasi semi-open, quasi preopen, quasi

semipreopen, quasi α-open) sets of a bitopological space (X, τ1, τ2) is denoted byQO(X ), τ1τ2(X ) or (1, 2) O(X ) (resp.QSO(X ),QPO(X ),QSPO(X ),Q α(X )).

Definition 5.3. Let (X, τ1, τ2) be a bitopological space and m1

X(resp. m2

X) an

m-structure on the topological space (X, τ1) (resp. (X, τ2)). The family

qmX = A ⊂ X : A = B ∪ C, where B ∈ m1

Xand C ∈ m2

X

is a minimal structure on X and hence is called a quasi m-structure on X . Each

member of qmX is said to be quasi mX -open (or briey quasi m-open). The

complement of a quasi mX -open set is said to be quasi mX -closed (or briey

quasi m-closed).


Remark 5.2. Let (X, τ1, τ2) be a bitopological space.

(1) If m1

Xand m2

Xhave property B, then qmX is an m-structure with property

B.

(2) If m1

X= τ1 and m2

X= τ2 (resp. SO(X, τ1) and SO(X, τ2), PO(X, τ1)

and PO(X, τ2), SPO(X, τ1) and SPO(X, τ2), α(X, τ1) and α(X, τ2)), thenqmX = QO(X ), τ1τ2(X ) or (1, 2) O(X ) (resp. QSO(X ), QPO(X ),QSPO(X ), Q αO(X )).

(3) Since SO(X, τi ) (resp. PO(X, τi ), SPO(X, τi ) and α(X, τi )) has propertyB for i = 1, 2, QSO(X ) (resp. QPO(X ), QSPO(X ) and Q αO(X )) hasproperty B.

Definition 5.4. Let (X, τ1, τ2) be a bitopological space. For a subset A of X , the

quasi mX -closure of A and the quasi mX -interior of A are dened as follows:

(1) qmCl(A) = ∩F : A ⊂ F, X − F ∈ qmX ,

(2) qmInt(A) = ∪U : U ⊂ A,U ∈ qmX .

Remark 5.3. Let (X, τ1, τ2) be a bitopological space and A a subset of X . If

qmX = QO(X ) (resp.QSO(X ),QPO(X ),QSPO(X ),Q αO(X )), then we have

(1) qmCl(A) = qCl(A) [37] (resp. qsCl(A) [12], qpCl(A) [28], qspCl(A) [40],qαCl(A) [39]),

(2) qmInt(A) = qInt(A) (resp. qsInt(A), qpInt(A), qspInt(A), q α Int(A)).

The notations qCl(A) and qInt(A) are also denoted by τ1τ2 Cl(X ) (or

(1, 2) Cl(A)) and τ1τ2 Int(X ) (or (1, 2) Int(A)), respectively.

C. (1, 2)∗-mX -open sets


(1) (1, 2)∗-semi-open [35] if A ⊂ τ1τ2 Cl(τ1τ2 Int(A)),(2) (1, 2)∗-preopen [35] if A ⊂ τ1τ2 Int(τ1τ2 Cl(A)),(3) (1, 2)∗-α-open [35] if A ⊂ τ1τ2 Int(τ1τ2 Cl(τ1τ2 Int(A))),(4) (1, 2)∗-semi-preopen [35] if A ⊂ τ1τ2 Cl(τ1τ2 Int(τ1τ2 Cl(A))).

The complement of a (1, 2)∗-semi-open (resp. (1, 2)∗-preopen, (1, 2)∗-α-open, (1, 2)∗-semi-preopen) set is said to be (1, 2)∗-semi-closed (resp. (1, 2)∗-preclosed, (1, 2)∗-α-closed, (1, 2)∗-semi-preclosed).

The family of all (1, 2)∗-semi-open (resp. (1, 2)∗-preopen, (1, 2)∗-α-open,(1, 2)∗-semi-preopen) sets is denoted by (1, 2)∗ SO(X ) (resp. (1, 2)∗ PO(X ),(1, 2)∗α(X ), (1, 2)∗ SPO(X )).


Remark 5.4. Let (X, τ1, τ2) be a bitopological space and A a subset of X .

(1) The families (1, 2)∗ SO(X ), (1, 2)∗ PO(X ), (1, 2)∗α(X ), and

(1, 2)∗ SPO(X ) are all m-structures with property B.

(2) By (1, 2)∗mX , we denote each member of the above ve families and call it

an m-structure determinded by τ1 and τ2. Let (1, 2)∗mX = τ12O(X ) (resp.(1, 2)∗ SO(X ), (1, 2)∗ PO(X ), (1, 2)∗α(X ), (1, 2)∗ SPO(X )), then we have(i) (1, 2)∗mCl(A) = τ1τ2 Cl(A) (resp. (1, 2)∗ sCl(A), (1, 2)∗ pCl(A),

(1, 2)∗αCl(A), (1, 2)∗ spCl(A)),(ii) (1, 2)∗mInt(A) = τ1τ2 Int(A) (resp. (1, 2)∗ sInt(A), (1, 2)∗ pInt(A),

(1, 2)∗α Int(A), (1, 2)∗ spInt(A)).

D. (1, 2)mX -open sets


(1) (1,2)-semi-open [13] if A ⊂ τ1τ2 Cl(τ1 Int(A)),(2) (1,2)-preopen [13] if A ⊂ τ1 Int(τ1τ2 Cl(A)),(3) (1,2)-α-open [13] if A ⊂ τ1 Int(τ1τ2 Cl(τ1 Int(A))),(4) (1,2)-semi-preopen [14] if A ⊂ τ1τ2 Cl(τ1 Int(τ1τ2 Cl(A))).

The complement of (1, 2)-semi-open (resp. (1, 2)-preopen, (1, 2)-α-open,(1, 2)-semi-preopen) set of X is said to be (1, 2)-semi-closed (resp. (1, 2)-preclosed, (1, 2)-α-closed, (1, 2)-semi-preclosed). The intersection of all (1, 2)-semi-closed (resp. (1, 2)-preclosed, (1, 2)-α-closed, (1, 2)-semi-preclosed) sets

containing A is called the (1, 2)-semi-closure (resp. (1, 2)-preclosure, (1, 2)-α-closure, (1, 2)-semi-preclosure) of A and is denoted by (1, 2) sCl(A) (resp.

(1, 2) pCl(A), (1, 2)αCl(A), (1, 2) spCl(A)). The union of (1, 2)-semi-open

(resp. (1, 2)-preopen, (1, 2)-α-open, (1, 2)-semi-preopen) sets of X contained

in A is called the (1, 2)-semi-interior (resp. (1, 2)-preinterior, (1, 2)-α-interior,(1, 2)-semi-preinterior) of A and is denoted by (1, 2) sInt(A) (resp. (1, 2) pInt(A),(1, 2)α Int(A), (1, 2) spInt(A)).

The collection of all (1, 2)-semi-open (resp. (1, 2)-preopen, (1, 2)-α-open,(1, 2)-semi-preopen) sets of X is denoted by (1, 2) SO(X ) (resp. (1, 2) PO(X ),(1, 2)α(X ), (1, 2) SPO(X )).

Remark 5.5. Let (X, τ1, τ2) be a bitopological space and A a subset of X .

(1) The families τ1τ2O(X ), (1, 2) SO(X ), (1, 2) PO(X ), (1, 2)α(X ) and

(1, 2) SPO(X ) are all m-structures on X having property B.

(2) By (1, 2)mX , we denote each one of the above families and call it an

m-structure determined by the topologies τ1 and τ2 on X . If (1, 2)mX =


τ1τ2O(X ) (resp. (1, 2) SO(X ), (1, 2) PO(X ), (1, 2)α(X ), (1, 2) SPO(X )),then we have

(i) (1, 2) mCl(A) = τ1τ2 Cl(A) (resp. (1, 2) sCl(A), (1, 2) pCl(A),(1, 2)αCl(A), (1, 2) spCl(A)),

(ii) (1, 2) mInt(A) = τ1τ2 Int(A) (resp. (1, 2) sInt(A), (1, 2) pInt(A),(1, 2)α Int(A), (1, 2) spInt(A)).

Remark 5.6. It follows from A, B, C and D that if (X, τ1, τ2) is a bitopologicalspace then someminimal structures on X determined by τ1 and τ2 are introduced.In the sequel, by m(τ1, τ2) (simply m12) we denote a minimal structure on Xdetermined by τ1 and τ2, that is, (i, j)mX , qm, (1, 2)∗mX or (1, 2)mX .

6. Generalizations of ultra weak continuity

Definition 6.1. A multifunction F : (X, τ) → (Y, σ1, σ2) is said to be

(1) ultra upper weakly continuous at a point x ∈ X [34] if for each (1, 2)α-openset V containing F (x), there exists an open set U containing x such that

F (U) ⊂ (1, 2)αCl(V ),(2) ultra lower weakly continuous at a point x ∈ X [34] if for each (1, 2)α-open

set V such that F (x) ∩V , ∅, there exists an open set U containing x such

that F (u) ∩ (1, 2)αCl(V ) , ∅ for every u ∈ U,

(3) ultra upper/lower weakly continuous if F has this property at each x ∈ X .

Hence, it turns out that F : (X, τ) → (Y, σ1, σ2) is ultra upper/lower weaklycontinuous at a point x ∈ X if and only if F : (X, τ) → (Y, (1, 2)α(Y )) is

upper/lower weakly (τ,m)-continuous at a point x ∈ X .

Definition 6.2. Let (X, τ) be a topological space, (Y, σ1, σ2) a bitopological

space and m12 = m(σ1, σ2) an minimal structure on Y determined by σ1 and

σ2. A multifunction F : (X, τ) → (Y, σ1, σ2) is said to be upper/lower weakly

(τ,m12)-continuous at a point x ∈ X (resp. on X) if F : (X, τ) → (Y,m12) is

upper/lower weakly (τ,m)-continuous at x ∈ X (resp. on X).

Hence a multifunction F : (X, τ) → (Y, σ1, σ2) is said to be

(1) upper weakly (τ,m12)-continuous at x ∈ X if for each m12-open set

V containing F (x), there exists an open set U containing x such that

F (U) ⊂ m12 Cl(V ),


(2) lower weakly (τ,m12)-continuous at x ∈ X if for each m12-open set Vsuch that F (x) ∩ V , ∅, there exists an open set U containing x such that

F (u) ∩ m12 Cl(V ) , ∅ for every u ∈ U ,

(3) upper/lower weakly (τ,m12)-continuous if F has this property at each

x ∈ X .

Remark 6.1. If m12 = (1, 2)α(Y ), then we obtain Denition 6.1.

By Theorems 4.1 and 4.2, for the families m12 = (i, j)mY , qm, (1, 2)∗mY

and (1, 2)mY , we obtain the following characterizations.

Theorem 6.1. For a multifunction F : (X, τ) → (Y, σ1, σ2) and a minimalstructure m12 = m(σ1, σ2) on Y , the following properties are equivalent:

(1) F is upper weakly (τ,m12)-continuous at x ∈ X ;(2) x ∈ Int(F+(m12 Cl(V ))) for each m12-open set V of Y containing F (x);(3) x ∈ F−(K ) for every m12-closed set K of Y such that x ∈

∈ Cl(F−(m12 Int(K )));(4) x ∈ F−(m12 Cl(B)) for every subset B of Y such that x ∈

∈ Cl(F−(m12 Int(m12 Cl(B))));(5) x ∈ Int(F+(m12 Cl(m12 Int(B)))) for every subset B of Y such that x ∈∈ F+(m12 Int(B)).

Theorem 6.2. For a multifunction F : (X, τ) → (Y, σ1, σ2) and a minimalstructure m12, the following properties are equivalent:

(1) F is lower weakly (τ,m12)-continuous at x ∈ X ;(2) x ∈ Int(F−(m12 Cl(V ))) for each m12-open set V of Y meeting F (x);(3) x ∈ F+(K ) for every m12-closed set K of Y such that x ∈

∈ Cl(F+(m12 Int(K )));(4) x ∈ F+(m12 Cl(B)) for every subset B of Y such that x ∈

∈ Cl(F+(m12 Int(m12 Cl(B))));(5) x ∈ Int(F−(m12 Cl(m12 Int(B)))) for every subset B of Y such that x ∈∈ F−(m12 Int(B)).

If we put m1;2 = (1, 2)α(Y ), then by Theorems 6.1 and 6.2 we obtain the

following two corollaries.

Corollary 6.1. For a multifunction F : (X, τ) → (Y, σ1, σ2) and a minimalstructure (1, 2)α(Y ) on Y , the following properties are equivalent:

(1) F is upper weakly (τ, (1, 2)α(Y ))-continuous at x ∈ X ;


(2) x ∈ Int(F+((1, 2)αCl(V ))) for each (1, 2)α(Y )-open set V of Y containingF (x);

(3) x ∈ F−(K ) for every (1, 2)α(Y )-closed set K of Y such that x ∈∈ Cl(F−((1, 2)α Int(K )));

(4) x ∈ F−((1, 2)αCl(B)) for every subset B of Y such that

x ∈ Cl(F−((1, 2)α Int((1, 2)αCl(B))));

(5) x ∈ Int(F+((1, 2)αCl((1, 2)α Int(B)))) for every subset B of Y such thatx ∈ F+((1, 2)α Int(B)).

Corollary 6.2. For a multifunction F : (X, τ) → (Y, σ1, σ2) and a minimalstructure (1, 2)α(Y ), the following properties are equivalent:

(1) F is lower weakly (τ, (1, 2)α(Y ))-continuous at x ∈ X ;(2) x ∈ Int(F−((1, 2)αCl(V ))) for each (1, 2)α(Y )-open set V of Y meeting

F (x);(3) x ∈ F+(K ) for every (1, 2)α(Y )-closed set K of Y such that x ∈∈ Cl(F+((1, 2)α Int(K )));

(4) x ∈ F+((1, 2)αCl(B)) for every subset B of Y such that

x ∈ Cl(F+((1, 2)α Int((1, 2)αCl(B))));

(5) x ∈ Int(F−((1, 2)αCl((1, 2)α Int(B)))) for every subset B of Y such thatx ∈ F−((1, 2)α Int(B)).

For a multifunction F : (X, τ) → (Y, σ1, σ2) and a minimal structure m12

onY determined byσ1 andσ2, we dene D+wm12

(F) and D−wm12(F) as follows:

D+

wm12(F) = x ∈ X : F is not upper weakly (τ,m12)-continuous at x ∈ X ,

D−wm12(F) = x ∈ X : F is not lower weakly (τ,m12)-continuous at x ∈ X .

The following theorems follow immediately from Theorems 4.3 and 4.4.

Theorem 6.3. For a multifunction F : (X, τ) → (Y, σ1, σ2) and a minimalstructure m12 on Y determined by σ1 and σ2, the following properties hold:

D+

wm12(F) =

⋃G∈m12

F+(G) − Int(F+(m12 Cl(G))) =

=⋃

B∈P(Y )

F+(m12 Int(B)) − Int(F+(m12 Cl(m12 Int(B)))) =

=⋃

B∈P(Y )

Cl(F−(m12 Int(m12 Cl(B)))) − F−(m12 Cl(B)) =


=⋃K ∈F

Cl(F−(m12 Int(K ))) − F−(K ),

where P(Y ) is the family of all subsets of Y and F is the family of all m12-closedsets of (Y, σ1, σ2).

Theorem 6.4. For a multifunction F : (X, τ) → (Y, σ1, σ2) and a minimalstructure m12 on Y determined by σ1 and σ2, the following properties hold:

D−wm12(F) =

⋃G∈m12

F−(G) − Int(F−(m12 Cl(G))) =

=⋃

B∈P(Y )

F−(m12 Int(B)) − Int(F−(m12 Cl(m12 Int(B)))) =

=⋃

B∈P(Y )

Cl(F+(m12 Int(m12 Cl(B)))) − F+m12 Cl(B) =

=⋃K ∈F

Cl(F+(m12 Int(K ))) − F+(K ),

where P(Y ) is the family of all subsets of Y and F is the family of all m12-closedsets of Y .



Corollary 6.3. For a multifunction F : (X, τ) → (Y, σ1, σ2) and a minimalstructure (1, 2)α(Y ) on Y determined by σ1 and σ2, the following propertieshold:

D+

w (1;2)(Y ) (F) =⋃

G∈(1;2)(Y )

F+(G) − Int(F+((1, 2)αCl(G))) =

=⋃

B∈P(Y )

F+((1, 2)α Int(B)) − Int(F+((1, 2)αCl((1, 2)α Int(B)))) =

=⋃

B∈P(Y )

Cl(F−((1, 2)α Int((1, 2)αCl(B)))) − F−((1, 2)αCl(B)) =

=⋃K ∈F

Cl(F−((1, 2)α Int(K ))) − F−(K ),

where P(Y ) is the family of all subsets of Y and F is the family of all(1, 2)α(Y )-closed sets of (Y, σ1, σ2).


Corollary 6.4. For a multifunction F : (X, τ) → (Y, σ1, σ2) and a minimalstructure (1, 2)α(Y ) on Y determined by σ1 and σ2, the following propertieshold:

D−w (1;2)(Y ) (F) =

⋃G∈(1;2)(Y )

F−(G) − Int(F−((1, 2)αCl(G))) =

=⋃

B∈P(Y )

F−((1, 2)α Int(B)) − Int(F−((1, 2)αCl((1, 2)α Int(B)))) =

=⋃

B∈P(Y )

Cl(F+((1, 2)α Int((1, 2)αCl(B)))) − F+(1, 2)αCl(B) =

=⋃K ∈F

Cl(F+((1, 2)α Int(K ))) − F+(K ),

where P(Y ) is the family of all subsets of Y and F is the family of all(1, 2)α(Y )-closed sets of Y .

Theorem 6.5. For a multifunction F : (X, τ) → (Y, σ1, σ2) and a minimalstructure m12 on Y determined by σ1 and σ2, D+

wm12(F) (resp. D−wm12

(F)) isidentical with the union of the frontiers of the upper (resp. lower) inverse imagesof the m12-closures of m12-open sets containing (resp. meeting) F(x).

Corollary 6.5. For a multifunction F : (X, τ) → (Y, σ1, σ2) and a minimalstructure (1, 2)α(Y ) on Y determined by σ1 and σ2, D+

w (1;2)(Y ) (F) (resp.D−w (1;2)(Y ) (F)) is identical with the union of the frontiers of the upper (resp.

lower) inverse images of the (1, 2)α(Y )-closures of (1, 2)α(Y )-open sets con-taining (resp. meeting) F(x).

By Theorems 4.6 and 4.7, we obtain the following two theorems.

Theorem 6.6. For a multifunction F : (X, τ) → (Y, σ1, σ2) and a minimalstructure m12 on Y determined by σ1 and σ2, the following properties areequivalent:

(1) F is upper weakly (τ,m12)-continuous;(2) F+(V ) ⊂ Int(F+(m12 Cl(V ))) for every V ∈ m12;(3) Cl(F−(m12 Int(K ))) ⊂ F−(K ) for every m12-closed set K ;(4) Cl(F−(m12 Int(m12 Cl(B)))) ⊂ F−(m12 Cl(B)) for every subset B of Y ;(5) F+(m12 Int(B)) ⊂ Int(F+(m12 Cl(m12 Int(B)))) for every subset B of Y .

Theorem 6.7. For a multifunction F : (X, τ) → (Y, σ1, σ2) and a minimalstructure m12 on Y determined by σ1 and σ2, the following properties areequivalent:


(1) F is lower weakly (τ,m12)-continuous;(2) F−(V ) ⊂ Int(F−(m12 Cl(V ))) for every V ∈ m12;(3) Cl(F+(m12 Int(K ))) ⊂ F+(K ) for every m12-closed set K ;(4) Cl(F+(m12 Int(m12 Cl(B)))) ⊂ F+(m12 Cl(B)) for every subset B of Y ;(5) F−(m12 Int(B)) ⊂ Int(F−(m12 Cl(m12 Int(B)))) for every subset B of Y .

Remark 6.2. In Theorems 6.6 and 6.7, if we put m12 = (1, 2)α(Y ), then we

obtain Theorems 2.6 and 2.7 in [34].

Definition 6.3. Let (X, τ1, τ2) be a bitopological space and m12 a minimal

structure on X determined by τ1 and τ2. A bitopological space (X, τ1, τ2) is saidto be m12-regular if the m-space (X,m12) is m-regular.


Theorem 6.8. Let (Y,m12) be an m-regular space. Then, for a multifunctionF : (X, τ) → (Y, σ1, σ2), the following properties are equivalent:

(1) F is upper weakly (τ,m12)-continuous;(2) Cl(F−(m12 Int(m12 Cl (B)))) ⊂ F−(m12 Cl (B)) for every subset B of Y ;(3) Cl(F−(m12 Int(m12 Cl(B)))) ⊂ F−(m12 Cl (B)) for every subset B of Y .

Theorem 6.9. Let (Y,m12) be an m-regular space. Then, for a multifunctionF : (X, τ) → (Y, σ1, σ2), the following properties are equivalent:

(1) F is lower weakly (τ,m12)-continuous;(2) Cl(F+(m12 Int(m12 Cl (B)))) ⊂ F+(m12 Cl (B)) for every subset B of Y ;(3) Cl(F+(m12 Int(m12 Cl(B)))) ⊂ F+(m12 Cl (B)) for every subset B of Y .



Corollary 6.6. Let (Y, (1, 2)α(Y )) be an m-regular space. Then, for a multi-function F : (X, τ) → (Y, σ1, σ2), the following properties are equivalent:

(1) F is upper weakly (τ, (1, 2)α(Y ))-continuous;(2) Cl(F−((1, 2)α Int((1, 2)αCl (B)))) ⊂ F−((1, 2)αCl (B)) for every sub-

set B of Y ;(3) Cl(F−((1, 2)α Int((1, 2)αCl(B)))) ⊂ F−((1, 2)αCl (B)) for every subset

B of Y .

Corollary 6.7. Let (Y, (1, 2)α(Y )) be an m-regular space. Then, for a multi-function F : (X, τ) → (Y, σ1, σ2), the following properties are equivalent:

(1) F is lower weakly (τ, (1, 2)α(Y ))-continuous;


(2) Cl(F+((1, 2)α Int((1, 2)αCl (B)))) ⊂ F+((1, 2)αCl (B)) for every sub-set B of Y ;

(3) Cl(F+((1, 2)α Int((1, 2)αCl(B)))) ⊂ F+((1, 2)αCl (B)) for every subsetB of Y .


Theorem 6.10. Let (Y, σ1, σ2) be an m12-regular space, where m12 is a minimalstructure on Y determined by σ1 and σ2. Then, for a multifunction F : (X, τ) →→ (Y, σ1, σ2), the following properties are equivalent:

(1) F is upper (τ,m12)-continuous;(2) F−(m12 Cl (B)) is closed in X for every subset B of Y ;(3) F−(K ) is closed in X for every m12-θ-closed set K of Y ;(4) F+(V ) is open in X for every m12-θ-open set V of Y .

Theorem 6.11. Let (Y, σ1, σ2) be an m12-regular space, where m12 is a minimalstructure on Y determined by σ1 and σ2. Then, for a multifunction F : (X, τ) →→ (Y, σ1, σ2), the following properties are equivalent:

(1) F is lower (τ,m12)-continuous;(2) F+(m12 Cl (B)) is closed in X for every subset B of Y ;(3) F+(K ) is closed in X for every m12-θ-closed set K of Y ;(4) F−(V ) is open in X for every m12-θ-open set V of Y ;(5) F is lower weakly (τ,m12)-continuous.

If we put m12 = (1, 2)α(Y ), then by Theorems 6.10 and 6.11, we obtain the

following two corolaries.

Corollary 6.8. Let (Y, σ1, σ2) be a (1, 2)α(Y )-regular space, where (1, 2)α(Y )is a minimal structure on Y determined by σ1 and σ2. Then, for a multifunctionF : (X, τ) → (Y, σ1, σ2), the following properties are equivalent:

(1) F is upper (τ, (1, 2)α(Y ))-continuous;(2) F−((1, 2)αCl (B)) is closed in X for every subset B of Y ;(3) F−(K ) is closed in X for every (1, 2)α(Y )-θ-closed set K of Y ;(4) F+(V ) is open in X for every (1, 2)α(Y )-θ-open set V of Y .

Corollary 6.9. Let (Y, σ1, σ2) be a (1, 2)α(Y )-regular space, where (1, 2)α(Y )is a minimal structure on Y determined by σ1 and σ2. Then, for a multifunctionF : (X, τ) → (Y, σ1, σ2), the following properties are equivalent:

(1) F is lower (τ, (1, 2)α(Y ))-continuous;(2) F+((1, 2)αCl (B)) is closed in X for every subset B of Y ;


(3) F+(K ) is closed in X for every (1, 2)α(Y )-θ-closed set K of Y ;(4) F−(V ) is open in X for every (1, 2)α(Y )-θ-open set V of Y ;(5) F is lower weakly (τ, (1, 2)α(Y ))-continuous.

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Takashi Noiri

2949-1 Shiokita-cho, Hinagu

Yatsushiro-shi, Kumamoto-ken

869-5142 JAPAN

[email protected]

Valeriu Popa


Univ. Vasile Alecsandri of Bacau

600115 Bacau, RUMANIA

[email protected]


EQUIVALENCE BETWEEN SOME MODIFIED SHIFT MAPS AND

GENERALISATIONS OF β-TRANSFORMATIONS

By

ISRAEL NCUBE


Abstract. We establish equivalence between some shift maps dened on the

space of one-sided sequences and a certain class of measure-preserving transformations,

namely: the so-called β-transformations and their generalisations.

1. Introduction

We consider deterministic iterative maps of the general form

(1) T : xn+1 = βxn mod 1,

where β > 1, xn ∈ R, and n = 0, 1, 2, . . .. In the literature, these maps are

often referred to as β-transformations. Part of the problem with handling maps

of the form (1) stems from the general level of diculty encountered in trying

to characterise their long-term behaviour, for any given initial point x0 ∈ R. Inparticular, it is dicult to accurately characterise the asymptotic behaviour of the

number of the periodic points of (1) (see [1], [3], [4], [6], [9], [10], [14], [21], and

[20] for details). Giving an accurate formula for the number of periodic points

of (1) is dicult, primarily because the transformation erratically sheds periodic

points as n is increased.We refer to this process as the pruning of periodic points.

The extent of the pruning process is unpredictable in the limit as n → ∞. Theproblem partially stems from the lack of knowledge of the particular sequence

Tn

(1), n ∈ N. The pioneering work related to the T3=2(·) transformation is due

to Mahler [9, 10]. He looked at this problem from a number-theoretic viewpoint,

by essentially noting that it is equivalent to the well-known problem of powers

of 3/2. The connection between the latter and (1) is clear, since the iterates of

T3=2(·) involve powers of 3/2, for any given x0 ∈ R. That is, Tn

3=2(·) retains the

60 ISRAEL NCUBE

fractional part of (1.5x0)n , for each n. In particular, when x0 = 1, we have the

sequence Tn

3=2(1), n ∈ N. In his seminal work, Mahler [10] asked whether there

exist real numbers α > 0 such thatα (3/2)n

<

1

2

for all integers n ≥ 0, where x denotes the fractional part of x. To the best

of our knowledge, the proof, or refutation, of this remains an open problem to

this day. The real numbers α > 0, if they exist, are termed Z numbers [10].

