Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 459024 9 pageshttpdxdoiorg1011552013459024

Research ArticleThe Numerical Semigroup of Phrasesrsquo Lengths ina Simple Alphabet

Aureliano M Robles-Peacuterez1 and Joseacute Carlos Rosales2

1 Departamento de Matematica Aplicada Universidad de Granada 18071 Granada Spain2Departamento de Algebra Universidad de Granada 18071 Granada Spain

Correspondence should be addressed to Aureliano M Robles-Perez aroblesugres

Received 10 August 2013 Accepted 25 September 2013

Academic Editors R Esteban-Romero and J Rada

Copyright copy 2013 A M Robles-Perez and J C Rosales This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

LetA be an alphabet with two elements Considering a particular class of words (the phrases) over such an alphabet we connectwith the theory of numerical semigroups We study the properties of the family of numerical semigroups which arise from thisstarting point

1 Introduction

LetA be a nonempty finite set called the alphabet Elementsof A are called letters or symbols A word is a sequence ofletters which can be finite or infinite We denote by Alowast

(resp A120596) the set of all finite (resp infinite) words over AThe sequence of zero letters is called the empty word and isdenoted by 120576 Any subset L sube Alowast is called a language overA The length of a word 119906 is denoted by |119906| If 119906 V are wordswe define their product or concatenation as the word 119906V Wesay that a word 119906 is a factor of a word V if there exist twowords119909 119910 such that V = 119909119906119910 If 119906 is a factor of V with 119909 = 120576 (resp119910 = 120576) then 119906 is a prefix (resp suffix) of V

We have taken these definitions from [1] In this book(and the references given therein) the authors study prob-lems related to Combinatorics on Words However we aregoing to consider a different point of view We are interestedin a very particular type of words (the phrases) and morespecifically their length

Definition 1 Let us takeA = 119886S We say that 119891 isin Alowast is aphrase if it fulfills the following conditions

(1) S is not a prefix or suffix of 119891(2) SS is not a factor of 119891

We denoteAF= 119891 isin Alowast | 119891 is a phrase

If we consider that S represents a gap between twowords then we have a suitable justification for the abovedefinition

Let C be a language over A such that C sube AF We willdenote by ℓ(C) = |119888| | 119888 isin C In this work we are going todeal with the structure of the set ℓ(C) for particular choices ofC In fact let 119908

1 119908

119899 sube Alowast be a finite set of words such

thatS is not a factor of 119908119894 1 le 119894 le 119899 ThenC(119908

1 119908

119899) sube

AF is the language in which each phrase 119891 is obtained asproduct of factors belonging to 119908

1 119908

119899cupS Moreover

in order to achieve the results of this paper we assume that120576 isin C(119908

1 119908

119899)

Example 2 If we take 119886119886119886119886 119886119886119886119886119886 then 1198911= 119886119886119886119886119886119886119886119886

1198912

= 119886119886119886119886119886119886119886119886119886 and 1198913

= 119886119886119886119886119886S119886119886119886119886 belong toC(119886119886119886119886 119886119886119886119886119886) However 119891

4= 119886119886119886119886S119886119886119886119886119886119886 1198915 = 119886119886119886

and 1198916= 119886119886S119886 do not belong toC(119886119886119886119886 119886119886119886119886119886)

Let N be the set of nonnegative integers A numericalsemigroup is a subset 119878 of (N +) that is closed under additioncontains the zero element and such that N 119878 is finite

In Section 2 we will show that ℓ(C(1199081 119908

119899)) is a

numerical semigroupWewill also see that there exist numer-ical semigroups that cannot be obtained by this procedureThis fact allows us to give the following definition

2 The Scientific World Journal

Definition 3 Let A be the alphabet given by the set 119886SA numerical semigroup 119878 is the set of lengths of a language ofphrases (PL-semigroup for abbreviation) if there exists C =

C(1199081 119908

119899) sube AF such that 119878 = ℓ(C)

The next aim of Section 2 will be to characterize PL-semigroups Concretely we will show that a numericalsemigroup 119878 is a PL-semigroup if and only if 119909+119910+ 1 isin 119878 forall 119909 119910 isin 119878 0

Let 119878 be a numerical semigroup Since N 119878 is a finite setwe can consider two notable invariants of 119878 (see [2]) On theone hand the Frobenius number of 119878 is the maximum ofN 119878and is denoted by F(119878) On the other hand the genus of 119878 isthe cardinality of N 119878 and is denoted by g(119878)

A Frobenius variety is a nonempty familyV of numericalsemigroups that fulfills the following conditions

(1) if 119878 119879 isinV then 119878 cap 119879 isinV(2) if 119878 isinV and 119878 =N then 119878 cup F(119878) isinV

Let us denote SPL = 119878 | 119878 is a PL-semigroup InSection 3 we will show that SPL is a Frobenius variety Thisfact together with the results of [3] will allow us to showorderly the elements of SPL in a tree with root N Moreoverwe will also characterize the sons of a vertex in order to buildrecursively such a tree

The multiplicity of a numerical semigroup 119878 denotedby m(119878) is the minimum of 119878 0 We will study the setSPL(119898) = 119878 isin SPL | m(119878) = 119898 in Section 4 In particularwe will show that this set is finite and has maximum andminimum with respect to the inclusion order Furthermorewe will determine the sets F(119878) | 119878 isin SPL(119898) and g(119878) |119878 isin SPL(119898) We will also see that the elements of SPL(119898)can be ordered in a tree with root the numerical semigroup119878 = 0119898 rarr (where the symbol rarr means that everyinteger greater than119898 belongs to 119878)

In Section 5 we will see that a PL-semigroup is deter-mined perfectly by a nonempty finite set of positive inte-gers In addition we will show explicitly the smallest PL-semigroup that contains a given nonempty finite set ofpositive integers

We finish this introduction pointing out that this workadmits different generalizations Some of them are inworkingprocess and other ones have already been developed (see [4])

2 PL-Semigroups

If 119883 is a nonempty subset of N we denote by ⟨119883⟩ thesubmonoid of (N +) generated by119883 that is

⟨119883⟩ = 12058211199091+ sdot sdot sdot + 120582

119899119909119899| 119899 isin N 0

1199091 119909

119899isin 119883 120582

1 120582

119899isin N

(1)

It is a well-known fact (see for instance [5 Lemma 21]) that⟨119883⟩ is a numerical semigroup if and only if gcd119883 = 1

(as usual gcd means greatest common divisor) On the otherhand every numerical semigroup 119878 is finitely generated andtherefore there exists a finite subset119883 of 119878 such that 119878 = ⟨119883⟩In addition if no proper subset of119883 generates 119878 then we say

that 119883 is a minimal system of generators of 119878 In [5 Theorem27] it is proved that every numerical semigroup 119878 has aunique (finite) minimal system of generators The elementsof such a system are calledminimal generators of 119878

LetA be an alphabet and let 1199081 119908

119899 sube Alowast be a finite

set of words If C = C(1199081 119908

119899) sube Alowast is the language in

which each word is obtained as product of factors belongingto 1199081 119908

119899 then it is easy to see that ℓ(C) is a submonoid

of (N +) In addition if gcd|1199081| |119908

119899| = 1 then ℓ(C) is

a numerical semigroup Moreover it is a simple exercise toshow that we can get any numerical semigroup in this way

As we indicated in the introduction we are going to studythe particular case in which we consider lengths of phrasesConsequently we will focus our attention in a particularfamily of numerical semigroups

Proposition 4 Let A = 119886S be an alphabet If C =

C(1199081 119908

119899) sube AF then ℓ(C) is a numerical semigroup

Proof We proceed in three steps

(i) First of all being that 120576 isin C and |120576| = 0 we have that0 isin ℓ(C)

(ii) Now let us see that if 1198971 1198972isin ℓ(C) then 119897

1+1198972isin ℓ(C)

In effect let 1198911 1198912isin C such that |119891

1| = 1198971and |119891

2| =

1198972 Then the concatenation 119891

11198912(of 1198911and 119891

2) is an

element ofC with |11989111198912| = 1198971+ 1198972

(iii) Finally let 119891 isin C with |119891| = 0 (ie 119891 = 120576) Since119891S119891 isin C we have that |119891| 2|119891| + 1 sube ℓ(C) Bythe previous step we know that C is closed underaddition and consequently ⟨|119891| 2|119891| + 1⟩ sube ℓ(C)As gcd|119891| 2|119891| + 1 = 1 we have ⟨|119891| 2|119891| + 1⟩ is anumerical semigroup Therefore N ℓ(C) is finite

We conclude that ℓ(C) is a numerical semigroup

From now on unless another thing is stated we takeA =

119886S As in the introduction we say that a numerical semi-group 119878 is a PL-semigroup if there existsC = C(119908

1 119908

119899) sube

AF such that 119878 = ℓ(C) From Proposition 4 we deduce thatif 119878 is a PL-semigroup and 119909 isin 119878 0 then 2119909 + 1 isin 119878Consequently there exist numerical semigroups which arenot of this type For example 119878 = ⟨5 7 9⟩ is not a PL-semigroup because 2 sdot 5 + 1 = 11 notin 119878

In the next result we give a characterization of PL-semigroups

Theorem 5 Let 119878 be a numerical semigroup The followingconditions are equivalent

(1) 119878 is a 119875119871-semigroup(2) If 119909 119910 isin 119878 0 then 119909 + 119910 + 1 isin 119878

Proof (1 rArr 2) By hypothesis 119878 = ℓ(C(1199081 119908

119899)) for some

nonempty finite set 1199081 119908

119899 sube Alowast If 119909 119910 isin 119878 0 then

there exist 119891 119892 isin C(1199081 119908

119899) 120576 such that |119891| = 119909 and

|119892| = 119910 It is clear that 119891S119892 isin C(1199081 119908

119899) and |119891S119892| =

119909 + 119910 + 1 Therefore 119909 + 119910 + 1 isin 119878(2 rArr 1) Let 119899

1 119899

119901 be the minimal system of

generators of 119878 Let us take the set 1199081 119908

119901| |119908119894| = 119899

119894

The Scientific World Journal 3

1 le 119894 le 119901 Our aim is to show that if C = C(1199081 119908

119901)

then 119878 = ℓ(C)Since 119899

1 119899

119901 sube ℓ(C) by applying Proposition 4 we

have 119878 = ⟨1198991 119899

119901⟩ sube ℓ(C) Now let 119897 isin ℓ(C) In order to

prove that 119897 isin 119878 we are going to use induction over 119897 If 119897 = 0then the result is trivially true Let us assume that 119897 gt 0 andlet 119891 isin C such that |119891| = 119897 IfS is not a factor of 119891 then theresult follows immediately In other case there exist 119891

1 1198912isin

C 120576 such that 119891 = 1198911S1198912 By hypothesis of induction

|1198911| |1198912| isin 119878 Thereby 119897 = |119891| = |119891

1| + |1198912| + 1 isin 119878

Remark 6 The previous theorem leads to the concept ofnumerical semigroup that admit a linear nonhomogeneouspattern For a general study of this family of numericalsemigroups see for instance [6 7]

Let 119878 be a numerical semigroup with minimal system ofgenerators given by 119899

1 119899

119901 Following [8] if 119904 isin 119878 then

we define the order of 119904 (in 119878) by

ord (119904 119878) = max 1205721+ sdot sdot sdot + 120572

119901| 12057211198991+ sdot sdot sdot + 120572

119901119899119901= 119904

with1205721 120572

119901isin N

(2)

If no ambiguity is possible then we write ord(119904)

Remark 7 From [5 Lemma 23 and Theorem 27] we havethat if 119883 is the minimal system of generators of 119878 thenevery system of generators of 119878 contains 119883 Consequentlythe definition of ord(119904 119878) does not depend on the consideredsystem of generators that is it only depends on 119904 and 119878

Lemma 8 Let 119878 be a numerical semigroup with minimalsystem of generators given by 119899

1 119899

119901 and let 119904 isin 119878

(1) If 119894 isin 1 119901 and 119904 minus 119899119894isin 119878 then ord(119904 minus 119899

119894) le

ord(119904) minus 1(2) If 119904 = 120572

11198991+ sdot sdot sdot + 120572

119901119899119901 with ord(119904) = 120572

1+ sdot sdot sdot + 120572

119901

and 120572119894= 0 then ord(119904 minus 119899

119894) = ord(119904) minus 1

Proof (1)Assume that 119904minus119899119894= 12057311198991+sdot sdot sdot+120573

119901119899119901 with120573

1+sdot sdot sdot+

120573119901= ord(119904 minus 119899

119894) Then 119904 = 120573

11198991+ sdot sdot sdot + (120573

119894+ 1)119899119894+ sdot sdot sdot + 120573

119901119899119901

and thus ord(119904minus119899119894)+1 = 120573

1+ sdot sdot sdot+ (120573

119894+1)+ sdot sdot sdot+120573

119901le ord(119904)

(2) Since 119904minus119899119894= 12057211198991+sdot sdot sdot+(120572

119894minus1)119899119894+sdot sdot sdot+120572

119901119899119901 we have

ord(119904minus119899119894) ge 1205721+ sdot sdot sdot + (120572

119894minus1)+ sdot sdot sdot +120572

119901= ord(119904)minus1 Thereby

ord(119904) minus 1 le ord(119904 minus 119899119894) Now by applying the previous item

we conclude that ord(119904 minus 119899119894) = ord(119904) minus 1

In item (2) of the next proposition it is shown a charac-terization of PL-semigroups in terms of minimal systems ofgeneratorsThus we can decide if a numerical semigroup is aPL-semigroup in an easier way

Proposition 9 Let 119878 be a numerical semigroup with minimalsystem of generators given by 119899

1 119899

119901 The following

conditions are equivalent

(1) 119878 is a 119875119871-semigroup(2) If 119894 119895 isin 1 119901 then 119899

119894+ 119899119895+ 1 isin 119878

(3) If 119904 isin 119878 0 1198991 119899

119901 then 119904 + 1 isin 119878

(4) If 119904 isin 119878 0 then 119904 + 0 119900119903119889(119904) minus 1 sube 119878

Proof (1 rArr 2) It is an immediate consequence ofTheorem 5(2 rArr 3) If 119904 isin 119878 0 119899

1 119899

119901 then it is clear that there

exist 119894 119895 isin 1 119901 and 1199041015840 isin 119878 such that 119904 = 119899119894+119899119895+1199041015840Thus

119904 + 1 = (119899119894+ 119899119895+ 1) + 119904

1015840

isin 119878(3 rArr 4)We reason by induction over ord(119904) If ord(119904) =

1 then the result is trivially true Now let us assume thatord(119904) ge 2 and that 120572

1 120572

119901are nonnegative integers such

that 119904 = 12057211198991+ sdot sdot sdot + 120572

119901119899119901 ord(119904) = 120572

1+ sdot sdot sdot + 120572

119901 and

119886119894= 0 for some 119894 isin 1 119901 By Lemma 8 we know that

ord(119904 minus 119899119894) = ord(119904) minus 1 Then by hypothesis of induction

we have that 119904 minus 119899119894+ 0 ord(119904) minus 2 sube 119878 Therefore

119904+0 ord(119904)minus2 sube 119878Moreover (119904minus119899119894+ord(119904)minus2)+119899

119894+1 isin

119878 Thereby 119904 + 0 ord(119904) minus 1 sube 119878(4 rArr 1) If 119909 119910 isin 1198780 then it is clear that ord(119909+119910) ge 2

Thus we get that 119909 + 119910 + 1 isin 119878 By applying Theorem 5 wecan conclude that 119878 is a PL-semigroup

Example 10 Let 119878 = ⟨4 5 6⟩ that is let 119878 be the numericalsemigroup with minimal system of generators given by4 5 6 It is obvious that 4 + 4 + 1 = 9 4 + 5 + 1 = 104+6+1 = 11 5+5+1 = 11 5+6+1 = 12 and 6+6+1 = 13are elements of 119878 Therefore by applying Proposition 9 wehave that 119878 is a PL-semigroup

3 The Frobenius Variety of thePL-Semigroups

The following result is straightforward to prove and appearsin [5]

Lemma 11 Let 119878 119879 be numerical semigroups

(1) 119878 cap 119879 is a numerical semigroup(2) If 119878 =N then 119878 cup F(119878) is a numerical semigroup

Having inmind the definition of Frobenius variety whichwas given in the introduction we get the next result

Proposition 12 The set S119875119871= 119878 | 119878 is a 119875119871-semigroup is a

Frobenius variety

Proof First of all let us observe that N isin SPL and thereforeSPL is a nonempty set

Let 119878 119879 isin SPL In order to show that 119878 cap 119879 isin SPL weare going to use Theorem 5 So if 119909 119910 isin (119878 cap 119879) 0 then119909 119910 isin 119878 0 and 119909 119910 isin 119879 0 Therefore 119909 + 119910 + 1 isin 119878 cap 119879Consequently 119878 cap 119879 isin SPL

Now let 119878 isin SPL such that 119878 =N By applying Theorem 5again we are going to see that 119878 cup F(119878) isin SPL Let 119909 119910 isin

(119878 cup F(119878)) 0 If 119909 119910 isin 119878 then 119909 + 119910 + 1 isin 119878 sube 119878 cup F(119878)On the other hand if F(119878) isin 119909 119910 then 119909 + 119910 + 1 gt F(119878)and thereby 119909 + 119910 + 1 isin 119878 sube 119878 cup F(119878) We conclude that119878 cup F(119878) isin SPL

A graph 119866 is a pair (119881 119864) where 119881 is a nonempty set and119864 is a subset of (V 119908) isin 119881 times 119881 | V = 119908 The elements of

4 The Scientific World Journal

119881 are called vertices of 119866 and the elements of 119864 are callededges of 119866 A path (of length 119899) connecting the vertices 119909and 119910 of 119866 is a sequence of different edges of the form(V0 V1) (V1 V2) (V

119899minus1 V119899) such that V

0= 119909 and V

119899= 119910

We say that a graph 119866 is a tree if there exist a vertex Vlowast

(known as the root of 119866) such that for every other vertex 119909 of119866 there exists a unique path connecting 119909 and Vlowast If (119909 119910) isan edge of the tree then we say that 119909 is a son of 119910

We define the graph G(S119875119871) in the following way

(i) S119875119871

is the set of vertices of G(SPL)(ii) (119878 1198781015840) isin S

119875119871times S119875119871

is an edge of G(S119875119871) if 1198781015840 = 119878 cup

F(119878)As a consequence of [3 Proposition 24 Theorem 27] we

have the next result

Theorem 13 The graph G(S119875119871) is a tree with root equal to N

Moreover the sons of a vertex 119878 isin S119875119871

are 119878 1199091 119878

119909119903 where 119909

1 119909

119903are the minimal generators of 119878 that are

greater than F(119878) and such that 119878 1199091 119878 119909

119903 isin S119875119871

Let us observe that if 119878 is a numerical semigroup and119909 isin 119878 then 119878 119909 is a numerical semigroup if and only if119909 is a minimal generator of 119878 In fact 119878 119909 is a numericalsemigroup whenever 119909 isin 119878 0 and 119909 = 119910 + 119911 for all 119910 119911 isin119878 0 As a consequence if we denote by msg(119878) the minimalsystem of generators of 119878 then msg(119878) = (119878 0) ((119878 0) +(119878 0)) (see [5 Lemma 23] for other proof of this result)In the following proposition we obtain an analogous for PL-semigroups of the first commented fact in this paragraph

Proposition 14 Let 119878 be a 119875119871-semigroup and let 119909 be aminimal generator of 119878 Then 119878 119909 is a PL-semigroup if andonly if 119909 minus 1 isin 0 cup (N 119878) cup (msg(119878))

Proof (Necessity) If 119909 minus 1 notin 0 cup (N 119878) cup (msg(119878)) then119909 minus 1 isin 119878 (0 cup (msg(119878))) Accordingly there exist 119910 119911 isin119878 0 such that 119909 minus 1 = 119910 + 119911 In fact it is clear that 119910 119911 isin119878 119909 0 Therefore by applyingTheorem 5 and that 119878 119909 isa PL-semigroup we have 119909 = 119910 + 119911 + 1 isin 119878 119909 which is acontradiction