Subsequently, many researchers have worked on this and other related problems

(see [1], [3], [20], [21], and references therein) over the years. Flatto [3] has

noted that the distribution of the fractional parts (3/2)n , 0 ≤ n < ∞, overthe interval [0, 1] is an intriguing and mysterious subject. According to him, by

numerical experiments, it seems that (3/2)n is uniformly distributed on [0, 1].This suggests that the sequence Tn

3=2(1) is uniformly distributed mod1, and is

thus ergodic.

In this paper, we establish equivalence between a certain class of shift maps

and expanding measure-preserving maps of the form

(2) xn+1 = 21+n xn + hn mod 1,

where hn ∈ R, and n, νn ∈ N. Our study in this direction is motivated by the

well-known fact that the expanding shift map

(3) T : xn+1 = 2xn mod 1

is equivalent to the Bernoulli shift map

(4) S : Ω→ Ω : ai → ai+1,

where Ω is the space of one-sided sequences and the ai is the ith element of a

sequence.

The paper is organised as follows. Sections 2-5 are studies of equivalence

between modied expanding shifts of the form (2) and generalisations of the

Bernoulli shift map S. In particular, Section 5 takes a closer look at the β-transformation T3=2 with a view to constructing its equivalent shift map, if it

exists. Section 6 is the conclusion.

2. Equivalence of Tνn and Sνn

In view of the analysis to come, we shall need the following well-known

denition of the notion of equivalence taken from the work of [11].

EQUIVALENCE BETWEEN SHIFT MAPS AND GENERALISED -TRANSFORMATIONS 61

Definition 2.1. Let (Xi,Λi, µi ), i = 1, 2, be two measure spaces, and Ti : Xi →

→ Xi , i = 1, 2, be two endomorphisms. We say that T1 is equivalent to T2 if

there exist maps F : X1 → X2 and G : X2 → X1 such that

(a) For every A2 ∈ Λ2, F−1(A2) ∈ Λ1 (mod 0) and µ1(F−1(A2)) = µ2(A2),(b) For every A1 ∈ Λ1, G−1(A1) ∈ Λ2 (mod 0) and µ2(G−1(A1)) = µ1(A1),(c) I = QF = FQ almost everywhere, where Q is G acting on a single point,

(d) T2F = FT1 almost everywhere, where F acts at a point.

The following denitions are standard, and will be useful in the discussion

to follow.

Definition 2.2. The Bernoulli shift S : Ω → Ω is given by S(ω0, ω1, . . .) == (ω1, ω2, . . .). It may also be dened by S : Y → Y such that S(ωi ) → ωi+1 for

each term ωi in ω ∈ Ω, i > 0, where Y is the alphabet of symbols.

The shift map S simply forgets the rst entry in a one-sided sequence, and

shifts all other entries one place to the left. Clearly, it is a k-to-one map of Ω, as

ω1 may be either y1 or y2 or . . . or yk . Dene the following metric on Ω [2]:

(5) dΩ(ω, t) =∞∑i=0

|ωi − ti |2i

,

where ω := ω0, ω1, . . . and t := t0, t1, . . . are two sequences in Ω. Since

|ωi − ti | is either y1 or y2 or . . . or yk , it follows that the innite series (5) is

dominated by the geometric series

(6)

∞∑i=0

1

2i,

and hence converges. The fact that dΩ is a metric on Ω is straightforward.

Definition 2.3. Consider the shift map T : xn+1 = bxn mod 1, b ∈ N, b > 1,

where b is the integer base. The transformation T : I → I is called a shift map

because T : 0.x0x1 . . . → 0.x1x2 . . . in base 2.

We now introduce the measure spaces on which our dynamical systems

will be dened. Let ω := (ω0, ω1, . . .) be any one-sided sequence on k symbols

Y = yi , i = 1, . . . , k. Denote the space of all one-sided sequences by Ω. We

dene the notion of measure m on Ω as follows [12]. Assign each yi a normed

measure u(yi ) > 0 such that∑

i u(yi ) = 1. The product measure m is obtained

as follows. Call a set [7]

(7) Kn = ω : ω1 = ω0

1, . . . , ωn = ω

0

n ∈ Ω

62 ISRAEL NCUBE

a cylinder set if Kn consists of all those sequences having a specied entry ω0

i

for the rst n symbols. Then the m-measure of Kn , denoted as m(Kn ), is theproduct of u-measures of the specied entries ωi , that is,

(8) m(Kn ) =n∏i=1

u(ωi ),

where ω = (ω0, ω1, . . . , ωn, . . .) ∈ Ω, and u(yi ) is the probability of occurrenceof y0 or y1 or . . . or yk in the ith position of ω. IfΩ is the space of all sequences,

then the symbols yi occurwith equal probability in each position andwe have that

u(yi ) = 1/k, resulting in m(Kn ) = 1/kn . Now, choose the σ-algebra associatedwith Ω to be

(9) ΛΩ :=8><>:

N⋃n

Kn : N ≥ 0 is nite9>=>; .

Noting that Ω = K0, we arrive at the measure space (Ω,ΛΩ,m). Let I = [0, 1),and consider the sequences of transformations

(10) Tn : I → I : xn+1 = 21+n xn mod 1,

and

(11) Sn : Ω→ Ω : ai → ai+n+1,

where νn ≥ 0, and νn is an arbitrary sequence of integers, not necessarily

increasing or one-to-one. Now let xn = 0.x0x1x2 . . . ∈ I, a number in base 2. It

is clear that multiplying xn by 2k , k ∈ Z, shifts the decimal point by k places to

the right. Thus, themap (10) shifts the decimal point of xn by (1+νn ) places to theright on each iteration, for any νn ∈ N.We deneTn

n(xn ) = Tn−1 . . .T1T0 (xn ).

The equivalence of Sn and Tn is straightforward.

Let S : Ω → Ω and T : I → I denote the well-known Bernoulli shift map

dened on the space of symbols and the expanding shift map dened on the

interval I = [0, 1), respectively. Furthermore, consider the dynamical systems

(Ω,ΛΩ,m, S) and (I,ΛI, `,T ), where k = 2, y1 = 0, and y2 = 1 for the sequence

space Ω and the real numbers x ∈ I are given in base 2 [7]. Following [7], we

denote by T−1(x) the preimage of x. Then, for any measurable set C, we have

that `(T−1C

)= `(C), which implies that the Lebesgue measure `(dx) = dx is

invariant under the action of T . Let us consider the measurable partition

(12) ζ := A0 = [0, 1/2), A1 = [1/2, 1),


and let

(13) Q(ω) :=

∞⋂k=0

T−k A!k

(where A!k= A0 or A1 since ωk = 0 or 1) be a function of a sequence

ω = ω0, ω1, . . . , ωn, . . . ∈ Ω. Now consider any subsets A ⊂ Ω and B ⊂ I.Dene the maps F : Ω→ I and G : I → Ω by

(14) F (A) :=

n−1⋂k=0

T−k A!k, ∀ω ∈ A,

and

(15) G(B) :=

n−1⋂k=0

S−kKxk, ∀x ∈ B,

where xk = 0 or 1 is the k th digit of x. If x = 0.x0x1x2 . . ., then xk =

= b1010k−1xc, where x is the fractional part of x, and bxc is the integral partof x. It is essential to note that both F and G are nite intersections of intervals

of I and Ω, respectively, since (n − 1) is nite; and so themselves are intervals

of I and Ω, respectively. In addition, we note that F → Q in the limit as n → ∞.In view of the analysis to come, the following result will be needed. The

proof is straightforward.

Lemma 2.1. F maps elements of ΛΩ to elements of ΛI .

In regard to the functions F, G, and Q dened above, the properties

(a)-(c) of Denition 2.1 follow unchanged. It remains only to prove (d). Let

ω := (a0, a1, a2, . . .) ∈ Ω. Then the shift (11) drops (1 + νn ) digits of ω on each

application, for any νn . That is, Sn (ω) = (ar, ar+1, . . .) ∈ Ω, r = 1 + νn; andthus

(16) FSn (ω) = 0.arar+1 . . . ∈ I .

Following Lemma 2.1, we have that F (ω) = 0.a0a1a2 . . . ∈ I, and (10) gives

(17) Tn F (ω) = 0.arar+1 . . . ∈ I .

Hence, following (16) and (17), we have thatTn F (ω) = FSn (ω) for anyω ∈ Ω,as required for (d). This completes the proof.

Remark 2.1. As the maps Tn are everywhere 21+n to 1, the measure puts

uniform conditional expectations on the inverse images. Thismakes it impossible

for there to be a conjugacy unless νn = 0. In fact, this shows that if S′n were

equivalent to Tn , then ν′n = νn . For two-sided actions, where the maps are

64 ISRAEL NCUBE

invertible, the question goes back to Halmos [24, 25] asking if the Bernoulli

2-shift were conjugate to the Bernoulli 3-shift. The Kolmogorov-Sinai theory

[26, 27, 28] of entropy answers this in the negative.

We now investigate conditions under which Tn is equivalent to S. Employ-

ing the usual notation, we see that S(ω) = a1, a2, a3, . . . ∈ Ω and FS(ω) == 0.a1a2a3 . . . ∈ I. Additionally, it is evident that F (ω) = 0.a0a1a2 . . . ∈ I andTn F (ω) = 0.arar+1 . . . ∈ I, where r = 1 + νn . Clearly, Tn is equivalent to S if

νn = 0 almost everywhere.

Finally, let us consider the sensitivity to initial conditions of Tn and Sn . Itis straightforward to show that

(18) Tn

n(x) = [2n+

∑n

i=1i ]xn ,

where x is the fractional part of x. Thus, the Lyapunov exponent [2, 13] is

given by

λ := limn→∞

1

nln

"d

dxn

Tn

n(xn )

#= (1 + ν) ln 2 > 0,

where ν := limn→∞1

n

∑n

i=1 νi . This implies that Tn exhibits sensitivity to initialconditions; that is, it is expansive.

Shifting our attention to the Ω space, we consider two sequences ω1, ω2 ∈

∈ KN+1, N 1. It is a straightforward exercise to show that

(19) dΩ(Sk

nω1, Sk

nω2) = (2k+

∑k

i=1i )d0,

where d0 = dΩ(ω1, ω2) = 2−N . As a consequence, the corresponding Lyapunov

exponent is given by

λ := limk→∞

"1

kln

1

d0

dΩ(Sk

nω1, Sk

nω2)

#= (1 + ν) ln 2 > 0,

where ν = limk→∞1

k

∑k

i=1 νi . We conclude that Sn is expansive, just like Tn .


3. Equivalence of Thn and Shn

Let xn = 0.a0a1a2 . . . ∈ I and ω = a0, a1, a2, . . . ∈ Ω. We consider the

maps

(20) Thn: I → I : xn+1 = 2xn + hn, mod1, hn ∈ R, mod1

and

(21) Shn: Y → Y : ai → a′

i,

where cia′i = ai+1+hi +ci+1, i ∈ N; and ai, hi, ci are symbols inY with meaning

as dened below. The map Thnis a modied form of the expanding shift map

(22) T : I → I : xn+1 = 2xn, mod1.

Thus, apart from shifting the decimal point by one place to the right when

multiplying xn by 2, we also add an arbitrary factor, hn ∈ R, to 2xn and then

take the fractional part of the result. Let us dene

(23) Tn

hn(x) = Thn

. . .Th2Th1

(x).

If we take hn to be any random sequence, this shows how spatially uniform

noise can aect T . The modied shift (21) on two symbols is designed to be

consistent with the addition of the digits of hn = 0.h0h1h2 . . . to those of

2xn = 0.a1a2a3 . . .. As a result, in (21), the hi are digits of hn and the ci arecarries from the addition of digits to the right. Now, consider the sum 2xn + hn .

The digit a′iof this sum is ai+1 + hi if the carry is zero and ai+1 + hi + 1 if the

carry is 1. Since ai+1 + hi can have at most two digits (in base 2), we see that the

sum is at most a two-digit number and can be expressed as

(24) cia′i = ai+1 + hi + ci+1, i ∈ N0,

where on the left we must interpret a′ias the `unit digit' and ci as the `two's

digit'. The term ci+1 is clearly the carry from the right and (24) determines the

carry ci to be passed on to the left. In addition, we note the following important

points.

• If ai ∈ A = 0, 1, we interpret + and 2xn as in ordinary arithmetic; if

A = r, s (abstract symbols), we have to encode these operations (i.e. +

and 2xn) for these symbols.

• While i increases from the left, carries are brought down from the right.

Where any of xn , hn are irrationals, this has to be done from innity,

which is computationally impossible, but is of course taken for granted

in real arithmetic. When each is a rational number (at least), this can be

done computationally, and we may list the digits for i = 0, 1, 2, . . .. Where

66 ISRAEL NCUBE

the calculation is computable, we adopt the convention that (24) holds for

i = 0, 1, 2, . . .; and where the calculation is not computable, we adopt the

convention that (24) holds ∀i ∈ N0. This is the most general case.

• This modied shift is such that a sequence ω ∈ Ω is shifted once to the left

and then overwritten by new symbols according to the addition of hn .

Again, assume F, G, and Q as dened above so that the equivalence of Thn

and Thndepends on (d) of Denition 2.1. Following Lemma 2.1, if ω =

= (a0, a1, a2, . . .), we have that

(25) F (ω) = 0.a0a1a2 . . . ∈ I .

Denition 2.3 and equation (24) imply that

(26)Thn

F (ω) = 0.a1a2 . . . + hn mod 1

= 0.b1b2b3 . . . , say,

for all i ∈ N0, where bi = a′i. From (21), we have that

(27) Shn(ω) = (a′

0, a′

1, a′

2, . . .),

where a′iis such that cia′i = ai+1 + hi + ci+1, for all i ∈ N0. Thus, we obtain

(28) FShn(ω) = 0.a′

0a′1. . . , ∀i ∈ N0.

Consequently, following (26) and (28), we have that ThnF (ω) = FShn

(ω) forany ω ∈ Ω. This completes the equivalence proof.

It is a simple exercise to show thatThnis expansive, with Lyapunov exponent

[2, 13] ln 2 > 0. To establish that Shnhas the same property, we proceed as

follows. Consider ω1, ω2 ∈ KN , N 1 as before. Using (21), we obtain

(29) ciai =8><>:ai+1 + hi, or

ai+1 + hi + 1, ∀i ∈ N0

for ω1 = a0, a1, a2, . . ., and

dibi =8><>:bi+1 + hi, or

bi+1 + hi + 1, ∀i ∈ N0

for ω2 = (b0, b1, b2, . . .). Hence, from (5), we obtain

dΩ(skhnω1, Sk

hnω2) = 2kd0,

where d0 := dΩ(ω1, ω2) = 2−N , implying that Shnis expansive, with Lyapunov

exponent [2, 13] λ = ln 2. This exponent is identical to the one obtained for Thn

and S. We can conclude, therefore, that adding hn to S has no eect on the

e-rate. It only translates the graph Thn: I → I by hn in the x direction without

changing its slope.


Finally, we investigate conditions under which Thnis equivalent to S.

We adopt the usual notation. Thus, S(ω) = a1, a2, . . . ∈ Ω and FS(ω) == 0.a1a2 . . . ∈ I. In addition, we have that F (ω) = 0.a0a1a2 . . . ∈ I and

ThnF (ω) = 0.a1a2a3 . . . + hn,mod1. Commutativity occurs when 0.a1a2 . . . +

+ hn = 0.a1a2 . . ., which is true only when hn = k (n) mod 1, k (n) ∈ Neverywhere, or when hn = 0. For hn = k (n) mod 1, together with (a)-(c) of

Denition 2.1, we see that Thnis equivalent to S (and thus Shn

is equivalent

to S). For hn , k (n) mod 1, we do not know whether there is any equivalence

ϕ : Ω→ I such that ϕS(ω) = Thnϕ(ω).

4. Equivalence of Thnνn and Shnνn

As usual, let xn = 0.a0a1a2 . . . ∈ I and ω = a0, a1, a2, . . . ∈ Ω, andconsider the maps

(30) Thnn : I → I : xn+1 = 21+n xn + hn mod 1,

and

(31) Shnn : Y → Y : ai → a′i,

where cia′i = ai+n+1 + hi + ci+1, for all i ∈ N0; and ai, hn, νn are dened as

before. It is important to note that in (31), the operations + and 21+n xn on

what are abstract symbols are understood as before. Since we are working in

base 2, the map (31)) shifts the decimal point of xn by 1 + νn places to the right,adds hn to the result, and then retains the fractional part of this. Let us dene

(32) Tn

hnn(x) = Thnn . . .Th22Th11 (x).

Now we see that for hn = 0 and νn arbitrary, we obtain the map Tn considered

above. The graphs of Tn

0n(x) sample the space of those of Tn (x), n ∈ N. When

hn , 0, νn = 0, we obtain the map Thnabove. The modied shift (31) is

constructed by designing the new sequences to mirror addition of the digits of

hn = 0.h0h1h2 . . . to those of 21+n xn = 0.arar+1ar+2 . . ., where r = 1 + νn .Consider the sum 21+n xn + hn . The digit a′

iof this sum is ai+1+n + hi if the

carry is zero and ai+n+1 + hi + 1 if the carry is 1. Since ai+1+n + hi can have

at most two digits (in base 2), we see that the sum, which is at most a two-digit

term, can be expressed as

(33) cia′i = ai+1+n + hi + ci+1, ∀i ∈ N0,

68 ISRAEL NCUBE

where ci, ai and ci+1 are interpreted as before. It is also worth noting that the

expression (33) generates all of the digits of Shnn (ω). Following Lemma 2.1,

we have that

(34) F (ω) = 0.a0a1a2 . . . ∈ I,

and Denition 2.3 means that

(35)Thnn F (ω) = 0.arar+1ar+2 . . . + hn mod 1

= 0.d1d2d3 . . . say,

where, from (33), di = a′iand r = 1 + νn . From (21), we have that

(36) Shnn (ω) = (a′0, a′

1, . . .),

where a′i: cia′i = ai+1+n + hi + ci+1, for all i ∈ N0; and thus

(37) FShnn (ω) = 0.a′0a′1a′2. . . ∈ I .

Whence, following (35) and (37), we have that

(38) Thnn F (ω) = FShnn (ω),

for any ω ∈ Ω. This completes the equivalence proof.

It is straightforward to show that the e-rate of Thnn is (1 + ν) ln 2 > 0,

where ν = limk→∞1

k

∑k

i=1 νi > 0. To show that Shnn is expansive as well, we

proceed as follows. Consider ω1, ω2 ∈ KN , N 1, as before. Then, from (31),

we obtain

(39) ai =8><>:ai+n+1 + hi, or

ai+n+1 + hi + 1, ∀i ∈ N0

for ω1 = a0, a1, . . ., and

(40) bi =8><>:bi+n+1 + hi, or

bi+n+1 + hi + 1, ∀i ∈ N0

for ω2 = b0, b1, b2, . . .. Hence, from (5), we arrive at

(41) dΩ(Sk

hnnω1, Sk

hnnω2) = [2k+

∑k

i=1i ]d0,

which means that dΩ grows exponentially at the e-rate λ = (1 + ν) ln 2 > 0,

where ν = limk→∞( 1k

∑k

i=1 νi ). This is the same value of λ we obtained for Snand Tn . Once again, Shnn is sensitive to initial conditions, and the addition of

uniform noise, hn , to Sn does not alter the exponent λ. It only has the eect oftranslating the graph of Tn by hn in the x direction without changing its slope.


We now consider conditions for which Thnn is equivalent to S under the

action of F. Using the usual notation, we have

(42) FS(ω) = 0.a1a2 . . . ∈ I,

and

(43) Thnn F (ω) = 0.arar+1ar+2 . . . + hn,mod1,

where r = 1 + νn . It is clear that FS(ω) = Thnn F (ω) = 0.arar+1ar+2 . . . hn

only when hn = k (n) mod 1, where k (n) ∈ N0, and νn = 0.

Remark 4.1. For hn = 0 = νn , together with (a)-(c) of Denition 2.1, we see

thatThnn is equivalent to S. First, as the action of S is stationary, i.e. one is acting

by a sequence of maps which are all the same, the map Thnn must likewise be

stationary, i.e. hn = h is a constant and, for the reasons given above, νn = 0.

Now we have an interesting question. It has nothing to do with S and T as we

know that Shnn and Thnn are conjugate. It is simply a question of whether

x → 2x+h mod 1 and x → 2x mod 1 are conjugate. Homan and Rudolph [23]

have characterised when any uniformly 2− 1maps, such as these, are conjugate,

and these two will be. It is a rather easy application of their work.

We now consider the map T3=2, and attempt to nd a modied shift map that

is equivalent to it.

5. Equivalence of T3/2 and S3/2?

Let xn = 0.a0a1a2 . . . ∈ I and ω = (a0, a1, a2, . . .) ∈ Ω, and consider the

maps

(44) T3

2

: I → I : xn+1 =3

2xn mod 1 = 2xn + hn (xn ) mod 1,

and

(45) S 3

2

: Y → Y : ai → a′i: a′

i= biai+1 − bi+1 − ai−1, ∀i ∈ N0.

In (44), hn (xn ) = − 1

2xn = −0.0a0a1a2 . . . ∈ I, ai any digit of xn , i ∈ N. Note

that (44) is reminiscent to (20), except that hn (xn ) = − 1

2xn in this instance. That

is, hn here depends on every point xn ∈ I and the addition of hn is no longer

uniform. This is an indication of the peculiarities of β-transformations.

The modied shift (45) generates all of the symbols of S3=2, and is designedto mirror the borrows involved in the calculation of 2xn −

1

2xn mod 1. Here,

70 ISRAEL NCUBE

the digit ai+1 arises from a shift left of the sequence ω in the usual sense. The

digit ai−1 is a shift right of the sequence ω, and is designed to model division by

2 in (44). Let αi = ai+1 − bi+1 − ai−1 and let the ith digit of 2xn −1

2xn mod 1,

for any xn , be a′i. Therefore, a′

i= αi + 2bi , where

(46)8><>:bi = 0 if αi ≥ 0

bi = 1 if αi < 0

As before, this must be true for all i ∈ N or for i = 1, 2, . . ., where the former

implies working in the realm of real arithmetic and the latter implies working

in the realm of computable arithmetic. From Lemma 2.1 and Denition 2.3, we

have that

(47)

T3

2

F (ω) = 0.a1a2 . . . −1

2xn mod 1

= 0.a1a2 . . . − 0.0a0a1a2 . . . mod 1

= 0.a′0a′1a′2. . . ∈ I .

Additionally, much as before,

(48) S 3

2

(ω) = (a′0, a′

1, a′

2, . . .),

and thus

(49) FS 3

2

(ω) = 0.a′0a′1a′2. . . ∈ I .

Hence, following (47) and (49), we obtain

(50) T3

2

F (ω) = FS 3

2

(ω), for any ω ∈ Ω.

This then, together with (a)-(c) of Denition 2.1, completes the equivalence

proof.

Consider T3=2 : I → I : xn+1 =3

2xn mod 1 and S3=2 : Ω → Ω : ai → a′

i,

where a′i+ 2bi = ai + ai−1 + bi+1 in order to mirror the action on a symbol in

the sequence space of T3=2 (here it is convenient to multiply xn by 3

2rather than

shifting and subtracting − 1

2xn). We now calculate the Lyapunov exponents in

the spaces I andΩ. In the former, it is straightforward to show that the Lyapunov

exponent of T3=2 is λ = ln(3

2

)> 0.

In the space Ω, the calculation of the e-rate is subtle, as we now show.

As before, consider two sequences ω1, ω2 ∈ KN+1, N 1, where KN+1 is a

cylinder set. Following (5), ifω1 = a0, a1, a2, . . . andω2 = c0, c1, c2, . . ., then

(51) dΩ(ω1, ω2) =∞∑i=0

|ai − ci |2i

≤1

2N.


From the preceding, it is clear that ω′1= S3=2ω1 and ω

′2= S3=2ω2 are dened by

digit sequences ω′1= a′

0, a′

1, a′

2, . . . and ω′

2= c′

0, c′

1, c′

2, . . ., where8><>:a′

i+ 2bi = ai + ai−1 + bi+1

c′i+ 2di = ai + ai−1 + di+1

, i ≤ N

since ω1, ω2 ∈ KN+1 and ai = ci for i ≤ N . Two cases arise from this, namely:

(a) bi+1 = di+1, and (b) bi+1 , di+1. We consider both cases in turn. For case (a),

we have that a′i+ 2bi = c′

i+ 2di . By equating coecients of 20, 21, we obtain8><>:a′

i= c′

i

bi = di

, i ≤ N .

For case (b), we consider bi+1 = 0 and di+1 = 1 without loss of generality. Then

we have that a′i+ 2bi + 1 = c′

i+ 2di . Equating the coecients of 20, 21 yields

two subcases, namely:

(Subcase 1)8><>:a′

i+ 1 = c′

i

bi = di,

which implies that a′i= 0, c′

i= 1 and bi = di ; and

(Subcase 2)8><>:a′

i= c′

i

2bi + 1 = 2di,

which cannot be satised by digits bi , di . In particular, from Subcase 2, we see

that if bN+1 = dN+1, then 8><>:a′i= c′

i

bi = di

, ∀i ≤ N .

Similarly, Subcase 1 implies that if bN+1 = 0, dN+1 = 1, then8><>:a′N= 0, c′

N= 1

bN = dN

, ∀i ≤ N − 1.

Hence, from (5), if bN+1 = dN+1, then dΩ(ω′1, ω′

2) = 2−N . If bN+1 = 0 and

dN+1 = 1, then dΩ(ω′1, ω′

2) = 2−(N−1) since now bN = dN is the carry, and

consequently we have that ai = ci , i < N . We note that there are two subcases of

bN+1 = dN+1, namely; bN+1 = 0 = dN+1 and bN+1 = 1 = dN+1. Thus, in the two

subcases, we have that d ′Ω= 2−N = d0

Ω, say. Similarly, there are two subcases of

bN+1 , dN+1, namely; bN+1 = 0, dN+1 = 1 and bN+1 = 1, dN+1 = 0; so that in

72 ISRAEL NCUBE

the two subcases we have d ′Ω= 2d0

Ω. The average displacement in the preceding

four subcases is given by

(52) d ′ =3

2d0

Ω.

The Lyapunov exponent for S3=2 may be written as

(53) λ = limn→∞

*,1n∞∑n=0

lndn+1

dn

+- ,and, on average, two orbits of KN+1 diverge on each iteration. Thus, λ == ln(3/2) > 0, which is in agreement with the e-rate obtained for T3=2.