(Sufficiency) Let 119910 119911 isin 119878 119909 0 Since 119878 is a PL-semigroup by Theorem 5 we have 119910 + 119911 + 1 isin 119878 As 119909 minus 1 isin0 cup (N 119878) cup (msg(119878)) we deduce that 119910 + 119911 + 1 = 119909 Thus119910+119911+1 isin 119878 119909 By applyingTheorem 5 again we concludethat 119878 119909 is a PL-semigroup

As a consequence of the previous proposition we have thenext result

Corollary 15 Let 119878 be a119875119871-semigroup such that 119878 =N and let119909 be a minimal generator of 119878 greater than F(119878) Then 119878 119909 isa 119875119871-semigroup if and only if 119909 minus 1 isin msg(119878) cup F(119878)

By applying Theorem 13 together with Corollary 15 wecan get the sons of a vertex of G(SPL) as is shown in thefollowing example

Example 16 It is clear that 119878 = ⟨4 6 7 9⟩ is a PL-semigroupwith Frobenius number equal to 5 From Theorem 13 and

Corollary 15 we deduce that the sons of 119878 are 119878 6 =

⟨4 7 9 10⟩ and 119878 7 = ⟨4 6 9 11⟩

Let us observe that we can build recursively a tree fromthe root if we know the sons of each vertexTherefore we canbuild the tree G(SPL) such as it is shown in Figure 1

In order to have an easier making of the tree G(SPL)we are going to study the relation between the minimalgenerators of a numerical semigroup 119878 and the minimalgenerators of 119878 119909 where 119909 is a minimal generator of119878 that is greater than F(119878) First of all let us observe thatif 119878 is minimally generated by 119898119898 + 1 2119898 minus 1 (ie119878 = 0119898 rarr ) then 119878 119898 = 0119898 + 1 rarr is minimallygenerated by 119898+1119898+2 2119898+1 In other case we havethe following result

Proposition 17 Let 119878 be a numerical semigroup withmsg(119878) = 119899

1 119899

119901 If m(119878) = 119899

1lt 119899119901and 119899

119901gt F(S)

then 119878 119899119901 = ⟨119899

1 119899

119901minus1 119899119901+ 1198991⟩

Proof Let us take 119894 isin 2 119901 Since 119899119901gt F(119878) and 119899

1lt 119899119894

we have that 119899119901+ 119899119894minus 1198991isin 119878 Thus 119899

119901+ 119899119894minus 1198991= 12057211198991+

sdot sdot sdot + 120572119901119899119901for some 120572

1 120572

119901isin N Thereby 119899

119901+ 119899119894= (1205721+

1)1198991+ sdot sdot sdot + 120572

119901119899119901 By applying that 119899

1 119899

119901 is a minimal

system of generators we have that 120572119901= 0Therefore 119899

119901+119899119894isin

⟨1198991 119899

119901minus1⟩ In particular 2119899

119901isin ⟨1198991 119899

119901minus1⟩

Now let 119904 isin 119878 119899119901 Then 119904 isin 119878 and thus there exist

1205731 120573

119901isin N such that 119904 = 120573

11198991+ sdot sdot sdot + 120573

119901119899119901 Since 2119899

119901isin

⟨1198991 119899

119901minus1⟩ we can assume that 120573

119901isin 0 1 On the one

hand if 120573119901= 0 then 119904 isin ⟨119899

1 119899

119901minus1⟩ On the other hand if

120573119901= 1 then there exists 119894 isin 1 119901minus1 such that120573

119894= 0 If 119894 =

1 then it is obvious that 119904 isin ⟨1198991 119899

119901minus1 119899119901+1198991⟩ And if 119894 = 1

since 119899119901+ 119899119894isin ⟨1198991 119899

119901minus1⟩ we have that 119904 isin ⟨119899

1 119899

119901minus1⟩

In any case we conclude that 119878 119899119901 = ⟨119899

1 119899

119901minus1 119899119901+

1198991⟩

Corollary 18 Let 119878 be a numerical semigroup withmsg(119878) =1198991 119899

119901 Ifm(119878) = 119899

1lt 119899119901and 119899119901gt F(119878) then

msg (119878 119899119901)

=

1198991 119899

119901minus1

119894119891 119905ℎ119890119903119890 119890119909119894119904119905119904 119894 isin 2 119901 minus 1

119904119906119888ℎ 119905ℎ119886119905 119899119901+ 1198991minus 119899119894isin 119878

1198991 119899

119901minus1 119899119901+ 1198991 119894119899 119900119905ℎ119890119903 119888119886119904119890

(3)

Proof From Proposition 17 we deduce that msg(119878 119899119901) is

1198991 119899

119901minus1 or 119899

1 119899

119901minus1 119899119901+ 1198991 In addition msg(119878

119899119901) = 119899

1 119899

119901minus1 if and only if 119899

119901+ 1198991isin ⟨1198991 119899

119901minus1⟩

If 119899119901+ 1198991isin ⟨1198991 119899

119901minus1⟩ then there exist 120572

1 120572

119901minus1isin

N such that 119899119901+1198991= 12057211198991+ sdot sdot sdot + 120572

119901minus1119899119901minus1

Since 1198991 119899

119901

is a minimal system of generators we get that 1205721= 0 Thus

there exists 119894 isin 2 119901 minus 1 such that 120572119894= 0 Consequently

119899119901+ 1198991minus 119899119894isin 119878

Conversely if there exists 119894 isin 2 119901 minus 1 such that 119899119901+

1198991minus 119899119894isin 119878 then 119899

119901+ 1198991minus 119899119894= 12057311198991+ sdot sdot sdot + 120573

119901119899119901for some

1205731 120573

119901isin NThereby 119899

119901+1198991= 12057311198991+sdot sdot sdot+ (120573

119894+1)119899119894+sdot sdot sdot+

120573119901119899119901 Since 119899

1 119899

119901 is a minimal system of generators we

have that 120573119901= 0 and therefore 119899

119901+ 1198991isin ⟨1198991 119899

119901minus1⟩

The Scientific World Journal 5

⟨5 6 7 8 9⟩ ⟨4 6 7 9⟩ ⟨4 5 7⟩ ⟨4 5 6⟩ ⟨3 7 8⟩

⟨3 5 7⟩

⟨3 4 5⟩

⟨3 4⟩

⟨2 3⟩

⟨2 5⟩

⟨4 5 6 7⟩

⟨1⟩ = N

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

Figure 1

We finish this section with an illustrative example aboutthe above corollary

Example 19 Let 119878 be the numerical semigroupwithmsg(119878) =3 5 7 It is obvious that F(119878) = 4 By Proposition 17 weknow that 1198785 = ⟨3 7 8⟩ In addition 8minus7 = 1 notin 119878TherebyapplyingCorollary 18 we have thatmsg(1198785) = 3 7 8 Onthe other hand applying Proposition 17 again we have that119878 7 = ⟨3 5 10⟩ Finally since 10 minus 5 = 5 isin 119878 we concludethat msg(119878 7) = 3 5

4 PL-Semigroups with a Fixed Multiplicity

Let 119898 be a positive integer We will denote by Δ(119898) =

0119898 rarr It is clear that Δ(119898) is the greatest (with respectto set inclusion) PL-semigroup with multiplicity119898 Our firstaim in this section will be to show that there also exists thesmallest (with respect to set inclusion) PL-semigroup withmultiplicity119898

As an immediate consequence of item (4) inProposition 9 we have the next result

Lemma20 If 119878 is a119875119871-semigroup119898 isin 1198780 and 119896 isin N0then 119896119898 + 119894 isin 119878 for all 119894 isin 0 119896 minus 1

Proposition 21 Let 119898 isin N 0 Then the numericalsemigroup generated by (119894 + 1)119898 + 119894 | 119894 isin 0 119898 minus 1 isthe smallest (with respect to set inclusion) 119875119871-semigroup withmultiplicity119898

Proof Let 119878 = ⟨119898 2119898+1 1198982+(119898minus1)⟩ FromLemma 20we know that any PL-semigroup with multiplicity 119898 has tocontain 119878 In order to conclude the proof we will show that 119878is a PL-semigroup For this purpose since (119894 + 1)119898 + 119894 | 119894 isin

0 119898minus1 is a system of generators of 119878 it will be enoughto check item (2) of Proposition 9 that is if 119894 119895 isin 0 119898 minus

1 then (119894 + 1)119898+ 119894 + (119895 + 1)119898+ 119895+ 1 isin 119878 We distinguish twocases

(1) If 119894+ 119895+1 le 119898minus1 then (119894+1)119898+119894+ (119895+1)119898+119895+1 =(119894 + 119895 + 2)119898 + (119894 + 119895 + 1) isin 119878

(2) If 119894 + 119895 + 1 ge 119898 then (119894 + 1)119898 + 119894 + (119895 + 1)119898 + 119895 + 1 =(119894 + 119895 + 2)119898 + (119894 + 119895 + 1) = (119898+ 1)119898+ (119894 + 119895 minus119898+ 2)119898

+ (119894 + 119895 minus 119898 + 1) isin 119878

We will denote byΘ(119898) = ⟨119898 2119898 + 1 1198982

+ (119898 minus 1)⟩

and bySPL(119898) the set of all PL-semigroups with multiplicityequal to 119898 Let us recall that Δ(119898) = max(SPL(119898)) andΘ(119898) = min(SPL(119898))

As an application of the above comment we have the nextresult

Corollary 22 The set SPL(119898) is finite

Proof If 119878 isin SPL(119898) thenΘ(119898) sube 119878 sube Δ(119898) SinceΔ(119898) andΘ(119898) are numerical semigroups we have that Δ(119898) Θ(119898) isfinite Consequently SPL(119898) is also finite

Remark 23 The previous result can be considered a particu-lar case of [6 Theorem 66]

Now we are interested in computing the Frobeniusnumber and the genus ofΘ(119898) For that several concepts andresults are introduced

If 119878 is a numerical semigroup and 119898 isin 119878 0 then theApery set of119898 in 119878 (see [9]) is Ap(119878 119898) = 119904 isin 119878 | 119904 minus 119898 notin 119878It is clear (see for instance [5 Lemma 24]) that Ap(119878 119898) =120596(0) = 0 120596(1) 120596(119898 minus 1) where 120596(119894) is for each 119894 isin0 119898 minus 1 the least element of 119878 that is congruent with 119894modulo119898

The next result is [5 Proposition 212]

Lemma24 Let 119878 be a numerical semigroup and let119898 isin 1198780Then

(1) F(119878) = max(Ap(119878 119898)) minus 119898(2) g(119878) = (1119898)(sum

119908isinAp(119878119898) 119908) minus ((119898 minus 1)2)

If 119886 119887 are integers with 119887 = 0 we denote by 119886 mod 119887 theremainder of the division of 119886 by 119887 The following result is [5Proposition 35]

6 The Scientific World Journal

Lemma 25 Let 119898 isin N 0 and let 119883 = 120596(0) =

0 120596(1) 120596(119898minus1) sube N such that for each 119894 isin 1 119898minus1120596(119894) is congruent with 119894 modulo 119898 Let 119878 be the numericalsemigroup generated by 119883 cup 119898 The following conditions areequivalent

(1) Ap(119878 119898) = 119883(2) 120596(119894)+120596(119895) ge 120596((119894+119895) mod 119898) for all 119894 119895 isin 1 119898minus

1

Proposition 26 If119898 isin N 0 then

Ap (Θ (119898) 119898)

= 120596 (0) = 0 120596 (1) = 2119898 + 1 120596 (119898 minus 1) = 1198982

+ 119898 minus 1

(4)

Proof It is clear that 120596(119894) = (119894 + 1)119898 + 119894 is congruent with 119894modulo 119898 for all 119894 isin 1 119898 minus 1 Let us see now that if119894 119895 isin 1 119898 minus 1 then 120596(119894) + 120596(119895) ge 120596((119894 + 119895) mod 119898)Indeed 120596(119894) +120596(119895) = (119894 + 1)119898 + 119894 + (119895 + 1)119898 + 119895 gt (119894 + 119895 + 1)119898+ (119894 + 119895) ge ((119894 + 119895) mod 119898 + 1)119898 + (119894 + 119895) mod 119898 = 120596((119894 +119895) mod 119898) The proof follows from Lemma 25

Corollary 27 If119898 isin N 0 1 then

(1) F(Δ(119898)) = 119898 minus 1(2) g(Δ(119898)) = 119898 minus 1(3) F(Θ(119898)) = 1198982 minus 1(4) g(Θ(119898)) = (119898 minus 1)(119898 + 2)2

Proof Items (1) and (2) are trivial On the other hand items(3) and (4) are immediate consequences of Lemma 24 andProposition 26

Remark 28 Thenumerical semigroupΘ(119898) can be rewrittenas

Θ (119898)

= 119898119898+ (119898+ 1) 119898 +2 (119898 + 1) 119898 + (119898 minus 1) (119898 + 1)

(5)

Thus Θ(119898) is a numerical semigroup generated by an arith-metic sequencewith first term119898 and commondifference119898+1(see [2 10])

If 119878 is a numerical semigroup then the cardinality of theminimal system of generators of 119878 is called the embeddingdimension of 119878 and is denoted by e(119878) It is well known (see [5Proposition 212]) that e(119878) le m(119878) We say that a numericalsemigroup 119878 has maximal embedding dimension if e(119878) =m(119878) It is clear that 119898119898 + 1 2119898 minus 1 is the minimalsystem of generators of Δ(119898) Therefore Δ(119898) is a numericalsemigroupwithmaximal embedding dimensionNowwewillshow that Θ(119898) has also maximal embedding dimension

The next result is [5 Corollary 36]

Lemma 29 Let 119878 be a numerical semigroup with multiplicity119898 and assume thatAp(119878 119898) = 120596(0) = 0 120596(1) 120596(119898minus1)

Then 119878 has maximal embedding dimension if and only if 120596(119894)+120596(119895) gt 120596((119894 + 119895) mod 119898) for all 119894 119895 isin 1 119898 minus 1

Let us observe that in the proof of Proposition 26 wehave shown that 120596(119894) + 120596(119895) gt 120596((119894 + 119895) mod 119898) for all119894 119895 isin 1 119898 minus 1 Therefore by applying Lemma 29 weget the following result

Corollary 30 If 119898 isin N 0 then Θ(119898) is a numericalsemigroup with maximal embedding dimension

As a consequence of this corollary we have that 119898 2119898 +

1 1198982

+119898minus1 is theminimal systemof generators ofΘ(119898)Now we want to show orderly the elements of SPL(119898)

Thus we define the graph G(SPL(119898)) in the following way

(i) SPL(119898) is the set of vertices of G(SPL(119898))

(ii) (119878 1198781015840) isin SPL(119898) timesSPL(119898) is an edge of G(SPL(119898)) if1198781015840

= 119878 cup F(119878)

The next result is analogous to Theorem 13

Theorem 31 The graph G(S119875119871(119898)) is a tree with root equal

to Δ(119898) Moreover the sons of a vertex 119878 isin S119875119871(119898) are 119878

1199091 119878119909

119903 with 119909

1 119909

119903= 119909 isin msg(119878) | 119909 =119898 119909 gt

F(119878) 119886119899119889 119878 119909 isin S119875119871

By applyingTheorem 31 and Corollaries 15 and 18 we canget easily G(SPL(119898)) such as is shown in the next example

Example 32 We are going to depict G(SPL(3)) that is thetree of the PL-semigroups with multiplicity equal to 3

⟨3 7 8⟩

⟨3 5 7⟩ ⟨3 4⟩

⟨3 4 5⟩ = Δ(3)

⟨3 7 11⟩ = Θ(3)

If119879 = (119881 119864) is a tree then the height of119879 is themaximumof the lengths of the paths that connect each vertex with theroot Let us observe that the height of G(SPL(3)) is 3 Ingeneral the height of the tree G(SPL(119898)) is equal to

g (Θ (119898)) minus g (Δ (119898)) = (119898 minus 1) (119898 + 2)

2

minus (119898 minus 1)

=

(119898 minus 1)119898

2

(6)

Let us study now the possible values of the Frobeniusnumber and the genus for PL-semigroups with multiplicity119898

The Scientific World Journal 7

Proposition 33 If119898 isin N 0 then

(1) g(119878) | 119878 isin SPL(119898) = 119909 isin N | 119898 minus 1 le 119909 le

(119898 minus 1)(119898 + 2)2

(2) F(119878) | 119878 isin SPL(119898) = (Δ(119898) Θ(119898)) cup 119898 minus 1

Proof Let us assume that Δ(119898) Θ(119898) = 1199091gt 1199092gt sdot sdot sdot gt

119909120588(1) By Corollary 27 we know that

g (119878) | 119878 isin SPL (119898) sube 119898 minus 1

(119898 minus 1) (119898 + 2)

2

(7)

For the opposite inclusion it is enough to observe thatΘ(119898)cup119909119894 rarr isin SPL(119898) and that g(Θ(119898)cup119909119894 rarr ) = (119898minus1)(119898+

2)2 minus 119894(2) It is clear that Θ(119898) cup 119909

119894+ 1 rarr isin SPL(119898) with

Frobenius number equal to 119909119894 Thus (Δ(119898) Θ(119898)) cup 119898 minus

1 sube F(119878) | 119878 isin SPL(119898) For the other inclusion let ustake 119878 isin SPL(119898) such that 119878 = Δ(119898) Then F(119878) gt 119898 andthereby F(119878) isin Δ(119898) Since Θ(119898) sube 119878 we have F(119878) notin Θ(119878)Therefore we conclude that F(119878) isin Δ(119898) Θ(119878)

Example 34 By Proposition 33 g(119878) | 119878 isin SPL(3) =

2 3 4 5 Since Δ(3) = ⟨3 4 5⟩ and Θ(3) = ⟨3 7 11⟩ wehave that (Δ(3) Θ(3)) cup 2 = 2 4 5 8 Therefore byapplying Proposition 33 again we conclude that F(119878) | 119878 isinSPL(3) = 2 4 5 8

5 The Smallest PL-Semigroup That Contains aGiven Set of Positive Integers

Let us observe that in general the infinite intersection ofelements of SPL is not a numerical semigroup For instance⋂119899isinN0 119899 rarr = 0 On the other hand it is clear that the

(finite or infinite) intersection of numerical semigroups isalways a submonoid of (N +)

Let 119872 be a submonoid of (N +) We will say that 119872 isa SPL-monoid if it can be expressed like the intersection ofelements of SPL

The next lemma has an immediate proof

Lemma35 The intersection ofS119875119871-monoids is aS

119875119871-monoid

In view of this result we can give the following definition

Definition 36 Let 119883 be a subset of N The SPL-monoidgenerated by119883 (denoted bySPL(119883)) is the intersection of allSPL-monoids containing119883

If 119872 = SPL(119883) then we will say that 119883 is a SPL-systemof generators of119872 In addition if no proper subset of 119883 is aSPL-system of generators of 119872 then we will say that 119883 is aminimal SPL-system of generators of119872

Let us recall that by Proposition 12 we know that S119875119871

isa Frobenius variety Therefore by applying [3 Corollary 19]we have the next result

Proposition 37 Every S119875119871-monoid has a unique minimal

S119875119871-system of generators which in addition is finite

The proof of the following lemma is also immediate

Lemma 38 If 119883 sube N then S119875119871(119883) is the intersection of all

119875119871-semigroups that contain 119883

Proposition 39 If 119883 is a nonempty subset of N 0 thenS119875119871(119883) is a 119875119871-semigroup

Proof We know that SPL(119883) is a submonoid of (N +)Therefore in order to show that SPL(119883) is a numericalsemigroup it will be enough to see that N SPL(119883) is a finiteset