Calculation of dynamical properties in Ω is given by the rules ai → a′ifor

components of the sequence and can be quite dicult. Suppose we now proceed

to the second iteration of S3=2, which we shall denote by ′′. We begin with the

state after the rst iteration, where

bN+1 = dN+1 ⇒ bN = dN , a′N= c′

N(2 cases),(54)

and

bN+1 , dN+1 ⇒ bN = dN , a′N, c′

N(2 cases).(55)

In (54), we have that a′i= c′

ifor i ≤ N ; and in (55) we have that a′

i= c′

ifor

i ≤ N − 1. We now consider (54) and (55) in turn, starting with (54). On the

second iteration, we have

(56)8><>:a′′

i+ 2b′

i= a′

i+ a′

i−1+ b′

i+1

c′′i+ 2d ′

i= a′

i+ a′

i−1+ d ′

i+1

, i ≤ N,

which, as before, implies that8><>:a′′N= c′′

N

b′N= d ′

N

and a′′i= c′′

i, i ≤ N,(57)

or 8><>:a′′N= 0, c′′

N= 1

b′N= d ′

N

and a′′i= c′′

i, i ≤ N − 1.(58)

From this, we nd that

(59) d2

Ω= d0

Ω, (2 cases)

or

(60) d2

Ω= 2d0

Ω(2 cases).


Now we consider (55). On the second iteration, we have that a′N= 0, c′

N= 1,

a′i= c′

i, i ≤ N − 1. Hence, we obtain

(61)8><>:a′′

N−1+ 2b′

N−1= 0 + a′

N−2+ b′

N

c′′N−1

+ 2d ′N−1= 1 + a′

N−2+ d ′

N

Once again, two cases are possible; namely, b′N= d ′

Nand b′

N, d ′

N. We begin

with the former, which yields

(62) a′′N−1 + 2b′

N−1 + 1 = c′′N−1 + 2d ′

N−1,

implying that

(63)8><>:a′′

N−1= 0, c′′

N−1= 1

b′N−1= d ′

N−1

only.

We have that a′′i= c′′

ifor i ≤ N − 2, which implies that d2

Ω= 22d0

Ω(2 cases).

Similarly, for a′N= 1, c′

N= 0, it can be shown that d2

Ω= 22d0

Ω(2 cases).

For the case b′N, d ′

N, we have two possibilities; namely, b′

N= 1 and

d ′N= 0 or b′

N= 0 and d ′

N= 1. For the former, we obtain that d2

Ω= 2d0

Ω

(2 cases), and for the latter we get d2

Ω= 23d0

Ω(1 case). As before, we see that

the average displacement in these 2 × 8 cases for S2

3=2is

(64) d2 =

(3

2

)2d0

Ω.

The preceding analysis suggests that dn = ( 32)nd0

Ω, so that the Lyapunov

exponent in the Ω space is

(65) λ = limn→∞

1

nln

dn

d0

Ω

= ln3

2> 0.

We note that d1, . . . , d5 may be expressed as

(66)

8>>>>>>>>>><>>>>>>>>>>:

d1 = 1

21(1 + 2)d0

Ω

d2 = 1

22(2 + 3 + 4)d0

Ω

d3 = 1

23(2 + 3 + 4 + 5 + 6 + 7)d0

Ω

d4 = 1

24(5 + 6 + 7 + . . . + 13)d0

Ω

d5 = 1

25(5 + 6 + 7 + 8 + . . . + 18)d0

Ω

Based on the above, we conjecture that

(67) dn =1

2n[k + (k + 1) + (k + 2) + . . . + (k + r)]d0

Ω,

74 ISRAEL NCUBE

where

(68) [k + (k + 1) + (k + 2) + . . . + (k + r)] = 3n .

This suggests that we have an arithmetic series, Lk , such that Lk+r − Lk−1 = 3n ,

which gives

(69)k + r2

[2 + (k + r − 1)] −k − 1

2[2 + (k − 2)] = 3n,

which simplies to

(70) r (r + 2k + 1) + 2k = 2 · 3n .

We see that (70) is a diophantine equation; solving it involves nding the values

r, k for some n. For example,

(71)8><>:n = 1⇒ k = 0, r = 2,

n = 2⇒ k = 2, r = 2.

Diophantine equations are notoriously dicult to solve.As can be seen from (70),

it becomes extremely dicult to determine r, k in the limit as n → ∞. Solutionsdo exist for the rst ve values of n, as demonstrated above. The foregoing

analysis indicates that calculation of dynamical properties (such as Lyapunov

exponents) within (Ω,ΛΩ, dΩ, S), that is, within instruction for changing symbols

in a sequence, will be very dicult in general. Progress can be made with the

Bernoulli shift only because it is such a simple instruction S : Y → Y : ai → ai+1.

Finally, it is essential to note that the average over displacements dn assumes

that the possibilities bi+1 = di+1 (2 cases) and bi+1 , di+1 (2 cases) come up

equally often for almost all sequences.

6. Conclusion

As usual, where transformations are equivalent, properties of one may be

used to nd properties of the other. The modied shifts Sn , Shn, and Shnn

have invariant measures and are ergodic since Tn , Thn, and Thnn have such

measures and are ergodic. TheT3=2map of course has the invariant Parrymeasure

h3=2 and is ergodic. As a result, the modied shift map S3=2 has these properties.Calculation in the spaceΩ of S3=2 is to be done via rules for transforming symbols

ai of ω = (a0, a1, . . .) to symbols a′iof Sω = (a′

0, a′

1, . . .), where S : Ω → Ω is

any endomorphism. Here, it seems simpler to work in terms of T3=2.


Although the Bernoulli shift has a central role to play in dynamical systems

theory, this apparently is the case only because of the simple rule ai → ai+1

for such shifts and because dieomorphisms and dierential equations have

been found which are isomorphic to it. Clearly, the problem of determining the

Lyapunov exponent for S3=2 introduces deep problems in the digit distribution of

numbers (recall our assumption that λ is an average of possible displacements)

and in diophantine equations. As an example, are we justied in dening dn as an

average in the above sense when the Parry measure for T3=2 is non-uniform and

thus the invariant measure for S3=2 is non-uniform? Further pursuit of conjecture

(67) could lead to interesting number-theoretic problems and their possible

solution when, for example, various calculations of Lyapunov exponent are to

agree. Finally, the problem of equivalence of the maps introduced in this paper

to Bernoulli shifts S is of real interest, but has not been pursued except to note

that when hn or νn are zero they reduce to S.

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(1992), 181201.

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University Press (1974).

76 ISRAEL NCUBE

[13] E. Ott, Chaos in Dynamical Systems, Cambridge University Press (1993).

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(1960), 401416.

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Springer-Verlag (1992).

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Math. Acad. Sci. Hung., 8, 472493.

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ical Society, 73 (1967), 747817.

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2problem, K. norske Vidensk. Selesk. Skr., 16

(1972), 14.

[21] I. Ncube, β-Transformations and Modied Shift Maps, M. Sc. thesis, University

of the Witwatersrand, 1996.

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[23] C. Hoffman and D. Rudolph, Uniform endomorphisms which are isomorphic to

a Bernoulli shift, Annals of Mathematics, 156 (2002), 79101.

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Israel Ncube

College of Engineering, Technology,& Physical Sciences


Alabama A &M University

Huntsville, AL 35762, U.S.A.

[email protected]


ON α, SEMI, β-OPEN SETS IN MINIMAL IDEAL TOPOLOGICAL

SPACES

By

M. PARIMALA AND A. SELVAKUMAR

(Received May 30, 2017)

Abstract. The purpose of this paper is to introduce α-mI-open sets, semi-mI-open sets and β-mI-open sets in ideal minimal spaces and to investigate the relationships

between the sets. Furthermore, decompositions of continuous functions are established.

1. Introduction

An ideal [7] I on a nonempty set X is a nonempty collection of subsets

of X which satises (i) A ∈ I and B ⊂ A implies B ∈ I and (ii) A ∈ I and

B ∈ I implies A ∪ B ∈ I. Given a topological space (X, τ) with an ideal I on

X and if P(X ) is the set of all subsets of X , a set operator (.)∗ : P(X ) → P(X ),called a local function [6] of A with respect to τ and I is dened as follows:

for A ⊂ X , A∗(I, τ) = x ∈ X : U ∩ A < I for every U ∈ τ(x) where τ(x) == U ∈ τ : x ∈ U . A Kuratowski closure operator cl∗(.) for a topology τ∗(I, τ),called the ∗-topology, ner than τ is dened by cl∗(A) = A∪A∗(I, τ) [13]. When

there is no chance for confusion, we will simply write A∗ for A∗(I, τ) and τ∗ forτ∗(I, τ). If I is an ideal on X , then the space (X, τ, I) is called an ideal space. Asubset A of an ideal space is said to be ∗-dense in itself [4](resp. ∗-closed [6]) if

A ⊂ A∗ (resp. A∗ ⊂ A). By a space (X, τ), we always mean a topological space

(X, τ) with no separation properties assumed. If A ⊂ X , cl(A) and int(A) will,respectively, denote the closure and interior of A in (X, τ) and int∗(A) will denotethe interior of A in (X, τ∗). The notion of I-open sets was introduced by Jankovicet al. [5], further it was investigated by Abd El-Monsef [1]. In 1965, Njastad [9]

initiated the investigation of α-open sets in topological spaces. As a modication

2000 Mathematics Subject Classication 54A05, 54C10

78 M. PARIMALA, A. SELVAKUMAR

of α-open sets, Hatir and Noiri [3] introduced the notion of α-I-open sets in anideal topological spaces (X, τ, I), where τ is a topology and I is an ideal.

In [8], Maki, Umehara and Noiri introduced the notion of minimal structure

andminimal spaces as a generalization of topological spaces on a given nonempty

set. Also, genaralized topologies which are other generalization of topology

were dened by Császár [2]. Further, it was studied by Popa and Noiri in [11].

A subfamily M of the power set P(X ) of a non empty set X is a minimal

structure, if φ, X ∈ M. (X,M) is called a minimal space (m-space). A subset

A of X is said to be m-open [8] if A ∈ M. The complement of a m-open

set is called a m-closed set. Dene m- int(A) = ∪U : U ⊂ A,U ∈ M andm- cl(A) = ∩F : A ⊂ F, X − F ∈ M. A minimal space (X,M) has the

property [U ] if the arbitrary union of m-open sets is m-open. (X, M) has theproperty [I] if the any nite intersection of m-open sets is m-open [11].

In 2009, Ozbakir and Yildirim [10] have dened the minimal local function

A∗min an ideal minimal space (X,M, I). In this paper, by using the local function

A∗m

we introduce and investigate the notion of α-mI-open set, semi-mI-openset, β-mI-open set in (X,M, I). Furthermore, decompositions of continuous

functions are established.

2. Preliminaries

Definition 2.1 ([10]). Let (X,M) be a minimal space with an ideal I on X and

(.)∗m be a set operator from P(X ) to P(X ) (P(X ) is the set of all subsets of X).

For a subset A ⊂ X , A∗m

(I,M) = x ∈ X : Um ∩ A < I; for every Um ∈ Um (x)is called the minimal local function of Awith respect to I and M . We will simply

write A∗mfor A∗

m(I,M).

Theorem 2.2 ([10]). Let (X,M) be a minimal space with I, I ′ ideals on X andA, B be subsets of X . Then(i) A ⊂ B ⇒ A∗

m⊂ B∗

m,

(ii) I ⊂ I ′ ⇒ A∗m

(I ′) ⊂ A∗m

(I),(iii) A∗

m= m- cl(A∗

m) ⊂ m- cl(A),

(iv) A∗m∪ B∗

m⊂ (A ∪ B)∗

m,

(v) (A∗m

)∗m⊂ A∗

m.

Remark 2.3 ([10]). If (X,M) has property [I], then A∗m∪ B∗

m= (A ∪ B)∗

m.

ON ; SEMI; -OPEN SETS IN MINIMAL IDEAL TOPOLOGICAL SPACES 79

Definition 2.4 ([10]). Let (X,M) be a minimal space with an ideal I on X . The

set operator m- cl∗ is called a minimal ∗-closure and is dened as m- cl∗(A) == A ∪ A∗

mfor A ⊂ X . We will denote by M∗(I,M) the minimal structure

generated by m- cl∗, that is,M∗(I,M) = U ⊂ X : m- cl∗(X − U) = X − U .M∗(I,M) is called ∗-minimal structure which is ner thanM. The elements of

M∗(I,M) are called minimal ∗-open (briey, m∗-open) and the complement of

an m∗-open set is called minimal ∗-closed (briey, m∗-closed).

Throughout the paper we simply writeM∗ forM∗(I,M). If I is an ideal

on X , then (X,M, I) is called an ideal minimal space.

Proposition 2.5 ([10]). The set operator m- cl∗ satisfies the following condi-tions:(i) A ⊂ m- cl∗(A),(ii) m- cl∗(φ) = φ and m- cl∗(X ) = X ,(iii) If A ⊂ B, then m- cl∗(A) ⊂ m- cl∗(B),(iv) m- cl∗(A) ∪ m- cl(B) ⊂ m- cl∗(A ∪ B).

Remark 2.6. If (X,M) has property [I], then m- cl∗(m- cl∗(A)) = m- cl∗(A)and m- cl∗(A) ∪ m- cl∗(B) = m- cl∗(A ∪ B).

Lemma 2.7 ([12]). Let (X, τ, I) be an ideal space and A ⊂ X . If A ⊂ A∗, thenA∗ = cl(A∗) = cl(A) = cl∗(A).

3. α-mI-open set, semi-mI-open set and β-mI-open set

Definition 3.1. A subset A of an ideal minimal space (X,M, I) is said to be(i) α-mI-open set if A ⊂ m- int(m- cl∗(m- int(A))).(ii) semi-mI-open set if A ⊂ m- cl∗(m- int(A)).(iii) β-mI-open set if A ⊂ m- cl(m- int(m- cl∗(A))).(iv) mI-open [10] if A ⊂ m- int(A∗

m).

(v) pre-mI-open set if A ⊂ m- int(m- cl∗(A)).

Theorem 3.2. For a subset of an ideal minimal space, the following hold.(i) Every α-mI-open set is α-m-open.(ii) Every semi-mI-open set is semi-m-open.(iii) Every β-mI-open set is β-m-open.


Proof. (a) Let A be an α-mI-open set. Then we have

A ⊂ m- int(m- cl∗(m- int(A))) = m- int((m- int(A))∗ ∪ m- int(A)) ⊂⊂ m- int(m- cl(m- int(A)) ∪ m- int(A)) ⊂

⊂ m- int(m- cl(m- int(A)))

Therefore this shows that A is αm-open.

(b) Let A be a semi-mI-open set. Then we have

A ⊂ m- cl∗(m- int(A)) ⊂ (m- int(A))∗ ∪ (m- int(A)) ⊂

⊂ m- cl∗(m- int(A)) ∪ (m- int(A)) == m- cl(m- int(A)).

Therefore this shows that A is semi-m-open.

(c) Let A be a β-mI-open set. Then we have

A ⊂ m- cl(m- int(m- cl∗(A))) = m- cl(m- int(A∗m∪ A)) ⊂

⊂ m- cl(m- int(m- cl(A) ∪ A)) = m- cl(m- int(m- cl(A))).

Therefore this shows that A is β-m-open.

Theorem 3.3. For a subset of an ideal minimal space, the following hold.(i) Every α-mI-open set is pre-mI-open.(ii) Every α-mI-open set is semi-mI-open.(iii) Every pre-mI-open set is β-mI-open.

Proof. The proof is obvious.

The converse of the above theorem need not be true.

Example 3.4. (i) Let X = a, b, c, d, M = X, φ, a, b, b, c, c, d andI = φ. Let A = a, b, c is pre-mI-open but not α-mI-open set.

(ii) Let X = a, b, c, d, M = X, φ, a, b, a, b, c, b, c, a, c and I == a, φ. Let A = a, c is β-mI-open but not pre-mI-open set.

(iii) Let X = a, b, c, d,M = X, φ, a, b, a, b, c, b, c, c, d and I = φ.Let A = a, b, d is semi-mI-open but not α-mI-open set.

Theorem 3.5. Every m-open set of an ideal minimal space is α-mI-open.

Proof. Let A be any minimal open set. Then we have A = m- int(A) ⊂⊂ m- int((m- int(A))∗ ∪ m- int(A)) = m- int(m- cl∗(m- int(A))). Therefore Ais α-mI-open.


Theorem 3.6. Let (X,M, I) be an ideal minimal space. A subset A of X isα-mI-open if an only if it is semi-mI-open and pre-mI-open.

Proof. Necessity: This is obvious.

Suciency: Let A be semi-mI-open and pre-mI-open. Then we

have A ⊂ m- int(m- cl∗(A)) ⊂ m- int(m-(cl∗(m- cl∗(m- int(A))))) =

= m- int(m- cl∗(m- int(A))). This shows that A is α-mI-open.

Lemma 3.7. Let (X,M, I) be an ideal minimal space. Let A be a subset of X . IfO is open in (X,M, I), then O ∩ cl∗(A) ⊂ cl∗(O ∩ A).

Proof. It is stated in Theorem 2.2, that if O ∈ M then O ∩ A∗m⊂ (O ∩ A)∗ for

any subset A of X . Thus we have O∩cl∗(A) = O∩ (A∗m∪ A) = (O∩ A∗

m)∪ (O∩

∩ A) ⊂ (O∩ A)∗∪ (O∩ A) = cl∗(O∩ A). ThereforeO∩cl∗(A) ⊂ cl∗(O∩ A).

Theorem 3.8. Let (X,M, I) be an ideal minimal space.(i) If V ∈ smIO(X ) and A ∈ αmIO(X ), then V ∩ A ∈ smIO(X ).(ii) If V ∈ pmIO(X ) and A ∈ αmIO(X ), then V ∩ A ∈ pmIO(X ).

Proof. (i) Let V ∈ smIO(X ) and A ∈ αmIO(X ). By Lemma 3.7, we obtain

V ∩ A ⊂ cl∗(int(V )) ∩ int(cl∗(int(A))) ⊂

⊂ cl∗((int(V )) ∩ int(cl∗(int(A)))) ⊂

⊂ cl∗((int(V )) ∩ cl∗(int(A))) ⊂

⊂ cl∗(cl∗(int(V ) ∩ int(A))) ⊂

⊂ cl∗(int(V ∩ A))

Therefore, V ∩ A ∈ smIO(X ).(ii) Let V ∈ pmIO(X ) and A ∈ αmIO(X ). By Lemma ??, we obtain

V ∩ A ⊂ int(cl∗(V )) ∩ int(cl∗(int(A))) ⊂

⊂ int((int(cl∗(V )) ∩ cl∗(int(A)))) ⊂

⊂ int(cl∗(int(cl∗(V )) ∩ int(A))) ⊂

⊂ int(cl∗(cl∗(V ) ∩ int(A))) ⊂

⊂ int(cl∗(V ∩ int(A))) ⊂

⊂ int(cl∗(V ∩ A))

Therefore, V ∩ A ∈ pmIO(X ).

Theorem 3.9. Let (X,M, I) be an ideal minimal space.(i) If A, B ∈ αmIO(X ), then A ∩ B ∈ αmIO(X ).


(ii) If A ∈ αmIO(X ) for each γ ∈ σ, then ∪ ∈A ∈ αmIO(X ).

Proof. (i) Let A, B ∈ αmIO(X ) by Theorem 3.6, A is semi-mI-open and

pre-mI-open. By Theorem ??Theorem 3.7?? , A ∩ B is semi-mI-open and pre-mI-open. ByTheorem 3.6, A ∩ B is αmIO(X ).

(ii) Let A ∈ αmIO(X ) for each γ ∈ σ. Then we have

A ⊂ m- int(m- cl(m- int(A ))) ⊂ m- int(m- cl(m- int(∪A ))).

Hence ∪ ∈A ⊂ m- int(m- cl(m- int(∪ ∈A ))).

4. Decomposition of continuity via minimal ideals

Definition 4.1. A function f : (X,M, I) −→ (Y, σ) is said to be α-mI-continuous (resp. semi-mI-continuous, pre-mI-continuous, β-mI-continuous)if for every V ∈ σ, f −1(V ) is an α-mI-open set (resp, semi-mI-open set,

pre-mI-open set, β-mI-open set) of (X,M, I).

Definition 4.2. A function f : (X,M) −→ (Y, σ) is said to be α-m-continuous

(resp. semi-m-continuous, pre-m-continuous, β-m-continuous) if for every V ∈∈ σ, f −1(V ) is an α-m-open set (resp, semi-m-open set, pre-m-open set,

β-m-open set) of (X,M).

Theorem 4.3. If a function f : (X,M, I) −→ (Y, σ) is α-mI-continuous(resp. semi-mI-continuous, pre-mI-continuous, β-mI-continuous) then f isα-m-continuous(resp. semi-m-continuous, pre-m-continuous, β-m-continuous).

Proof. The proof is obvious.

Theorem 4.4. Let f : (X,M, I) −→ (Y, σ) be a function, then the followingstatements are equivalent:(i) f is α-mI-continuous;(ii) For each x ∈ X and each open set V ⊂ Y containing f (x), there exists

W ∈ αmIO(X ) such that x ∈ W , f (W ) ⊂ V ;(iii) The inverse image of each closed set in Y is α-mI-closed;(iv) m- cl(m- int∗(m- cl( f −1(B)))) ⊂ f −1(m- cl(B)) for each B ⊂ Y ;(v) f (m- cl(m- int∗(m- cl(A)))) ⊂ m- cl( f (A)) for each A ⊂ X .

Proof. (i)⇒ (ii). Let x ∈ X and V be any open set of Y containing f (x). SetW = f −1(V ), then by Denition 4.1, W is an α-mI-open set containing x and

f (W ) ⊂ V .


(ii)⇒ (iii). Let F be a closed set of Y . Set V = Y − F, then V is open in Y .Let x ∈ f −1(V ), by (ii), there exists an α-mI-open set W of X containing x such

that f (W ) ⊂ V . Thus, we obtain

x ∈ W ⊂ m- int(m- cl∗(m- int(W ))) ⊂

⊂ m- int(m- cl∗(m- int( f −1(V ))))

and hence

f −1(V ) ⊂ m- int(m- cl∗(m- int( f −1(V )))).This shows that f −1(V ) is α-mI-open in X . Hence f −1(F) = X − f −1(Y − F) == X − f −1(V ) is α-mI-closed in X .

(iii)⇒ (iv). Let B be any subset of Y . Since m- cl(B) is closed in Y , by (iii),f −1(m- cl(B)) is α-mI-closed and X − f −1(m- cl(B)) is α-mI-open. Thus

X − f −1(m- cl(B)) ⊂ m- int(m- cl∗(m- int(X − f −1(m- cl(B))))) =

= X − m- cl(m- int∗(m- cl( f −1(m- cl(B))))).

Hence we obtain

m- cl(m- int∗(m- cl( f −1(m- cl(B))))) ⊂ f −1(m- cl(B)).

(iv)⇒ (v). Let A be any subset of X . By (iv), we have

m- cl(m- int∗(m- cl(A))) ⊂

⊂ m- cl(m- int∗(m- cl( f −1( f (A))))) ⊂ f −1(m- cl( f (A)))

and hence f (m- cl(m- int∗(m- cl(A)))) ⊂ m- cl( f (A))(v)⇒ (i). Let V be any open set of Y . Then, by (v),

f (m- cl(m- int∗(m- cl( f −1(Y − V ))))) ⊂

⊂ m- cl( f ( f −1(Y − V ))) ⊂ m- cl(Y − V ) = Y − V .

Therefore, we have

m- cl(m- int∗(m- cl( f −1(Y − V )))) ⊂ f −1(Y − V ) ⊂ X − f −1(V ).

Consequently, we obtain that f −1(V ) ⊂ m- int(m- cl∗(m- int( f −1(V )))). Thisshows that f −1(V ) is α-mI-open. Thus, f is α-mI-continuous.

Theorem 4.5. A function f : (X,M, I) → (Y, σ) is α-mI-continuous if andonly if it is semi-mI-continuous and pre-mI-continuous.

Proof. This is an immediate consequence of Theorem 3.6.


References

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I-continuous functions, Kyungpook Math. J., 32 (1992), 2130.

[2] Á. Császár, Generalized topology, generalized continuity, Acta Math. Hungar.,

96 (2002), 351357.

[3] E. Hatir and T. Noiri, On decompositions of continuity via idealization, Acta

Math. Hungar., 96 (2002), 341349.

[4] E. Hayashi, Topologies dened by local properties, Math. Ann., 156 (1964),

205215.

[5] D. Jankovi¢ and T. R. Hamlett, Compatible extensions of ideals, Boll. Un. Mat.

Ital., B(7)6 (1992), 453-465.

[6] D. Jankovi¢ and T. R. Hamlett, New Topologies from old via Ideals, Amer.

Math. Monthly, 97 (1990), 295310.

[7] K. Kuratowski, Topology, Vol. I, Academic Press (New York, 1966).

[8] H. Maki, J. Umehara and T. Noiri, Every topological space is pre T1=2, Mem.

Fac. Sci. Kochi Univ. Ser. A Math., 17 (1996), 3342.

[9] O. Njastad, On some classes of nearly open sets, Pacic J. Math., 15 (1965),

961970.

[10] O. B. Ozbakiri and E. D. Yildirim, On some closed sets in ideal minimal spaces,

Acta Math. Hungar., 125 (2009), 227235.

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Galati, Ser. Mat. Fiz. Mec. Teor., 18 (2000), 3141.

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Acad. Sci., 20 (1945), 5161.

M. Parimala


Bannari Amman Institute of Technology

Sathyamangalam-638401

[email protected]

A. Selvakumar


SNS College of Engineering

Coimbatore [email protected]


FEEBLY INVO-CLEAN UNITAL RINGS

By

PETER V. DANCHEV

(Received June 3, 2017

Revised October 18, 2017)

Abstract. We dene the class of feebly invo-clean rings and give their compre-

hensive study. Our main result is the necessary and sucient condition that a ring Ris strongly feebly invo-clean if, and only if, the Jacobson radical J (R) is nil and R is

decomposable as R1 × R2 × R3, where R1 is a strongly feebly invo-clean ring which is

nil-clean with bounded index of nilpotence, R2 is a subdirect product of the elds Z3,and R3 is a subdirect product of the elds Z5.

1. Introduction and background

Everywhere in the text of the current paper, all our rings are assumed to

be associative, containing the identity element 1, which diers from the zero

element 0. Our terminology and notations are mainly in agreement with [5]. For

instance, J (R) stands for the Jacobson radical of R, Inv(R) for the multiplicative

set of all involutions in R, Nil(R) for the set of all nilpotents in R and Id(R) forthe set of all idempotents in R. All other specic things concerning our work aregiven in detail in the sequel.

Remember that a ring R is called invo-clean if every element can be

expressed as the sum of an involution and an idempotent, and weakly invo-clean

if every element can be expressed as the sum or the dierence of an involution

and an idempotent (see, e.g., [1] and [2], respectively). The major role of this

investigation, motivated by [1] and [2], is to enlarge in a natural way these two

notions to the following ones:

2000 Mathematics Subject Classication 16D60, 16S34, 16U60

86 PETER V. DANCHEV

Definition 1.1. We shall say that a ring R is feebly invo-clean if, for each r ∈ R,there are v ∈ Inv(R) and e, f ∈ Id(R) with e f = f e such that r = v + e − f .