Let 119909 isin 119883 If 119878 is a PL-semigroup containing 119883 then (byTheorem 5) we know that 119909 2119909 + 1 sube 119878 and in this way⟨119909 2119909 + 1⟩ sube 119878 From Lemma 38 we have that ⟨119909 2119909 + 1⟩ subeSPL(119883) Since gcd119909 2119909 + 1 = 1 we get that ⟨119909 2119909 + 1⟩is a numerical semigroup and thus N ⟨119909 2119909 + 1⟩ is finiteConsequently N SPL(119883) is finite

Now let us see that SPL(119883) is a PL-semigroup Let 119909 119910 isinSPL(119883) 0 If 119878 is a PL-semigroup containing 119883 fromLemma 38 we deduce that 119909 119910 isin 1198780 and fromTheorem 5we have that 119909 + 119910 + 1 isin 119878 By applying again Lemma 38we have that 119909 + 119910 + 1 isin SPL(119883) Therefore by applyingTheorem 5 once more we can assert that SPL(119883) is a PL-semigroup

Remark 40 Let us observe that in general Proposition 39 isnot true for Frobenius varieties In fact let S be the set of allnumerical semigroups It is clear thatS is a Frobenius varietyIf we take 119883 = 2 then the intersection of all elements of Scontaining 2 is exactly119872 = ⟨2⟩ which is not a numericalsemigroup

The next result will be key for our last purpose in thissection

Theorem 41 S119875119871

= S119875119871(119883) | 119883 is a nonempty finite

subset 119900119891 N 0

Proof By Proposition 39 we have that

SPL (119883) | 119883 is a nonempty finite subset ofN 0 sube SPL(8)

For the other inclusion it is enough to observe that if 119878 isin SPLthen (by Proposition 37) there exists a nonempty finite subset119883 of N 0 such that 119878 = SPL(119883)

Since SPL is a Frobenius variety by applying [3 Proposi-tion 24] we get the next result

Proposition 42 Let119872 be aS119875119871-monoid and let 119909 isin 119872Then

119872119909 is aS119875119871-monoid if and only if 119909 belongs to the minimal

S119875119871-system of generators of119872

As an immediate consequence of this propositionwe havethe following result

8 The Scientific World Journal

Corollary 43 Let 119883 be a nonempty subset of N 0 Thenthe minimal S

119875119871-system of generators of S

119875119871(119883) is 119909 isin 119883 |

S119875119871(119883) 119909 is a PL-semigroup

Example 44 By Proposition 21 119878 = ⟨3 7 11⟩ is a PL-semigroup By applying Proposition 14 we easily deduce that

119909 isin 3 7 11 | 119878 119909 is a PL-semigroup = 3 (9)

Therefore 119878 = SPL(3) and 3 is the minimalSPL-system ofgenerators of 119878

Now we want to describe SPL(119883) when 119883 is a fixednonempty finite set of positive integers Let us observe thatby Theorem 41 we know that every PL-semigroup can beobtained in this way

Let 1198991 119899

119901be positive integers We will denote by

119878(1198991 119899

119901) the set 120572

11198991+ sdot sdot sdot + 120572

119901119899119901+ 119903 | 119903 120572

1 120572

119901isin

N 119903 lt 1205721+ sdot sdot sdot + 120572

119901 cup 0 Our next purpose will be to show

that 119878(1198991 119899

119901) = SPL(1198991 119899119901)

Lemma 45 Let 119878 be a numerical semigroup let 1199041 119904

119905isin

119878 0 and let 1205721 120572

119905isin N Then ord(120572

11199041+ sdot sdot sdot + 120572

119905119904119905) ge

1205721+ sdot sdot sdot + 120572

119905

Proof Let 1198991 119899

119901 be the minimal system of generators

of 119878 Then for each 119894 isin 1 119905 there exist 1205731198941 120573

119894119901isin N

such that 119904119894= 12057311989411198991+ sdot sdot sdot + 120573

119894119901119899119901 Moreover since 119904

119894= 0 we

have that 1205731198941+ sdot sdot sdot + 120573

119894119901ge 1 Thus

12057211199041+ sdot sdot sdot + 120572

119905119904119905

= 1205721(120573111198991+ sdot sdot sdot + 120573

1119901119899119901) + sdot sdot sdot + 120572

119905(12057311990511198991+ sdot sdot sdot + 120573

119905119901119899119901)

= (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) 1198991+ sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901) 119899119901

(10)

Therefore

ord (12057211199041+ sdot sdot sdot + 120572

119905119904119905)

ge (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) + sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901)

= 1205721(12057311+ sdot sdot sdot + 120573

1119901) + sdot sdot sdot + 120572

119905(1205731199051+ sdot sdot sdot + 120573

119905119901)

ge 1205721+ sdot sdot sdot + 120572

119905

(11)

Theorem 46 If 1198991 119899

119901are positive integers then

119878(1198991 119899

119901) is the smallest (with respect to set inclusion)

119875119871-semigroup containing 1198991 119899

119901

Proof We divide the proof into five steps

(i) Let us see that if 119909 119910 isin 119878(1198991 119899

119901) 0 then 119909 +

119910 isin 119878(1198991 119899

119901) In effect we know that there exist

1205721 120572

119901 1205731 120573

119901 119903 1199031015840 nonnegative integers such

that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903 119910 = 120573

11198991+sdot sdot sdot+120573

119901119899119901+1199031015840

119903 lt 1205721+ sdot sdot sdot + 120572

119901 and 1199031015840 lt 120573

1+ + 120573

119901 Therefore

119909 + 119910 = (1205721+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840 with119903 + 1199031015840

lt (1205721+ 1205731) + sdot sdot sdot + (120572

119901+ 120573119899) Consequently

119909 + 119910 isin 119878(1198991 119899

119901)

(ii) Let us see that N 119878(1198991 119899

119901) is finite Since 119899

1=

1sdot1198991+0sdot1198992+sdot sdot sdot+0sdot119899

119901+0 and 2119899

1+1 = 2sdot119899

1+0sdot1198992+sdot sdot sdot+0sdot

119899119901+1 we have that 119899

1 21198991isin 119878(1198991 119899

119901) By applying

the first step we get that ⟨1198991 21198991⟩ sube 119878(119899

1 119899

119901)

Using the same reasoning as we did in the proof ofProposition 39 we have the result

(iii) From the previous steps we know that 119878(1198991 119899

119901)

is a numerical semigroup Let us see now that119878(1198991 119899

119901) is a PL-semigroup In order to do that

it is enough (by Theorem 5) to show that if 119909 119910 isin

119878(1198991 119899

119901) 0 then 119909 + 119910 + 1 isin 119878(119899

1 119899

119901)

Indeed arguing as in the first step we have that 119909 +119910 + 1 = (120572

1+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840

+ 1

with 119903 + 1199031015840 + 1 lt (1205721+1205731) + sdot sdot sdot + (120572

119901+120573119899) Therefore

119909 + 119910 + 1 isin 119878(1198991 119899

119901)

(iv) Following the proof of the second step it is clear that1198991 119899

119901 sube 119878(119899

1 119899

119901)

(v) Finally let us see that 119878(1198991 119899

119901) is the small-

est PL-semigroup that contains 1198991 119899

119901 In fact

we will show that if 119879 is PL-semigroup containing1198991 119899

119901 then 119878(119899

1 119899

119901) sube 119879 Thus let 119909 isin

119878(1198991 119899

119901) 0 Then there exist 120572

1 120572

119901 119903 isin N

such that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903with 119903 lt 120572

1+sdot sdot sdot+120572

119901

Since 1198991 119899

119901 sube 119879 by Proposition 9 we have that

12057211198991+sdot sdot sdot+120572

119901119899119901+0 ord(120572

11198991+sdot sdot sdot+120572

119901119899119901)minus1 sube 119879

By applying Lemma 45 we have that 119903 lt ord(12057211198991+

sdot sdot sdot + 120572119901119899119901) and therefore 119909 isin 119879

In this way we have proved the statement

The next result is an immediate consequence of theprevious theorem

Corollary 47 If 1198991 119899

119901are positive integers then

SPL(1198991 119899119901) = 119878(1198991 119899119901)

We finish this section with an example that illustrates itscontent

Example 48 It is clear that 119878(4 7) = 0 4 7 8 9 11 rarr =

⟨4 7 9⟩ Therefore SPL(4 7) = ⟨4 7 9⟩

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referee for hisher usefulcomments and suggestions that helped to improve this workBoth of the authors are supported by FQM-343 (Junta deAndalucıa) MTM2010-15595 (MICINN Spain) and FEDERfunds The second author is also partially supported by Juntade AndalucıaFeder Grant no FQM-5849

The Scientific World Journal 9

References

[1] M Lothaire Applied Combinatorics on Words vol 105 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press 2005

[2] J L Ramırez Alfonsın The Diophantine Frobenius ProblemOxford University Press 2005

[3] J C Rosales ldquoFamilies of numerical semigroups closed underfinite intersections and for the Frobenius numberrdquo HoustonJournal of Mathematics vol 34 no 2 pp 339ndash348 2008

[4] J C Rosales M B Branco and D Torrao ldquoBracelet monoidsand numerical semigroupsrdquo preprint

[5] J C Rosales and P A Garcıa-Sanchez Numerical Semigroupsvol 20 of Developments in Mathematics Springer New YorkNY USA 2009

[6] M Bras-Amoros P A Garcıa-Sanchez and A Vico-OtonldquoNonhomogeneous patterns on numerical semigroupsrdquo Inter-national Journal of Algebra and Computation vol 23 no 6 pp1469ndash1483 2013

[7] K Stokes and M Bras-Amoros ldquoLinear non-homogeneoussymmetric patterns and prime power generators in numeri-cal semigroups associated to combinatorial congurationsrdquo toappear in Semigroup Forum 2013

[8] L Bryant ldquoGoto numbers of a numerical semigroup ring andthe gorensteiness of associated graded ringsrdquo Communicationsin Algebra vol 38 no 6 pp 2092ndash2128 2010

[9] R Apery ldquoSur les branches superlineaires des courbesalgebriquesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol222 pp 1198ndash1200 1946

[10] J B Roberts ldquoNote on linear formsrdquo Proceedings of the Ameri-can Mathematical Society vol 7 pp 465ndash469 1956

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2 The Scientific World Journal

Definition 3 Let A be the alphabet given by the set 119886SA numerical semigroup 119878 is the set of lengths of a language ofphrases (PL-semigroup for abbreviation) if there exists C =

C(1199081 119908

119899) sube AF such that 119878 = ℓ(C)

The next aim of Section 2 will be to characterize PL-semigroups Concretely we will show that a numericalsemigroup 119878 is a PL-semigroup if and only if 119909+119910+ 1 isin 119878 forall 119909 119910 isin 119878 0

Let 119878 be a numerical semigroup Since N 119878 is a finite setwe can consider two notable invariants of 119878 (see [2]) On theone hand the Frobenius number of 119878 is the maximum ofN 119878and is denoted by F(119878) On the other hand the genus of 119878 isthe cardinality of N 119878 and is denoted by g(119878)

A Frobenius variety is a nonempty familyV of numericalsemigroups that fulfills the following conditions

(1) if 119878 119879 isinV then 119878 cap 119879 isinV(2) if 119878 isinV and 119878 =N then 119878 cup F(119878) isinV

Let us denote SPL = 119878 | 119878 is a PL-semigroup InSection 3 we will show that SPL is a Frobenius variety Thisfact together with the results of [3] will allow us to showorderly the elements of SPL in a tree with root N Moreoverwe will also characterize the sons of a vertex in order to buildrecursively such a tree

The multiplicity of a numerical semigroup 119878 denotedby m(119878) is the minimum of 119878 0 We will study the setSPL(119898) = 119878 isin SPL | m(119878) = 119898 in Section 4 In particularwe will show that this set is finite and has maximum andminimum with respect to the inclusion order Furthermorewe will determine the sets F(119878) | 119878 isin SPL(119898) and g(119878) |119878 isin SPL(119898) We will also see that the elements of SPL(119898)can be ordered in a tree with root the numerical semigroup119878 = 0119898 rarr (where the symbol rarr means that everyinteger greater than119898 belongs to 119878)

In Section 5 we will see that a PL-semigroup is deter-mined perfectly by a nonempty finite set of positive inte-gers In addition we will show explicitly the smallest PL-semigroup that contains a given nonempty finite set ofpositive integers

We finish this introduction pointing out that this workadmits different generalizations Some of them are inworkingprocess and other ones have already been developed (see [4])

2 PL-Semigroups

If 119883 is a nonempty subset of N we denote by ⟨119883⟩ thesubmonoid of (N +) generated by119883 that is

⟨119883⟩ = 12058211199091+ sdot sdot sdot + 120582

119899119909119899| 119899 isin N 0

1199091 119909

119899isin 119883 120582

1 120582

119899isin N

(1)

It is a well-known fact (see for instance [5 Lemma 21]) that⟨119883⟩ is a numerical semigroup if and only if gcd119883 = 1

(as usual gcd means greatest common divisor) On the otherhand every numerical semigroup 119878 is finitely generated andtherefore there exists a finite subset119883 of 119878 such that 119878 = ⟨119883⟩In addition if no proper subset of119883 generates 119878 then we say

that 119883 is a minimal system of generators of 119878 In [5 Theorem27] it is proved that every numerical semigroup 119878 has aunique (finite) minimal system of generators The elementsof such a system are calledminimal generators of 119878

LetA be an alphabet and let 1199081 119908

119899 sube Alowast be a finite

set of words If C = C(1199081 119908

119899) sube Alowast is the language in

which each word is obtained as product of factors belongingto 1199081 119908

119899 then it is easy to see that ℓ(C) is a submonoid

of (N +) In addition if gcd|1199081| |119908

119899| = 1 then ℓ(C) is

a numerical semigroup Moreover it is a simple exercise toshow that we can get any numerical semigroup in this way

As we indicated in the introduction we are going to studythe particular case in which we consider lengths of phrasesConsequently we will focus our attention in a particularfamily of numerical semigroups

Proposition 4 Let A = 119886S be an alphabet If C =

C(1199081 119908

119899) sube AF then ℓ(C) is a numerical semigroup

Proof We proceed in three steps

(i) First of all being that 120576 isin C and |120576| = 0 we have that0 isin ℓ(C)

(ii) Now let us see that if 1198971 1198972isin ℓ(C) then 119897

1+1198972isin ℓ(C)

In effect let 1198911 1198912isin C such that |119891

1| = 1198971and |119891

2| =

1198972 Then the concatenation 119891

11198912(of 1198911and 119891

2) is an

element ofC with |11989111198912| = 1198971+ 1198972

(iii) Finally let 119891 isin C with |119891| = 0 (ie 119891 = 120576) Since119891S119891 isin C we have that |119891| 2|119891| + 1 sube ℓ(C) Bythe previous step we know that C is closed underaddition and consequently ⟨|119891| 2|119891| + 1⟩ sube ℓ(C)As gcd|119891| 2|119891| + 1 = 1 we have ⟨|119891| 2|119891| + 1⟩ is anumerical semigroup Therefore N ℓ(C) is finite

We conclude that ℓ(C) is a numerical semigroup

From now on unless another thing is stated we takeA =

119886S As in the introduction we say that a numerical semi-group 119878 is a PL-semigroup if there existsC = C(119908

1 119908

119899) sube

AF such that 119878 = ℓ(C) From Proposition 4 we deduce thatif 119878 is a PL-semigroup and 119909 isin 119878 0 then 2119909 + 1 isin 119878Consequently there exist numerical semigroups which arenot of this type For example 119878 = ⟨5 7 9⟩ is not a PL-semigroup because 2 sdot 5 + 1 = 11 notin 119878

In the next result we give a characterization of PL-semigroups

Theorem 5 Let 119878 be a numerical semigroup The followingconditions are equivalent

(1) 119878 is a 119875119871-semigroup(2) If 119909 119910 isin 119878 0 then 119909 + 119910 + 1 isin 119878

Proof (1 rArr 2) By hypothesis 119878 = ℓ(C(1199081 119908

119899)) for some

nonempty finite set 1199081 119908

119899 sube Alowast If 119909 119910 isin 119878 0 then

there exist 119891 119892 isin C(1199081 119908

119899) 120576 such that |119891| = 119909 and

|119892| = 119910 It is clear that 119891S119892 isin C(1199081 119908

119899) and |119891S119892| =

119909 + 119910 + 1 Therefore 119909 + 119910 + 1 isin 119878(2 rArr 1) Let 119899

1 119899

119901 be the minimal system of

generators of 119878 Let us take the set 1199081 119908

119901| |119908119894| = 119899

119894

The Scientific World Journal 3

1 le 119894 le 119901 Our aim is to show that if C = C(1199081 119908

119901)

then 119878 = ℓ(C)Since 119899

1 119899

119901 sube ℓ(C) by applying Proposition 4 we

have 119878 = ⟨1198991 119899

119901⟩ sube ℓ(C) Now let 119897 isin ℓ(C) In order to

prove that 119897 isin 119878 we are going to use induction over 119897 If 119897 = 0then the result is trivially true Let us assume that 119897 gt 0 andlet 119891 isin C such that |119891| = 119897 IfS is not a factor of 119891 then theresult follows immediately In other case there exist 119891

1 1198912isin

C 120576 such that 119891 = 1198911S1198912 By hypothesis of induction

|1198911| |1198912| isin 119878 Thereby 119897 = |119891| = |119891

1| + |1198912| + 1 isin 119878

Remark 6 The previous theorem leads to the concept ofnumerical semigroup that admit a linear nonhomogeneouspattern For a general study of this family of numericalsemigroups see for instance [6 7]

Let 119878 be a numerical semigroup with minimal system ofgenerators given by 119899

1 119899

119901 Following [8] if 119904 isin 119878 then

we define the order of 119904 (in 119878) by

ord (119904 119878) = max 1205721+ sdot sdot sdot + 120572

119901| 12057211198991+ sdot sdot sdot + 120572

119901119899119901= 119904

with1205721 120572

119901isin N

(2)

If no ambiguity is possible then we write ord(119904)

Remark 7 From [5 Lemma 23 and Theorem 27] we havethat if 119883 is the minimal system of generators of 119878 thenevery system of generators of 119878 contains 119883 Consequentlythe definition of ord(119904 119878) does not depend on the consideredsystem of generators that is it only depends on 119904 and 119878

Lemma 8 Let 119878 be a numerical semigroup with minimalsystem of generators given by 119899

1 119899

119901 and let 119904 isin 119878

(1) If 119894 isin 1 119901 and 119904 minus 119899119894isin 119878 then ord(119904 minus 119899

119894) le

ord(119904) minus 1(2) If 119904 = 120572

11198991+ sdot sdot sdot + 120572

119901119899119901 with ord(119904) = 120572

1+ sdot sdot sdot + 120572

119901

and 120572119894= 0 then ord(119904 minus 119899

119894) = ord(119904) minus 1

Proof (1)Assume that 119904minus119899119894= 12057311198991+sdot sdot sdot+120573

119901119899119901 with120573

1+sdot sdot sdot+

120573119901= ord(119904 minus 119899

119894) Then 119904 = 120573

11198991+ sdot sdot sdot + (120573

119894+ 1)119899119894+ sdot sdot sdot + 120573

119901119899119901

and thus ord(119904minus119899119894)+1 = 120573

1+ sdot sdot sdot+ (120573

119894+1)+ sdot sdot sdot+120573

119901le ord(119904)

(2) Since 119904minus119899119894= 12057211198991+sdot sdot sdot+(120572

119894minus1)119899119894+sdot sdot sdot+120572

119901119899119901 we have

ord(119904minus119899119894) ge 1205721+ sdot sdot sdot + (120572

119894minus1)+ sdot sdot sdot +120572

119901= ord(119904)minus1 Thereby

ord(119904) minus 1 le ord(119904 minus 119899119894) Now by applying the previous item

we conclude that ord(119904 minus 119899119894) = ord(119904) minus 1

In item (2) of the next proposition it is shown a charac-terization of PL-semigroups in terms of minimal systems ofgeneratorsThus we can decide if a numerical semigroup is aPL-semigroup in an easier way

Proposition 9 Let 119878 be a numerical semigroup with minimalsystem of generators given by 119899

1 119899

119901 The following

conditions are equivalent

(1) 119878 is a 119875119871-semigroup(2) If 119894 119895 isin 1 119901 then 119899

119894+ 119899119895+ 1 isin 119878

(3) If 119904 isin 119878 0 1198991 119899

119901 then 119904 + 1 isin 119878

(4) If 119904 isin 119878 0 then 119904 + 0 119900119903119889(119904) minus 1 sube 119878

Proof (1 rArr 2) It is an immediate consequence ofTheorem 5(2 rArr 3) If 119904 isin 119878 0 119899

1 119899

119901 then it is clear that there

exist 119894 119895 isin 1 119901 and 1199041015840 isin 119878 such that 119904 = 119899119894+119899119895+1199041015840Thus

119904 + 1 = (119899119894+ 119899119895+ 1) + 119904

1015840

isin 119878(3 rArr 4)We reason by induction over ord(119904) If ord(119904) =

1 then the result is trivially true Now let us assume thatord(119904) ge 2 and that 120572

1 120572

119901are nonnegative integers such

that 119904 = 12057211198991+ sdot sdot sdot + 120572

119901119899119901 ord(119904) = 120572

1+ sdot sdot sdot + 120572

119901 and

119886119894= 0 for some 119894 isin 1 119901 By Lemma 8 we know that

ord(119904 minus 119899119894) = ord(119904) minus 1 Then by hypothesis of induction

we have that 119904 minus 119899119894+ 0 ord(119904) minus 2 sube 119878 Therefore

119904+0 ord(119904)minus2 sube 119878Moreover (119904minus119899119894+ord(119904)minus2)+119899

119894+1 isin

119878 Thereby 119904 + 0 ord(119904) minus 1 sube 119878(4 rArr 1) If 119909 119910 isin 1198780 then it is clear that ord(119909+119910) ge 2

Thus we get that 119909 + 119910 + 1 isin 119878 By applying Theorem 5 wecan conclude that 119878 is a PL-semigroup

Example 10 Let 119878 = ⟨4 5 6⟩ that is let 119878 be the numericalsemigroup with minimal system of generators given by4 5 6 It is obvious that 4 + 4 + 1 = 9 4 + 5 + 1 = 104+6+1 = 11 5+5+1 = 11 5+6+1 = 12 and 6+6+1 = 13are elements of 119878 Therefore by applying Proposition 9 wehave that 119878 is a PL-semigroup

3 The Frobenius Variety of thePL-Semigroups

The following result is straightforward to prove and appearsin [5]

Lemma 11 Let 119878 119879 be numerical semigroups

(1) 119878 cap 119879 is a numerical semigroup(2) If 119878 =N then 119878 cup F(119878) is a numerical semigroup

Having inmind the definition of Frobenius variety whichwas given in the introduction we get the next result

Proposition 12 The set S119875119871= 119878 | 119878 is a 119875119871-semigroup is a

Frobenius variety

Proof First of all let us observe that N isin SPL and thereforeSPL is a nonempty set

Let 119878 119879 isin SPL In order to show that 119878 cap 119879 isin SPL weare going to use Theorem 5 So if 119909 119910 isin (119878 cap 119879) 0 then119909 119910 isin 119878 0 and 119909 119910 isin 119879 0 Therefore 119909 + 119910 + 1 isin 119878 cap 119879Consequently 119878 cap 119879 isin SPL

Now let 119878 isin SPL such that 119878 =N By applying Theorem 5again we are going to see that 119878 cup F(119878) isin SPL Let 119909 119910 isin

(119878 cup F(119878)) 0 If 119909 119910 isin 119878 then 119909 + 119910 + 1 isin 119878 sube 119878 cup F(119878)On the other hand if F(119878) isin 119909 119910 then 119909 + 119910 + 1 gt F(119878)and thereby 119909 + 119910 + 1 isin 119878 sube 119878 cup F(119878) We conclude that119878 cup F(119878) isin SPL

A graph 119866 is a pair (119881 119864) where 119881 is a nonempty set and119864 is a subset of (V 119908) isin 119881 times 119881 | V = 119908 The elements of

4 The Scientific World Journal

119881 are called vertices of 119866 and the elements of 119864 are callededges of 119866 A path (of length 119899) connecting the vertices 119909and 119910 of 119866 is a sequence of different edges of the form(V0 V1) (V1 V2) (V

119899minus1 V119899) such that V

0= 119909 and V

119899= 119910

We say that a graph 119866 is a tree if there exist a vertex Vlowast

(known as the root of 119866) such that for every other vertex 119909 of119866 there exists a unique path connecting 119909 and Vlowast If (119909 119910) isan edge of the tree then we say that 119909 is a son of 119910

We define the graph G(S119875119871) in the following way

(i) S119875119871

is the set of vertices of G(SPL)(ii) (119878 1198781015840) isin S

119875119871times S119875119871

is an edge of G(S119875119871) if 1198781015840 = 119878 cup

F(119878)As a consequence of [3 Proposition 24 Theorem 27] we

have the next result

Theorem 13 The graph G(S119875119871) is a tree with root equal to N

Moreover the sons of a vertex 119878 isin S119875119871

are 119878 1199091 119878

119909119903 where 119909

1 119909

119903are the minimal generators of 119878 that are

greater than F(119878) and such that 119878 1199091 119878 119909

119903 isin S119875119871

Let us observe that if 119878 is a numerical semigroup and119909 isin 119878 then 119878 119909 is a numerical semigroup if and only if119909 is a minimal generator of 119878 In fact 119878 119909 is a numericalsemigroup whenever 119909 isin 119878 0 and 119909 = 119910 + 119911 for all 119910 119911 isin119878 0 As a consequence if we denote by msg(119878) the minimalsystem of generators of 119878 then msg(119878) = (119878 0) ((119878 0) +(119878 0)) (see [5 Lemma 23] for other proof of this result)In the following proposition we obtain an analogous for PL-semigroups of the first commented fact in this paragraph

Proposition 14 Let 119878 be a 119875119871-semigroup and let 119909 be aminimal generator of 119878 Then 119878 119909 is a PL-semigroup if andonly if 119909 minus 1 isin 0 cup (N 119878) cup (msg(119878))

Proof (Necessity) If 119909 minus 1 notin 0 cup (N 119878) cup (msg(119878)) then119909 minus 1 isin 119878 (0 cup (msg(119878))) Accordingly there exist 119910 119911 isin119878 0 such that 119909 minus 1 = 119910 + 119911 In fact it is clear that 119910 119911 isin119878 119909 0 Therefore by applyingTheorem 5 and that 119878 119909 isa PL-semigroup we have 119909 = 119910 + 119911 + 1 isin 119878 119909 which is acontradiction

(Sufficiency) Let 119910 119911 isin 119878 119909 0 Since 119878 is a PL-semigroup by Theorem 5 we have 119910 + 119911 + 1 isin 119878 As 119909 minus 1 isin0 cup (N 119878) cup (msg(119878)) we deduce that 119910 + 119911 + 1 = 119909 Thus119910+119911+1 isin 119878 119909 By applyingTheorem 5 again we concludethat 119878 119909 is a PL-semigroup

As a consequence of the previous proposition we have thenext result

Corollary 15 Let 119878 be a119875119871-semigroup such that 119878 =N and let119909 be a minimal generator of 119878 greater than F(119878) Then 119878 119909 isa 119875119871-semigroup if and only if 119909 minus 1 isin msg(119878) cup F(119878)

By applying Theorem 13 together with Corollary 15 wecan get the sons of a vertex of G(SPL) as is shown in thefollowing example

Example 16 It is clear that 119878 = ⟨4 6 7 9⟩ is a PL-semigroupwith Frobenius number equal to 5 From Theorem 13 and

Corollary 15 we deduce that the sons of 119878 are 119878 6 =

⟨4 7 9 10⟩ and 119878 7 = ⟨4 6 9 11⟩

Let us observe that we can build recursively a tree fromthe root if we know the sons of each vertexTherefore we canbuild the tree G(SPL) such as it is shown in Figure 1

In order to have an easier making of the tree G(SPL)we are going to study the relation between the minimalgenerators of a numerical semigroup 119878 and the minimalgenerators of 119878 119909 where 119909 is a minimal generator of119878 that is greater than F(119878) First of all let us observe thatif 119878 is minimally generated by 119898119898 + 1 2119898 minus 1 (ie119878 = 0119898 rarr ) then 119878 119898 = 0119898 + 1 rarr is minimallygenerated by 119898+1119898+2 2119898+1 In other case we havethe following result

Proposition 17 Let 119878 be a numerical semigroup withmsg(119878) = 119899

1 119899

119901 If m(119878) = 119899

1lt 119899119901and 119899

119901gt F(S)

then 119878 119899119901 = ⟨119899

1 119899

119901minus1 119899119901+ 1198991⟩

Proof Let us take 119894 isin 2 119901 Since 119899119901gt F(119878) and 119899

1lt 119899119894

we have that 119899119901+ 119899119894minus 1198991isin 119878 Thus 119899

119901+ 119899119894minus 1198991= 12057211198991+

sdot sdot sdot + 120572119901119899119901for some 120572

1 120572

119901isin N Thereby 119899

119901+ 119899119894= (1205721+

1)1198991+ sdot sdot sdot + 120572

119901119899119901 By applying that 119899

1 119899

119901 is a minimal

system of generators we have that 120572119901= 0Therefore 119899

119901+119899119894isin

⟨1198991 119899

119901minus1⟩ In particular 2119899

119901isin ⟨1198991 119899

119901minus1⟩

Now let 119904 isin 119878 119899119901 Then 119904 isin 119878 and thus there exist

1205731 120573

119901isin N such that 119904 = 120573

11198991+ sdot sdot sdot + 120573

119901119899119901 Since 2119899

119901isin

⟨1198991 119899

119901minus1⟩ we can assume that 120573

119901isin 0 1 On the one

hand if 120573119901= 0 then 119904 isin ⟨119899

1 119899

119901minus1⟩ On the other hand if

120573119901= 1 then there exists 119894 isin 1 119901minus1 such that120573

119894= 0 If 119894 =

1 then it is obvious that 119904 isin ⟨1198991 119899

119901minus1 119899119901+1198991⟩ And if 119894 = 1

since 119899119901+ 119899119894isin ⟨1198991 119899

119901minus1⟩ we have that 119904 isin ⟨119899

1 119899

119901minus1⟩

In any case we conclude that 119878 119899119901 = ⟨119899

1 119899

119901minus1 119899119901+

1198991⟩

Corollary 18 Let 119878 be a numerical semigroup withmsg(119878) =1198991 119899

119901 Ifm(119878) = 119899

1lt 119899119901and 119899119901gt F(119878) then

msg (119878 119899119901)

=

1198991 119899

119901minus1

119894119891 119905ℎ119890119903119890 119890119909119894119904119905119904 119894 isin 2 119901 minus 1

119904119906119888ℎ 119905ℎ119886119905 119899119901+ 1198991minus 119899119894isin 119878

1198991 119899

119901minus1 119899119901+ 1198991 119894119899 119900119905ℎ119890119903 119888119886119904119890

(3)

Proof From Proposition 17 we deduce that msg(119878 119899119901) is

1198991 119899

119901minus1 or 119899

1 119899

119901minus1 119899119901+ 1198991 In addition msg(119878

119899119901) = 119899

1 119899

119901minus1 if and only if 119899

119901+ 1198991isin ⟨1198991 119899

119901minus1⟩

If 119899119901+ 1198991isin ⟨1198991 119899

119901minus1⟩ then there exist 120572

1 120572

119901minus1isin

N such that 119899119901+1198991= 12057211198991+ sdot sdot sdot + 120572

119901minus1119899119901minus1

Since 1198991 119899

119901

is a minimal system of generators we get that 1205721= 0 Thus

there exists 119894 isin 2 119901 minus 1 such that 120572119894= 0 Consequently

119899119901+ 1198991minus 119899119894isin 119878

Conversely if there exists 119894 isin 2 119901 minus 1 such that 119899119901+

1198991minus 119899119894isin 119878 then 119899

119901+ 1198991minus 119899119894= 12057311198991+ sdot sdot sdot + 120573

119901119899119901for some

1205731 120573

119901isin NThereby 119899

119901+1198991= 12057311198991+sdot sdot sdot+ (120573

119894+1)119899119894+sdot sdot sdot+

120573119901119899119901 Since 119899

1 119899

119901 is a minimal system of generators we

have that 120573119901= 0 and therefore 119899

119901+ 1198991isin ⟨1198991 119899

119901minus1⟩

The Scientific World Journal 5

⟨5 6 7 8 9⟩ ⟨4 6 7 9⟩ ⟨4 5 7⟩ ⟨4 5 6⟩ ⟨3 7 8⟩

⟨3 5 7⟩

⟨3 4 5⟩

⟨3 4⟩

⟨2 3⟩

⟨2 5⟩

⟨4 5 6 7⟩

⟨1⟩ = N

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

Figure 1

We finish this section with an illustrative example aboutthe above corollary

Example 19 Let 119878 be the numerical semigroupwithmsg(119878) =3 5 7 It is obvious that F(119878) = 4 By Proposition 17 weknow that 1198785 = ⟨3 7 8⟩ In addition 8minus7 = 1 notin 119878TherebyapplyingCorollary 18 we have thatmsg(1198785) = 3 7 8 Onthe other hand applying Proposition 17 again we have that119878 7 = ⟨3 5 10⟩ Finally since 10 minus 5 = 5 isin 119878 we concludethat msg(119878 7) = 3 5

4 PL-Semigroups with a Fixed Multiplicity

Let 119898 be a positive integer We will denote by Δ(119898) =

0119898 rarr It is clear that Δ(119898) is the greatest (with respectto set inclusion) PL-semigroup with multiplicity119898 Our firstaim in this section will be to show that there also exists thesmallest (with respect to set inclusion) PL-semigroup withmultiplicity119898

As an immediate consequence of item (4) inProposition 9 we have the next result

Lemma20 If 119878 is a119875119871-semigroup119898 isin 1198780 and 119896 isin N0then 119896119898 + 119894 isin 119878 for all 119894 isin 0 119896 minus 1

Proposition 21 Let 119898 isin N 0 Then the numericalsemigroup generated by (119894 + 1)119898 + 119894 | 119894 isin 0 119898 minus 1 isthe smallest (with respect to set inclusion) 119875119871-semigroup withmultiplicity119898

Proof Let 119878 = ⟨119898 2119898+1 1198982+(119898minus1)⟩ FromLemma 20we know that any PL-semigroup with multiplicity 119898 has tocontain 119878 In order to conclude the proof we will show that 119878is a PL-semigroup For this purpose since (119894 + 1)119898 + 119894 | 119894 isin

0 119898minus1 is a system of generators of 119878 it will be enoughto check item (2) of Proposition 9 that is if 119894 119895 isin 0 119898 minus

1 then (119894 + 1)119898+ 119894 + (119895 + 1)119898+ 119895+ 1 isin 119878 We distinguish twocases

(1) If 119894+ 119895+1 le 119898minus1 then (119894+1)119898+119894+ (119895+1)119898+119895+1 =(119894 + 119895 + 2)119898 + (119894 + 119895 + 1) isin 119878

(2) If 119894 + 119895 + 1 ge 119898 then (119894 + 1)119898 + 119894 + (119895 + 1)119898 + 119895 + 1 =(119894 + 119895 + 2)119898 + (119894 + 119895 + 1) = (119898+ 1)119898+ (119894 + 119895 minus119898+ 2)119898

+ (119894 + 119895 minus 119898 + 1) isin 119878

We will denote byΘ(119898) = ⟨119898 2119898 + 1 1198982

+ (119898 minus 1)⟩

and bySPL(119898) the set of all PL-semigroups with multiplicityequal to 119898 Let us recall that Δ(119898) = max(SPL(119898)) andΘ(119898) = min(SPL(119898))

As an application of the above comment we have the nextresult

Corollary 22 The set SPL(119898) is finite

Proof If 119878 isin SPL(119898) thenΘ(119898) sube 119878 sube Δ(119898) SinceΔ(119898) andΘ(119898) are numerical semigroups we have that Δ(119898) Θ(119898) isfinite Consequently SPL(119898) is also finite

Remark 23 The previous result can be considered a particu-lar case of [6 Theorem 66]

Now we are interested in computing the Frobeniusnumber and the genus ofΘ(119898) For that several concepts andresults are introduced

If 119878 is a numerical semigroup and 119898 isin 119878 0 then theApery set of119898 in 119878 (see [9]) is Ap(119878 119898) = 119904 isin 119878 | 119904 minus 119898 notin 119878It is clear (see for instance [5 Lemma 24]) that Ap(119878 119898) =120596(0) = 0 120596(1) 120596(119898 minus 1) where 120596(119894) is for each 119894 isin0 119898 minus 1 the least element of 119878 that is congruent with 119894modulo119898

The next result is [5 Proposition 212]

Lemma24 Let 119878 be a numerical semigroup and let119898 isin 1198780Then

(1) F(119878) = max(Ap(119878 119898)) minus 119898(2) g(119878) = (1119898)(sum

119908isinAp(119878119898) 119908) minus ((119898 minus 1)2)

If 119886 119887 are integers with 119887 = 0 we denote by 119886 mod 119887 theremainder of the division of 119886 by 119887 The following result is [5Proposition 35]

6 The Scientific World Journal

Lemma 25 Let 119898 isin N 0 and let 119883 = 120596(0) =

0 120596(1) 120596(119898minus1) sube N such that for each 119894 isin 1 119898minus1120596(119894) is congruent with 119894 modulo 119898 Let 119878 be the numericalsemigroup generated by 119883 cup 119898 The following conditions areequivalent

(1) Ap(119878 119898) = 119883(2) 120596(119894)+120596(119895) ge 120596((119894+119895) mod 119898) for all 119894 119895 isin 1 119898minus

1

Proposition 26 If119898 isin N 0 then

Ap (Θ (119898) 119898)

= 120596 (0) = 0 120596 (1) = 2119898 + 1 120596 (119898 minus 1) = 1198982

+ 119898 minus 1

(4)

Proof It is clear that 120596(119894) = (119894 + 1)119898 + 119894 is congruent with 119894modulo 119898 for all 119894 isin 1 119898 minus 1 Let us see now that if119894 119895 isin 1 119898 minus 1 then 120596(119894) + 120596(119895) ge 120596((119894 + 119895) mod 119898)Indeed 120596(119894) +120596(119895) = (119894 + 1)119898 + 119894 + (119895 + 1)119898 + 119895 gt (119894 + 119895 + 1)119898+ (119894 + 119895) ge ((119894 + 119895) mod 119898 + 1)119898 + (119894 + 119895) mod 119898 = 120596((119894 +119895) mod 119898) The proof follows from Lemma 25

Corollary 27 If119898 isin N 0 1 then

(1) F(Δ(119898)) = 119898 minus 1(2) g(Δ(119898)) = 119898 minus 1(3) F(Θ(119898)) = 1198982 minus 1(4) g(Θ(119898)) = (119898 minus 1)(119898 + 2)2

Proof Items (1) and (2) are trivial On the other hand items(3) and (4) are immediate consequences of Lemma 24 andProposition 26

Remark 28 Thenumerical semigroupΘ(119898) can be rewrittenas

Θ (119898)

= 119898119898+ (119898+ 1) 119898 +2 (119898 + 1) 119898 + (119898 minus 1) (119898 + 1)

(5)

Thus Θ(119898) is a numerical semigroup generated by an arith-metic sequencewith first term119898 and commondifference119898+1(see [2 10])

If 119878 is a numerical semigroup then the cardinality of theminimal system of generators of 119878 is called the embeddingdimension of 119878 and is denoted by e(119878) It is well known (see [5Proposition 212]) that e(119878) le m(119878) We say that a numericalsemigroup 119878 has maximal embedding dimension if e(119878) =m(119878) It is clear that 119898119898 + 1 2119898 minus 1 is the minimalsystem of generators of Δ(119898) Therefore Δ(119898) is a numericalsemigroupwithmaximal embedding dimensionNowwewillshow that Θ(119898) has also maximal embedding dimension