Since it is easily checked that e − f = e(1 − f ) − f (1 − e) as well as thate(1 − f ) and f (1 − e) are orthogonal idempotents, we hereafter may without

loss of generality assume that e f = f e = 0.

Besides, it is self-evident that if 2 = 0, then e − f = e + f is always an

idempotent, so that feebly invo-clean rings are just invo-clean.

Definition 1.2. A ring R is said to be strongly feebly invo-clean if, for every

r ∈ R, there are v ∈ Inv(R) and e, f ∈ Id(R) with e f = f e = 0 such that

r = v + e − f and ve = ev.

Definition 1.3. A ring R is said to be super strongly feebly invo-clean if, for

any r ∈ R, there are v ∈ Inv(R) and e, f ∈ Id(R) with e f = f e = 0 such that

r = v + e − f and ve = ev, v f = f v.

Obvious commutative examples of such rings are Z2, Z3, Z4, Z5 and

Z6 ∼= Z2 × Z3, but simple calculations show that Z7 is not so.

Moreover, the direct product Z5 × Z5 is a feebly invo-clean ring that is not

weakly invo-clean. Even more, as established in [2], any weakly invo-clean ring

R with 5 ∈ J (R) is isomorphic to Z5. Here we state a little more conceptual

conrmation to this fact: Since 5 ∈ J (R) and 1 + J (R) ⊆ U (R), it followsthat 6 ∈ U (R), that is, 2 ∈ U (R) and 3 ∈ U (R). According to [2, Proposition

4.9], R has to be reduced and hence abelian. If now Id(R) = 0, 1, then [2,

Proposition 4.10] enables us that R ∼= Z5, and we are set. In the other case whenId(R) , 0, 1, one can decompose R = R1×R2, where in view of [2, Proposition

4.15] the direct factor R1 is weakly invo-clean and R2 is invo-clean. But it is

readily veried that 2 ∈ U (R2) and 3 ∈ U (R2). However, this contradicts [1,Lemma 2.2], because 23.3 , 0 in R2, which substantiates our claim.

The paper is organized as follows: Here we stated the fundamentals needed

for our successful presentation. In the next section, we present our basic results.

In the nal section we ask the validity of two problems of some interest and

importance.

FEEBLY INVO-CLEAN UNITAL RINGS 87

2. Main results

We begin in this section with the following two technicalities.

Lemma 2.1. If R is a feebly invo-clean ring, then 30 ∈ Nil(R) and, in particular,R ∼= R1×R2×R3, where R1, R2, R3 are feebly invo-clean rings with 2 ∈ Nil(R1),3 ∈ Nil(R2) and 5 ∈ Nil(R3).

Proof. Writing 3 = v + e − f , we deduce that (3 − v)2 = e + f and thus

(10 − 6v)2 = (e + f )2 = e + f = 10 − 6v. This implies that 114v = 126 and,

by squaring, we obtain that 2880 = 0. Multiplying this by 34.55, we get that

306 = 0, as asserted.

Furthermore, since (2, 3, 5) = 1, the nal part for decomposing R into the

direct product of three rings is standard, and so we omit the details. Finally, one

trivially sees that a homomorphic image and respectively a direct factor of a

feebly invo-clean ring is again feebly invo-clean.

Remark 2.2. It was proved in [2, Theorem 3.5] that if P is a ring of characteristic

2, then P is invo-clean if, and only if, P is nil-clean of index of nilpotence 2.

Thus the direct factor R1 in Lemma 2.1 can easily be described, because it is

obvious that the quotient R1/J (R1) is feebly invo-clean of characteristic 2 and

hence it is an invo-clean ring.

Proposition 2.3. The Jacobson radical J (R) of a strongly feebly invo-cleanring R is nil with index of nilpotence at most 8.

Proof. For an arbitrary z ∈ J (R) we write that z = v + e− f , where v ∈ Inv(R)and e, f ∈ Id(R) with ve = ev and e f = f e = 0. Since z − v = e − f is a

unit and (e − f )3 = e − f , it follows that (e − f )2 = e + f = 1 and, therefore,

z = v + (2e − 1). But v and 2e − 1 do commute, so that by squaring we have

z2 − 2 = 2v(2e − 1). A new squaring allows us to conclude that z4 − 4z2 = 0.

Replacing now z by 2z, we detect that 16z2(z2 − 1) = 0 whence 16z2 = 0

because z2 − 1 inverts. Thus multiplying the equality z4 − 4z2 = 0 by 4, we nd

4z4 = 0. Again replacing z by z2 in the equation z4 − 4z2 = 0, we nally derive

that z8 − 4z4 = z8 = 0, as required.

The next statement is pivotal.

Proposition 2.4. Strongly feebly invo-clean rings of characteristic 3 are re-duced.

88 PETER V. DANCHEV

Proof. For any q ∈ R with q2 = 0wewrite q = v+e− f , where v ∈ Inv(R) ande, f ∈ Id(R) satisfying ve = ev and e f = f e = 0. Furthermore, q − v = e − fforces that (q− v)3 = q− v which amounts to vqv − qvq = −q. Multiplying by qon the left and on the right, we deduce that vqvq = qvqv = 0. Multiplying also

by v on the left or on the right, it follows that vq + qv = 0.

On the other side, squaring q−v = e− f , we infer that 1 = 1−qv−vq = e+ f .Summarizing these two equalities, one concludes that 1+q−v = 2e = −e, whichgives that q = (v − 1) − e, where one checks that v − 1 ∈ Id(R). Consequently,we obtain that 0 = q3 = [(v − 1) − e]3 = (v − 1) − e = q, as required.

The last assertion can be extended like this:

Proposition 2.5. Let R be a strongly feebly invo-clean ring whose 2 is invertible.Then

(1) Nil(R) = J (R) = 0.(2) If Id(R) = 0, 1, then either R ∼= Z3 or R ∼= Z5.

Proof. (1) Since by virtue of Proposition 2.3 it must be that J (R) ⊆ Nil(R), itis enough to show only that R is reduced. To that aim, given q ∈ R with q2 = 0,

we write q = v + e − f where v2 = 1, ve = ev and e2 = e, f 2 = f , e f = f e = 0.

Same in the proof of Proposition 2.4, we infer that q = v + (2e − 1) = v + uis a sum of two involutions. We claim that q = 0 by using the trick: Squaring

q = v +u, it follows that 2vu = −2, i.e., vu = −1. Hence v = −u and so v +u = 0,

as needed.

As for (2), since all elements in R are of the type r = v or r = v + 1 or

r = v − 1, the proof goes on in the same manner as in [2, Proposition 4.10].

Mimicking [4], let us recall that a ring R is said to be nil-clean if each its

element is presentable as the sum of a nilpotent and an idempotent.

So, we are have now accumulated all the ingredients necessary to proceed

by proving the following.

Theorem 2.6. Suppose that R is a ring. Then R is strongly feebly invo-clean if,and only if, R ∼= R1× R2× R3, where R1 is a strongly feebly invo-fine ring whichis nil-clean with bounded index of nilpotence, R2 can be embedded in

∏ Z3

and R3 (is a commutative feebly invo-clean ring which) can be embedded in∏ Z5, where λ, µ are some ordinals.

Proof. ⇒. With Lemma 2.1 at hand, one writes that R ∼= R1×R2×R3, where

R1, R2, R3 are strongly feebly invo-clean rings with 2 ∈ Nil(R1), 3 ∈ Nil(R2)and 5 ∈ Nil(R3).


As for R1, owing to Remark 2.2 accomplished with Proposition 2.3, one

concludes that R1 is a nil-clean ring (see, e.g., [4]) whose index of nilpotence

has to be bounded.

As for R2 and R3, we have that 2 is an invertible element, and so Proposi-

tion 2.5 works to get that they are semiprimitive reduced rings of characteristics

3 and 5, respectively. This yields at once that R2 and R3 are both abelian.

However, all elements in R2 and R3 are expressible only by idempotents. In-

deed, if r = v + e − f is the standard record utilized above and 3 = 0, then

r = (v − 1) + e + (1 − f ) is a sum of three idempotents. If now 5 = 0, then

3v +3 is always an idempotent whenever v is an involution and so 2r = v + e− fensures that r = (3v + 3) + 3e − 3 − 3 f is also a function of idempotents only,

bearing in mind 2.3 = 1. This sustains our assertion. That is why, both rings R2

and R3 are commutative of characteristics 3 and 5, respectively. Furthermore, a

well-known trick applies to obtain that R2∼= R2/J (R2) can be embedded in the

direct product∏M ∈Max(R2) (R2/M), where M is a maximal ideal of R2 and hence

R2/M is a eld of characteristic 3 that is simultaneously a feebly invo-clean ring.

Thus R2/M ∼= Z3, and we are done.Similarly can be processed R3 to get that R3/M ∼= Z5 and so the wanted

claim.

⇐. According to either [1] or [2], we deduce that R2 is a commutative

invo-clean ring and thus it is itself feebly invo-clean. Since the direct product of

strongly feebly invo-clean rings is again a strongly feebly invo-clean ring, what

remains to apply is that, by default, R3 is feebly invo-clean.

It easily follows that an arbitrary direct product∏Z5 is always feebly

invo-clean. In fact, to prove feebly nil-cleanness, one observes that each element

in Z5 is an idempotent (for instance, 0, 1) or a unit (for instance, 1, 2, 3, 4) as 1, 4are involutions. Moreover, every unit is the sum or the dierence of an involution

and an idempotent, say 1 = 1 + 0 = 1− 0, 2 = 1 + 1, 3 = 4− 1, 4 = 4 + 0 = 4− 0.

Likewise, 0 = 1 − 1 = 4 + 1. These presentations of all of the elements in Z5unambiguously show that each element of this subdirect product is presentable

in the wanted form.

However,whether or not the subdirect product of theZ5's is feebly invo-cleanis still in question; let us however notice that a ring is a subdirect product of

copies of the eld Z5 only when 5 = 0 and each element satises the equation

x5 = x (see, e.g., [5]).

The next consequence follows now directly from the preceding key theorem.

Corollary 2.7. Strongly feebly invo-clean rings are clean.

90 PETER V. DANCHEV

By the same token super strongly feebly clean rings could be described, too.

We now can state a common generalization of a major result from [3].

First, recall that a ring R is said to be there a feebly nil-clean ring if, for each

r ∈ R, there are q ∈ Nil(R) and e, f ∈ Id(R) with e f = f e = 0 such that

r = q+ e− f . It was established in [3] that if R is a ring in which either 2 ∈ J (R)or 3 ∈ J (R), then R is feebly nil-clean if, and only if, J (R) is nil and R/J (R) isfeebly nil-clean. However, these two restrictions on the elements 2 and 3 could

be dropped o. Specically, the following statement is true:

Theorem 2.8. A ring R is feebly nil-clean if, and only if, J (R) is nil and R/J (R)is feebly nil-clean.

Proof. The left-to-right implication follows in the same manner as in [3].

As for the right-to-left implication, suppose that R/J (R) is feebly nil-cleanand J (R) is nil. Take r in R. In the factor ring R/J (R), there is a decomposition

(∗) r + J (R) = χ + π1 − π2

where π1, π2 are orthogonal idempotents in the quotient R/J (R) and χ is a

nilpotent in the quotient R/J (R). Now we use [6, Proposition 7] to get that J (R)is strongly lifting. Hence, by [6, Proposition 11(1)], orthogonal idempotents

π1, π2 lift to orthogonal idempotents e1, e2 in R (even more, they lift strongly).

Therefore, formula (∗) becomes

r − e1 + e2 + J (R) = χ,

resulting in (r − e1 + e2)n ∈ J (R) for some n, whence r − e1 + e2 is a nilpotent,as required.

It is worth noting that, according to results from [3] in connection with the

last criterion in Theorem 2.8, it follows as a valuable consequence that the center

of a feebly nil-clean ring is again feebly nil-clean.

3. Left-open problems

In closing we pose the following two questions:

Problem3.1. Characterize the isomorphism structure of feebly invo-clean rings.

Problem 3.2. Characterize the isomorphism structure of feebly nil-clean rings.


We end the work with the following technical corrigendum.

Correction. In [2, Theorem 3.5] the letter J (R1) should be written as

Nil(R1) as well as in [2, Theorem 4.18] the letter J (R′′) should be written

as Nil(R′′).

References

[1] P. V. Danchev, Invo-clean unital rings, Commun. Korean Math. Soc., 32 (2017),

1927.

[2] P. V. Danchev, Weakly invo-clean unital rings, Afr. Mat., 28 (2017), 12851295.

[3] P. V. Danchev, Feebly nil-clean unital rings, Proc. Jangjeon Math. Soc., in press.

[4] A. J. Diesl, Nil clean rings, J. Algebra, 383 (2013), 197211.

[5] T. Y. Lam, A First Course in Noncommutative Rings, Second Edition, Graduate

Texts in Math., Vol. 131, Springer-Verlag, Berlin-Heidelberg-New York, 2001.

[6] W. K. Nicholson and Y. Zhou, Strong lifting, J. Algebra, 285 (2005), 795818.

Peter V. Danchev

Department of Mathematics and Informatics

University of Plovdiv

Bulgaria

[email protected]


A GENERALIZATION OF HOPFIAN ABELIAN GROUPS

By

ANDREY CHEKHLOV AND PETER DANCHEV

(Received July 1, 2017)

Abstract. We dene the concept of R-Hopan abelian groups, which is a non-

trivial generalization of the classical Hopan groups. A systematic study of these groups

is given in some dierent ways.

1. Introduction and background

All groups into consideration, unless specied something else, are assumed

to be abelian. Almost all notions and notations are classical as the unexplained

explicitly ones follow those from [10] and [11]. For instance, G[pn] = g ∈∈ G : png = 0 is the pn-socle of G, and pnG = png : g ∈ G is the n-thpower subgroup of G, where n ∈ N.

Recall that a group G is said to be Hopan if each epimorphism G → G is

an automorphism. Also, it is well known that a group is Hopan if and only if it

does not have proper isomorphic quotient groups to itself.

The rst known in the literature study of Hopan groups is that in [2]

(see [15], too) as there certain examples are provided.

Some other obvious examples of such groups are these (compare also

with [12, Example 2.4] in the case of unbounded p-groups):

• Finitely generated (in particular, nite) groups.

• All torsion-free groups of nite rank.

• Every group whose non-zero endomorphisms are monomorphisms.

• Every group G with endomorphism ring E(G) ∼= Z; in particular, the groupof integers Z.

2000 Mathematics Subject Classication 20K10, 20K20, 20K21

94 ANDREY CHEKHLOV, PETER DANCHEV

It was established in [15] that a reduced Hopan p-group G has cardinality

|G | ≤ 2ℵ0 . Even more, in [12] and [4] was proven that there is a reduced Hopan

p-group of cardinality ℵ0 < |G | ≤ 2ℵ0 with nite Ulm-Kaplansky invariants

fk (G) for k ∈ N, because Hopan groups which are direct sums of cyclic

p-groups must be nite.

An interesting way to generalize Hopcity is to consider a group G such

that for any surjective endomorphism f : G → G there exists an injective

endomorphism φ : G → Gwhose composition f φ is an automorphism.However,

we may simplify this a bit more as follows:

Definition 1.1. AgroupG is called R-Hopan if each surjective endomorphism

of G is right invertible, that is, for each surjection f : G → G there is an

endomorphism ϕ : G → G with f ϕ = 1.

It easily follows that ϕ is an injection.

To keep a record straight, notice that Denition 1.1 was suggested to the

second named author by B. Goldsmith in a private communication; in fact,

his suggestion was motivated by the original proposal stated above before

Denition 1.1, which is due to the second named author, for generalizing

Hopcity.

Moreover, one observes that Hopan groups are always R-Hopan. Forthe converse, recall that a ring is called Dedekind nite (or, in other terms,

directly nite) if for each two elements a, b of the ring, the condition ab = 1

implies ba = 1. Thus, it is obviously true that if E(G) is Dedekind nite (in

particular, Engel (cf. [6]) or commutative), then the R-Hopan property forces

the Hopan one (compare with Proposition 1.6 below). The same holds when Gis indecomposable as well (compare with Corollary 1.7 below).

However, the two notions of R-Hopcity and Hopcity dier one to other asthe following folklore construction demonstrates (again suggested by B. Gold-

smith to the second named author).

Example 1.2. There is an R-Hopan group which is not Hopan.

Proof. Let V be a vector space. If f is a surjection of V , then one can write

V = T ⊕ ker( f ) and T ∼= V/ ker( f ) ∼= Im( f ) = V . So, there is an isomorphism

g : T → V , where g is the restriction of f . Hence, there exists an inverse

isomorphism φ : V → T and hence gφ(y) = y for all y ∈ V , so that f φ is the

identity map on V , whence it is an automorphism of V . Furthermore, since φ is

injective, the R-Hopan property holds for any f . However, if V is of innite

dimension, then it is certainly not Hopan.

GENERALIZED HOPFIAN ABELIAN GROUPS 95

Similarly, we may also dene

Definition 1.3. AgroupG is called L-Hopan if each surjective endomorphism

of G is left invertible, that is, for each surjection f : G → G there is an

endomorphism ψ : G → G with ψ f = 1.

One trivially sees that L-Hopan = Hopan.

The proof of Example 1.2 suggests that a group G is R-Hopan if, and onlyif, for any surjection f : G → G, the kernel ker( f ) is a direct summand of G,

that is, any surjection of G splits.

Specically, one may state the following improvement:

Theorem 1.4. A group G is R-Hopfian if, and only if, for any subgroup H ofG, the isomorphism G ∼= G/H implies that H is a direct summand of G.

Proof. Firstly, let G be R-Hopan and, suppose that G/H ∼= G for some

arbitrary subgroup H ≤ G. So, there is an epimorphism f ofG with ker( f ) = H .

Since f ϕ = 1 for some ϕ ∈ E(G), the element ϕ f is an idempotent of E(G) withker(ϕ f ) = ker( f ) = H . Therefore, H is a direct summand of G, as required.

Secondly, for any subgroup H of G assume that G/H being isomorphic to

G will imply that H is a direct summand of G. If now f is an epimorphism of G,

then G/H ∼= G, where H = ker( f ). But, by assumption, H is a direct summand

of G and thus we may repeat the same trick as in Example 1.2.

Resulted, R-Hopcity appears to be a special case of the well-known

condition: If G/H is isomorphic to a direct summand of G, then H is a direct

summand of G.

In other words, if a group G is R-Hopan but non Hopan, then it has a

proper (, G) isomorphic direct summand but, however, the converse fails. In fact,

if f is a surjective endomorphism with f ϕ = 1, then, because G is not Hopan,

ϕ f , 1 will be an idempotent with ker(ϕ f ) = ker( f ) , 0, as required. That the

converse implication is untrue can be illustrated by every divisible p-group of

innite rank, which group possesses such a proper direct summand. However,

manifestly, this group is not R-Hopan in conjunction with Example 2.3 below,

which gives the claim.

Corollary 1.5. A p-group with finite Ulm-Kaplansky invariants is R-Hopfianif, and only if, it is Hopfian.

Proof. Assuming that G/H ∼= G for an arbitrary subgroup H of G, we may

write with the help of Theorem 1.4 that H is a direct summand of G with


complement isomorphic to G. But since G has nite Ulm-Kaplansky invariants,

it follows that H = 0, i.e., G is Hopan, as asserted.

The following statement shows to what extent R-Hopcity and Hopcity

dier each to other.

Proposition 1.6. Suppose G is an R-Hopfian group. Then the following threeitems are equivalent:

(i) G is not Hopfian;(ii) E(G) is not Dedekind finite;(iii) E(G) contains an epimorphism which is left zero-divisor.

Proof. The equivalence (i) ⇐⇒ (ii) is self-evident. To prove (i) ⇒ (iii), we

observe that if G is non Hopan, then it has a surjection f with f ϕ = 1 for some

ϕ ∈ E(G). But ϕ f , 1, which is a non-zero idempotent of the ring E(G), andso f (ϕ f − 1) = 0, as needed. Conversely, to show (iii)⇒ (ii), given a surjective

endomorphism f of the R-Hopan groupG, we have f ϕ = 1 for some ϕ ∈ E(G).However, f δ = 0 for some 0 , δ ∈ E(G) whence ϕ f , 1, as required.

We close this section with the following consequence of interest.

Corollary 1.7. Each indecomposable R-Hopfian group is Hopfian.

Proof. For any epimorphism f of G we have f ϕ = 1 for some ϕ ∈ E(G).Since G is indecomposable, it must be that ϕ f = 1 because this element is an

idempotent. Thus f is an automorphism, as desired.

In this connection, in [3] were discovered groups possessing proper isomor-

phic direct summands.Moreover, in [7] were examined certain generalizations of

those groups which do not have proper pure subgroups isomorphic themselves.

Finally, in [8] were explored those groups having all (proper) fully invariant

subgroups isomorphic.

So, the purpose of the present paper is to give a comprehensive investigation

of the discussed subject by nding suitable results, thus showing the complicated

structure of R-Hopan groups in various dierent aspects.

2. Non-trivial examples

Here we shall demonstrate certain constructions of R-Hopcity.


• Every free group is R-Hopan.

In fact, for such a group G, if the rank r0(G) of G is nite, then, as showed

above,G being torsion-free has to beHopan and hence R-Hopan. If now r0(G)is innite and f ∈ E(G) is surjective, then ker( f ) must be a direct summand of

G whence G/ ker( f ) ∼= G. We, furthermore, apply the same method of proof as

that in Example 1.2.

• Every divisible torsion-free group is R-Hopan.

This follows in the same manner as in the case of free groups.

• Any direct sum of cyclic groups of the same order pn for some n ∈ N and

prime p is R-Hopan.

In fact, assume that G/H ∼= G for some subgroup H . Writing G/H ==

⊕i∈I〈xi〉 with o(xi ) = pn , one can infer that

(⊕i∈I〈xi〉

)∩ H = 0 and

G =(⊕

i∈I〈xi〉

)⊕ H , where xi = xi + H . Henceforth, we may repeat the same

idea as that in Example 1.2.

• Letting Bn be the direct sum of cyclic groups with the same rank of order

pn , then G =⊕

nBn is not R-Hopan.

Indeed, G/G[p] ∼= pG ∼= G and G[p] is fully invariant in G.

Lemma 2.1. Each R-Hopfian group does not contain non-zero fully invariantsubgroups in the kernels of its epimorphisms.

Proof. Assume that G is an R-Hopan group with a fully invariant subgroup

B , 0 of ker( f ) where f is an epimorphism of G. Hence B ⊆ ker( f ϕ) and thusthe equality f ϕ = 1 is impossible for every ϕ ∈ E(G). This contradiction givesthe claim.

Another example of non-R-Hopan group is the following one:

Example 2.2. Let B be a group such that Hom(B,Z) = 0, and let C be a free

group of innite rank r (C) ≥ |B |. Then A = B ⊕ C is not an R-Hopan group.

Proof. Decomposing C = C1 ⊕ C2 with r (C1) = r (C2) (and hence they are

equal to r (C) and |C1 | = |C2 |), then there exist an epimorphism f : C1 → Band an isomorphism ϕ : C2 → C. Setting f | C2 = 0, f | B = 0 and ϕ | C1 = 0,

ϕ | B = 0, we may assume that f , ϕ ∈ E(A). Therefore, f +ϕ is an epimorphism

of A with B ⊆ ker( f + ϕ). Hereafter, we may apply Lemma 2.1 to get the

assertion.


In addition, in contrast to one of the bullets above, it follows that not each

direct sum of cyclic groups is R-Hopan (compare with Corollary 3.12 below).

Example 2.3. The quasi-cyclic group Z(p∞) is not R-Hopan. In particular,

any divisible p-group is not R-Hopan.

Proof. If we have a homomorphism f : G → K for some group K such that

ker( f ) is essential in G, then for every map g : L → G, where L is a group, the

composition f g is not monic.

Since by [10] or [11] each divisible p-group is a direct sum of isomorphic

copies of the quasi-cyclic group Z(p∞), the second part follows appealing to

Theorem 3.4 below.

Notice that, however, the group Z(p∞) has a commutative and hence

Dedekind nite endomorphism ring.

Example 2.4. Every reduced torsion-free algebraically compact group is R-Hopan.

Proof. According to [10], such a group G can be presented as G =∏

p Gp ,

where Gp is the p-adic algebraically compact component of G which is fully

invariant in G. Therefore, owing to Corollary 3.8 below, it is enough to show

that all Gp are R-Hopan. To this goal, if the p-rank of Gp is nite, then all its

epimorphisms are monomorphisms. Otherwise, if the p-rank of Gp is innite,

then ker( f ) is always a direct summand of G whenever f is its epimorphism,

whence G/ ker( f ) ∼= G. The further arguments go on as in Example 1.2 above

to conclude the claim.

Referring to Example 3.3 below, the condition reduced in Example 2.4 is

essential.

In contrast to [13] and [14], we deduce the following:

Example 2.5. If C is a free group of innite rank and F is a nite group, then

C ⊕ F is not R-Hopan.

Proof. There exists an epimorphism f : C → C ⊕ F such that f can also be

considered as an epimorphism of the group C ⊕ F with F ⊆ ker( f ). But thiscontradicts Lemma 2.1 since F is fully invariant in C ⊕ F.

It is worth noticing that if however r0(C) is nite, then C ⊕ F is necessarily

Hopan.


Example 2.6. There exists an R-Hopan group G such that G ⊕G is R-Hopanbut G ⊕ G is not Hopan.

Proof. If G is a free group of innite rank r0(G), then G ⊕ G is also free and

thus, by what we have shown above, G ⊕ G is R-Hopan. However, this squareis not Hopan because of the innity of the rank r0(G).

3. Main results

Subgroups of R-Hopan groups are, in general, not again R-Hopan.However, one may derive:

Proposition 3.1. If G is R-Hopfian, then nG is R-Hopfian for any n ∈ N.

Proof. If f : nG → nG is an epimorphism of nG, then in view of [10, Propo-

sition 113.3] there exists an epimorphism φ of G whose restriction φ | nG = f .Since φϕ = 1G , we conclude that (φ | nG)(ϕ | nG) = 1nG , as required.

Remark 1. The converse does not hold as the following construction shows:

Let C be a free group of innite rank r (C) ≥ |B |, where B is a bounded group,

say nB = 0. An appeal to Example 2.2 gives that the direct sum C ⊕ B is not

R-Hopan, while n(C ⊕ B) = nC remains R-Hopan.

The following is one of our chief results.

Theorem 3.2. If A = B ⊕ C is a group, where B is an R-Hopfian subgroupwhich is fully invariant in A and C is a Hopfian subgroup, then A is R-Hopfian.