The next result is [5 Corollary 36]

Lemma 29 Let 119878 be a numerical semigroup with multiplicity119898 and assume thatAp(119878 119898) = 120596(0) = 0 120596(1) 120596(119898minus1)

Then 119878 has maximal embedding dimension if and only if 120596(119894)+120596(119895) gt 120596((119894 + 119895) mod 119898) for all 119894 119895 isin 1 119898 minus 1

Let us observe that in the proof of Proposition 26 wehave shown that 120596(119894) + 120596(119895) gt 120596((119894 + 119895) mod 119898) for all119894 119895 isin 1 119898 minus 1 Therefore by applying Lemma 29 weget the following result

Corollary 30 If 119898 isin N 0 then Θ(119898) is a numericalsemigroup with maximal embedding dimension

As a consequence of this corollary we have that 119898 2119898 +

1 1198982

+119898minus1 is theminimal systemof generators ofΘ(119898)Now we want to show orderly the elements of SPL(119898)

Thus we define the graph G(SPL(119898)) in the following way

(i) SPL(119898) is the set of vertices of G(SPL(119898))

(ii) (119878 1198781015840) isin SPL(119898) timesSPL(119898) is an edge of G(SPL(119898)) if1198781015840

= 119878 cup F(119878)

The next result is analogous to Theorem 13

Theorem 31 The graph G(S119875119871(119898)) is a tree with root equal

to Δ(119898) Moreover the sons of a vertex 119878 isin S119875119871(119898) are 119878

1199091 119878119909

119903 with 119909

1 119909

119903= 119909 isin msg(119878) | 119909 =119898 119909 gt

F(119878) 119886119899119889 119878 119909 isin S119875119871

By applyingTheorem 31 and Corollaries 15 and 18 we canget easily G(SPL(119898)) such as is shown in the next example

Example 32 We are going to depict G(SPL(3)) that is thetree of the PL-semigroups with multiplicity equal to 3

⟨3 7 8⟩

⟨3 5 7⟩ ⟨3 4⟩

⟨3 4 5⟩ = Δ(3)

⟨3 7 11⟩ = Θ(3)

If119879 = (119881 119864) is a tree then the height of119879 is themaximumof the lengths of the paths that connect each vertex with theroot Let us observe that the height of G(SPL(3)) is 3 Ingeneral the height of the tree G(SPL(119898)) is equal to

g (Θ (119898)) minus g (Δ (119898)) = (119898 minus 1) (119898 + 2)

2

minus (119898 minus 1)

=

(119898 minus 1)119898

2

(6)

Let us study now the possible values of the Frobeniusnumber and the genus for PL-semigroups with multiplicity119898

The Scientific World Journal 7

Proposition 33 If119898 isin N 0 then

(1) g(119878) | 119878 isin SPL(119898) = 119909 isin N | 119898 minus 1 le 119909 le

(119898 minus 1)(119898 + 2)2

(2) F(119878) | 119878 isin SPL(119898) = (Δ(119898) Θ(119898)) cup 119898 minus 1

Proof Let us assume that Δ(119898) Θ(119898) = 1199091gt 1199092gt sdot sdot sdot gt

119909120588(1) By Corollary 27 we know that

g (119878) | 119878 isin SPL (119898) sube 119898 minus 1

(119898 minus 1) (119898 + 2)

2

(7)

For the opposite inclusion it is enough to observe thatΘ(119898)cup119909119894 rarr isin SPL(119898) and that g(Θ(119898)cup119909119894 rarr ) = (119898minus1)(119898+

2)2 minus 119894(2) It is clear that Θ(119898) cup 119909

119894+ 1 rarr isin SPL(119898) with

Frobenius number equal to 119909119894 Thus (Δ(119898) Θ(119898)) cup 119898 minus

1 sube F(119878) | 119878 isin SPL(119898) For the other inclusion let ustake 119878 isin SPL(119898) such that 119878 = Δ(119898) Then F(119878) gt 119898 andthereby F(119878) isin Δ(119898) Since Θ(119898) sube 119878 we have F(119878) notin Θ(119878)Therefore we conclude that F(119878) isin Δ(119898) Θ(119878)

Example 34 By Proposition 33 g(119878) | 119878 isin SPL(3) =

2 3 4 5 Since Δ(3) = ⟨3 4 5⟩ and Θ(3) = ⟨3 7 11⟩ wehave that (Δ(3) Θ(3)) cup 2 = 2 4 5 8 Therefore byapplying Proposition 33 again we conclude that F(119878) | 119878 isinSPL(3) = 2 4 5 8

5 The Smallest PL-Semigroup That Contains aGiven Set of Positive Integers

Let us observe that in general the infinite intersection ofelements of SPL is not a numerical semigroup For instance⋂119899isinN0 119899 rarr = 0 On the other hand it is clear that the

(finite or infinite) intersection of numerical semigroups isalways a submonoid of (N +)

Let 119872 be a submonoid of (N +) We will say that 119872 isa SPL-monoid if it can be expressed like the intersection ofelements of SPL

The next lemma has an immediate proof

Lemma35 The intersection ofS119875119871-monoids is aS

119875119871-monoid

In view of this result we can give the following definition

Definition 36 Let 119883 be a subset of N The SPL-monoidgenerated by119883 (denoted bySPL(119883)) is the intersection of allSPL-monoids containing119883

If 119872 = SPL(119883) then we will say that 119883 is a SPL-systemof generators of119872 In addition if no proper subset of 119883 is aSPL-system of generators of 119872 then we will say that 119883 is aminimal SPL-system of generators of119872

Let us recall that by Proposition 12 we know that S119875119871

isa Frobenius variety Therefore by applying [3 Corollary 19]we have the next result

Proposition 37 Every S119875119871-monoid has a unique minimal

S119875119871-system of generators which in addition is finite

The proof of the following lemma is also immediate

Lemma 38 If 119883 sube N then S119875119871(119883) is the intersection of all

119875119871-semigroups that contain 119883

Proposition 39 If 119883 is a nonempty subset of N 0 thenS119875119871(119883) is a 119875119871-semigroup

Proof We know that SPL(119883) is a submonoid of (N +)Therefore in order to show that SPL(119883) is a numericalsemigroup it will be enough to see that N SPL(119883) is a finiteset

Let 119909 isin 119883 If 119878 is a PL-semigroup containing 119883 then (byTheorem 5) we know that 119909 2119909 + 1 sube 119878 and in this way⟨119909 2119909 + 1⟩ sube 119878 From Lemma 38 we have that ⟨119909 2119909 + 1⟩ subeSPL(119883) Since gcd119909 2119909 + 1 = 1 we get that ⟨119909 2119909 + 1⟩is a numerical semigroup and thus N ⟨119909 2119909 + 1⟩ is finiteConsequently N SPL(119883) is finite

Now let us see that SPL(119883) is a PL-semigroup Let 119909 119910 isinSPL(119883) 0 If 119878 is a PL-semigroup containing 119883 fromLemma 38 we deduce that 119909 119910 isin 1198780 and fromTheorem 5we have that 119909 + 119910 + 1 isin 119878 By applying again Lemma 38we have that 119909 + 119910 + 1 isin SPL(119883) Therefore by applyingTheorem 5 once more we can assert that SPL(119883) is a PL-semigroup

Remark 40 Let us observe that in general Proposition 39 isnot true for Frobenius varieties In fact let S be the set of allnumerical semigroups It is clear thatS is a Frobenius varietyIf we take 119883 = 2 then the intersection of all elements of Scontaining 2 is exactly119872 = ⟨2⟩ which is not a numericalsemigroup

The next result will be key for our last purpose in thissection

Theorem 41 S119875119871

= S119875119871(119883) | 119883 is a nonempty finite

subset 119900119891 N 0

Proof By Proposition 39 we have that

SPL (119883) | 119883 is a nonempty finite subset ofN 0 sube SPL(8)

For the other inclusion it is enough to observe that if 119878 isin SPLthen (by Proposition 37) there exists a nonempty finite subset119883 of N 0 such that 119878 = SPL(119883)

Since SPL is a Frobenius variety by applying [3 Proposi-tion 24] we get the next result

Proposition 42 Let119872 be aS119875119871-monoid and let 119909 isin 119872Then

119872119909 is aS119875119871-monoid if and only if 119909 belongs to the minimal

S119875119871-system of generators of119872

As an immediate consequence of this propositionwe havethe following result

8 The Scientific World Journal

Corollary 43 Let 119883 be a nonempty subset of N 0 Thenthe minimal S

119875119871-system of generators of S

119875119871(119883) is 119909 isin 119883 |

S119875119871(119883) 119909 is a PL-semigroup

Example 44 By Proposition 21 119878 = ⟨3 7 11⟩ is a PL-semigroup By applying Proposition 14 we easily deduce that

119909 isin 3 7 11 | 119878 119909 is a PL-semigroup = 3 (9)

Therefore 119878 = SPL(3) and 3 is the minimalSPL-system ofgenerators of 119878

Now we want to describe SPL(119883) when 119883 is a fixednonempty finite set of positive integers Let us observe thatby Theorem 41 we know that every PL-semigroup can beobtained in this way

Let 1198991 119899

119901be positive integers We will denote by

119878(1198991 119899

119901) the set 120572

11198991+ sdot sdot sdot + 120572

119901119899119901+ 119903 | 119903 120572

1 120572

119901isin

N 119903 lt 1205721+ sdot sdot sdot + 120572

119901 cup 0 Our next purpose will be to show

that 119878(1198991 119899

119901) = SPL(1198991 119899119901)

Lemma 45 Let 119878 be a numerical semigroup let 1199041 119904

119905isin

119878 0 and let 1205721 120572

119905isin N Then ord(120572

11199041+ sdot sdot sdot + 120572

119905119904119905) ge

1205721+ sdot sdot sdot + 120572

119905

Proof Let 1198991 119899

119901 be the minimal system of generators

of 119878 Then for each 119894 isin 1 119905 there exist 1205731198941 120573

119894119901isin N

such that 119904119894= 12057311989411198991+ sdot sdot sdot + 120573

119894119901119899119901 Moreover since 119904

119894= 0 we

have that 1205731198941+ sdot sdot sdot + 120573

119894119901ge 1 Thus

12057211199041+ sdot sdot sdot + 120572

119905119904119905

= 1205721(120573111198991+ sdot sdot sdot + 120573

1119901119899119901) + sdot sdot sdot + 120572

119905(12057311990511198991+ sdot sdot sdot + 120573

119905119901119899119901)

= (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) 1198991+ sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901) 119899119901

(10)

Therefore

ord (12057211199041+ sdot sdot sdot + 120572

119905119904119905)

ge (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) + sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901)

= 1205721(12057311+ sdot sdot sdot + 120573

1119901) + sdot sdot sdot + 120572

119905(1205731199051+ sdot sdot sdot + 120573

119905119901)

ge 1205721+ sdot sdot sdot + 120572

119905

(11)

Theorem 46 If 1198991 119899

119901are positive integers then

119878(1198991 119899

119901) is the smallest (with respect to set inclusion)

119875119871-semigroup containing 1198991 119899

119901

Proof We divide the proof into five steps

(i) Let us see that if 119909 119910 isin 119878(1198991 119899

119901) 0 then 119909 +

119910 isin 119878(1198991 119899

119901) In effect we know that there exist

1205721 120572

119901 1205731 120573

119901 119903 1199031015840 nonnegative integers such

that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903 119910 = 120573

11198991+sdot sdot sdot+120573

119901119899119901+1199031015840

119903 lt 1205721+ sdot sdot sdot + 120572

119901 and 1199031015840 lt 120573

1+ + 120573

119901 Therefore

119909 + 119910 = (1205721+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840 with119903 + 1199031015840

lt (1205721+ 1205731) + sdot sdot sdot + (120572

119901+ 120573119899) Consequently

119909 + 119910 isin 119878(1198991 119899

119901)

(ii) Let us see that N 119878(1198991 119899

119901) is finite Since 119899

1=

1sdot1198991+0sdot1198992+sdot sdot sdot+0sdot119899

119901+0 and 2119899

1+1 = 2sdot119899

1+0sdot1198992+sdot sdot sdot+0sdot

119899119901+1 we have that 119899

1 21198991isin 119878(1198991 119899

119901) By applying

the first step we get that ⟨1198991 21198991⟩ sube 119878(119899

1 119899

119901)

Using the same reasoning as we did in the proof ofProposition 39 we have the result

(iii) From the previous steps we know that 119878(1198991 119899

119901)

is a numerical semigroup Let us see now that119878(1198991 119899

119901) is a PL-semigroup In order to do that

it is enough (by Theorem 5) to show that if 119909 119910 isin

119878(1198991 119899

119901) 0 then 119909 + 119910 + 1 isin 119878(119899

1 119899

119901)

Indeed arguing as in the first step we have that 119909 +119910 + 1 = (120572

1+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840

+ 1

with 119903 + 1199031015840 + 1 lt (1205721+1205731) + sdot sdot sdot + (120572

119901+120573119899) Therefore

119909 + 119910 + 1 isin 119878(1198991 119899

119901)

(iv) Following the proof of the second step it is clear that1198991 119899

119901 sube 119878(119899

1 119899

119901)

(v) Finally let us see that 119878(1198991 119899

119901) is the small-

est PL-semigroup that contains 1198991 119899

119901 In fact

we will show that if 119879 is PL-semigroup containing1198991 119899

119901 then 119878(119899

1 119899

119901) sube 119879 Thus let 119909 isin

119878(1198991 119899

119901) 0 Then there exist 120572

1 120572

119901 119903 isin N

such that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903with 119903 lt 120572

1+sdot sdot sdot+120572

119901

Since 1198991 119899

119901 sube 119879 by Proposition 9 we have that

12057211198991+sdot sdot sdot+120572

119901119899119901+0 ord(120572

11198991+sdot sdot sdot+120572

119901119899119901)minus1 sube 119879

By applying Lemma 45 we have that 119903 lt ord(12057211198991+

sdot sdot sdot + 120572119901119899119901) and therefore 119909 isin 119879

In this way we have proved the statement

The next result is an immediate consequence of theprevious theorem

Corollary 47 If 1198991 119899

119901are positive integers then

SPL(1198991 119899119901) = 119878(1198991 119899119901)

We finish this section with an example that illustrates itscontent

Example 48 It is clear that 119878(4 7) = 0 4 7 8 9 11 rarr =

⟨4 7 9⟩ Therefore SPL(4 7) = ⟨4 7 9⟩

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referee for hisher usefulcomments and suggestions that helped to improve this workBoth of the authors are supported by FQM-343 (Junta deAndalucıa) MTM2010-15595 (MICINN Spain) and FEDERfunds The second author is also partially supported by Juntade AndalucıaFeder Grant no FQM-5849

The Scientific World Journal 9

References

[1] M Lothaire Applied Combinatorics on Words vol 105 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press 2005

[2] J L Ramırez Alfonsın The Diophantine Frobenius ProblemOxford University Press 2005

[3] J C Rosales ldquoFamilies of numerical semigroups closed underfinite intersections and for the Frobenius numberrdquo HoustonJournal of Mathematics vol 34 no 2 pp 339ndash348 2008

[4] J C Rosales M B Branco and D Torrao ldquoBracelet monoidsand numerical semigroupsrdquo preprint

[5] J C Rosales and P A Garcıa-Sanchez Numerical Semigroupsvol 20 of Developments in Mathematics Springer New YorkNY USA 2009

[6] M Bras-Amoros P A Garcıa-Sanchez and A Vico-OtonldquoNonhomogeneous patterns on numerical semigroupsrdquo Inter-national Journal of Algebra and Computation vol 23 no 6 pp1469ndash1483 2013

[7] K Stokes and M Bras-Amoros ldquoLinear non-homogeneoussymmetric patterns and prime power generators in numeri-cal semigroups associated to combinatorial congurationsrdquo toappear in Semigroup Forum 2013

[8] L Bryant ldquoGoto numbers of a numerical semigroup ring andthe gorensteiness of associated graded ringsrdquo Communicationsin Algebra vol 38 no 6 pp 2092ndash2128 2010

[9] R Apery ldquoSur les branches superlineaires des courbesalgebriquesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol222 pp 1198ndash1200 1946

[10] J B Roberts ldquoNote on linear formsrdquo Proceedings of the Ameri-can Mathematical Society vol 7 pp 465ndash469 1956

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The Scientific World Journal 3

1 le 119894 le 119901 Our aim is to show that if C = C(1199081 119908

119901)

then 119878 = ℓ(C)Since 119899

1 119899

119901 sube ℓ(C) by applying Proposition 4 we

have 119878 = ⟨1198991 119899

119901⟩ sube ℓ(C) Now let 119897 isin ℓ(C) In order to

prove that 119897 isin 119878 we are going to use induction over 119897 If 119897 = 0then the result is trivially true Let us assume that 119897 gt 0 andlet 119891 isin C such that |119891| = 119897 IfS is not a factor of 119891 then theresult follows immediately In other case there exist 119891

1 1198912isin

C 120576 such that 119891 = 1198911S1198912 By hypothesis of induction

|1198911| |1198912| isin 119878 Thereby 119897 = |119891| = |119891

1| + |1198912| + 1 isin 119878

Remark 6 The previous theorem leads to the concept ofnumerical semigroup that admit a linear nonhomogeneouspattern For a general study of this family of numericalsemigroups see for instance [6 7]

Let 119878 be a numerical semigroup with minimal system ofgenerators given by 119899

1 119899

119901 Following [8] if 119904 isin 119878 then

we define the order of 119904 (in 119878) by

ord (119904 119878) = max 1205721+ sdot sdot sdot + 120572

119901| 12057211198991+ sdot sdot sdot + 120572

119901119899119901= 119904

with1205721 120572

119901isin N

(2)

If no ambiguity is possible then we write ord(119904)

Remark 7 From [5 Lemma 23 and Theorem 27] we havethat if 119883 is the minimal system of generators of 119878 thenevery system of generators of 119878 contains 119883 Consequentlythe definition of ord(119904 119878) does not depend on the consideredsystem of generators that is it only depends on 119904 and 119878

Lemma 8 Let 119878 be a numerical semigroup with minimalsystem of generators given by 119899

1 119899

119901 and let 119904 isin 119878

(1) If 119894 isin 1 119901 and 119904 minus 119899119894isin 119878 then ord(119904 minus 119899

119894) le

ord(119904) minus 1(2) If 119904 = 120572

11198991+ sdot sdot sdot + 120572

119901119899119901 with ord(119904) = 120572

1+ sdot sdot sdot + 120572

119901

and 120572119894= 0 then ord(119904 minus 119899

119894) = ord(119904) minus 1

Proof (1)Assume that 119904minus119899119894= 12057311198991+sdot sdot sdot+120573

119901119899119901 with120573

1+sdot sdot sdot+

120573119901= ord(119904 minus 119899

119894) Then 119904 = 120573

11198991+ sdot sdot sdot + (120573

119894+ 1)119899119894+ sdot sdot sdot + 120573

119901119899119901

and thus ord(119904minus119899119894)+1 = 120573

1+ sdot sdot sdot+ (120573

119894+1)+ sdot sdot sdot+120573

119901le ord(119904)

(2) Since 119904minus119899119894= 12057211198991+sdot sdot sdot+(120572

119894minus1)119899119894+sdot sdot sdot+120572

119901119899119901 we have

ord(119904minus119899119894) ge 1205721+ sdot sdot sdot + (120572

119894minus1)+ sdot sdot sdot +120572

119901= ord(119904)minus1 Thereby

ord(119904) minus 1 le ord(119904 minus 119899119894) Now by applying the previous item

we conclude that ord(119904 minus 119899119894) = ord(119904) minus 1

In item (2) of the next proposition it is shown a charac-terization of PL-semigroups in terms of minimal systems ofgeneratorsThus we can decide if a numerical semigroup is aPL-semigroup in an easier way

Proposition 9 Let 119878 be a numerical semigroup with minimalsystem of generators given by 119899

1 119899

119901 The following

conditions are equivalent

(1) 119878 is a 119875119871-semigroup(2) If 119894 119895 isin 1 119901 then 119899

119894+ 119899119895+ 1 isin 119878

(3) If 119904 isin 119878 0 1198991 119899

119901 then 119904 + 1 isin 119878

(4) If 119904 isin 119878 0 then 119904 + 0 119900119903119889(119904) minus 1 sube 119878

Proof (1 rArr 2) It is an immediate consequence ofTheorem 5(2 rArr 3) If 119904 isin 119878 0 119899

1 119899

119901 then it is clear that there

exist 119894 119895 isin 1 119901 and 1199041015840 isin 119878 such that 119904 = 119899119894+119899119895+1199041015840Thus

119904 + 1 = (119899119894+ 119899119895+ 1) + 119904

1015840

isin 119878(3 rArr 4)We reason by induction over ord(119904) If ord(119904) =

1 then the result is trivially true Now let us assume thatord(119904) ge 2 and that 120572

1 120572

119901are nonnegative integers such

that 119904 = 12057211198991+ sdot sdot sdot + 120572

119901119899119901 ord(119904) = 120572

1+ sdot sdot sdot + 120572

119901 and

119886119894= 0 for some 119894 isin 1 119901 By Lemma 8 we know that

ord(119904 minus 119899119894) = ord(119904) minus 1 Then by hypothesis of induction

we have that 119904 minus 119899119894+ 0 ord(119904) minus 2 sube 119878 Therefore

119904+0 ord(119904)minus2 sube 119878Moreover (119904minus119899119894+ord(119904)minus2)+119899

119894+1 isin

119878 Thereby 119904 + 0 ord(119904) minus 1 sube 119878(4 rArr 1) If 119909 119910 isin 1198780 then it is clear that ord(119909+119910) ge 2

Thus we get that 119909 + 119910 + 1 isin 119878 By applying Theorem 5 wecan conclude that 119878 is a PL-semigroup

Example 10 Let 119878 = ⟨4 5 6⟩ that is let 119878 be the numericalsemigroup with minimal system of generators given by4 5 6 It is obvious that 4 + 4 + 1 = 9 4 + 5 + 1 = 104+6+1 = 11 5+5+1 = 11 5+6+1 = 12 and 6+6+1 = 13are elements of 119878 Therefore by applying Proposition 9 wehave that 119878 is a PL-semigroup

3 The Frobenius Variety of thePL-Semigroups

The following result is straightforward to prove and appearsin [5]

Lemma 11 Let 119878 119879 be numerical semigroups

(1) 119878 cap 119879 is a numerical semigroup(2) If 119878 =N then 119878 cup F(119878) is a numerical semigroup

Having inmind the definition of Frobenius variety whichwas given in the introduction we get the next result

Proposition 12 The set S119875119871= 119878 | 119878 is a 119875119871-semigroup is a

Frobenius variety

Proof First of all let us observe that N isin SPL and thereforeSPL is a nonempty set

Let 119878 119879 isin SPL In order to show that 119878 cap 119879 isin SPL weare going to use Theorem 5 So if 119909 119910 isin (119878 cap 119879) 0 then119909 119910 isin 119878 0 and 119909 119910 isin 119879 0 Therefore 119909 + 119910 + 1 isin 119878 cap 119879Consequently 119878 cap 119879 isin SPL

Now let 119878 isin SPL such that 119878 =N By applying Theorem 5again we are going to see that 119878 cup F(119878) isin SPL Let 119909 119910 isin

(119878 cup F(119878)) 0 If 119909 119910 isin 119878 then 119909 + 119910 + 1 isin 119878 sube 119878 cup F(119878)On the other hand if F(119878) isin 119909 119910 then 119909 + 119910 + 1 gt F(119878)and thereby 119909 + 119910 + 1 isin 119878 sube 119878 cup F(119878) We conclude that119878 cup F(119878) isin SPL

A graph 119866 is a pair (119881 119864) where 119881 is a nonempty set and119864 is a subset of (V 119908) isin 119881 times 119881 | V = 119908 The elements of

4 The Scientific World Journal

119881 are called vertices of 119866 and the elements of 119864 are callededges of 119866 A path (of length 119899) connecting the vertices 119909and 119910 of 119866 is a sequence of different edges of the form(V0 V1) (V1 V2) (V

119899minus1 V119899) such that V

0= 119909 and V

119899= 119910

We say that a graph 119866 is a tree if there exist a vertex Vlowast

(known as the root of 119866) such that for every other vertex 119909 of119866 there exists a unique path connecting 119909 and Vlowast If (119909 119910) isan edge of the tree then we say that 119909 is a son of 119910

We define the graph G(S119875119871) in the following way

(i) S119875119871

is the set of vertices of G(SPL)(ii) (119878 1198781015840) isin S

119875119871times S119875119871

is an edge of G(S119875119871) if 1198781015840 = 119878 cup

F(119878)As a consequence of [3 Proposition 24 Theorem 27] we

have the next result

Theorem 13 The graph G(S119875119871) is a tree with root equal to N

Moreover the sons of a vertex 119878 isin S119875119871

are 119878 1199091 119878

119909119903 where 119909

1 119909

119903are the minimal generators of 119878 that are

greater than F(119878) and such that 119878 1199091 119878 119909

119903 isin S119875119871

Let us observe that if 119878 is a numerical semigroup and119909 isin 119878 then 119878 119909 is a numerical semigroup if and only if119909 is a minimal generator of 119878 In fact 119878 119909 is a numericalsemigroup whenever 119909 isin 119878 0 and 119909 = 119910 + 119911 for all 119910 119911 isin119878 0 As a consequence if we denote by msg(119878) the minimalsystem of generators of 119878 then msg(119878) = (119878 0) ((119878 0) +(119878 0)) (see [5 Lemma 23] for other proof of this result)In the following proposition we obtain an analogous for PL-semigroups of the first commented fact in this paragraph

Proposition 14 Let 119878 be a 119875119871-semigroup and let 119909 be aminimal generator of 119878 Then 119878 119909 is a PL-semigroup if andonly if 119909 minus 1 isin 0 cup (N 119878) cup (msg(119878))

Proof (Necessity) If 119909 minus 1 notin 0 cup (N 119878) cup (msg(119878)) then119909 minus 1 isin 119878 (0 cup (msg(119878))) Accordingly there exist 119910 119911 isin119878 0 such that 119909 minus 1 = 119910 + 119911 In fact it is clear that 119910 119911 isin119878 119909 0 Therefore by applyingTheorem 5 and that 119878 119909 isa PL-semigroup we have 119909 = 119910 + 119911 + 1 isin 119878 119909 which is acontradiction

(Sufficiency) Let 119910 119911 isin 119878 119909 0 Since 119878 is a PL-semigroup by Theorem 5 we have 119910 + 119911 + 1 isin 119878 As 119909 minus 1 isin0 cup (N 119878) cup (msg(119878)) we deduce that 119910 + 119911 + 1 = 119909 Thus119910+119911+1 isin 119878 119909 By applyingTheorem 5 again we concludethat 119878 119909 is a PL-semigroup

As a consequence of the previous proposition we have thenext result

Corollary 15 Let 119878 be a119875119871-semigroup such that 119878 =N and let119909 be a minimal generator of 119878 greater than F(119878) Then 119878 119909 isa 119875119871-semigroup if and only if 119909 minus 1 isin msg(119878) cup F(119878)

By applying Theorem 13 together with Corollary 15 wecan get the sons of a vertex of G(SPL) as is shown in thefollowing example

Example 16 It is clear that 119878 = ⟨4 6 7 9⟩ is a PL-semigroupwith Frobenius number equal to 5 From Theorem 13 and

Corollary 15 we deduce that the sons of 119878 are 119878 6 =

⟨4 7 9 10⟩ and 119878 7 = ⟨4 6 9 11⟩

Let us observe that we can build recursively a tree fromthe root if we know the sons of each vertexTherefore we canbuild the tree G(SPL) such as it is shown in Figure 1

In order to have an easier making of the tree G(SPL)we are going to study the relation between the minimalgenerators of a numerical semigroup 119878 and the minimalgenerators of 119878 119909 where 119909 is a minimal generator of119878 that is greater than F(119878) First of all let us observe thatif 119878 is minimally generated by 119898119898 + 1 2119898 minus 1 (ie119878 = 0119898 rarr ) then 119878 119898 = 0119898 + 1 rarr is minimallygenerated by 119898+1119898+2 2119898+1 In other case we havethe following result

Proposition 17 Let 119878 be a numerical semigroup withmsg(119878) = 119899

1 119899

119901 If m(119878) = 119899

1lt 119899119901and 119899

119901gt F(S)

then 119878 119899119901 = ⟨119899

1 119899

119901minus1 119899119901+ 1198991⟩

Proof Let us take 119894 isin 2 119901 Since 119899119901gt F(119878) and 119899

1lt 119899119894

we have that 119899119901+ 119899119894minus 1198991isin 119878 Thus 119899

119901+ 119899119894minus 1198991= 12057211198991+

sdot sdot sdot + 120572119901119899119901for some 120572

1 120572

119901isin N Thereby 119899

119901+ 119899119894= (1205721+

1)1198991+ sdot sdot sdot + 120572

119901119899119901 By applying that 119899

1 119899

119901 is a minimal

system of generators we have that 120572119901= 0Therefore 119899

119901+119899119894isin

⟨1198991 119899

119901minus1⟩ In particular 2119899

119901isin ⟨1198991 119899

119901minus1⟩

Now let 119904 isin 119878 119899119901 Then 119904 isin 119878 and thus there exist

1205731 120573

119901isin N such that 119904 = 120573

11198991+ sdot sdot sdot + 120573

119901119899119901 Since 2119899

119901isin

⟨1198991 119899

119901minus1⟩ we can assume that 120573

119901isin 0 1 On the one

hand if 120573119901= 0 then 119904 isin ⟨119899

1 119899

119901minus1⟩ On the other hand if

120573119901= 1 then there exists 119894 isin 1 119901minus1 such that120573

119894= 0 If 119894 =

1 then it is obvious that 119904 isin ⟨1198991 119899

119901minus1 119899119901+1198991⟩ And if 119894 = 1

since 119899119901+ 119899119894isin ⟨1198991 119899

119901minus1⟩ we have that 119904 isin ⟨119899

1 119899

119901minus1⟩

In any case we conclude that 119878 119899119901 = ⟨119899

1 119899

119901minus1 119899119901+

1198991⟩

Corollary 18 Let 119878 be a numerical semigroup withmsg(119878) =1198991 119899

119901 Ifm(119878) = 119899

1lt 119899119901and 119899119901gt F(119878) then

msg (119878 119899119901)

=

1198991 119899

119901minus1

119894119891 119905ℎ119890119903119890 119890119909119894119904119905119904 119894 isin 2 119901 minus 1

119904119906119888ℎ 119905ℎ119886119905 119899119901+ 1198991minus 119899119894isin 119878

1198991 119899

119901minus1 119899119901+ 1198991 119894119899 119900119905ℎ119890119903 119888119886119904119890

(3)

Proof From Proposition 17 we deduce that msg(119878 119899119901) is

1198991 119899

119901minus1 or 119899

1 119899

119901minus1 119899119901+ 1198991 In addition msg(119878

119899119901) = 119899

1 119899

119901minus1 if and only if 119899

119901+ 1198991isin ⟨1198991 119899

119901minus1⟩

If 119899119901+ 1198991isin ⟨1198991 119899

119901minus1⟩ then there exist 120572

1 120572

119901minus1isin

N such that 119899119901+1198991= 12057211198991+ sdot sdot sdot + 120572

119901minus1119899119901minus1

Since 1198991 119899

119901

is a minimal system of generators we get that 1205721= 0 Thus

there exists 119894 isin 2 119901 minus 1 such that 120572119894= 0 Consequently

119899119901+ 1198991minus 119899119894isin 119878

Conversely if there exists 119894 isin 2 119901 minus 1 such that 119899119901+

1198991minus 119899119894isin 119878 then 119899

119901+ 1198991minus 119899119894= 12057311198991+ sdot sdot sdot + 120573

119901119899119901for some

1205731 120573

119901isin NThereby 119899

119901+1198991= 12057311198991+sdot sdot sdot+ (120573

119894+1)119899119894+sdot sdot sdot+

120573119901119899119901 Since 119899

1 119899

119901 is a minimal system of generators we

have that 120573119901= 0 and therefore 119899

119901+ 1198991isin ⟨1198991 119899

119901minus1⟩

The Scientific World Journal 5

⟨5 6 7 8 9⟩ ⟨4 6 7 9⟩ ⟨4 5 7⟩ ⟨4 5 6⟩ ⟨3 7 8⟩

⟨3 5 7⟩

⟨3 4 5⟩

⟨3 4⟩

⟨2 3⟩

⟨2 5⟩

⟨4 5 6 7⟩

⟨1⟩ = N

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

Figure 1

We finish this section with an illustrative example aboutthe above corollary

Example 19 Let 119878 be the numerical semigroupwithmsg(119878) =3 5 7 It is obvious that F(119878) = 4 By Proposition 17 weknow that 1198785 = ⟨3 7 8⟩ In addition 8minus7 = 1 notin 119878TherebyapplyingCorollary 18 we have thatmsg(1198785) = 3 7 8 Onthe other hand applying Proposition 17 again we have that119878 7 = ⟨3 5 10⟩ Finally since 10 minus 5 = 5 isin 119878 we concludethat msg(119878 7) = 3 5

4 PL-Semigroups with a Fixed Multiplicity

Let 119898 be a positive integer We will denote by Δ(119898) =

0119898 rarr It is clear that Δ(119898) is the greatest (with respectto set inclusion) PL-semigroup with multiplicity119898 Our firstaim in this section will be to show that there also exists thesmallest (with respect to set inclusion) PL-semigroup withmultiplicity119898

As an immediate consequence of item (4) inProposition 9 we have the next result

Lemma20 If 119878 is a119875119871-semigroup119898 isin 1198780 and 119896 isin N0then 119896119898 + 119894 isin 119878 for all 119894 isin 0 119896 minus 1

Proposition 21 Let 119898 isin N 0 Then the numericalsemigroup generated by (119894 + 1)119898 + 119894 | 119894 isin 0 119898 minus 1 isthe smallest (with respect to set inclusion) 119875119871-semigroup withmultiplicity119898

Proof Let 119878 = ⟨119898 2119898+1 1198982+(119898minus1)⟩ FromLemma 20we know that any PL-semigroup with multiplicity 119898 has tocontain 119878 In order to conclude the proof we will show that 119878is a PL-semigroup For this purpose since (119894 + 1)119898 + 119894 | 119894 isin

0 119898minus1 is a system of generators of 119878 it will be enoughto check item (2) of Proposition 9 that is if 119894 119895 isin 0 119898 minus

1 then (119894 + 1)119898+ 119894 + (119895 + 1)119898+ 119895+ 1 isin 119878 We distinguish twocases

(1) If 119894+ 119895+1 le 119898minus1 then (119894+1)119898+119894+ (119895+1)119898+119895+1 =(119894 + 119895 + 2)119898 + (119894 + 119895 + 1) isin 119878

(2) If 119894 + 119895 + 1 ge 119898 then (119894 + 1)119898 + 119894 + (119895 + 1)119898 + 119895 + 1 =(119894 + 119895 + 2)119898 + (119894 + 119895 + 1) = (119898+ 1)119898+ (119894 + 119895 minus119898+ 2)119898

+ (119894 + 119895 minus 119898 + 1) isin 119878

We will denote byΘ(119898) = ⟨119898 2119898 + 1 1198982

+ (119898 minus 1)⟩

and bySPL(119898) the set of all PL-semigroups with multiplicityequal to 119898 Let us recall that Δ(119898) = max(SPL(119898)) andΘ(119898) = min(SPL(119898))

As an application of the above comment we have the nextresult

Corollary 22 The set SPL(119898) is finite

Proof If 119878 isin SPL(119898) thenΘ(119898) sube 119878 sube Δ(119898) SinceΔ(119898) andΘ(119898) are numerical semigroups we have that Δ(119898) Θ(119898) isfinite Consequently SPL(119898) is also finite

Remark 23 The previous result can be considered a particu-lar case of [6 Theorem 66]

Now we are interested in computing the Frobeniusnumber and the genus ofΘ(119898) For that several concepts andresults are introduced

If 119878 is a numerical semigroup and 119898 isin 119878 0 then theApery set of119898 in 119878 (see [9]) is Ap(119878 119898) = 119904 isin 119878 | 119904 minus 119898 notin 119878It is clear (see for instance [5 Lemma 24]) that Ap(119878 119898) =120596(0) = 0 120596(1) 120596(119898 minus 1) where 120596(119894) is for each 119894 isin0 119898 minus 1 the least element of 119878 that is congruent with 119894modulo119898

The next result is [5 Proposition 212]

Lemma24 Let 119878 be a numerical semigroup and let119898 isin 1198780Then

(1) F(119878) = max(Ap(119878 119898)) minus 119898(2) g(119878) = (1119898)(sum

119908isinAp(119878119898) 119908) minus ((119898 minus 1)2)

If 119886 119887 are integers with 119887 = 0 we denote by 119886 mod 119887 theremainder of the division of 119886 by 119887 The following result is [5Proposition 35]

6 The Scientific World Journal

Lemma 25 Let 119898 isin N 0 and let 119883 = 120596(0) =

0 120596(1) 120596(119898minus1) sube N such that for each 119894 isin 1 119898minus1120596(119894) is congruent with 119894 modulo 119898 Let 119878 be the numericalsemigroup generated by 119883 cup 119898 The following conditions areequivalent

(1) Ap(119878 119898) = 119883(2) 120596(119894)+120596(119895) ge 120596((119894+119895) mod 119898) for all 119894 119895 isin 1 119898minus

1

Proposition 26 If119898 isin N 0 then

Ap (Θ (119898) 119898)

= 120596 (0) = 0 120596 (1) = 2119898 + 1 120596 (119898 minus 1) = 1198982

+ 119898 minus 1

(4)

Proof It is clear that 120596(119894) = (119894 + 1)119898 + 119894 is congruent with 119894modulo 119898 for all 119894 isin 1 119898 minus 1 Let us see now that if119894 119895 isin 1 119898 minus 1 then 120596(119894) + 120596(119895) ge 120596((119894 + 119895) mod 119898)Indeed 120596(119894) +120596(119895) = (119894 + 1)119898 + 119894 + (119895 + 1)119898 + 119895 gt (119894 + 119895 + 1)119898+ (119894 + 119895) ge ((119894 + 119895) mod 119898 + 1)119898 + (119894 + 119895) mod 119898 = 120596((119894 +119895) mod 119898) The proof follows from Lemma 25

Corollary 27 If119898 isin N 0 1 then

(1) F(Δ(119898)) = 119898 minus 1(2) g(Δ(119898)) = 119898 minus 1(3) F(Θ(119898)) = 1198982 minus 1(4) g(Θ(119898)) = (119898 minus 1)(119898 + 2)2

Proof Items (1) and (2) are trivial On the other hand items(3) and (4) are immediate consequences of Lemma 24 andProposition 26

Remark 28 Thenumerical semigroupΘ(119898) can be rewrittenas

Θ (119898)

= 119898119898+ (119898+ 1) 119898 +2 (119898 + 1) 119898 + (119898 minus 1) (119898 + 1)

(5)

Thus Θ(119898) is a numerical semigroup generated by an arith-metic sequencewith first term119898 and commondifference119898+1(see [2 10])