Proof. Let f be an epimorphism of the group A, and π, θ the corresponding

projections on B and C, respectively. Clearly, θ f θ is an epimorphism of C and

hence by assumption it has to be its isomorphism, putting u = (θ f θ)−1. We shall

show that π f π is an epimorphism of B. To that aim, for each b ∈ B there is

x = y + z with y ∈ B and z ∈ C such that

b = f (y + z) = (π f π + π f θ + θ f θ)(y + z) = π f π(y) + π f θ(z) + θ f θ(z),

taking into account that θ f π = 0 and f = π f π + π f θ + θ f θ. Since θ f θ(z) ∈ C,

it follows that z = 0 and b = π f π(y). Whence (π f π)g = π for some g ∈ E(B),


because π | B = 1B. Letting g | C = 0 and u | B = 0, we obtain

(π f π + π f θ + θ f θ)(g − g(π f θ)u + u) == (π f π)g − (π f π)g(π f θ)u + (π f θ)u + (θ f θ)u,

where we bear in mind that (π f π)u = 0, (π f θ)g = 0 and (θ f θ)g = 0.

Furthermore, since both (π f π)g = π and ((π f θ)u)A ⊆ B hold, it must

be that (π f π)g(π f θ)u = (π f θ)u. Thus, using that (θ f θ)u = θ, we deduce

f (g − g(π f θ)u + u) = 1A, because π + θ = 1A.

The following construction shows that the requirement on Hopcity in the

last theorem is really essential (compare also with one of the bullets above).

Example 3.3. The direct sum of two R-Hopan groups is not R-Hopan.

Proof. Suppose C is such a reduced group that there exists an epimorphism of

C in the divisible group D , 0 with C ⊕ C ∼= C; for instance, C can be chosen

to be a free group of rank r (C) ≥ |D |. Same as Example 2.2 above, the group

D ⊕ C ⊕ C ∼= D ⊕ C is not R-Hopan, as desired.

We now proceed by proving the direct summand question.

Theorem 3.4. A direct summand of an R-Hopfian group is R-Hopfian.

Proof. Write A = B ⊕ C, where A is R-Hopan. Let π and θ be projections ofthe group A on the summands B andC, respectively, and let f be an epimorphism

of the group B. Therefore, f π+θ is an epimorphism of A and thus ( f π+θ)g = 1Afor some g ∈ E(A). Observing that ( f π + θ)g = ( f π + θ)(π + θ)g = f πg + θg,we then obtain f πgπ + θgπ = π. Since f πgπ(B) ⊆ B and π(B) ⊆ B, we haveθgπ = 0, i.e., f (πgπ) = π = 1B, where πgπ ∈ E(B), as required.

As consequences, we obtain:

Corollary 3.5. If G is R-Hopfian, then G/dG is R-Hopfian.

Proof. Observe thatG/dG ∼= rG, where the latter is a direct summand ofG.

Corollary 3.6. A divisible group is R-Hopfian if, and only if, it is torsion-free.

Proof. As quoted above, any divisible torsion-free group is R-Hopan. So,assume that D is a divisible R-Hopan group. According to Theorem 3.4, each

p-component Dp is also R-Hopan as being a direct summand of D. Moreover,

Dp/Dp[p] ∼= Dp and the socle Dp[p] is fully invariant in Dp . Consequently,

Dp[p] = 0 which guarantees that Dp = 0, as expected.


Corollary 3.7. If the group A = (dA) ⊕ (r A) is R-Hopfian, then its divisibleand reduced parts dA and r A are R-Hopfian. Conversely, if dA is an R-Hopfiangroup and r A is a Hopfian group, then A is R-Hopfian.

Proof. Follows immediately by a simple combination of Theorems 3.2 and 3.4.

Corollary 3.8. Suppose A =∏

i∈I Ai is a group, where all subgroups Ai arefully invariant in A. Then A is R-Hopfian if, and only if, all Ai are R-Hopfian.

The following statement is pivotal.

Lemma 3.9. Suppose that G is a direct sum of cyclic p-groups. Then G isR-Hopfian if, and only if, G is presented as G = A ⊕ B, where A is a direct sumof cyclic groups of the same order pn for some n ∈ N, and either B = 0 or B isa finite group with orders of the generators of its cyclic direct summands > pn .

Proof. Necessity. Writing G =⊕

k ∈NGnk , where Gnk is a non-zero direct

sum of cyclic groups of order pnk with n1 < n2 < . . . < nk < . . .. ThusG contains a direct summand isomorphic to the group H =

⊕k ∈N Z(pnk ).

We, therefore, have that H/K ∼= H for some subgroup K ≤ H containing

H[p], which contradicts Lemma 2.1. Consequently, G has to be bounded, say

G =⊕m

k=1 Gnk . Assume for a moment that some Gns is innite, where s > 1,

and suppose C = 〈x〉 , 0 is a cyclic direct summand of Gns−1 . If now f is an

epimorphism of the group C ⊕ Gns satisfying the condition C ⊆ ker( f ), thenone observes that f ϕ(x) ∈ p(C ⊕ Gns ) for each ϕ ∈ E(C ⊕ Gns ), whence thevalidity of the equality f ϕ = 1 is impossible. This substantiates our assertion.

Suciency. Let f be an epimorphism of the group G = A⊕ B. We assert

that its kernel F = ker( f ) is a pure subgroup of G. In fact, if ptF , F ∩ ptG for

some maximal natural t, then pt y = x, where y ∈ G and x ∈ F \ ptF. Claim that

f (y) < pG. To show this, if assume in a way of contradiction that f (y) = pr z,for some r ≥ 1, then y − pr z1 = x1 ∈ F; f (z1) = z. So, x − pt+r z1 = pt x1.Since t is maximal, we have x − pt x1 = pt+r z1 = pt+r x2, where x2 ∈ F. Hencex = pt (x1 + pr x2) and x1 + pr x2 ∈ F, a contradiction which gives our claim that

f (y) < pG. Since pt f (y) = 0, then t ≥ n and, resultantly, x ∈ B. But then, sinceB is nite with orders > pn of the generators of its cyclic direct summands, we

deduce that f (G) cannot contain a (proper) subgroup isomorphic to B. SinceG/F is a direct sum of cyclic groups, by [10] or [11] the pure subgroup Fseparates in G as a direct summand. Henceforth, we may apply the technique

described in Example 1.2 to get our assertion after all.


As an immediate consequence, we derive:

Corollary 3.10. If an R-Hopfian group is a direct sum of cyclic p-groups, thenit is bounded. In particular, basic subgroups of R-Hopfian p-groups need not beR-Hopfian.

Proof. The rst part-half follows directly from Lemma 3.9. As for the second

one, since there is an unbounded reduced R-Hopan p-group, its basic subgroupsmust be unbounded as well, and thus they are certainly not R-Hopan.

The next assertion gives a suitable representation of an unbounded reduced

R-Hopan p-group in the following manner:

Theorem 3.11. Any unbounded reduced R-Hopfian p-group G can be presentedas G = A ⊕ B, where A is a direct sum of cyclic groups of the same order pn

for some n ∈ N, and B is a Hopfian group with finite Ulm-Kaplansky invariantsfk (B), k ∈ N, as f0(B) = . . . = fn−1(B) = 0. In particular, |B | ≤ 2ℵ0 and theorders of all cyclic direct summands of B are > pn .

Proof. It follows by a direct combination of Lemma 3.9 and Theorem 3.4

that such a decomposition exists with A as given in the text and an R-Hopansubgroup B. In order to prove that B is necessarily Hopan, we assume the

contrary and observe that as being R-Hopan the group B is isomorphic to its

proper direct summand, but this is impossible because of the niteness of its

Ulm-Kaplansky invariants.

In case of arbitrary direct sums of cyclic groups, we state:

Corollary 3.12. A direct sum of cyclic groups is R-Hopfian if, and only if, itis either free or is mixed whose torsion-free part is of finite rank and the torsionpart is R-Hopfian.

Proof. Necessity. In virtue of Theorem 3.4 a direct summand of an R-Hopan group is again R-Hopan, so that if it is not free, then it must be mixed

and hence Example 2.2 applies to get that its torsion-free rank is nite.

Suciency. It follows from Theorem 3.2 accomplished with the fact that

free groups of nite rank are Hopan.

Remark 2. Corollary 1.5 can also be viewed as a consequence of Theorem 3.11.

In fact, every bounded p-group with nite Ulm-Kaplansky invariants has to be

nite and thus Hopan. If it is, however, unbounded, then the direct summand

A in Theorem 3.11 must be nite, and since the direct sum of a nite group and

a Hopan group is also Hopan, we are set.


We are now ready to enlarge substantially Example 1.2 alluded to above as

follows:

Theorem 3.13. For any prime p there is an R-Hopfian p-group which is notHopfian.

Proof. Suppose that G is the R-Hopan group as in Theorem 3.11, and that f

is its arbitrary epimorphism. Write f =(α βγ δ

), where α ∈ E(A), δ ∈ E(B),

β ∈ Hom(B, A) and γ ∈ Hom(A, B).Furthermore, let A have innite rank and let B be such an unbounded

Hopan p-group that f0(B) = · · · = f2n−1(B) = 0 (as indicated in [12] and [4]

such a group B exists).

Next, choose an endomorphism ϕ =

(x y

z t

)of the group G with f ϕ = 1.

One sees that βz = 0; in fact, z(A) ⊆ pnB and β(pnB) ⊆ pn A = 0. Also, δis an epimorphism of B. In fact, γ(A) + δ(B) = B. Since pn A = 0, we have

δ(pnB) = pnB, that is, δ | pnB is an epimorphism of the group pnB. Butγ(A) ⊆ pnB and hence δ(B) = B, as required. Therefore, δ is an automorphism

because B is Hopan.

After that, choose x to be an endomorphism of the R-Hopan group A with

αx = 1, z = −δ−1γx and y = −x βt. What remains to show is that t ∈ E(B) canbe chosen such that γy + δt = −γx βt + δt = (−γx βδ−1 + 1)δt = 1. To that goal,

such a choice is really possible since γ(A) ⊆ pnB, and because pn A = 0 it must

be that (γx βδ−1)2 = 0. Consequently, (−γx βδ−1 + 1)δ is an automorphism of

the group B, so that t can be taken as its invertible.

The next statement demonstrates that fully invariant subgroups and factor-

groups modulo fully invariant subgroups of (unbounded) R-Hopan groups

generally need not be even R-Hopan.

Proposition 3.14. If G is an unbounded R-Hopfian p-group, then there existsa natural number n such that both G[pn] and G/pnG are not R-Hopfian.

Proof. For each positive integer k writeG = Bt1 ⊕ · · ·⊕Btk ⊕Gtk , where Btk is

a non-zero direct sum of cyclic groups of order ptk with t1 < t2 < . . . < tk < . . ..If n ≥ t2, we have G[pn] = Bt1 ⊕ (G[pn]∩Gt1 ), where the subgroup G[pn]∩Gt1

must be innite and the orders of the generators of its cyclic direct summands

are > pt1 . Utilizing Lemma 3.9 such G[pn] is not R-Hopan, as claimed.

Moreover, one sees that G/pnG ∼= Bt1 ⊕ (Gt1/pnG), where the factor-groupGt1/pnG is innite and again the orders of the generators of its cyclic direct


summands are > pt1 . Therefore, Lemma 3.9 applies again to get that G/pnG is

not R-Hopan, as asserted.

Reciprocally, if K is a fully invariant subgroup of a group G such that both

K and G/K are R-Hopan, then G is not necessarily R-Hopan. In fact, in

notations of Remark 1 above, we can take G = B ⊕ C and K = B = G[n] forsome integer n ≥ 1, so that G/G[n] ∼= nG = nC ∼= C being R-Hopan does notimply that G is R-Hopan.

We will now exhibit an important property which is always satised by

Hopan groups but not generally by R-Hopan groups, which once again showsthe discrepancy of these two group classes.

Proposition 3.15. A reduced p-group G satisfies the property that G = H ⊕ Vwith G ∼= H imply V = 0 if, and only if, all its Ulm-Kaplansky invariantsfk (G) are finite for k ∈ N.

Proof. Necessity. Assume in a way of contradiction that fn (G) is innite forsome n ≥ 1. HenceG = Bn+1⊕C, where Bn+1 is with rank fn (G) a direct sum of

cyclic groups of order pn+1. Therefore, Bn+1 = B′n+1⊕ B′′

n+1, where B′

n+1∼= B′′

n+1

and G = B′n+1⊕ (B′′

n+1⊕ C) with G ∼= B′′

n+1⊕ C, which contradicts the stated

property because B′n+1, 0.

Suciency. It is straightforward since G = H ⊕ V implies with the aid

of [10] that fk (G) = fk (H) + fk (V ) and thus G ∼= H assures that fk (G) == fk (H), i.e., fk (V ) = 0 which, in turn, gives that V = 0, as asserted.

Since for Hopan groups such Ulm-Kaplansky invariants are nite, these

groups have this decomposition property. However, in general, R-Hopan groupsneed not satisfy that, because in view of Theorem 3.11 the Ulm-Kaplansy

invariant fn−1(G) may be innite for some natural n > 2.

The last proposition can be extended to arbitrary groups with that property

like this (compare with Proposition 1.6):

Proposition 3.16. A group G satisfies the condition G = H ⊕ V with G ∼= Himplying V = 0 if, and only if, the ring E(G) is Dedekind finite.

Proof. The claim follows bearing in mind that the decompositionG = H ⊕V ∼=∼= H holds exactly when f h = 1 for some endomorphisms f , h of G. We further

can process as in Proposition 1.6.

As immediate consequences, we yield the following facts for groups satis-

fying Proposition 3.16:


(a) A reduced p-group is with Dedekind nite endomorphism ring if, and only

if, for every positive integer k the k-th Ulm Kaplansky invariants are nite

(cf. [5]).

(b) A divisible group is with Dedekind nite endomorphism ring if, and only

if, it has a nite torsion-free rank and nite p-rank for each prime p (cf. [5]).(c) A completely decomposable torsion-free group is with Dedekind nite

endomorphism ring if, and only if, its divisible part and its homogeneous

components have nite rank.

(d) An algebraically compact group is with Dedekind nite endomorphism ring

if, and only if, its divisible part and its p-components have nite rank.

The following result is major.

Theorem 3.17. Let G =⊕

t ∈ΩGt be a completely decomposable torsion-free

group, where Gt are its homogeneous components of the same type t, and letthe set Ω satisfy the minimality condition. Then G is R-Hopfian if, and only if,for each t ∈ Ω, if t is not maximal in Ω, then the rank of Gt is finite.

Proof. Necessity. Assume by a way of contradiction that the rank of Gt is

innite and that t < τ for some τ ∈ Ω, as take inG a rank one direct summand A.Since Gt has an innite rank, there is an epimorphism f : Gt → Gt ⊕ A. Settingf | A = 0, one sees that f ∈ E(Gt ⊕ A). In conjunction with Theorem 3.4,

the direct summand Gt ⊕ A must also be R-Hopan. However, this contradictsLemma 2.1, as expected.

Suciency. First of all, observe that any homogeneous completely de-

composable group has to be R-Hopan. Indeed, if its rank is nite, then it is

of necessity a Hopan group. If, otherwise, the rank is innite and f is an

epimorphism of the group, then ker( f ) separates as its direct summand (see,

e.g., [10, Proposition 86.5]), so that the proof of Example 1.2 works hereafter.

Furthermore, writeG = B⊕C, where B is the direct sum of allGt of innite

rank. The condition onΩ yields that each such Gt is fully invariant in G, whence

B is also so in G. Therefore, B is R-Hopan as the direct sum of fully invariant

R-Hopan groups (see Corollary 3.8, too).The other direct summand C is a completely decomposable group with a

nite number of homogeneous components, and the set of all types of its rank one

direct summands satises the minimality condition as well. Consequently, [17,

Theorem 1] implies thatC is Hopan. We nally apply Theorem 3.2 to conclude

that G is R-Hopan, as claimed.

Remark 3. Note that the limitation on minimality of the set of all types of the

components is essential and cannot be removed. In fact, in [17] was constructed a


non-Hopan completely decomposable torsion-free group Awhose components

of the same kind have nite rank, and the set of all types of its rank 1 summands

does not satisfy the minimality condition. This group A is even not R-Hopan. Infact, it has an epimorphism which is contained in the kernel of a fully invariant

subgroup of A. Hence, by Lemma 2.1, the group A need not be R-Hopan,indeed, as asserted.

Notice also that pure subgroups of nite rank of homogeneous completely

decomposable groups (eventually of idempotent type) are always direct sum-

mands.

Recall that a sp-group A is a reduced mixed group with an innite number

of non-zero p-components Ap such that the natural embedding⊕

pAp → A

can be extended to a pure embedding A→∏

p Ap .

Note that Hopcity of sp-groups and algebraically compact groups was

studied in [16], too.

In [1] was established the following criterion for a group to be a sp-group.

Specically, the following is true:

Theorem 3.18. The following three conditions are equivalent for a reducedmixed group A with an infinite number of non-zero p-components Ap :

(1) A is a sp-group, i.e., the pure embeddings ⊕p Ap ⊂ A ⊆∏

p Ap hold;(2) The embeddings ⊕p Ap ⊂ A ⊆

∏p Ap hold and A/(⊕p Ap ) is a divisible

torsion-free group;(3) For each prime p there is a group Bp such that A = Ap ⊕Bp with pBp = Bp .

So, we come to the following assertion.

Theorem 3.19. Suppose A is a sp-group which is fully invariant in∏

p Ap .Then A is R-Hopfian if, and only if, each Ap is R-Hopfian.

Proof. The necessity follows from Theorem 3.4 above, because Ap is a direct

summand of A.As for the suciency, let f be an epimorphism of A. Since the groups

Ap and Bp from Theorem 3.18 are fully invariant in A, then the restriction

f p = f | Ap is an epimorphism of Ap too. That is why, f pϕp = 1Apfor some

ϕp ∈ E(Ap ) and thus f p, ϕp generate endomorphisms f = (. . . , f p, . . .) andϕ = (. . . , ϕp, . . .) of the group

∏p Ap , respectively. Since the subgroup ⊕p Ap

is dense in A, then f = f on A; and since A is fully invariant in∏

p Ap , then

ϕ | A is an endomorphism of A. Furthermore, f ϕ = 1 is fullled in ⊕p Ap and,


since it is dense in A, this equality is fullled also in A, i.e., A is R-Hopan, asexpected.

4. Left-open problems

We state here some unanswered questions of some interest and importance.

Problem 1. If G is a Hopfian group, is it true that G ⊕ G is R-Hopfian?

It is worthwhile noticing that in [9] was constructed a torsion-free Hopan

group C such that all its non-zero endomorphisms are monomorphisms and

such that its square C ⊕ C is not Hopan, but as Corner pointed out on p. 306

this square C ⊕C has an isomorphic proper direct summand (compare also with

Example 2.6). However,C⊕C is necessarily R-Hopan byDenition 1.1 becausereferring to [9, Lemma 2] all endomorphisms ofG⊕G are right invertible; notice

that Corner had written all maps in a opposite way than that of us.

Moreover, we point out that if a Hopan p-group B with nite Ulm-

Kaplansky invariants exists such that B ⊕ B is not Hopan, and G = A ⊕ Bis the R-Hopan p-group from Theorem 3.11, then the group G ⊕ G is not

R-Hopan; if A is a nite group, then G must be Hopan and thus there is a

Hopan p-group with the property that its square G ⊕ G is not even R-Hopan.Following a way of similarity, we can adapt Denition 1.1 in case of

arbitrary (not necessarily commutative) groups and rings. And so, one can state

the following:

Problem 2. For a group G determine when the endomorphism ring E(G) as wellas its additive group End(G) and multiplicative (automorphism) group Aut(G)are R-Hopfian objects only in terms of G.

Problem 3. Characterize those groups G with the property that, for any sub-group H of G, the isomorphism G ∼= G/H implies that H is essential in a directsummand of G, i.e., those groups G with the property that, for each surjectionf : G → G, the kernel ker( f ) is essential in a direct summand of G.

Problem 4. Describe those groups G = H ⊕ V which along with G ∼= H implyV = 0.

Some partial cases were handled above.


References

[1] U. F. Albrecht, H.-P. Goeters and W. Wickless, The at dimension of mixed

abelian groups as E-modules, Rocky Mountain J. Math., 25 (1995), 569590.

[2] R. Baer, Groups without proper isomorphic quotient groups, Bull. Amer. Math.

Soc., 50 (1944), 267278.

[3] R. A. Beaumont andR. S. Pierce, Isomorphic direct summands of abelian groups,

Math. Ann., 153 (1964), 2137.

[4] G. Braun and L. Strüngmann, The independence of the notions of Hopan and

co-Hopan abelian p-groups, Proc. Amer. Math. Soc., 143 (2015), 33313341.

[5] S. Breaz, G. Calugareanu and P. Schultz, Modules with Dedekind nite

endomorphism rings,Mathematica (Cluj), 53 (2011), 1528.

[6] A. R. Chekhlov, Torsion-free weakly transitive E-Engel abelian groups, Math.

Notes, 94 (2013), 583589.

[7] A. R. Chekhlov, On abelian groups with right-invariant isometries, Sib. Math. J.,

55 (2014), 574577.

[8] A. R. Chekhlov and P. V. Danchev, On abelian groups having all proper fully

invariant subgroups isomorphic, Comm. Algebra, 43 (2015), 50595073.

[9] A. L.S. Corner, Three examples on hopcity in torsion-free abelian groups, Acta

Math. Acad. Sci. Hungar., 16 (1965), 303310.

[10] L. Fuchs, Innite Abelian Groups I and II, Acad. Press, London (1970 and 1973).

[11] L. Fuchs, Abelian Groups, Springer, Switzerland (2015).

[12] B. Goldsmith and K. Gong, A note on Hopan and co-Hopan abelian groups,

Contemp. Math., 576 (2012), 129136.

[13] R. Hirshon, On Hopan groups, Pac. J. Math., 32 (1970), 753766.

[14] R. Hirshon, Some theorems on Hopcity, Trans. Amer. Math. Soc., 141 (1969),

229244.

[15] J. M. Irwin and J. Takashi, A quasi-decomposable abelian group without proper

isomorphic quotient groups and proper isomorphic subgroups, Pacif. J. Math., 29

(1969), 151160.

[16] E. V. Kaigorodov, On two classes of hopan abelian groups, Vest. Tomsk Univer-

sity, 22 (2013), 2232.

[17] E. V. Kaigorodov, Hopan completely decomposable torsion-free groups, Vest.

Tomsk University, 24 (2013), 2428.

Andrey Chekhlov


University of Tomsk

Russia

[email protected]

Peter Danchev


University of Plovdiv

Bulgaria

[email protected]


RIEMANN MANIFOLD ADMITTING RICCI QUARTER

SYMMETRIC METRIC CONNECTION

By

KANAK KANTI BAISHYA


Abstract. Recently the present author introduced the notion of generalized quasi-

conformal curvature tensor which bridges Conformal curvature tensor, Concircular

curvature tensor, Projective curvature tensor and Conharmonic curvature tensor. The

object of the present paper is to study quasi-conformal like curvature tensor with respect

to Ricci quarter symmetric connections.

1. Introduction

Recently, in tune with Yano and Sawaki [6], the present authors [7] have

introduced and studied generalized quasi-conformal curvature tensor in the

frame of N (k, µ)-manifold. The generalized quasi-conformal curvature tensor

is dened for n-dimensional manifold as

(1.1)

W (X,Y )Z =n−2

n[(1 + (n−1)a−b)−1 + (n−1)(a + b)c]C(X,Y )Z+

+ [1 − b + (n − 1)a]E(X,Y )Z + (n − 1)(b − a)P(X,Y )Z+

+n − 2

n(c − 1)1 + 2n(a + b)C(X,Y )Z

for all X , Y & Z ∈ χ(M), the set of all vector eld of the manifold M ,

where scalar triple (a, b, c) are real constants. The beauty of such curvature

tensor lies in the fact that it has the avour of Riemann curvature tensor Rif the scalar triple (a, b, c) ≡ (0, 0, 0), conformal curvature tensor C [8] if

2000 Mathematics Subject Classication 53C15, 53C25

110 KANAK KANTI BAISHYA

(a, b, c) ≡(− 1

n−2,− 1

n−2, 1

), conharmonic curvature tensor C [11] if (a, b, c) ≡

≡(− 1

n−2,− 1

n−2, 0

), concircular curvature tensor E ([5], p. 84) if (a, b, c) ≡

≡ (0, 0, 1), projective curvature tensor P ([5], p. 84) if (a, b, c) ≡(− 1

n−1, 0, 0

)and m-projective curvature tensor H [4], if (a, b, c) ≡

(− 1

2n−2,− 1

2n−2, 0

). The

equation (1.1) can also be written as

(1.2)

W (X,Y )Z = R(X,Y )Z + a[S(Y, Z )X − S(X, Z )Y ]+

+ b[g(Y, Z )QX − g(X, Z )QY ]−

−crn

(1

n − 1+ a + b

)[g(Y, Z )X − g(X, Z )Y ].

The notion of Ricci Quarter-Symmetric metric connections in a Riemannian

manifold has been introduced by A. Ghosh [2]. Thereafter, many authors stud-

ied Ricci quarter-symmetric connection on metric and non-metric spaces. The

present paper deals with an Einstein manifold admitting a Ricci quarter sym-

metric metric connection. In this paper we have obtained an equivalence relation

between the locally symmetric and quasi conformal like symmetric manifold.

Furtherwe have studied an equivalence relation between the locally bi-symmetric

and quasi conformal like bi-symmetric manifold. Finally, we have shown that

a generalized conformally (resp. concircularly, projectively) 2-recurrent Ein-

stein manifold admitting a Ricci quarter symmetric metric connection is confor-

mally (resp. concircularly, projectively) at whereas a generalized conformally

2-recurrent Einstein manifold admitting a Ricci quarter symmetric metric con-

nection is not conharmonically at.

2. Ricci quarter-symmetric metric connection

Let (Mn, g) be a Riemannian manifold equipped with a Riemannian metricg, and ∇ be the Levi-Civita connection associated with g. Let χ(M) denote theset of all tangent vector elds on M .

A linear connection ∇ is called a metric connection if

(2.1) ( ∇Xg)(Y, Z ) = 0, ∀ X,Y, Z ∈ χ(M).

A linear connection ∇ in a Riemannian manifold (Mn, g) is said to be Ricciquarter-symmetric metric connection if the torsion tensor T satises [9]

(2.2) T (X,Y ) = π(Y )QX − π(X )QY,

RIEMANN MANIFOLD ADMITTING RICCI QUARTER SYMMETRIC METRIC CONNECTION 111

where π is a non-zero 1-form dened by π(X ) = g(X, ρ) and Q is the (1, 1)Ricci tensor dened by

(2.3) g(QX,Y ) = S(X,Y ),

S is the Ricci tensor of M .

If ∇ is the Levi-Civita connection of the manifold (Mn, g), then the Ricci

Quarter symmetric connection [9] is given by

(2.4) ∇XY − ∇XY = π(Y )QX − S(X,Y )ρ.

Taking a local coordinate system in M such that g, ∇, ∇, π, Q, T have the

local expression, respectively, g j i , Γhji, Γh

ji, πi , Qh

jand Th

ji, then, by a direct

computation, we have

(2.5) Thji = π jQ

hi − πiQ

hj .

Theorem 2.1. For a Ricci quarter-symmetric metric connection, in a localcoordinate, the expression holds

(2.6) Γhji = Γhji + π

hSi j − π jQhi − πiQ

hj ,

where π j = πigi j .

Proof. Since ∇ is a metric connection, then, ∀ X , Y , Z ∈ χ(M), we have

( ∇Xg)(Y, Z ) = X (g(Y, Z )) − g( ∇XY, Z ) − g(Y, ∇X Z ) = 0,(2.7)

( ∇Yg)(Z, X ) = Y (g(Z, X )) − g( ∇Y Z, X ) − g(Z, ∇Y X ) = 0,(2.8)

( ∇Zg)(X,Y ) = Z (g(X,Y )) − g( ∇Z X,Y ) − g(X, ∇ZY ) = 0.(2.9)

By using (2.7), (2.8) and (2.9), we get

(2.10) g( ∇XY, Z ) = g(∇XY, Z )+π(Z )S(X,Y )−π(Y )S(X, Z )−π(X )S(Y, Z ).

In a local coordinate (U, xi ), we can choose

X =∂

∂xi, Y =

∂

∂x j, Z =

∂

∂xk.

Substituting (2.10) into (2.9) above, we have locally the following

(2.11) Γhji = Γhji + πk Si jghk − π jSikghk − πiSjkghk .

Proposition 2.2. If Qhi

is proportional to the identity tensor δhi

, then theRicci quarter-symmetric metric connection is reduced into a semi-symmetricconnection, and the coefficient is given as

(2.12) Γhji = Γhji + π

hgi j − π jδ

hi − πiδ

hj .


3. The quasi-conformal like curvature tensor with respect to ∇

Let R and R be the curvature tensors of the connections ∇ and∇ respectively.

Then it can be shown that [9]

(3.1)

R(X,Y )Z = R(X,Y )Z − α(Y, Z )QX + α(X, Z )QY − S(Y, Z )LX+

+ S(X, Z )LY + π(Z )[(∇XQ)Y − (∇YQ)X]−

− [(∇X S)(Y, Z ) − (∇Y S)(X, Z )]ρ,

where α is a tensor eld of type (0, 2) dened by

(3.2) α(X,Y ) = g(LX,Y ) = (∇Xπ)Y − π(Y )π(QX ) +1

2π(ρ)S(X,Y )

and L is a tensor eld of type (1, 1) dened by

(3.3) LX = ∇X ρ − π(QX )ρ +1

2π(ρ)QX .

Here we shall consider Mn to be an Einstein manifold, that is,

(3.4) S(X,Y ) =rng(X,Y ),

where r is the scalar curvature of the manifold.

Considering (3.1), (3.4) and (1.2), we get

(3.5)R(X,Y )Z = R(X,Y )Z −

rn[α(Y, Z )X − α(X, Z )Y+

+ g(Y, Z )LX − g(X, Z )LY ].

Contracting (3.5), we get

(3.6) S(Y, Z ) =rn[(1 − m)g(Y, Z ) − ( n − 2)g(Y, Z )],

where S is the Ricci tensor of ∇ and m is the trace of M (Y, Z ). Now putting

Y = Z = ei , where ei ; i = 1, 2, . . . , n is an orthonormal basis of the tangent

space at any point, we get by taking the sum for 1 ≤ i ≤ n in the relation (3.6)

(3.7) r =rn[(n − 2(n − 1)m],

where r is the scalar curvature of ∇.

As a consequence of (3.4), equation (1.2) reduces to

W ′(X,Y, Z,U) = R′(X,Y, Z,U) + λ[g(Y, Z )g(X,U) − g(X, Z )g(Y,U)](3.8)

where λ =rn[(a + b) (1 − c) −

cn − 1

],(3.9)

whereW ′(X,Y, Z,U) = g(W (X,Y )Z,U) and R′(X,Y, Z,U) = g(R(X,Y )Z,U).


Let W ′ be the quasi-conformal like curvature tensor of the connection ∇.

Then from (3.8), we have

W (X,Y )Z = R(X,Y )Z + λ[g(Y, Z )X − g(X, Z )Y ](3.10)

where λ =rn[(a + b)(1 − c) −

cn − 1

].(3.11)

Taking the covariant derivative of (3.10), we obtain

(3.12) (∇U W )(X,Y )Z = (∇U R)(X,Y )Z

Hence, we can state

Theorem 3.1. In an Einstein manifold (Mn, g) equipped with Ricci Quartersymmetric metric connection, the quasi conformal like recurrent manifold is ageneralized recurrent.

Theorem 3.2. In an Einstein manifold (Mn, g) equipped with Ricci Quartersymmetric metric connection, the following conditions are equivalent

(a) M is locally symmetric.(b) M is quasi conformal like symmetric.

4. Quasi conformal like bi-symmetric manifold

A Riemann manifold is said to be bi-symmetric if it satises

(4.1) (∇U∇V R)(X,Y )Z = 0.

AnRiemannmanifoldwill be called quasi conformal like bi-symmetricmanifold,

if it satises

(4.2) (∇U∇VW )(X,Y )Z = 0.

Taking the covariant dierentiation on both sides of (3.12), we obtain

(4.3) (∇U∇V W )(X,Y )Z = (∇U∇V R)(X,Y )Z .

Hence we can state

Theorem 4.1. In an Einstein manifold (Mn, g) equipped with Ricci quartersymmetric metric connection, the following conditions are equivalent

(a) M is bi-symmetric.(b) M is conformal like bi-symmetric.


5. Generalized quasi conformal like 2-recurrent manifold

A non-at Riemannian manifold of dimension n is called generalized

2-recurrent Riemannian manifold [1] when the Riemannian curvature tensor

R satises the condition.

(5.1) (∇U∇VW )(X,Y )Z = A(U)(∇VW )(X,Y )Z + B(V,U)W (X,Y )Z,

where A is a 1-form, B is a non-zero (0, 2) tensor and ∇ is the Levi-Civita

connection. The (0, 2) tensor B is dened by

(5.2) B(X,Y ) = g(X, QY ),

where Q is linear transformation from the tangent space at p(∈ M) : TP (M) →→ TP (M). When the quasi conformal like curvature tensor W satisfy the

conditions

(5.3) (∇U∇VW )(X,Y )Z = A(U)(∇VW )(X,Y )Z + B(V,U)W (X,Y )Z,

then the manifold will be called generalized quasi conformal like 2-recurrent

manifold, where A, B are stated earlier. Using Biachi's identity we nd from

(3.12) that

(5.4) (∇U W )(X,Y )Z + (∇X W )(Y,U)Z + (∇X W )(U,Y )Z = 0.

Again, from (5.4) we nd that

(5.5) (∇V∇U W )(X,Y )Z + (∇V∇X W )(Y,U)Z + (∇V∇X W )(U,Y )Z = 0

Making use of (5.2) and (5.4) in (5.5), we get

(5.6) B(U,V ) W (X,Y )Z + B(Y,U) W (V, X )Z + B(X,U) W (Y,V )Z = 0.

Now contracting (5.6), we nd that

(5.7)

B( W (X,Y )Z,U)+B(Y,U)n∑i=1

W ′(ei, X, Z, ei )−B(X,U)n∑i=1

W ′(ei,Y, Z, ei )=0.

(5.8)

n∑i=1

W ′(ei, X, Z, ei ) =

= (1 − b + 2na)S(X, Z ) + r"b −

c 1 + (n − 1)(a + b)n

#g(X, Z ).


In view of (3.4), (5.8), (5.7) takes the form

(5.9)

B( W (X,Y )Z,U) =

=rn[(c − 1) 1 + (n − 1)(a + b)][B(Y,U)g(X, Z ) − B(X,U)g(Y, Z )].

For, an arbitrary choice of U we can state the following

Theorem 5.1. A generalized conformally (resp. concircularly, projectively)2-recurrent Einstein manifold equipped with Ricci quarter symmetric metricconnection is conformally (resp. concircularly, projectively) flat.

Remark 5.2. A generalized conharmonically 2-recurrent Einstein manifold

equipped with Ricci quarter symmetric metric connection is not a conharmoni-

cally at.

Acknowledgement. The author would like to thank UGC, ERO-Kolkata, for

their nancial support, File no. PSW-194/15-16.

References

[1] A. K. Ray, On generalised 2-recurrent tensor in Riemannian space, Bull. Acad.

Roy. Blg. Cl. Sc. 5 e. Ser. Tom. 58, 2 (1972), 220228.

[2] A. Ghosh, On Ricci Quarter-Symmetric metric connections in a Riemannian

manifold, Analele Stiintice Ale Universitatii AL. I. Cuza, IASI, Tomul XLIII,

s.I.a. Matematica, 1997.

[3] E. O. Canfes and I. Gul, Weyl Manifold with a Ricci Quarter-Symmetric Con-

nection, Iran J. Sci. Technol. Trans. Sci. 40 (2016), 171.

[4] G. K. Pokhariyal and R. S. Mishra, Curvatur tensors and their relativistics

signicance I., Yokohama Math. J., 18 (1970), 105108,.

[5] K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Math. Studies

32, Princeton University Press, 1953.

[6] K. Yano and S. Sawaki, Riemannian manifolds admitting a conformal transfor-

mation group, J. Di. Geom., 2(1968), 161184.

[7] K. K. Baishya and P. R. Chowdhury, On generalized quasi-conformal N (k, µ)-manifolds, Commun. Korean Math. Soc., 31 (2016), 163176.

[8] L. P. Eisenhart, Riemannian Geometry, Princeton University Press, 1949.

[9] R. S. Mishra and S. N. Pande, On Quarter symmetric metric F-connections,Tensor (N.S.), 34 (1980), 17.


[10] S. A. Demirbag, H. Bagdatliyilmaz, S. A. Uysal and F. Ozenzengin, On

quasi Einstein manifolds admitting a Ricci Quarter-Symmetric metric connection,

Bulletin of Mathematical Analysis and Applications, 4 (2011), 8491.

[11] Y. Ishii, On conharmonic transformations, Tensor (N.S.), 7 (1957), 7380.

Kanak Kanti Baishya


Kurseong college

Kurseong, Darjeeling-734203

West Bengal, India

[email protected]


ON PSEUDO CYCLIC PARALLEL RICCI SYMMETRIC

MANIFOLDS

By

S. K. SAHA


(Dedicated to the memory of Professor M.C. CHAKI on the occasion of his 105th birth

anniversary)

Abstract. Pseudo Ricci symmetric manifold was introduced and studied by

M. C. Chaki in 1988. The object of this paper is to dene and study a new type

of Riemannian manifold called pseudo cyclic parallel Ricci symmetric manifold. An

existence theorem is given and some properties are obtained.

Introduction

In a paper M. C. Chaki [5] introduced and studied a type of non-at

Riemannian manifold called pseudo Ricci symmetric manifold. According to

him, a non-at Riemannian manifold (Mn, g), (n > 2), is called pseudo Ricci

symmetric manifold if its Ricci tensor S of type (0, 2) is not identically zero andsatises the condition

(1) (∇X S)(Y, Z ) = 2A(X )S(Y, Z ) + A(Y )S(X, Z ) + A(Z )S(Y, X ),

for every vector eld X , Y , Z , where A is a non-zero 1-form dened by

g(X, ρ) = A(X ), ∇ denotes the operator of covariant dierentiation with respectto the metric tensor g. A is called the associated 1-form and ρ is called the

associated vector eld of the manifold. An n-dimensional manifold of this kind

was denoted by the symbol (PRS)n . Many works [8], [10], [15] etc. have been

done on (PRS)n and its generalizations by several authors.The aimof this paper is to dene and study a new type of non-atRiemannian

manifold called pseudo cyclic parallel Ricci symmetric manifold. A non-at

2000 Mathematics Subject Classication 53C25

118 S. K. SAHA

Riemannian manifold (Mn, g), (n > 2), is called pseudo cyclic parallel Ricci

symmetric manifold if its Ricci tensor S of type (0, 2) is not identically zero andsatises the condition

(2) (∇X S)(Y, Z ) = −2A(X )S(Y, Z ) + A(Y )S(Z, X ) + A(Z )S(X,Y ),

for every vector eld X , Y , Z , where A is a non-zero 1-form dened by

(3) g(X, ρ) = A(X ),

∇, A and ρ have the meaning already mentioned. An n-dimensional manifold

of this kind shall be denoted by the symbol (PCPRS)n . The main dierence of

these two manifolds is that in (PRS)n , the Ricci tensor is not cyclic parallel butin (PCPRS)n , the Ricci tensor is cyclic parallel. This can be easily veried fromthe denition of cyclic parallel Ricci tensor [1] as given below: The Ricci tensor

S of type (0, 2) is called cyclic parallel if it satises the condition

(4) (∇X S)(Y, Z ) + (∇Y S)(Z, X ) + (∇Z S)(X,Y ) = 0.

So the name pseudo cyclic parallel Ricci symmetric manifold has been chosen.

After preliminaries an existence theorem of (PCPRS)n has been proved. Inthe section 3 it is shown that in a (PCPRS)n , the scalar curvature r is the eigen

value of the Ricci tensor S of type (0, 2) corresponding to the eigen vector ρwhich is the associated vector eld of the manifold and the scalar curvature of

(PCPRS)n is constant. It is also shown that in a (PCPRS)n , either the scalarcurvature is zero or for non-zero scalar curvature the associated 1-form A is

closed. In the next section it is shown that in a (PCPRS)n , the Ricci curvature inthe direction of ρ is the scalar curvature of the manifold and the Ricci curvature

in the direction of any vector eld X is covariantly constant. In section 5 it is

shown that if in a (PCPRS)n , the associated vector eld is a unit torse forming

vector eld and a , −1, r , 0, then it is a quasi Einstein manifold and if in a

(PCPRS)n with zero scalar curvature the energy of the associated torse forming

vector eld ρ is constant, then the integral curves of ρ are geodesics. In the nextsection it is shown that in a conformally at (PCPRS)n , the Ricci tensor is aCodazzi tensor, the scalar curvature is a non-zero constant, the associated 1-form

A is closed and a conformally at (PCPRS)n is a manifold of quasi constant

curvature. Finally, it is shown that In a (PCPRS)4 perfect uid space time, the

energy density σ and the isotropic pressure p of the uid are both constants and

their values are − 1

2K(r + 2λ) and 1

2K(2λ − r) respectively, where r , λ and K are

scalar curvature, cosmological constant and gravitational constant of the space

time respectively.

ON PSEUDO CYCLIC PARALLEL RICCI SYMMETRIC MANIFOLDS 119

1. Preliminaries

In this section we shall obtain some formulas which will be used in sequel.

Let (Mn, g) be aRiemannianmanifold and ei , i = 1, 2, . . . , n be an orthonormal

basis of the tangent space at each point and i is summed for 1 ≤ i ≤ n. Let r be

the scalar curvature [3] of the manifold and it is dened by

(1.1) S(ei, ei ) = r .

Let L be the symmetric endomorphism corresponding to the Ricci tensor S of

type (0, 2) and is dened by

(1.2) g(LX,Y ) = S(X,Y ),

Putting Y = Z = ei in (2) and i is summed for 1 ≤ i ≤ n, we get,

(1.3) dr (X ) = −2A(X )r + 2S(X, ρ).

From (2) we get

(1.4) (∇X S)(Y, Z ) − (∇Z S)(Y, X ) = −3[A(X )S(Y, Z ) − A(Z )S(X,Y )].

2. Existence theorem of (PCPRS)n

To show the existence of (PCPRS)n we consider a Riemannian manifold

(Mn, g) which admits a linear connection ∇ dened by

(2.1) ∇XY = ∇XY + A(Y )X − A(X )Y,

such that the Ricci tensor S of type (0, 2) which corresponds to the connection

∇ is parallel with respect to the connection ∇ i.e.,

(2.2) ∇S = 0.

From (2.2) we have

(∇X S)(Y, Z ) = 0

which gives

(2.3) X S(Y, Z ) − S(∇XY, Z ) − S(Y,∇X Z ) = 0.

From (2.1) and (2.3) we have

(2.4) (∇X S)(Y, Z ) = −2A(X )S(Y, Z ) + A(Y )S(Z, X ) + A(Z )S(X,Y ).

The connection ∇ is not identical with ∇. Hence ∇S , 0.

120 S. K. SAHA

Thus (2.4) shows that the manifold is a (PCPRS)n . This leads to the

following theorem:

Theorem 2.1. If a Riemannian manifold (Mn, g), (n > 2), admits a linearconnection ∇ which satisfies (2.1) and (2.2), then the manifold is a (PCPRS)n .

This proves the existence (PCPRS)n .Conversely, let us suppose that (2.4) holds in a Riemannian manifold

(Mn, g). Now we get from (2.4)

(2.5)X S(Y, Z ) − S(∇XY, Z ) − S(Y,∇X Z ) =

= −2A(X )S(Y, Z ) + A(Y )S(Z, X ) + A(Z )S(X,Y ).


(2.6) (∇X S)(Y, Z ) = 0,

i.e. ∇S = 0, for all X , Y and Z . This leads to the following theorem:

Theorem 2.2. If a Riemannian manifold (Mn, g), (n > 2), is a (PCPRS)n ,then the Ricci tensor S of type (0, 2) which corresponds to the connection ∇ isparallel with respect to the connection ∇ defined by (2.1).

From Theorem 2.1 and Theorem 2.2 we have a necessary and sucient

condition for a Riemannian manifold (Mn, g) to be a (PCPRS)n as follows:

Theorem 2.3. A Riemannian manifold (Mn, g), (n > 2), is a (PCPRS)n if andonly if it admits a linear connection ∇ defined by (2.1) for which the Ricci tensorS of type (0, 2) which corresponds to the connection ∇ is parallel.

3. Scalar curvature of (PCPRS)n

In this section we shall investigate the nature of scalar curvature of

(PCPRS)n . Putting Y = Z = ei in (1.4) and i is summed for 1 ≤ i ≤ n,we get,

(3.1) dr (X ) = −6[A(X )r − S(X, ρ)].

From (1.3) and (3.1) we get

(3.2) A(X )r = S(X, ρ).


From (3) and (3.2) we nd that r is the eigen value of the Ricci tensor Sof type (0, 2) corresponding to the eigen vector ρ which is the associated vectoreld of (PCPRS)n . Hence we can state the following theorem:

Theorem 3.1. In a (PCPRS)n , the scalar curvature r is the eigen value of theRicci tensor S of type (0, 2) corresponding to the eigen vector ρ which is theassociated vector field of the manifold.


(3.3) dr (X ) = 0,

which shows that the scalar curvature of (PCPRS)n is constant. Hence we can

state the following theorem:

Theorem 3.2. The scalar curvature of (PCPRS)n is constant.


(3.4) (∇X S)(Y, ρ) − (∇Y S)(X, ρ) = rdA(X,Y ).

Putting Y = ρ in (1.4) and using (3.2) we get

(3.5) (∇X S)(Z, ρ) − (∇Z S)(X, ρ) = 0.


(3.6) rdA(X,Y ) = 0.

From (3.6) we see that either the scalar curvature is zero or for non-zero scalar

curvature the associated 1-form A is closed. Hence we can state the following

theorem:

Theorem 3.3. In a (PCPRS)n , either the scalar curvature is zero or for non-zeroscalar curvature the associated 1-form A is closed.

Note. In a (PRS)n , the following results [5] are proved:

S(X, ρ) = 0.(i)

r = 0,(ii)

if the scalar curvature r of (PRS)n is constant, while

(iii) r , 0,

the associated 1-form A is closed.

The above results are dierent from the results of (PCPRS)n .

122 S. K. SAHA

4. Ricci curvature (PCPRS)n

In this section we shall nd out the nature of Ricci curvature [3] of

(PCPRS)n . Putting X = ρ in (3.2) we get

(4.1)S(ρ, ρ)g(ρ, ρ)

= r .

Again if we replace Y , Z by X in (2) we get

(4.2) (∇X S)(X, X ) = 0.

From (4.1) we see that the Ricci curvature in the direction of ρ is the scalar

curvature of the manifold and from (4.2) we see that the Ricci curvature in the

direction of any vector eld X is covariantly constant. Hence we can state the

following theorem:

Theorem 4.1. In a (PCPRS)n , the Ricci curvature in the direction of ρ is thescalar curvature of the manifold and the Ricci curvature in the direction of anyvector field X is covariantly constant.

5. Associated vector eld ρ of (PCPRS)n

In this section we nd the consequences if the associated vector eld ρ of(PCPRS)n is a torse forming vector eld.

Putting Z = ρ in (2) and using (3.2) and (3.3) we get

(5.1) (∇X S)(Y, ρ) = −r A(X )A(Y ) + A(ρ)S(X,Y ).

Again from (3.2) and (3.3) we get

(5.2) (∇X S)(Y, ρ) = rg(Y,∇X ρ) − S(Y,∇X ρ).


(5.3) A(ρ)S(X,Y ) − r A(X )A(Y ) = rg(Y,∇X ρ) − S(Y,∇X ρ).

Let ρ be a torse forming vector eld, then

(5.4) ∇X ρ = aX + ω(X )ρ,

where a is a non-zero scalar and ω is a non-zero 1-form. From (5.3) and (5.4)

we have

(5.5) [a + A(ρ)]S(X,Y ) = rag(X,Y ) + r A(X )A(Y ).


The denition of quasi Einsteinmanifold [7] is as follows: A non-at Riemannian

manifold (Mn, g), (n > 2), is called quasi Einstein manifold if its Ricci tensor Sof type (0, 2) is not identically zero and satises the condition

(5.6) S(X,Y ) = ag(X,Y ) + bB(X )B(Y ),

where a, b are scalars b , 0 and B is a non-zero 1-form dened by B(X ) == g(X, µ), µ is a unit vector eld. Many works [6], [14], [16] etc. have been

done on quasi Einstein manifolds and its generalization by several authors. If ρis a unit vector eld, a + 1 , 0 and r , o from (5.5) we get

(5.7) S(X,Y ) =ar

a + 1g(X,Y ) +

ra + 1

A(X ) A(Y ),

which satises all the conditions of quasi Einstein manifolds [7]. Hence we can

state the following theorem:

Theorem 5.1. If in a (PCPRS)n the associated vector field is a unit torseforming vector field and a , −1, r , 0, then it is a quasi Einstein manifold.

Let

(5.8) f =1

2g(ρ, ρ)

be the energy of the torse forming vector eld ρ given by (5.4). From (5.8) we

get

(5.9) df (X ) = g(∇X ρ, ρ).

Again, if the scalar curvature vanishes, then from (5.5) we get

(5.10) A(ρ) = −a.

If the energy of the torse forming vector eld is constant, then from (5.5), (5.9)

and (5.10) we get

(5.11) A(X ) = ω(X ).

From (5.4), (5.10) and (5.11) we have

(5.12) ∇X ρ = −A(ρ)X + A(X )ρ.

Putting X = ρ in (5.12) we get

∇ ρ = 0,

which shows that the integral curves of the torse forming vector eld ρ are

geodesics. This leads to the following theorem:

124 S. K. SAHA

Theorem 5.2. If in a (PCPRS)n with zero scalar curvature the energy of theassociated torse forming vector field ρ is constant, then the integral curves of ρare geodesics.

6. Conformally at (PCPRS)n

In a conformally at manifold [4] we have

(6.1) (∇X S)(Y, Z )−(∇Z S)(Y, X ) =1

2(n − 1)[dr (X )g(Y, Z )−dr (Z )g(Y, X )].

In a (PCPRS)n we have (3.3), i.e.

(6.2) dr (X ) = 0.


(6.3) (∇X S)(Y, Z ) − (∇Z S)(Y, X ) = 0,

which shows that S is a Codazzi tensor. Hence we have the following theorem:

Theorem 6.1. In a conformally flat (PCPRS)n , the Ricci tensor is a Codazzitensor.

From (2) and (6.3) we get

(6.4) A(X )S(Y, Z ) = A(Z )S(Y, X ).

Putting Z = ρ in (6.4) and using (3.2) we get

(6.5) S(X,Y ) = rT (X )T (Y ),

where T (X ) = g(X, φ), φ =

A() is a unit vector eld. If r = 0, then S = 0

which is inadmissible by denition of (PCPRS)n . Hence and from (6.2) we can

conclude that r is a non-zero constant. This leads us to the following theorem:

Theorem 6.2. In a conformally flat (PCPRS)n , the scalar curvature is a non-zero constant.

From Theorem 3.3 and Theorem 6.2 we see that in a conformally at

(PCPRS)n , the associated 1-form A is closed. Thus we have the following

theorem:

Theorem 6.3. In a conformally flat (PCPRS)n , the associated 1-form A isclosed.


Again, (6.5) satises all the condition of quasi Einstein manifolds as given

in (5.6). Hence we can state the following theorem:

Theorem 6.4. A conformally flat (PCPRS)n is a special type of quasi Einsteinmanifold.

Now we have the denition of the manifold of quasi constant curvature as

follows:

A Riemannian manifold (Mn, g), (n > 2), is called a manifold of quasi

constant curvature [12], [13] if it satises the following condition

(6.6)

R(X,Y, Z,W ) = c1[g(Y, Z )g(X,W ) − g(X, Z )g(Y,W )]++c2[g(Y, Z )D(X )D(W ) − g(X, Z )D(Y )D(W )++g(X,W )D(Y )D(Z ) − g(Y,W )D(X )D(Z )],

where D is a non-zero 1-form and c1, c2 are scalars of which c2 , o. Again, weknow that if a Riemannian manifold (Mn, g), (n > 3), is conformally at, then

(6.7)

R(X,Y, Z,W ) =1

n − 2[S(Y, Z )g(X,W ) − S(X, Z )g(Y,W )+

+S(X,W )g(Y, Z ) − S(Y,W )g(X, Z )]−

−r

(n − 1)(n − 2)[g(Y, Z )g(X,W ) − g(X, Z )g(Y,W )].


(6.8)

R(X,Y, Z,W ) =r

n − 2[g(X,W )T (Y )T (Z ) − g(Y,W )T (X )T (Z )+

+g(Y, Z )T (X )T (W ) − g(X, Z )T (Y )T (W )]−

−r

(n − 1)(n − 2)[g(Y, Z )g(X,W ) − g(X, Z )g(Y,W )],

which can be written as

(6.9)

R(X,Y, Z,W ) = a1[g(Y, Z )g(X,W ) − g(X, Z )g(Y,W )]++a2[g(X,W )T (Y )T (Z ) − g(Y,W )T (X )T (Z )++g(Y, Z )T (X )T (W ) − g(X, Z )T (Y )T (W )],

where a1 = − r

(n−1)(n−2) , a2 = r

n−2are non-zero scalars.

Comparing (6.6) and (6.9) we see that a conformally at (PCPRS)n is a

manifold of quasi constant curvature. Thus we have the following theorem:

Theorem 6.5. A conformally flat (PCPRS)n is a manifold of quasi constantcurvature.

126 S. K. SAHA

Since a 3-dimentional Riemannian manifold is always conformally at, the

relation (6.9) holds in (PCPRS)3. Hence we can state the following corollary:

Corollary 6.1. Every (PCPRS)3 is a manifold of quasi constant curvature.