If 119878 is a numerical semigroup then the cardinality of theminimal system of generators of 119878 is called the embeddingdimension of 119878 and is denoted by e(119878) It is well known (see [5Proposition 212]) that e(119878) le m(119878) We say that a numericalsemigroup 119878 has maximal embedding dimension if e(119878) =m(119878) It is clear that 119898119898 + 1 2119898 minus 1 is the minimalsystem of generators of Δ(119898) Therefore Δ(119898) is a numericalsemigroupwithmaximal embedding dimensionNowwewillshow that Θ(119898) has also maximal embedding dimension

The next result is [5 Corollary 36]

Lemma 29 Let 119878 be a numerical semigroup with multiplicity119898 and assume thatAp(119878 119898) = 120596(0) = 0 120596(1) 120596(119898minus1)

Then 119878 has maximal embedding dimension if and only if 120596(119894)+120596(119895) gt 120596((119894 + 119895) mod 119898) for all 119894 119895 isin 1 119898 minus 1

Let us observe that in the proof of Proposition 26 wehave shown that 120596(119894) + 120596(119895) gt 120596((119894 + 119895) mod 119898) for all119894 119895 isin 1 119898 minus 1 Therefore by applying Lemma 29 weget the following result

Corollary 30 If 119898 isin N 0 then Θ(119898) is a numericalsemigroup with maximal embedding dimension

As a consequence of this corollary we have that 119898 2119898 +

1 1198982

+119898minus1 is theminimal systemof generators ofΘ(119898)Now we want to show orderly the elements of SPL(119898)

Thus we define the graph G(SPL(119898)) in the following way

(i) SPL(119898) is the set of vertices of G(SPL(119898))

(ii) (119878 1198781015840) isin SPL(119898) timesSPL(119898) is an edge of G(SPL(119898)) if1198781015840

= 119878 cup F(119878)

The next result is analogous to Theorem 13

Theorem 31 The graph G(S119875119871(119898)) is a tree with root equal

to Δ(119898) Moreover the sons of a vertex 119878 isin S119875119871(119898) are 119878

1199091 119878119909

119903 with 119909

1 119909

119903= 119909 isin msg(119878) | 119909 =119898 119909 gt

F(119878) 119886119899119889 119878 119909 isin S119875119871

By applyingTheorem 31 and Corollaries 15 and 18 we canget easily G(SPL(119898)) such as is shown in the next example

Example 32 We are going to depict G(SPL(3)) that is thetree of the PL-semigroups with multiplicity equal to 3

⟨3 7 8⟩

⟨3 5 7⟩ ⟨3 4⟩

⟨3 4 5⟩ = Δ(3)

⟨3 7 11⟩ = Θ(3)

If119879 = (119881 119864) is a tree then the height of119879 is themaximumof the lengths of the paths that connect each vertex with theroot Let us observe that the height of G(SPL(3)) is 3 Ingeneral the height of the tree G(SPL(119898)) is equal to

g (Θ (119898)) minus g (Δ (119898)) = (119898 minus 1) (119898 + 2)

2

minus (119898 minus 1)

=

(119898 minus 1)119898

2

(6)

Let us study now the possible values of the Frobeniusnumber and the genus for PL-semigroups with multiplicity119898

The Scientific World Journal 7

Proposition 33 If119898 isin N 0 then

(1) g(119878) | 119878 isin SPL(119898) = 119909 isin N | 119898 minus 1 le 119909 le

(119898 minus 1)(119898 + 2)2

(2) F(119878) | 119878 isin SPL(119898) = (Δ(119898) Θ(119898)) cup 119898 minus 1

Proof Let us assume that Δ(119898) Θ(119898) = 1199091gt 1199092gt sdot sdot sdot gt

119909120588(1) By Corollary 27 we know that

g (119878) | 119878 isin SPL (119898) sube 119898 minus 1

(119898 minus 1) (119898 + 2)

2

(7)

For the opposite inclusion it is enough to observe thatΘ(119898)cup119909119894 rarr isin SPL(119898) and that g(Θ(119898)cup119909119894 rarr ) = (119898minus1)(119898+

2)2 minus 119894(2) It is clear that Θ(119898) cup 119909

119894+ 1 rarr isin SPL(119898) with

Frobenius number equal to 119909119894 Thus (Δ(119898) Θ(119898)) cup 119898 minus

1 sube F(119878) | 119878 isin SPL(119898) For the other inclusion let ustake 119878 isin SPL(119898) such that 119878 = Δ(119898) Then F(119878) gt 119898 andthereby F(119878) isin Δ(119898) Since Θ(119898) sube 119878 we have F(119878) notin Θ(119878)Therefore we conclude that F(119878) isin Δ(119898) Θ(119878)

Example 34 By Proposition 33 g(119878) | 119878 isin SPL(3) =

2 3 4 5 Since Δ(3) = ⟨3 4 5⟩ and Θ(3) = ⟨3 7 11⟩ wehave that (Δ(3) Θ(3)) cup 2 = 2 4 5 8 Therefore byapplying Proposition 33 again we conclude that F(119878) | 119878 isinSPL(3) = 2 4 5 8

5 The Smallest PL-Semigroup That Contains aGiven Set of Positive Integers

Let us observe that in general the infinite intersection ofelements of SPL is not a numerical semigroup For instance⋂119899isinN0 119899 rarr = 0 On the other hand it is clear that the

(finite or infinite) intersection of numerical semigroups isalways a submonoid of (N +)

Let 119872 be a submonoid of (N +) We will say that 119872 isa SPL-monoid if it can be expressed like the intersection ofelements of SPL

The next lemma has an immediate proof

Lemma35 The intersection ofS119875119871-monoids is aS

119875119871-monoid

In view of this result we can give the following definition

Definition 36 Let 119883 be a subset of N The SPL-monoidgenerated by119883 (denoted bySPL(119883)) is the intersection of allSPL-monoids containing119883

If 119872 = SPL(119883) then we will say that 119883 is a SPL-systemof generators of119872 In addition if no proper subset of 119883 is aSPL-system of generators of 119872 then we will say that 119883 is aminimal SPL-system of generators of119872

Let us recall that by Proposition 12 we know that S119875119871

isa Frobenius variety Therefore by applying [3 Corollary 19]we have the next result

Proposition 37 Every S119875119871-monoid has a unique minimal

S119875119871-system of generators which in addition is finite

The proof of the following lemma is also immediate

Lemma 38 If 119883 sube N then S119875119871(119883) is the intersection of all

119875119871-semigroups that contain 119883

Proposition 39 If 119883 is a nonempty subset of N 0 thenS119875119871(119883) is a 119875119871-semigroup

Proof We know that SPL(119883) is a submonoid of (N +)Therefore in order to show that SPL(119883) is a numericalsemigroup it will be enough to see that N SPL(119883) is a finiteset

Let 119909 isin 119883 If 119878 is a PL-semigroup containing 119883 then (byTheorem 5) we know that 119909 2119909 + 1 sube 119878 and in this way⟨119909 2119909 + 1⟩ sube 119878 From Lemma 38 we have that ⟨119909 2119909 + 1⟩ subeSPL(119883) Since gcd119909 2119909 + 1 = 1 we get that ⟨119909 2119909 + 1⟩is a numerical semigroup and thus N ⟨119909 2119909 + 1⟩ is finiteConsequently N SPL(119883) is finite

Now let us see that SPL(119883) is a PL-semigroup Let 119909 119910 isinSPL(119883) 0 If 119878 is a PL-semigroup containing 119883 fromLemma 38 we deduce that 119909 119910 isin 1198780 and fromTheorem 5we have that 119909 + 119910 + 1 isin 119878 By applying again Lemma 38we have that 119909 + 119910 + 1 isin SPL(119883) Therefore by applyingTheorem 5 once more we can assert that SPL(119883) is a PL-semigroup

Remark 40 Let us observe that in general Proposition 39 isnot true for Frobenius varieties In fact let S be the set of allnumerical semigroups It is clear thatS is a Frobenius varietyIf we take 119883 = 2 then the intersection of all elements of Scontaining 2 is exactly119872 = ⟨2⟩ which is not a numericalsemigroup

The next result will be key for our last purpose in thissection

Theorem 41 S119875119871

= S119875119871(119883) | 119883 is a nonempty finite

subset 119900119891 N 0

Proof By Proposition 39 we have that

SPL (119883) | 119883 is a nonempty finite subset ofN 0 sube SPL(8)

For the other inclusion it is enough to observe that if 119878 isin SPLthen (by Proposition 37) there exists a nonempty finite subset119883 of N 0 such that 119878 = SPL(119883)

Since SPL is a Frobenius variety by applying [3 Proposi-tion 24] we get the next result

Proposition 42 Let119872 be aS119875119871-monoid and let 119909 isin 119872Then

119872119909 is aS119875119871-monoid if and only if 119909 belongs to the minimal

S119875119871-system of generators of119872

As an immediate consequence of this propositionwe havethe following result

8 The Scientific World Journal

Corollary 43 Let 119883 be a nonempty subset of N 0 Thenthe minimal S

119875119871-system of generators of S

119875119871(119883) is 119909 isin 119883 |

S119875119871(119883) 119909 is a PL-semigroup

Example 44 By Proposition 21 119878 = ⟨3 7 11⟩ is a PL-semigroup By applying Proposition 14 we easily deduce that

119909 isin 3 7 11 | 119878 119909 is a PL-semigroup = 3 (9)

Therefore 119878 = SPL(3) and 3 is the minimalSPL-system ofgenerators of 119878

Now we want to describe SPL(119883) when 119883 is a fixednonempty finite set of positive integers Let us observe thatby Theorem 41 we know that every PL-semigroup can beobtained in this way

Let 1198991 119899

119901be positive integers We will denote by

119878(1198991 119899

119901) the set 120572

11198991+ sdot sdot sdot + 120572

119901119899119901+ 119903 | 119903 120572

1 120572

119901isin

N 119903 lt 1205721+ sdot sdot sdot + 120572

119901 cup 0 Our next purpose will be to show

that 119878(1198991 119899

119901) = SPL(1198991 119899119901)

Lemma 45 Let 119878 be a numerical semigroup let 1199041 119904

119905isin

119878 0 and let 1205721 120572

119905isin N Then ord(120572

11199041+ sdot sdot sdot + 120572

119905119904119905) ge

1205721+ sdot sdot sdot + 120572

119905

Proof Let 1198991 119899

119901 be the minimal system of generators

of 119878 Then for each 119894 isin 1 119905 there exist 1205731198941 120573

119894119901isin N

such that 119904119894= 12057311989411198991+ sdot sdot sdot + 120573

119894119901119899119901 Moreover since 119904

119894= 0 we

have that 1205731198941+ sdot sdot sdot + 120573

119894119901ge 1 Thus

12057211199041+ sdot sdot sdot + 120572

119905119904119905

= 1205721(120573111198991+ sdot sdot sdot + 120573

1119901119899119901) + sdot sdot sdot + 120572

119905(12057311990511198991+ sdot sdot sdot + 120573

119905119901119899119901)

= (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) 1198991+ sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901) 119899119901

(10)

Therefore

ord (12057211199041+ sdot sdot sdot + 120572

119905119904119905)

ge (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) + sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901)

= 1205721(12057311+ sdot sdot sdot + 120573

1119901) + sdot sdot sdot + 120572

119905(1205731199051+ sdot sdot sdot + 120573

119905119901)

ge 1205721+ sdot sdot sdot + 120572

119905

(11)

Theorem 46 If 1198991 119899

119901are positive integers then

119878(1198991 119899

119901) is the smallest (with respect to set inclusion)

119875119871-semigroup containing 1198991 119899

119901

Proof We divide the proof into five steps

(i) Let us see that if 119909 119910 isin 119878(1198991 119899

119901) 0 then 119909 +

119910 isin 119878(1198991 119899

119901) In effect we know that there exist

1205721 120572

119901 1205731 120573

119901 119903 1199031015840 nonnegative integers such

that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903 119910 = 120573

11198991+sdot sdot sdot+120573

119901119899119901+1199031015840

119903 lt 1205721+ sdot sdot sdot + 120572

119901 and 1199031015840 lt 120573

1+ + 120573

119901 Therefore

119909 + 119910 = (1205721+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840 with119903 + 1199031015840

lt (1205721+ 1205731) + sdot sdot sdot + (120572

119901+ 120573119899) Consequently

119909 + 119910 isin 119878(1198991 119899

119901)

(ii) Let us see that N 119878(1198991 119899

119901) is finite Since 119899

1=

1sdot1198991+0sdot1198992+sdot sdot sdot+0sdot119899

119901+0 and 2119899

1+1 = 2sdot119899

1+0sdot1198992+sdot sdot sdot+0sdot

119899119901+1 we have that 119899

1 21198991isin 119878(1198991 119899

119901) By applying

the first step we get that ⟨1198991 21198991⟩ sube 119878(119899

1 119899

119901)

Using the same reasoning as we did in the proof ofProposition 39 we have the result

(iii) From the previous steps we know that 119878(1198991 119899

119901)

is a numerical semigroup Let us see now that119878(1198991 119899

119901) is a PL-semigroup In order to do that

it is enough (by Theorem 5) to show that if 119909 119910 isin

119878(1198991 119899

119901) 0 then 119909 + 119910 + 1 isin 119878(119899

1 119899

119901)

Indeed arguing as in the first step we have that 119909 +119910 + 1 = (120572

1+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840

+ 1

with 119903 + 1199031015840 + 1 lt (1205721+1205731) + sdot sdot sdot + (120572

119901+120573119899) Therefore

119909 + 119910 + 1 isin 119878(1198991 119899

119901)

(iv) Following the proof of the second step it is clear that1198991 119899

119901 sube 119878(119899

1 119899

119901)

(v) Finally let us see that 119878(1198991 119899

119901) is the small-

est PL-semigroup that contains 1198991 119899

119901 In fact

we will show that if 119879 is PL-semigroup containing1198991 119899

119901 then 119878(119899

1 119899

119901) sube 119879 Thus let 119909 isin

119878(1198991 119899

119901) 0 Then there exist 120572

1 120572

119901 119903 isin N

such that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903with 119903 lt 120572

1+sdot sdot sdot+120572

119901

Since 1198991 119899

119901 sube 119879 by Proposition 9 we have that

12057211198991+sdot sdot sdot+120572

119901119899119901+0 ord(120572

11198991+sdot sdot sdot+120572

119901119899119901)minus1 sube 119879

By applying Lemma 45 we have that 119903 lt ord(12057211198991+

sdot sdot sdot + 120572119901119899119901) and therefore 119909 isin 119879

In this way we have proved the statement

The next result is an immediate consequence of theprevious theorem

Corollary 47 If 1198991 119899

119901are positive integers then

SPL(1198991 119899119901) = 119878(1198991 119899119901)

We finish this section with an example that illustrates itscontent

Example 48 It is clear that 119878(4 7) = 0 4 7 8 9 11 rarr =

⟨4 7 9⟩ Therefore SPL(4 7) = ⟨4 7 9⟩

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referee for hisher usefulcomments and suggestions that helped to improve this workBoth of the authors are supported by FQM-343 (Junta deAndalucıa) MTM2010-15595 (MICINN Spain) and FEDERfunds The second author is also partially supported by Juntade AndalucıaFeder Grant no FQM-5849

The Scientific World Journal 9

References

[1] M Lothaire Applied Combinatorics on Words vol 105 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press 2005

[2] J L Ramırez Alfonsın The Diophantine Frobenius ProblemOxford University Press 2005

[3] J C Rosales ldquoFamilies of numerical semigroups closed underfinite intersections and for the Frobenius numberrdquo HoustonJournal of Mathematics vol 34 no 2 pp 339ndash348 2008

[4] J C Rosales M B Branco and D Torrao ldquoBracelet monoidsand numerical semigroupsrdquo preprint

[5] J C Rosales and P A Garcıa-Sanchez Numerical Semigroupsvol 20 of Developments in Mathematics Springer New YorkNY USA 2009

[6] M Bras-Amoros P A Garcıa-Sanchez and A Vico-OtonldquoNonhomogeneous patterns on numerical semigroupsrdquo Inter-national Journal of Algebra and Computation vol 23 no 6 pp1469ndash1483 2013

[7] K Stokes and M Bras-Amoros ldquoLinear non-homogeneoussymmetric patterns and prime power generators in numeri-cal semigroups associated to combinatorial congurationsrdquo toappear in Semigroup Forum 2013

[8] L Bryant ldquoGoto numbers of a numerical semigroup ring andthe gorensteiness of associated graded ringsrdquo Communicationsin Algebra vol 38 no 6 pp 2092ndash2128 2010

[9] R Apery ldquoSur les branches superlineaires des courbesalgebriquesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol222 pp 1198ndash1200 1946

[10] J B Roberts ldquoNote on linear formsrdquo Proceedings of the Ameri-can Mathematical Society vol 7 pp 465ndash469 1956

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

119881 are called vertices of 119866 and the elements of 119864 are callededges of 119866 A path (of length 119899) connecting the vertices 119909and 119910 of 119866 is a sequence of different edges of the form(V0 V1) (V1 V2) (V

119899minus1 V119899) such that V

0= 119909 and V

119899= 119910

We say that a graph 119866 is a tree if there exist a vertex Vlowast

(known as the root of 119866) such that for every other vertex 119909 of119866 there exists a unique path connecting 119909 and Vlowast If (119909 119910) isan edge of the tree then we say that 119909 is a son of 119910

We define the graph G(S119875119871) in the following way

(i) S119875119871

is the set of vertices of G(SPL)(ii) (119878 1198781015840) isin S

119875119871times S119875119871

is an edge of G(S119875119871) if 1198781015840 = 119878 cup

F(119878)As a consequence of [3 Proposition 24 Theorem 27] we

have the next result

Theorem 13 The graph G(S119875119871) is a tree with root equal to N

Moreover the sons of a vertex 119878 isin S119875119871

are 119878 1199091 119878

119909119903 where 119909

1 119909

119903are the minimal generators of 119878 that are

greater than F(119878) and such that 119878 1199091 119878 119909

119903 isin S119875119871

Let us observe that if 119878 is a numerical semigroup and119909 isin 119878 then 119878 119909 is a numerical semigroup if and only if119909 is a minimal generator of 119878 In fact 119878 119909 is a numericalsemigroup whenever 119909 isin 119878 0 and 119909 = 119910 + 119911 for all 119910 119911 isin119878 0 As a consequence if we denote by msg(119878) the minimalsystem of generators of 119878 then msg(119878) = (119878 0) ((119878 0) +(119878 0)) (see [5 Lemma 23] for other proof of this result)In the following proposition we obtain an analogous for PL-semigroups of the first commented fact in this paragraph

Proposition 14 Let 119878 be a 119875119871-semigroup and let 119909 be aminimal generator of 119878 Then 119878 119909 is a PL-semigroup if andonly if 119909 minus 1 isin 0 cup (N 119878) cup (msg(119878))

Proof (Necessity) If 119909 minus 1 notin 0 cup (N 119878) cup (msg(119878)) then119909 minus 1 isin 119878 (0 cup (msg(119878))) Accordingly there exist 119910 119911 isin119878 0 such that 119909 minus 1 = 119910 + 119911 In fact it is clear that 119910 119911 isin119878 119909 0 Therefore by applyingTheorem 5 and that 119878 119909 isa PL-semigroup we have 119909 = 119910 + 119911 + 1 isin 119878 119909 which is acontradiction

(Sufficiency) Let 119910 119911 isin 119878 119909 0 Since 119878 is a PL-semigroup by Theorem 5 we have 119910 + 119911 + 1 isin 119878 As 119909 minus 1 isin0 cup (N 119878) cup (msg(119878)) we deduce that 119910 + 119911 + 1 = 119909 Thus119910+119911+1 isin 119878 119909 By applyingTheorem 5 again we concludethat 119878 119909 is a PL-semigroup

As a consequence of the previous proposition we have thenext result

Corollary 15 Let 119878 be a119875119871-semigroup such that 119878 =N and let119909 be a minimal generator of 119878 greater than F(119878) Then 119878 119909 isa 119875119871-semigroup if and only if 119909 minus 1 isin msg(119878) cup F(119878)