7. Semi Riemannian (PCPRS)n

Let a semi Riemannian (PCPRS)4 be a general relativistic space time

(M4, g), where g is a Lorentz metric with signature (+, +, +.−). We consider

general relativistic perfect uid space time (M4, g) with unit time like velocity

vector eld as the associated vector eld ρ of (PCPRS)4 i.e.,

(7.1) g(ρ, ρ) = −1.

The sources of any gravitational eld (matter and energy) are represented in

relativity by a type of (0, 2) symmetric tensor T called the energy momentum

tensor [2], [9], [11]. T is given by

(7.2) T (X,Y ) = (σ + p) A(X )A(Y ) + pg(X,Y ),

where σ and p are the energy density and the isotropic pressure of the uid

respectively, while A is dened by

(7.3) g(X, ρ) = A(X ).

The Einstein equation with cosmological constant is as follows:

(7.4) S(X,Y ) −1

2rg(X,Y ) + λg(X,Y ) = KT (X,Y ),

where K is a gravitational constant. From (7.2) and (7.4) we have

(7.5) S(X,Y ) −1

2rg(X,Y ) + λg(X,Y ) = K[(σ + p)A(X ) A(Y ) + pg(X,Y )].

Putting Y = ρ in (7.5) and using (3.2) we get

(7.6) σ = −1

2K(r + 2λ).

Let ei , i = 1, 2, 3, 4 be an orthonormal basis of the frame eld at a point of the

space time and contracting (7.5) we get

(7.7) 3p − σ =1

K(4λ − r).



(7.8) p =1

2K(2λ − r).

Since the scalar curvature is constant by Theorem 3.1, from (7.6) and (7.8) we see

that the energy density and the isotropic pressure of the uid are both constants.

Thus we have the following theorem:

Theorem 7.1. In a (PCPRS)4 perfect fluid space time, the energy density σand the isotropic pressure p of the fluid are both constants and their values are− 1

2K(r + 2λ) and 1

2K(2λ − r) respectively.

References

[1] A. Gray, Einsteinlike manifolds which are not Einstein, Geom. Dedicata, 7

(1978), 259280.

[2] B. O'Neil, Semi Riemannian Geometry with application to relativity, Academic

Press, Inc. (1983), 399340.

[3] K. Yano, Integral formulas in Riemannian Geometry, Mercel Deker,Inc., New

York, 1970.

[4] L. P. Eisenhart, Riemannian Geometry, Princeton Univ. Press, 1926.

[5] M. C. Chaki, On pseudo Ricci symmetric manifolds, Bulg. J. Phys., 15 (1988),

526531.

[6] M. C. Chaki, On generalized quasi Einstein manifolds, Publ. Math. Debrecen, 58

(2001), 683691.

[7] M.C.Chaki andA.K.Maity, On quasi Einsteinmanifolds,Publ.Math.Debrecen,

57 (2000), 297306.

[8] M. C. Chaki and S. Koley, On generalized pseudo Ricci symmetric manifolds,

Periodica Mathematica Hungarica, 28 (1994), 123129.

[9] M. C. Chaki and S. Ray Guha, On a type of space time of general relativity,

Tensor N. S., 64 (2003), 227231.

[10] P. Chakrabarti and S. K. Saha, On generalized almost pseudo Ricci symmetric

manifolds, Tensor N. S., 73 (2011), 198206.

[11] R.K. Sachs andW.Hu,General relativity formathematicians, NewYork, Springer

Verlag, 1977.

[12] T.Adati,Manifolds of quasi constant curvature, II, Quasi-umbilical hyper surfaces,

TRU Math., 21, (1985), 221226.

[13] T. Adati, Manifolds of quasi constant curvature, III, A manifold admitting a

concircular vector eld, Tensor N. S., 45 (1987), 189194.

[14] U. C. De and B. K. De, On Quasi Einstein manifolds, Comm. Korean Math.Soc.,

23 (2008), 413420.

128 S. K. SAHA

[15] U. C. De and B. K. De, On conformally at generalized pseudo Ricci symmetric

manifolds, Soochow J. Math., 23 (1997), 381389.

[16] U. C. De and G. C. Ghosh, On generalized quasi Einstein manifolds, Kyunpook

Math. J., 44 (2004), 607615.

S. K. Saha

Member, Calcutta Mathematical Society

18/348, Kumar lane, Chinsurah

Hooghly - 712101, W. B., India

[email protected]


WEAK INSERTION OF A CONTRA-γ-CONTINUOUS FUNCTION∗

By

MAJID MIRMIRAN


Abstract. A sucient condition in terms of lower cut sets are given for the

weak insertion of a contra-γ-continuous function between two comparable real-valued

functions.

1. Introduction

The concept of a preopen set in a topological space was introduced by

H.H. Corson and E.Michael in 1964 [5]. A subset A of a topological space (X, τ)is called preopen or locally dense or nearly open if A ⊆ Int(Cl(A)). A set A is

called preclosed if its complement is preopen or equivalently if Cl(Int(A)) ⊆ A.The term `preopen' was used for the rst time by A. S. Mashhour, M. E. Abd

El-Monsef and S. N. El-Deeb [21], while the concept of a `locally dense' set was

introduced by H. H. Corson and E. Michael [5].

The concept of a semi-open set in a topological space was introduced

by N. Levine in 1963 [18]. A subset A of a topological space (X, τ) is called

semi-open [11] if A ⊆ Cl(Int(A)). A set A is called semi-closed if its complement

is semi-open or equivalently if Int(Cl(A)) ⊆ A.Recall that a subset A of a topological space (X, τ) is called γ-open if A∩S is

preopen, whenever S is preopen [2]. A set A is called γ-closed if its complement

is γ-open or equivalently if A ∪ S is preclosed, whenever S is preclosed.

We have that if a set is γ-open, then it is semi-open and preopen.

2000 Mathematics Subject Classication 54C08, 54C10, 54C50; 26A15, 54C30∗ This work was supported by University of Isfahan and Centre of Excellence for Mathematics

(University of Isfahan).

130 MAJID MIRMIRAN

A generalized class of closed sets was considered by Maki in [20]. He

investigated the sets that can be represented as union of closed sets and called

them V−sets. Complements of V−sets, i.e., sets that are intersection of open setsare called Λ-sets [20].

Recall that a real-valued function f dened on a topological space X is

called A-continuous [24] if the preimage of every open subset of R belongs

to A, where A is a collection of subsets of X . Most of the denitions of

function used throughout this paper are consequences of the denition of A-continuity. However, for unknown concepts the reader may refer to [6, 12]. In

the recent literature many topologists had focused their research in the direction

of investigating dierent types of generalized continuity.

J. Dontchev in [7] introduced a new class of mappings called contra-

continuity. A good number of researchers have also initiated dierent types

of contra-continuous like mappings in the papers [1, 4, 9, 10, 11, 13, 14, 23].

Hence, a real-valued function f dened on a topological space X is called

contra-γ-continuous (resp. contra-semi-continuous, contra-precontinuous) if

the preimage of every open subset of R is γ−closed (resp. semi−closed, pre-closed) in X [7].

Results of Kat¥tov [15, 16] concerning binary relations and the concept of

an indenite lower cut set for a real-valued function, which is due to Brooks [3],

are used in order to give a necessary and sucient conditions for the insertion of

a contra-γ-continuous function between two comparable real-valued functions.

If g and f are real-valued functions dened on a space X , we write g ≤ fin case g(x) ≤ f (x) for all x in X .

The following denitions aremodications of conditions considered in [17].

A property P dened relative to a real-valued function on a topological space

is a cγ-property provided that any constant function has property P and provided

that the sum of a function with property P and any contra-γ-continuous functionalso has property P. If P1 and P2 are cγ-property, the following terminology

is used: A space X has the weak cγ-insertion property for (P1, P2) if and only

if for any functions g and f on X such that g ≤ f , g has property P1 and fhas property P2, then there exists a contra-γ-continuous function h such that

g ≤ h ≤ f .In this paper, is given a sucient condition for the weak cγ-insertion

property. Also, several insertion theorems are obtained as corollaries of these

results.

WEAK INSERTION OF A CONTRA- -CONTINUOUS FUNCTION 131

2. The Main Result

Before giving a sucient condition for insertability of a contra-γ-continuousfunction, the necessary denitions and terminology are stated.

Let (X, τ) be a topological space, the family of all γ-open, γ-closed,semi-open, semi-closed, preopen and preclosed will be denoted by γO(X, τ),γC(X, τ), sO(X, τ), sC(X, τ), pO(X, τ) and pC(X, τ), respectively.

Definition 2.1. Let A be a subset of a topological space (X, τ). We dene

the subsets AΛ and AV as follows: AΛ = ∩O : O ⊇ A,O ∈ (X, τ) andAV = ∪F : F ⊆ A, Fc ∈ (X, τ).

In [8, 19, 22], AΛ is called the kernel of A.We dene the subsets γ(AΛ), γ(AV ), p(AΛ), p(AV ), s(AΛ) and s(AV ) as

follows:

γ(AΛ) = ∩O : O ⊇ A,O ∈ γO(X, τ)γ(AV ) = ∪F : F ⊆ A, F ∈ γC(X, τ),p(AΛ) = ∩O : O ⊇ A,O ∈ pO(X, τ),p(AV ) = ∪F : F ⊆ A, F ∈ pC(X, τ),s(AΛ) = ∩O : O ⊇ A,O ∈ sO(X, τ) ands(AV ) = ∪F : F ⊆ A, F ∈ sC(X, τ).

γ(AΛ) (resp. p(AΛ), s(AΛ)) is called the γ-kernel (resp. prekernel, semi-kernel)

of A.

The following rst two denitions are modications of conditions consid-

ered in [15, 16].

Definition 2.2. If ρ is a binary relation in a set S then ρ is dened as follows:

x ρ y if and only if y ρ v implies x ρ v and u ρ x implies u ρ y for any u and v

in S.

Definition 2.3. A binary relation ρ in the power set P(X ) of a topological

space X is called a strong binary relation in P(X ) in case ρ satises each of thefollowing conditions:

1) If Ai ρ Bj for any i ∈ 1, . . . ,m and for any j ∈ 1, . . . , n, then there existsa set C in P(X ) such that Ai ρ C and C ρ Bj for any i ∈ 1, . . . ,m and anyj ∈ 1, . . . , n.

2) If A ⊆ B, then A ρ B.3) If A ρ B, then γ(AΛ) ⊆ B and A ⊆ γ(BV ).

132 MAJID MIRMIRAN

The concept of a lower indenite cut set for a real-valued function was

dened by Brooks [3] as follows:

Definition 2.4. If f is a real-valued function dened on a space X and if

x ∈ X : f (x) < ` ⊆ A( f , `) ⊆ x ∈ X : f (x) ≤ ` for a real number `, thenA( f , `) is called a lower indenite cut set in the domain of f at the level `.

We now give the following main result:

Theorem 2.1. Let g and f be real-valued functions on the topological space X ,in which γ-kernel sets are γ-open, with g ≤ f . If there exists a strong binaryrelation ρ on the power set of X and if there exist lower indefinite cut sets A( f , t)and A(g, t) in the domain of f and g at the level t for each rational number t suchthat if t1 < t2 then A( f , t1) ρ A(g, t2), then there exists a contra-γ-continuousfunction h defined on X such that g ≤ h ≤ f .

Proof. Let g and f be real-valued functions dened on the X such that g ≤ f .By hypothesis there exists a strong binary relation ρ on the power set of X and

there exist lower indenite cut sets A( f , t) and A(g, t) in the domain of f and g atthe level t for each rational number t such that if t1 < t2 then A( f , t1) ρ A(g, t2).

Dene functions F and G mapping the rational numbers Q into the power

set of X by F (t) = A( f , t) and G(t) = A(g, t). If t1 and t2 are any elements

of Q with t1 < t2, then F (t1) ρ F (t2),G(t1) ρ G(t2), and F (t1) ρ G(t2). ByLemmas 1 and 2 of [16] it follows that there exists a function H mapping Q into

the power set of X such that if t1 and t2 are any rational numbers with t1 < t2,then F (t1) ρ H (t2), H (t1) ρ H (t2) and H (t1) ρ G(t2).

For any x in X , let h(x) = inft ∈ Q : x ∈ H (t).We rst verify that g ≤ h ≤ f : If x is in H (t) then x is inG(t ′) for any t ′ > t;

since x is in G(t ′) = A(g, t ′) implies that g(x) ≤ t ′, it follows that g(x) ≤ t.Hence g ≤ h. If x is not in H (t), then x is not in F (t ′) for any t ′ < t; since x is

not in F (t ′) = A( f , t ′) implies that f (x) > t ′, it follows that f (x) ≥ t. Henceh ≤ f .

Also, for any rational numbers t1 and t2 with t1 < t2, we have h−1(t1, t2) == γ(H (t2)V ) \ γ(H (t1)Λ). Hence h−1(t1, t2) is γ-closed in X , i.e., h is a

contra-γ-continuous function on X .

The above proof used the technique of Theorem 1 in [15].


3. Applications

The abbreviations cγc, cpc and csc are used for contra-γ-continuous,contra-precontinuous and contra-semi-continuous, respectively.

Before stating the consequences of Theorems 2.1, we suppose that X is a

topological space whose γ-kernel sets are γ-open.

Corollary3.1. If for each pair of disjoint preopen (resp. semi-open) sets G1,G2

of X , there exist γ-closed sets F1 and F2 of X such that G1 ⊆ F1, G2 ⊆ F2 andF1 ∩ F2 = ∅ then X has the weak cγ-insertion property for (cpc, cpc) (resp.(csc, csc)).

Proof. Let g and f be real-valued functions dened on X , such that f and g

are cpc (resp. csc), and g ≤ f . If a binary relation ρ is dened by A ρ B in case

p(AΛ) ⊆ p(BV ) (resp. s(AΛ) ⊆ s(BV )), then by hypothesis ρ is a strong binaryrelation in the power set of X . If t1 and t2 are any elements of Q with t1 < t2,then

A( f , t1) ⊆ x ∈ X : f (x) ≤ t1 ⊆ x ∈ X : g(x) < t2 ⊆ A(g, t2);

since x ∈ X : f (x) ≤ t1 is a preopen (resp. semi-open) set and since

x ∈ X : g(x) < t2 is a preclosed (resp. semi-closed) set, it follows that

p(A( f , t1)Λ) ⊆ p(A(g, t2)V ) (resp. s(A( f , t1)Λ) ⊆ s(A(g, t2)V )). Hence t1 < t2implies that A( f , t1) ρ A(g, t2). The proof follows from Theorem 2.1.

Corollary 3.2. If for each pair of disjoint preopen (resp. semi-open) setsG1,G2, there exist γ-closed sets F1 and F2 such that G1 ⊆ F1, G2 ⊆ F2 andF1 ∩ F2 = ∅ then every contra-precontinuous (resp. contra-semi-continuous)function is contra-γ-continuous.

Proof. Let f be a real-valued contra-precontinuous (resp. contra-semi-

continuous) function dened on X . Set g = f , then by Corollary 3.1, there

exists a contra-γ-continuous function h such that g = h = f .

Corollary 3.3. If for each pair of disjoint subsets G1,G2 of X , such that G1 ispreopen and G2 is semi-open, there exist γ-closed subsets F1 and F2 of X suchthat G1 ⊆ F1, G2 ⊆ F2 and F1 ∩ F2 = ∅ then X have the weak cγ-insertionproperty for (cpc, csc) and (csc, cpc).

Proof. Let g and f be real-valued functions dened on X , such that g is cpc(resp. csc) and f is csc (resp. cpc), with g ≤ f . If a binary relation ρ is denedby A ρ B in case s(AΛ) ⊆ p(BV ) (resp. p(AΛ) ⊆ s(BV )), then by hypothesis ρ

134 MAJID MIRMIRAN

is a strong binary relation in the power set of X . If t1 and t2 are any elements of

Q with t1 < t2, then

A( f , t1) ⊆ x ∈ X : f (x) ≤ t1 ⊆ x ∈ X : g(x) < t2 ⊆ A(g, t2);

since x ∈ X : f (x) ≤ t1 is a semi-open (resp. preopen) set and since

x ∈ X : g(x) < t2 is a preclosed (resp. semi-closed) set, it follows that

s(A( f , t1)Λ) ⊆ p(A(g, t2)V ) (resp. p(A( f , t1)Λ) ⊆ s(A(g, t2)V )). Hence t1 < t2implies that A( f , t1) ρ A(g, t2). The proof follows from Theorem 2.1.

Acknowledgement. This research was partially supported by Centre of Excel-

lence for Mathematics(University of Isfahan).

References

[1] A. Al-Omari and M. S. Md Noorani, Some properties of contra-b-continuousand almost contra-b-continuous functions, European J. Pure. Appl. Math., 2

(2009), 231220.

[2] D. Andrijevic and M. Ganster, On PO-equivalent topologies, IV International

Meeting on Topology in Italy (Sorrento, 1988), Rend. Circ. Mat. Palermo (2)

Suppl., 24 (1990), 251256.

[3] F. Brooks, Indenite cut sets for real functions, Amer. Math. Monthly, 78 (1971),

10071010.

[4] M. Caldas and S. Jafari, Some properties of contra-β-continuous functions,

Mem. Fac. Sci. Kochi. Univ., 22 (2001), 1928.

[5] H. H. Corson and E. Michael, Metrizability of certain countable unions, Illinois

J. Math., 8 (1964), 351360.

[6] J. Dontchev, The characterization of some peculiar topological space via α− andβ−sets, Acta Math. Hungar., 69 (1995), 6771.

[7] J. Dontchev, Contra-continuous functions and strongly S-closed space, Intrnat.

J. Math. Math. Sci., 19 (1996), 303310.

[8] J. Dontchev, and H. Maki, On sg-closed sets and semi-λ-closed sets, Questions

Answers Gen. Topology, 15 (1997), 259266.

[9] E. Ekici, On contra-continuity, Annales Univ. Sci. Budapest., 47 (2004), 127137.

[10] E. Ekici, New forms of contra-continuity, Carpathian J. Math., 24 (2008), 3745.

[11] A. I. El-Magbrabi, Some properties of contra-continuous mappings, Int. J. Gen-

eral Topol., 3 (2010), 5564.

[12] M. Ganster and I. Reilly, A decomposition of continuity, Acta Math. Hungar.,

56 (1990), 299301.

[13] S. Jafari and T. Noiri, Contra-continuous function between topological spaces,

Iranian Int. J. Sci., 2 (2001), 153167.


[14] S. Jafari and T. Noiri, On contra-precontinuous functions, Bull. Malaysian Math.

Sc. Soc., 25 (2002), 115128.

[15] M. Kat¥tov, On real-valued functions in topological spaces, Fund. Math., 38

(1951), 8591.

[16] M. Kat¥tov, Correction to, On real-valued functions in topological spaces,

Fund. Math., 40 (1953), 203205.

[17] E. Lane, Insertion of a continuous function, Pacic J. Math., 66 (1976), 181190.

[18] N. Levine, Semi-open sets and semi-continuity in topological space, Amer. Math.

Monthly, 70 (1963), 3641.

[19] S. N. Maheshwari and R. Prasad, On ROs-spaces, Portugal. Math., 34 (1975),

213217.

[20] H.Maki, GeneralizedΛ-sets and the associated closure operator, The special Issue

in commemoration of Prof. Kazuada IKEDA's Retirement, (1986), 139146.

[21] A. S. Mashhour,M. E. Abd El-Monsef and S. N. El-Deeb, On pre-continuous

and weak pre-continuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982),

4753.

[22] M. Mrsevic, On pairwise R and pairwise R1 bitopological spaces, Bull. Math.

Soc. Sci. Math. R. S. Roumanie, 30 (1986), 141145.

[23] A. A. Nasef, Some properties of contra-continuous functions, Chaos Solitons

Fractals, 24 (2005), 471477.

[24] M. Przemski, A decomposition of continuity and α-continuity, Acta Math. Hun-

gar., 61 (1993), 9398.

Majid Mirmiran


University of Isfahan

Isfahan 81746-73441, Iran

[email protected]


PARTIAL COVERING OF THE UNIT CIRCLE BY FOUR EQUAL

CIRCLES∗

ByZS. GÁSPÁR, T. TARNAI, AND K. HINCZ

(Received October 13, 2017)

Abstract. How must n equal circles of given radius be placed so that they cover

as great a part of the area of the unit circle as possible? In this paper the case of n = 4 is

investigated. It is known that the centres of the four equal circles form a square in both

the maximum packing and the minimum covering congurations, and it is expected that

for radii between the maximum packing radius and the minimum covering radius the

circle centres also form squares. It will be shown that this expectation is not fullled in

a region of the radii.

1. Introduction

Consider n equal circles of given radius r such that r (n)max ≤ r ≤ R(n)

min, where

r (n)max is the maximum radius of n equal circles that can be packed in the unit circle

without overlapping, and R(n)min is the minimum radius of n equal circles which

can cover the unit circle without interstices. Here the following problem can be

posed [10]: How must these circles be placed so that the area of the part of the

unit circle covered by the n circles will be a maximum? It is obvious that if rvaries continuously from r (n)

max to R(n)min, a transition from the maximum packing

to the minimum covering is obtained. The question is: How does this transition

happen?

Zahn [10] established the solution for n = 2 (for 0.5 < r < 1). He also

provided conjectural solutions for 3 ≤ n ≤ 10 (for between three and six selected

values of r), but in some cases the accuracy of his calculations was not sucient

2000 Mathematics Subject Classication 52C15; 05B40∗ Supported by the National Research, Development and Innovation Fund (NKFIA, K119440)

138 ZS. GÁSPÁR, T. TARNAI, K. HINCZ

to determine the correct circle conguration and to identify its symmetry. We

developed a method [3] by which, for n = 5, we produced improved conjectural

solutions and numerically determined the transition frompacking to covering [4].

The cases of n = 3 and 4 are particularly interesting, because the centres of the

equal circles form an equilateral triangle and a square, respectively, in both

the maximum packing and the minimum covering congurations. Therefore

one may think that, in the transition from packing to covering, the structure

of the partial coverings with maximum area remains unchanged, that is, the

centres of the circles form equilateral triangles and squares for all maximum

area congurations. For n = 3 Szalkai [8] recently proved our conjecture [4] that

this is indeed the case: for varying r , the threefold symmetry of the congurations

is maintained. In the case of n = 4, however, contrary to expectation, something

dierent happens.

The aim of this paper is to show how the maximum area conguration

changes for n = 4, when r varies from the maximum packing radius to the

minimum covering radius, and to provide conjectures to the solution of the

problem.

2. Packing and covering

To simplify the notation, rmax and Rmin will stand for r (4)max and R(4)

min,

respectively, throughout the paper. Additionally, D2 and D4 denote dihedral

symmetry groups in the plane (D2 consists of 2 rotations and 2 axial reections,

D4 consists of 4 rotations and 4 axial reections) [1, 9] In the case n = 4,

the radius of the largest circles that can be packed in the unit circle without

overlapping is rmax =√2 − 1, and the radius of the smallest circles by which

the unit circle can be covered without interstices is Rmin = 1/√2 The maximum

packing was determined by Pirl [6], and the minimum covering came out as a

special case of a theorem of Molnár [5]. Both congurations have D4 symmetry

in the plane (Figs. 1a and d).

3. Main results

Let us suppose that, for the circle radius r such that rmax < r < Rmin, the

optimal circle arrangement has only D2 symmetry. This means that the centres

PARTIAL COVERING OF THE UNIT CIRCLE BY FOUR EQUAL CIRCLES 139

rmax

r1

(a) (b)

r2Rmin

(c) (d)

Figure 1. Special optimal arrangements of four equal circles:

(a) maximum packing, rmax =√2 − 1,

(b) at the point of bifurcation (catastrophe point), r1 = 0.5502505523,(c) two circles become tangent to each other, r2 = 0.5539433380,

(d) minimum covering, Rmin =1√2

O1, O2, O3, O4 of the equal circles (discs) C1, C2, C3, C4 form the four vertices

of a rhombus, and the centre of the rhombus coincides with the centre O of the

unit circle (disc) C (Fig. 2). (Symmetry D2 would also be fullled if the centres

of the four circles formed a rectangle, but according to the relationships of the

stress interpretation of the problem [3] this conguration cannot occur.) If circles

Ci and Cj intersect, Ci ∩ Cj , ∅, i ≤ 4, j ≤ 4, i , j, then they have a double

overlap, the area of which will be denoted by A1. If Ci and the exterior of the

unit circle C intersect, Ci ∩ CC , ∅, i ≤ 4, then they have an external overlap,

the area of which will be denoted by A2. We introduce the term coverage which

is dened as the ratio of the area of the part of the unit circle covered by the

equal circles to the area of the unit circle. Overlaps determine parts of the equal

circles that could be removed without changing the coverage of the unit circle.


O3 O1

O2

O4

Obc

bc

bc

bc

bca a

b

b

A2(b)

A2(a)

A1(L)

Figure 2. Layout and notation

The sum of their areas is called surplus area, and is denoted by A. For agiven radius r of the equal circles, the mathematical problem of maximizing

the coverage of the unit circle is replaced with minimizing the surplus area. Let

the half diagonals of the rhombus be denoted by a and b. Then the distance Lbetween centres of adjacent circles (the edge length of the rhombus) is

(1) L =√

a2 + b2.

Then we have the surplus area in the form

(2) A(a, b) = 4A1 (L(a, b)) + 2A2(a) + 2A2(b)

which is a function of two variables: a and b. It has a critical (or stationary) pointif grad A = 0, that is,

(3)4dA1

dLdLda

+2dA2

da= 0,

4dA1

dLdLdb

+2dA2

db= 0.

According to Csikós's theorem [2] the derivative of the areas above with respect

to the distance between the centres of two circles is equal to the length of the

chord which the two circles have in common. In the case of two equal circles,

the overlap area decreases with an increase in the distance; because of this the

derivative has a negative sign, therefore relationships in (3) are in detail

(4a) − 2√4r2 − L2

aL+

√4 −

(1 + a2 − r2

a

)2= 0,


(4b) − 2√4r2 − L2

bL+

√4 −

(1 + b2 − r2

b

)2= 0.

Solutions to the equations (4a,b) form two sets in R3 of the variables r , a,b. The rst solution set of (4a,b) has points for the radii r ∈ (rmax, Rmin):

(5) a = b =1

3

√−3 + 9r2 + 6

√1 − 3r2 + 3r4.

The circle arrangements determined by these solutions have symmetry D4. For

given r , the surplus area function (2) has a minimum at the values given by

(5) only if its Hessian matrix is positive denite. We found that, starting with

rmax, for increasing values of r the Hessian matrix is positive denite up to

r1 = 0.5502505227 where it becomes singular. For radius r > r1, the circle

arrangement with symmetry D4 is not even locally optimal, since the function

(2) has a saddle point. (The Hessian is positive in the interval [rmax, r1), changessign at r1, then is negative in the interval (r1, Rmin), and at r = Rmin, the Hessian

as a function has only the limit, which is negative.)

The second solution set of (4a,b) has points only for the radii r ∈ (r1, Rmin):

(6a) a =1

3

√21r2 − 3 ± 6

√(7r6 + 20r4 − 10r2 + 1

)/(1 + r2

),

(6b) b =1

3

√21r2 − 3 ∓ 6

√(7r6 + 20r4 − 10r2 + 1

)/(1 + r2

).

The circle arrangements determined by these solutions have symmetry D2. By

using the terminology of catastrophe theory [7] we can say that the function (2)

has a point of standard cusp catastrophe at radius r1. Therefore it is advantageousto apply a coordinate transformation such that the points of the optimal circle

arrangements with symmetry D4 lie in a coordinate plane, and in this way

a projection onto a plane perpendicular to this coordinate plane shows the

bifurcation of the solution. With the transformation

(7) x =a + b2, y =

a − b2,

the points of the circle arrangements with symmetry D4 will be in the plane

xr , and the projection onto the plane yr shows that the pitchfork bifurcation is

symmetric. However, (6a,b) shows the surplus area correctly only if a ≥ r and

b ≥ r . This is satised only in the interval (r1, r2) where r2 = 0.5539433380. Inthe case if r = r2 and a > b, then b = r , that is, the circles C2 and C4 are tangent


to each other (Fig. 1c), thus from the projection of the second set onto the plane

yr only that part is the real solution.

In the case r ∈ (r2, Rmin), provided a > b, the surplus area is:

(8) A(a, b) = 4A1 (L(a, b)) + A1 (2b) + 2A2 (a) + 2A2(b).

This has a stationary point if

(9)4dA1

dLdLda

+2dA2

da= 0,

4dA1

dLdLdb

+2dA1

d (2b)+2dA2

db= 0,

that is, after reducing by 2:

(10a) − 2√4r2 − L2

aL+

√4 −

(1 + a2 − r2

a

)2= 0,

(10b) − 2√4r2 − L2

bL−

√4r2 − 4b2 +

√4 −

(1 + b2 − r2

b

)2= 0.

The set of equations (10a,b) can be solved only numerically. In case a < b we

have the same optimal arrangements but they are rotated by 90 degrees. Since

a , b the circle conguration has symmetry D2. The values of y characterizing

the optimal arrangements of the four equal circles are shown in Fig. 3, where the

maximum coverage as a function of r is also presented.

In order to conrm our results it should be checked whether we obtain the

same circle arrangements as optimal congurations if we do not suppose any

symmetry. By using a technique based on stress interpretation of the partial

covering problem [3] we executed this numerical check and obtained the same

results.

Perhaps it seems to be a contradiction that, in the case of r = Rmin, the

optimal solution has symmetry D4 but according to a short remark above the

limit of the Hessian is negative there, fromwhich we could think that the solution

cannot be even locally optimal. This seeming contradiction can be dissolved by

the fact that, in the neighbourhood of this position, with an innitesimal change

of the circle conguration, multiple overlaps can appear, therefore function (2)

is not valid at r = Rmin.

In conclusion, we can establish that for r = rmax, that is, for the maximum

packing, the centres of the four circles form the vertices of a square. For increasing

r , rst, the square shape is maintained until r = r1, where the square starts tobecome a rhombus. Then, the rhombus becomes more and more elongated (the

ratio a/b increases) until b = r = r2, that is, the opposite circlesC2 andC4 will be

tangent to each other. Finally, the deformation of the rhombus takes the opposite


(b)

r

coverage

| | | | |

||

b c

bc bc

bc

D4

D2

D2

D4

rmax Rmin0.5 r1 r2 0.6

0.8

0.9

1

(a)

r

||

−0.1

0

0.1

a − b

rmax Rmin

D4

r1

r2

r2

D4

D2

D2

D2

D2

Figure 3. (a) Relationship between a − b and r for optimal circle

arrangements, (b) optimum coverage of the unit circle by 4 equal

circles as a function of r

direction (the ratio a/bdecreases), whileC2 andC4 intersect but their intersection

and C1 and C3 are pairwise disjoint (no triple overlaps occur), the conguration

tends to regain the square shape, which does happen at r = Rmin, that is, for

the minimum covering. It should be noted, as the bifurcation showed, that the

development of the rhombus can happen not only with horizontal elongation but

also with vertical elongation.

References

[1] H. S. M. Coxeter, Introduction to Geometry. Wiley, New York 1961.

[2] B. Csikós, On the volume of the union of balls, Discrete Comput. Geom., 20

(1998), 449461.


[3] Zs. Gáspár, T. Tarnai, K. Hincz, Partial covering of a circle by equal circles, Part

I: The mechanical models, J. Comput. Geom., 5 (2014),104125.

[4] Zs. Gáspár, T. Tarnai, K. Hincz, Partial covering of a circle by equal circles, Part

II: The case of 5 circles, J. Comput. Geom., 5 (2014), 126149.

[5] J. Molnár, On an extremal problem in elementary geometry (in Hungarian),Mat.

Fiz. Lapok, 48 (1942), 249253.

[6] U. Pirl, Der Mindestabstand von n in der Einheitskreisscheibe gelagenen Punkten,

Math. Nachr., 40 (1969), 111124.

[7] T. Poston, I. Stewart, Catastrophe Theory and its Applications, Pitman, London

1978.

[8] B. Szalkai, Optimal cover of a disk with three smaller congruent disks, Adv.

Geom., 16 (2016), 465476.

[9] H. WEIL, Symmetry, Princeton University Press 1952.

[10] C. T. Zahn, Black box maximization of circular coverage, J. Res. Nat. Bur.

Standards Sect., B 66 (1962), 181216.

Zsolt Gáspár, Tibor Tarnai, Krisztián Hincz

Department of Structural Mechanics

Budapest University of Technology and Economics

M¶egyetem rkp. 3.

H-1521, Budapest, Hungary

gaspar, [email protected]@epito.bme.hu


A REMARK ON EXTENSION OF A MONOIDAL STRUCTURE

By

NEHA GUPTA AND PRADIP KUMAR

(Received October 17, 2017)

Abstract. In this article we explore a possibility to extend a monoidal structure

of a category to a category containing it such that the former becomes a monoidal

subcategory of the later. Moreover we give a set of sucient conditions on a subcategory

such that dualizablility of the subcategory makes the entire category with extended

monoidal structure dualizable as well.

1. Introduction

Let B be a subcategory of a given category D. Let B has a monoidal

structure. In this article, we discuss when can one extend the monoidal structure

ofB toD such thatB becomes amonoidal subcategory ofmonoidal categoryD.

The extension of a monoidal structure that we discuss is not the same as the

graded extension of the monoidal category discussed in [1]. In [4], Ponto Kate

and Shulman Michael have discussed a normal lax symmetric monoidal functor

between two monoidal categories; and using such a functor from a subcategory

to the category containing it, we are able to extend a monoidal structure.

In section 2, we recall the basics about the monoidal category, dual objects

and strong lax symmetric monoidal functor. In section 3 and 4, we discuss

various properties of the induced (extended) monoidal structure.

2000 Mathematics Subject Classication 18D10,19D23

146 NEHA GUPTA, PRADIP KUMAR

2. Preliminaries

2.1. Monoidal category

We rst set the terminology that we use throughout the article which

can be found in [6],[4],[2], etc. A monoidal category is a collection of data

(C, ⊗, a, I, λ, ρ) where C is a category, ⊗ is a functor from C × C to C called the

tensor product, an associator a for ⊗ which is a family of isomorphisms

(1) aX;Y;Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z ),

an unit object I of C, and a left unitor λ and a right unitor ρ with respect to Iwhich are also a family of isomorphisms

(2) λX : I ⊗ X → X

(3) ρX : X ⊗ I→ X

such that a, λ and ρ) satisfy the pentagon identity and the triangle identities (fordetails we refer [2]).

Let X and Y be objects in C. We say X has a left dual Y (or Y has a right

dual X) if there is

• a morphism ev: Y ⊗ X → I (called evaluation) and• a morphism coev: I→ X ⊗ Y (called coevaluation)

such that the compositions

(4) X−1X

−−−→ I⊗Xcoev⊗IdX−−−−−−−→ (X⊗Y ) ⊗ X

aX;Y ;X

−−−−−−→ X⊗(Y⊗X )IdX ⊗ev−−−−−−→ X⊗I

X−−→ X

and

(5) Y−1Y

−−−→ I⊗YIdY ⊗coev−−−−−−−→ Y ⊗ (X⊗Y )

a−1Y ;X;Y

−−−−−−→ (Y ⊗X )⊗Yev⊗IdY−−−−−−→ I⊗Y

Y−−→ Y

are both identity. If X has a left (right) dual, we say that X is left (right) dualizable.

If left and right duals of X are isomorphic in C, we say X is dualizable, and in

general the dual of X is denoted as X∗. A category is called dualizable if it is amonoidal category where every object X is dualizable, that is, has a dual X∗.

Let C andD be symmetric monoidal categories. In [4], Ponto has discussed

a special functor called lax symmetric monoidal functor and we discuss it below.

A REMARK ON EXTENSION OF A MONOIDAL STRUCTURE 147

Definition 2.1 (Lax Symmetric monoidal functor). Lax symmetric

monoidal functor F between symmetric monoidal categories C and D consists

of natural transformations

c : F (M) ⊗ F (N ) → F (M ⊗ N )i : ID → F (IC)

satisfying appropriate coherence axioms given in [4].

F is said to be normal if i is a natural isomorphism and F is said to be

strong if both c and i are natural isomorphisms.

Proposition 2.2 (Proposiiton 6.1, [4]). Let F : C → D be a normal laxsymmetric monoidal functor. Let M be a dualizable object in C with dual M∗

and assume that c : F (M) ⊗ F (M∗) → F (M ⊗ M∗) is an isomorphism thenF (M) is dualizable with dual F (M∗).

3. Monoidal structure on Image of F

Let C and D be two categories and let C has a monoidal structure given

by (⊗, a, I, λ, ρ). Let F : C → D be any functor such that Im(F ) which is

the collection of the images of all objects and morphisms in C becomes a

subcategory of D. With this set up we wish to give a monoidal structure to the

category Im(F ), which we shall denote by B. If F satises certain conditions,

discussed below, then the category Im(F ) gets a monoidal structure induced

from C.

Let the functor F saties following conditions:

(1) For objects a1, a2, a3, a4 in C, such that whenever we have F (a1) == F (a2) and F (a3) = F (a4) then F (a1 ⊗ a3) = F (a2 ⊗ a4) and

(2) For morphisms f , f ′, g, g′ in C, such that whenever we have F ( f ) == F ( f ′) and F (g) = F (g′) then we have F ( f ⊗ g) = F ( f ′ ⊗ g

′).We dene the monoidal functor ⊗F onB where ⊗F : B×B → B is dened

as follows. On objects,

⊗F (b1, b2) := F (a1 ⊗ a2)

where b1 = F (a1) and b2 = F (a2). On morphisms,

⊗F ( f1, f2) := F (h1 ⊗ h2)


where f1 = F (h1) and f2 = F (h2). The conditions imposed on the functor F

ensures that ⊗F is well dened. The compatibility of ⊗F with composition in

B will be followed from the compatibility of the tensor ⊗ with composition in

C. Similar is the compatibility of the identity maps of B. Explicitly we have

Idb1

⊗F Idb2

= IdF (a1)

⊗F IdF (a2)

= F

(Ida1

⊗ Ida2

)= IdF (a1⊗a2)

= Idb1⊗b2

.

Therefore the functor ⊗F denes a tensor product onB. ForB to have amonoidal

structure B must also have a associator as well as left and right unitors for ⊗F .

These are F (a), F (λ), F (ρ) where a, λ and ρ are the respective associator, aleft unitor and right unitor C and they will satisfy triangle and pentagon axioms

since these holds in C. Thus we have proved the following.

Theorem 3.1. If F is a functor from a monoidal category (C, ⊗) to a categoryDsuch that Im(F ) is a sub category ofD and F satisfies the following conditions(1) for any objects a1, a2, a3, a4 in C such that whenever F (a1) =

= F (a2) and F (a3) = F (a4), we have F (a1 ⊗ a3) = F (a2 ⊗ a4)(2) for any morphisms f , g, f ′, g′ in C such that whenever F ( f ) = F ( f ′) andF (g) = F (g′) then F ( f ⊗ g) = F ( f ′ ⊗ g

′),then F gives a monoidal structure on Im(F ) induced from C.

Let D be a monoidal category with tensor ⊗D and C = D × D, then C is

also a monoidal category with obvious tensor, dened component-wise. Dene

a functor F : C → D such that it associates each object (X,Y ) in C to X in D

and each morphism ( f , g) ∈ Hom(X1,Y1), (X2,Y2)) to f ∈ Hom(X1,Y1).One can think of the functor F as a projection onto the rst component.

Further, F also satises both the condition of Theorem 3.1. Therefore we now

have the monoidal structure ⊗F on D. Both the monoidal structures ⊗D and

⊗F happen to be same this case.

Corollary 3.2. Let F : C → Im(F ) be a functor as in Theorem 3.1. If (C, ⊗)is symmetric then the category (Im(F ), ⊗F ) is symmetric as well.

Proof. We need to give a symmetric structure in Im(F ) and this will be

the image of the symmetry of C under the functor F . Explicitly, if D1, D2

are objects in the category Im(F ), then there are C1,C2 are objects in C

such that F (C1) = D1 and F (C2) = D2. Since C is symmetric, therefore

there is a symmetric map, say sC1;C2: C1 ⊗ C2 → C2 ⊗ C1 which is an

isomorphism. Consider F (sC1;C2) : F (C1 ⊗ C2) → F (C2 ⊗ C1). As F is a

functor thereforeF (sC1;C2) is an isomorphism between D1⊗F D2 = F (C1⊗C2)

and D2 ⊗F D1 = F (D1 ⊗ D1).


Corollary 3.3. A functor F between a symmetric monoidal category C toIm(F ) which satisfies the conditions of Theorem 3.1 is a strong lax symmetricmonoidal functor.

Proof. This is direct from the denition of the induced monoidal structure on

Im(F ) described in the Theorem 3.1, which is, D1 ⊗F D2 = F (C1 ⊗ C2). Thusby denition itself we get

F (C1) ⊗F F (C2) = F (C1 ⊗ C2)

and

IIm(F ) = F (IC).

Corollary 3.4. Let C be a symmetric category and if an object A is dualizablein (C, ⊗), then F (A) is dualizable in (Im(F ), ⊗F ) provided F satisfies theconditions of Theorem 3.1.

Proof. Since by Corollary 3.3 F is a strong lax symmetric monoidal functor.

Then by proposition 2.2, we have the result directly.

4. Product category and Base

4.1. Product Category

Here we recall the product category in a way we need for our purpose.

Denition of product category is motivated from [6].

LetΛ be a non empty indexing set. Abusing the use of notation, we dene an

indexing category, also denoted by Λ, with objects as elements of the indexing

set and morphisms as

Hom(λ1, λ2) =8><>:

Id if λ1 = λ2 = λ

∅ otherwise

Composition is the composition of identity map.

Let B be a category. We dene the product category of B over an indexing

category Λ as follows

(1) an object is a functor F from Λ to B with F (λ) being an object in B and

F (Hom(λ, λ)) = IdF () .

(2) amorphism between two objectsF ,G is a natural transformation η : F → Gwhich is a family of morphisms η : F (λ) → G(λ) in B,


(3) composition ofmorphisms is the vertical composition of two natural transfor-

mations. That is, if η ∈ Hom(F ,G) and ζ ∈ Hom(G,H ) are two verticallycomposable natural transformations, then the composition ζ η is a natural

transformation between F andH given by ζ η(λ) = ζ (λ) η(λ) for everyλ ∈ Λ.

We denote the product category of B over an index category Λ as ΠΛ(B). Inparticular if the indexing set is a nite set, then the product category of a category

B as dened above is isomorphic to B × ... × B.

4.2. Base

LetD be a category. We dene a base forD as the data (B,Λ, ∗,G), whereB is a category, Λ some index category, ∗ and G are functors dened from

ΠΛ(B) to D and D to ΠΛ(B) respectively such that ∗G ∼= IdD . We call B as

base category forD, ∗ as the pasting functor and G as the decomposition functor.

In particular we see that every category is a trivial base of itself. Simply

take B = D, index set Λ = 1 with G and ∗ as identity functors.

Proposition 4.1. If the base of a product category is monoidal then so will bethe product category.

Proof. Let ΠΛ(B) be the product category of B over Λ with base (B,Λ, ∗,G).Let (B, ⊗, a, I, λ, ρ) be the monoidal structure of B. A tensor product

⊗′ : ΠΛ(B) × ΠΛ(B) → ΠΛ(B) is dened as follows(1) For objects F ,G in ΠΛ(B), F ⊗′ G is a functor from Λ→ B dened as

(a) On objects(F ⊗′ G

)(λ) = F (λ) ⊗B G(λ) for λ ∈ Λ.

(b) On morphisms(F ⊗′ G

)(Id

) = F (Id

) ⊗B G(Id

) = IdF ()

⊗B IdG()= IdF ()⊗BG()

(2) For i = 1, 2 if ηi are two natural transformations from Fi → Gi then

η1 ⊗′ η2 : F1 ⊗

′ F2 → G1 ⊗′ G2 is a natural transformation which is given

by the collection of morphisms(η1 ⊗

′ η2) : F1(λ) ⊗B F2(λ) → G1(λ) ⊗B G2(λ)

where each of these morphisms are dened as (η1) ⊗B (η2) for λ ∈ Λ.(3) For two pair of composable transformations

F11

−−→ G11−−→ H1 and F2

2

−−→ G22−−→ H2


their composition

(η1 ⊗′ η2) (ζ1 ⊗′ ζ2) = (η1 ζ1) ⊗′ (η2 ζ2)

holds in ΠΛ(B) using the composition of morphisms in B.

(4) IdF ⊗′G = IdF ⊗′ IdG is true using the monoidal structure of B.

The associator a′ of ΠΛ(B) is a natural transformation

a′ :(F ⊗′ G

)⊗ H → F ⊗′

(G ⊗′ H

)which is dened for each λ ∈ Λ by the corresponding associator a inB. Similarly

the left unitor λ ′ and the right unitor ρ′ for ⊗′ in ΠΛ(B) are dened using the

corresponding unitors of B. The unit I′ in ΠΛ(B) is a functor from Λ to B given

by λ maps to IB for each λ ∈ Λ. They satisfy triangle and pentagon axioms since

they are true in B.

IfB has a monoidal structure (⊗, a, I, λ, ρ) then we will denote the monoidal

structure on ΠΛB as in Proposition 4.1 by (⊗′, a′, I′, λ ′, ρ′).

Remark 4.2. It is easy to see that (ΠΛ(B), ⊗′) is symmetric if and only if (B, ⊗)is symmetric.

Theorem 4.3. There exists a monoidal structure on a categoryD if it has a base(B,Λ, ∗,G) such that(1) the base category B has a monoidal structure and(2) the pasting functor ∗ satisfies both the conditions of Theorem 3.1. Explicitly

for objectsH1,H2,H3,H4 in ΠΛ(B), such that whenever we have

∗(H1) = ∗(H2) and ∗ (H3) = ∗(H4)

then∗(H1 ⊗

′H3) = ∗(H2 ⊗′H4)

And, a similar condition on morphisms. That is, for morphisms η, η ′, ζ, ζ ′

in ΠΛ(B), such that

∗(η) = ∗(η ′) and ∗ (ζ ) = ∗(ζ ′)

then, we must have

∗(η ⊗′ ζ ) = ∗(η ′ ⊗′ ζ ′).

Moreover if B is symmetric then with induced monoidal structure D will alsobe symmetric.


Proof. Image of the pasting functor ∗ is the category D and it satises the

conditions of Theorem 3.1. Therefore we get an induced monoidal structure on

the category D inherited from ΠΛB. More precisely it is the composition of

functors ∗ ⊗′ (G × G). We denote it by ⊗G . Explicitly on any pair of objects Xand Y , and any pair of morphisms f : X1 → Y1 and g : X2 → Y2 in D, the tensor

is dened as:

X ⊗G Y = ∗(G(X ) ⊗′ G(Y )

), and

f ⊗G g = ∗(G( f ) ⊗′ G(g)

).

The associativity aG , the unit object IG , the left unitor λG and the right unitor

ρG are dened as ∗(a′), ∗(I′), ∗(λ ′) and ∗(ρ′) respectively. These morphisms

are isomorphisms which will be preserved under the action of functor ∗ hence

they will satisfy the pentagon and triangle axioms.

If B is symmetric then ΠΛB is symmetric. Therefore by Corollary 3.2

symmetric structure of ΠΛB is carried to D by the functor ∗.

In particular, suppose B is a subcategory of a category D such that we

have the same setup as in above theorem. That is, if B has a monoidal structure

⊗ along with two functors ∗ and G as dened above, then (B, ⊗) becomes a

monoidal subcategory of (D, ⊗G).

Definition 4.4. Let (B,Λ, ∗,G) be a base for a category D such that B has a

monoidal structure. We call the monoidal structure ⊗G on D as in Theorem 4.3

as induced or extended monoidal structure on D from its base B.

Let B be the category of nite dimensional vector spaces over R with usual

tensor product ⊗R of vector spaces. By Theorem 4.3, takingD = B we can think

of D as a base of itself with Λ = 1, and the functors G and ∗ as the identity

functor. We see that ⊗G = ⊗B .

Proposition 4.5. B is a dualizable category if and only if ΠΛB is a dualizablecategory.

Proof. Firstly, let us assume that B is a dualizable category with

(B, ⊗B, a, IB, l, r) as its monoidal structure. That means every object has a dual

in B. Let F : Λ → B be an object in ΠΛB. We dene a functor F ∗ : Λ → B

such that

(1) For any λ ∈ Λ, we have F ∗(λ) := F (λ)∗, here F (λ)∗ is the dual of F (λ)in B.

(2) F (Id ) = IdF ()∗


Clearly F ∗ is an object in ΠΛB. We claim that F ∗ is the dual of F in the

category ΠΛB with monoidal structure as in proposition 4.1.

We have natrual transformations ev: F ⊗′ F ∗ → IΠΛB , dened by

ev: F (λ) ⊗B F (λ)∗ → IB .

Here ev is the evaluation map in the monoidal categoryB for F (λ) and F (λ)∗.Similarly, we have coevaluation map in the category ΠΛB, given by natural

transformation, coev: IΠΛB → F ⊗′ F ∗ . It is dened by coevaluation maps

coev in the monoidal category B, given explicitly for each F (λ) and F (λ)∗ as

coev

: IB → F (λ) ⊗B F (λ)∗.

For each λ, F (λ)∗ is left dual of F (λ) inB and so the compositions of following

morphisms

F (λ)l−1F ( )−−−−→ IB⊗BF (λ)

coev ⊗B IdF ( )−−−−−−−−−−−−→ (F (λ)⊗BF ∗(λ))⊗BF (λ)

aF ( );F ∗ ( );F ( )−−−−−−−−−−−−→

F (λ)⊗B (F ∗(λ)⊗BF (λ))IdF ( )⊗B ev

−−−−−−−−−−→ F (λ)⊗BIBrF ( )−−−−→ F (λ)

and

F ∗(λ)r−1F ∗ ( )−−−−→ IB⊗BF

∗(λ)IdF ∗ ( )⊗B coev

−−−−−−−−−−−→F ∗(λ)⊗B (F (λ)⊗F ∗(λ))a−1F ∗ ( );F ( );F ∗ ( )−−−−−−−−−−−−→

(F ∗(λ)⊗BF (λ))⊗BF ∗(λ)ev ⊗B IdF ∗ ( )−−−−−−−−−−−→ IB⊗F

∗(λ)lF ∗ ( )−−−−→ F ∗(λ)

will be identities. This proves that F ∗ is the dual of F in (ΠΛB, ⊗′).Conversely, let X be an object in B. Fix λ0 ∈ Λ. Let us construct a functor

FX : Λ→ B as

FX (λ) =8><>:

X ; λ = λ0

IB; otherwise

with FX (Id0) = IdX and FX (Id ) = IdIB for λ , λ0. Then FX is an object in

ΠΛB which is a dualizable category. Let FX∗ be the dual of FX in ΠΛB with

evaluation and coevaluation morphisms being ev and coev respectively. These

morphisms are basically natural transformations which represents families of

morphisms corresponding to each λ ∈ Λ. For λ = λ0, consider the object

FX∗(λ0) in B. We claim this is the dual of X in B. We have the evaluation and

coevaluation morphisms as

ev0

: FX∗(λ0) ⊗B FX (λ0) → I′(λ0)

coev0

: I′(λ0) → FX (λ0) ⊗B FX ∗(λ0)


in B where I′(λ0) = IB . The required composition of morphisms as mentioned

in (4) and (5) will be identitites since they are identities at the level of natural

transformations also.

Our last result relates dualizability condition of a category D and its base

(B,Λ, ∗,G). Let the pasting functor ∗ satises the condition of Theorem 4.3.

This gives a tensor ⊗G on D as in Theorem 4.3. Our following theorem gives

the condition when D becomes dualizable using the dualizability of B.

Theorem 4.6. If the pasting functor ∗ satisfies

∗(F1 ⊗′ F2) = ∗(F1) ⊗G ∗(F2); and ∗ (I′) = IG(6)

then if (B, ⊗B ) is dualizable then (D, ⊗G) is dualizable.

Proof. Let X be an object in D. Then G(X ) is an object in ΠΛB which is a

dualizable category by proposition 4.5. So, let G(X )∗ be the dual object of G(X )with following morphisms

ev: G(X )∗ ⊗′ G(X ) → I′

coev: I′ → G(X ) ⊗′ G(X ) → I′

in ΠΛB. Using the fact that the pasting functor ∗ is monoidal and ∗G ∼= IG , wehave following morohisms in D:

∗(ev) : Y ⊗G X → IG∗(coev) : IG → X ⊗G Y

where Y = ∗ (G(X )∗) is the image of G(X )∗ under the pasting functor ∗. The

objectY acts as the dual to X inD since the required compositions of morphisms

as mentioned in (4) and (5) will be both identities in D.

In particular, let B be a sub category of D such that there are functors ∗, G

and an indexing categoryΛwhich makes (B,Λ, ∗,G) a base forD. If the functor

∗ satises the conditions in Theorem 4.3 and 4.6, then the category D with the

extended tensor product ⊗G is dualizable if the sub category B is dualizable.

References

[1] A. M. Cegarra, A. R. Garzon, J. A. Ortega, Graded extensions of monoidal

categories, J. Algebra, 241 (2001), 620657.

[2] Ch. Kassel, Quantum Groups, Graduate texts in mathematics; vol I. 155.


[3] J. Kock, Frobenius Algebras and 2D Topological Quantum Field Theories, Cam-

bridge University Press 2003.

[4] K. Ponto, M. Shulman, Traces in symmetric monoidal categories, Expo. Math.,

32 (2014), 248273.

[5] B. Day, C. Pastro, Note on Frobenius monoidal functors, New York J. Math., 14

(2008) 733742.

[6] S. Mac Lane, Categories for the Working Mathematician (2nd ed.), Berlin, New

York: Springer-Verlag.

[7] W. Imrich, S. Klavzar, Product Graphs: Structure and Recognition, Wiley

(2000).

Neha Gupta, Pradip Kumar


Shiv Nadar University, Dadri

U. P. 201314, India

neha.gupta,[email protected]

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