By applying Theorem 13 together with Corollary 15 wecan get the sons of a vertex of G(SPL) as is shown in thefollowing example

Example 16 It is clear that 119878 = ⟨4 6 7 9⟩ is a PL-semigroupwith Frobenius number equal to 5 From Theorem 13 and

Corollary 15 we deduce that the sons of 119878 are 119878 6 =

⟨4 7 9 10⟩ and 119878 7 = ⟨4 6 9 11⟩

Let us observe that we can build recursively a tree fromthe root if we know the sons of each vertexTherefore we canbuild the tree G(SPL) such as it is shown in Figure 1

In order to have an easier making of the tree G(SPL)we are going to study the relation between the minimalgenerators of a numerical semigroup 119878 and the minimalgenerators of 119878 119909 where 119909 is a minimal generator of119878 that is greater than F(119878) First of all let us observe thatif 119878 is minimally generated by 119898119898 + 1 2119898 minus 1 (ie119878 = 0119898 rarr ) then 119878 119898 = 0119898 + 1 rarr is minimallygenerated by 119898+1119898+2 2119898+1 In other case we havethe following result

Proposition 17 Let 119878 be a numerical semigroup withmsg(119878) = 119899

1 119899

119901 If m(119878) = 119899

1lt 119899119901and 119899

119901gt F(S)

then 119878 119899119901 = ⟨119899

1 119899

119901minus1 119899119901+ 1198991⟩

Proof Let us take 119894 isin 2 119901 Since 119899119901gt F(119878) and 119899

1lt 119899119894

we have that 119899119901+ 119899119894minus 1198991isin 119878 Thus 119899

119901+ 119899119894minus 1198991= 12057211198991+

sdot sdot sdot + 120572119901119899119901for some 120572

1 120572

119901isin N Thereby 119899

119901+ 119899119894= (1205721+

1)1198991+ sdot sdot sdot + 120572

119901119899119901 By applying that 119899

1 119899

119901 is a minimal

system of generators we have that 120572119901= 0Therefore 119899

119901+119899119894isin

⟨1198991 119899

119901minus1⟩ In particular 2119899

119901isin ⟨1198991 119899

119901minus1⟩

Now let 119904 isin 119878 119899119901 Then 119904 isin 119878 and thus there exist

1205731 120573

119901isin N such that 119904 = 120573

11198991+ sdot sdot sdot + 120573

119901119899119901 Since 2119899

119901isin

⟨1198991 119899

119901minus1⟩ we can assume that 120573

119901isin 0 1 On the one

hand if 120573119901= 0 then 119904 isin ⟨119899

1 119899

119901minus1⟩ On the other hand if

120573119901= 1 then there exists 119894 isin 1 119901minus1 such that120573

119894= 0 If 119894 =

1 then it is obvious that 119904 isin ⟨1198991 119899

119901minus1 119899119901+1198991⟩ And if 119894 = 1

since 119899119901+ 119899119894isin ⟨1198991 119899

119901minus1⟩ we have that 119904 isin ⟨119899

1 119899

119901minus1⟩

In any case we conclude that 119878 119899119901 = ⟨119899

1 119899

119901minus1 119899119901+

1198991⟩

Corollary 18 Let 119878 be a numerical semigroup withmsg(119878) =1198991 119899

119901 Ifm(119878) = 119899

1lt 119899119901and 119899119901gt F(119878) then

msg (119878 119899119901)

=

1198991 119899

119901minus1

119894119891 119905ℎ119890119903119890 119890119909119894119904119905119904 119894 isin 2 119901 minus 1

119904119906119888ℎ 119905ℎ119886119905 119899119901+ 1198991minus 119899119894isin 119878

1198991 119899

119901minus1 119899119901+ 1198991 119894119899 119900119905ℎ119890119903 119888119886119904119890

(3)

Proof From Proposition 17 we deduce that msg(119878 119899119901) is

1198991 119899

119901minus1 or 119899

1 119899

119901minus1 119899119901+ 1198991 In addition msg(119878

119899119901) = 119899

1 119899

119901minus1 if and only if 119899

119901+ 1198991isin ⟨1198991 119899

119901minus1⟩

If 119899119901+ 1198991isin ⟨1198991 119899

119901minus1⟩ then there exist 120572

1 120572

119901minus1isin

N such that 119899119901+1198991= 12057211198991+ sdot sdot sdot + 120572

119901minus1119899119901minus1

Since 1198991 119899

119901

is a minimal system of generators we get that 1205721= 0 Thus

there exists 119894 isin 2 119901 minus 1 such that 120572119894= 0 Consequently

119899119901+ 1198991minus 119899119894isin 119878

Conversely if there exists 119894 isin 2 119901 minus 1 such that 119899119901+

1198991minus 119899119894isin 119878 then 119899

119901+ 1198991minus 119899119894= 12057311198991+ sdot sdot sdot + 120573

119901119899119901for some

1205731 120573

119901isin NThereby 119899

119901+1198991= 12057311198991+sdot sdot sdot+ (120573

119894+1)119899119894+sdot sdot sdot+

120573119901119899119901 Since 119899

1 119899

119901 is a minimal system of generators we

have that 120573119901= 0 and therefore 119899

119901+ 1198991isin ⟨1198991 119899

119901minus1⟩

The Scientific World Journal 5

⟨5 6 7 8 9⟩ ⟨4 6 7 9⟩ ⟨4 5 7⟩ ⟨4 5 6⟩ ⟨3 7 8⟩

⟨3 5 7⟩

⟨3 4 5⟩

⟨3 4⟩

⟨2 3⟩

⟨2 5⟩

⟨4 5 6 7⟩

⟨1⟩ = N

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

Figure 1

We finish this section with an illustrative example aboutthe above corollary

Example 19 Let 119878 be the numerical semigroupwithmsg(119878) =3 5 7 It is obvious that F(119878) = 4 By Proposition 17 weknow that 1198785 = ⟨3 7 8⟩ In addition 8minus7 = 1 notin 119878TherebyapplyingCorollary 18 we have thatmsg(1198785) = 3 7 8 Onthe other hand applying Proposition 17 again we have that119878 7 = ⟨3 5 10⟩ Finally since 10 minus 5 = 5 isin 119878 we concludethat msg(119878 7) = 3 5

4 PL-Semigroups with a Fixed Multiplicity

Let 119898 be a positive integer We will denote by Δ(119898) =

0119898 rarr It is clear that Δ(119898) is the greatest (with respectto set inclusion) PL-semigroup with multiplicity119898 Our firstaim in this section will be to show that there also exists thesmallest (with respect to set inclusion) PL-semigroup withmultiplicity119898

As an immediate consequence of item (4) inProposition 9 we have the next result

Lemma20 If 119878 is a119875119871-semigroup119898 isin 1198780 and 119896 isin N0then 119896119898 + 119894 isin 119878 for all 119894 isin 0 119896 minus 1

Proposition 21 Let 119898 isin N 0 Then the numericalsemigroup generated by (119894 + 1)119898 + 119894 | 119894 isin 0 119898 minus 1 isthe smallest (with respect to set inclusion) 119875119871-semigroup withmultiplicity119898

Proof Let 119878 = ⟨119898 2119898+1 1198982+(119898minus1)⟩ FromLemma 20we know that any PL-semigroup with multiplicity 119898 has tocontain 119878 In order to conclude the proof we will show that 119878is a PL-semigroup For this purpose since (119894 + 1)119898 + 119894 | 119894 isin

0 119898minus1 is a system of generators of 119878 it will be enoughto check item (2) of Proposition 9 that is if 119894 119895 isin 0 119898 minus

1 then (119894 + 1)119898+ 119894 + (119895 + 1)119898+ 119895+ 1 isin 119878 We distinguish twocases

(1) If 119894+ 119895+1 le 119898minus1 then (119894+1)119898+119894+ (119895+1)119898+119895+1 =(119894 + 119895 + 2)119898 + (119894 + 119895 + 1) isin 119878

(2) If 119894 + 119895 + 1 ge 119898 then (119894 + 1)119898 + 119894 + (119895 + 1)119898 + 119895 + 1 =(119894 + 119895 + 2)119898 + (119894 + 119895 + 1) = (119898+ 1)119898+ (119894 + 119895 minus119898+ 2)119898

+ (119894 + 119895 minus 119898 + 1) isin 119878

We will denote byΘ(119898) = ⟨119898 2119898 + 1 1198982

+ (119898 minus 1)⟩

and bySPL(119898) the set of all PL-semigroups with multiplicityequal to 119898 Let us recall that Δ(119898) = max(SPL(119898)) andΘ(119898) = min(SPL(119898))

As an application of the above comment we have the nextresult

Corollary 22 The set SPL(119898) is finite

Proof If 119878 isin SPL(119898) thenΘ(119898) sube 119878 sube Δ(119898) SinceΔ(119898) andΘ(119898) are numerical semigroups we have that Δ(119898) Θ(119898) isfinite Consequently SPL(119898) is also finite

Remark 23 The previous result can be considered a particu-lar case of [6 Theorem 66]

Now we are interested in computing the Frobeniusnumber and the genus ofΘ(119898) For that several concepts andresults are introduced

If 119878 is a numerical semigroup and 119898 isin 119878 0 then theApery set of119898 in 119878 (see [9]) is Ap(119878 119898) = 119904 isin 119878 | 119904 minus 119898 notin 119878It is clear (see for instance [5 Lemma 24]) that Ap(119878 119898) =120596(0) = 0 120596(1) 120596(119898 minus 1) where 120596(119894) is for each 119894 isin0 119898 minus 1 the least element of 119878 that is congruent with 119894modulo119898

The next result is [5 Proposition 212]

Lemma24 Let 119878 be a numerical semigroup and let119898 isin 1198780Then

(1) F(119878) = max(Ap(119878 119898)) minus 119898(2) g(119878) = (1119898)(sum

119908isinAp(119878119898) 119908) minus ((119898 minus 1)2)

If 119886 119887 are integers with 119887 = 0 we denote by 119886 mod 119887 theremainder of the division of 119886 by 119887 The following result is [5Proposition 35]

6 The Scientific World Journal

Lemma 25 Let 119898 isin N 0 and let 119883 = 120596(0) =

0 120596(1) 120596(119898minus1) sube N such that for each 119894 isin 1 119898minus1120596(119894) is congruent with 119894 modulo 119898 Let 119878 be the numericalsemigroup generated by 119883 cup 119898 The following conditions areequivalent

(1) Ap(119878 119898) = 119883(2) 120596(119894)+120596(119895) ge 120596((119894+119895) mod 119898) for all 119894 119895 isin 1 119898minus

1

Proposition 26 If119898 isin N 0 then

Ap (Θ (119898) 119898)

= 120596 (0) = 0 120596 (1) = 2119898 + 1 120596 (119898 minus 1) = 1198982

+ 119898 minus 1

(4)

Proof It is clear that 120596(119894) = (119894 + 1)119898 + 119894 is congruent with 119894modulo 119898 for all 119894 isin 1 119898 minus 1 Let us see now that if119894 119895 isin 1 119898 minus 1 then 120596(119894) + 120596(119895) ge 120596((119894 + 119895) mod 119898)Indeed 120596(119894) +120596(119895) = (119894 + 1)119898 + 119894 + (119895 + 1)119898 + 119895 gt (119894 + 119895 + 1)119898+ (119894 + 119895) ge ((119894 + 119895) mod 119898 + 1)119898 + (119894 + 119895) mod 119898 = 120596((119894 +119895) mod 119898) The proof follows from Lemma 25

Corollary 27 If119898 isin N 0 1 then

(1) F(Δ(119898)) = 119898 minus 1(2) g(Δ(119898)) = 119898 minus 1(3) F(Θ(119898)) = 1198982 minus 1(4) g(Θ(119898)) = (119898 minus 1)(119898 + 2)2

Proof Items (1) and (2) are trivial On the other hand items(3) and (4) are immediate consequences of Lemma 24 andProposition 26

Remark 28 Thenumerical semigroupΘ(119898) can be rewrittenas

Θ (119898)

= 119898119898+ (119898+ 1) 119898 +2 (119898 + 1) 119898 + (119898 minus 1) (119898 + 1)

(5)

Thus Θ(119898) is a numerical semigroup generated by an arith-metic sequencewith first term119898 and commondifference119898+1(see [2 10])

If 119878 is a numerical semigroup then the cardinality of theminimal system of generators of 119878 is called the embeddingdimension of 119878 and is denoted by e(119878) It is well known (see [5Proposition 212]) that e(119878) le m(119878) We say that a numericalsemigroup 119878 has maximal embedding dimension if e(119878) =m(119878) It is clear that 119898119898 + 1 2119898 minus 1 is the minimalsystem of generators of Δ(119898) Therefore Δ(119898) is a numericalsemigroupwithmaximal embedding dimensionNowwewillshow that Θ(119898) has also maximal embedding dimension

The next result is [5 Corollary 36]

Lemma 29 Let 119878 be a numerical semigroup with multiplicity119898 and assume thatAp(119878 119898) = 120596(0) = 0 120596(1) 120596(119898minus1)

Then 119878 has maximal embedding dimension if and only if 120596(119894)+120596(119895) gt 120596((119894 + 119895) mod 119898) for all 119894 119895 isin 1 119898 minus 1

Let us observe that in the proof of Proposition 26 wehave shown that 120596(119894) + 120596(119895) gt 120596((119894 + 119895) mod 119898) for all119894 119895 isin 1 119898 minus 1 Therefore by applying Lemma 29 weget the following result

Corollary 30 If 119898 isin N 0 then Θ(119898) is a numericalsemigroup with maximal embedding dimension

As a consequence of this corollary we have that 119898 2119898 +

1 1198982

+119898minus1 is theminimal systemof generators ofΘ(119898)Now we want to show orderly the elements of SPL(119898)

Thus we define the graph G(SPL(119898)) in the following way

(i) SPL(119898) is the set of vertices of G(SPL(119898))

(ii) (119878 1198781015840) isin SPL(119898) timesSPL(119898) is an edge of G(SPL(119898)) if1198781015840

= 119878 cup F(119878)

The next result is analogous to Theorem 13

Theorem 31 The graph G(S119875119871(119898)) is a tree with root equal

to Δ(119898) Moreover the sons of a vertex 119878 isin S119875119871(119898) are 119878

1199091 119878119909

119903 with 119909

1 119909

119903= 119909 isin msg(119878) | 119909 =119898 119909 gt

F(119878) 119886119899119889 119878 119909 isin S119875119871

By applyingTheorem 31 and Corollaries 15 and 18 we canget easily G(SPL(119898)) such as is shown in the next example

Example 32 We are going to depict G(SPL(3)) that is thetree of the PL-semigroups with multiplicity equal to 3

⟨3 7 8⟩

⟨3 5 7⟩ ⟨3 4⟩

⟨3 4 5⟩ = Δ(3)

⟨3 7 11⟩ = Θ(3)

If119879 = (119881 119864) is a tree then the height of119879 is themaximumof the lengths of the paths that connect each vertex with theroot Let us observe that the height of G(SPL(3)) is 3 Ingeneral the height of the tree G(SPL(119898)) is equal to

g (Θ (119898)) minus g (Δ (119898)) = (119898 minus 1) (119898 + 2)

2

minus (119898 minus 1)

=

(119898 minus 1)119898

2

(6)

Let us study now the possible values of the Frobeniusnumber and the genus for PL-semigroups with multiplicity119898

The Scientific World Journal 7

Proposition 33 If119898 isin N 0 then

(1) g(119878) | 119878 isin SPL(119898) = 119909 isin N | 119898 minus 1 le 119909 le

(119898 minus 1)(119898 + 2)2

(2) F(119878) | 119878 isin SPL(119898) = (Δ(119898) Θ(119898)) cup 119898 minus 1

Proof Let us assume that Δ(119898) Θ(119898) = 1199091gt 1199092gt sdot sdot sdot gt

119909120588(1) By Corollary 27 we know that

g (119878) | 119878 isin SPL (119898) sube 119898 minus 1

(119898 minus 1) (119898 + 2)

2

(7)

For the opposite inclusion it is enough to observe thatΘ(119898)cup119909119894 rarr isin SPL(119898) and that g(Θ(119898)cup119909119894 rarr ) = (119898minus1)(119898+

2)2 minus 119894(2) It is clear that Θ(119898) cup 119909

119894+ 1 rarr isin SPL(119898) with

Frobenius number equal to 119909119894 Thus (Δ(119898) Θ(119898)) cup 119898 minus

1 sube F(119878) | 119878 isin SPL(119898) For the other inclusion let ustake 119878 isin SPL(119898) such that 119878 = Δ(119898) Then F(119878) gt 119898 andthereby F(119878) isin Δ(119898) Since Θ(119898) sube 119878 we have F(119878) notin Θ(119878)Therefore we conclude that F(119878) isin Δ(119898) Θ(119878)

Example 34 By Proposition 33 g(119878) | 119878 isin SPL(3) =

2 3 4 5 Since Δ(3) = ⟨3 4 5⟩ and Θ(3) = ⟨3 7 11⟩ wehave that (Δ(3) Θ(3)) cup 2 = 2 4 5 8 Therefore byapplying Proposition 33 again we conclude that F(119878) | 119878 isinSPL(3) = 2 4 5 8

5 The Smallest PL-Semigroup That Contains aGiven Set of Positive Integers

Let us observe that in general the infinite intersection ofelements of SPL is not a numerical semigroup For instance⋂119899isinN0 119899 rarr = 0 On the other hand it is clear that the

(finite or infinite) intersection of numerical semigroups isalways a submonoid of (N +)

Let 119872 be a submonoid of (N +) We will say that 119872 isa SPL-monoid if it can be expressed like the intersection ofelements of SPL

The next lemma has an immediate proof

Lemma35 The intersection ofS119875119871-monoids is aS

119875119871-monoid

In view of this result we can give the following definition

Definition 36 Let 119883 be a subset of N The SPL-monoidgenerated by119883 (denoted bySPL(119883)) is the intersection of allSPL-monoids containing119883

If 119872 = SPL(119883) then we will say that 119883 is a SPL-systemof generators of119872 In addition if no proper subset of 119883 is aSPL-system of generators of 119872 then we will say that 119883 is aminimal SPL-system of generators of119872

Let us recall that by Proposition 12 we know that S119875119871

isa Frobenius variety Therefore by applying [3 Corollary 19]we have the next result

Proposition 37 Every S119875119871-monoid has a unique minimal

S119875119871-system of generators which in addition is finite

The proof of the following lemma is also immediate

Lemma 38 If 119883 sube N then S119875119871(119883) is the intersection of all

119875119871-semigroups that contain 119883

Proposition 39 If 119883 is a nonempty subset of N 0 thenS119875119871(119883) is a 119875119871-semigroup

Proof We know that SPL(119883) is a submonoid of (N +)Therefore in order to show that SPL(119883) is a numericalsemigroup it will be enough to see that N SPL(119883) is a finiteset

Let 119909 isin 119883 If 119878 is a PL-semigroup containing 119883 then (byTheorem 5) we know that 119909 2119909 + 1 sube 119878 and in this way⟨119909 2119909 + 1⟩ sube 119878 From Lemma 38 we have that ⟨119909 2119909 + 1⟩ subeSPL(119883) Since gcd119909 2119909 + 1 = 1 we get that ⟨119909 2119909 + 1⟩is a numerical semigroup and thus N ⟨119909 2119909 + 1⟩ is finiteConsequently N SPL(119883) is finite

Now let us see that SPL(119883) is a PL-semigroup Let 119909 119910 isinSPL(119883) 0 If 119878 is a PL-semigroup containing 119883 fromLemma 38 we deduce that 119909 119910 isin 1198780 and fromTheorem 5we have that 119909 + 119910 + 1 isin 119878 By applying again Lemma 38we have that 119909 + 119910 + 1 isin SPL(119883) Therefore by applyingTheorem 5 once more we can assert that SPL(119883) is a PL-semigroup

Remark 40 Let us observe that in general Proposition 39 isnot true for Frobenius varieties In fact let S be the set of allnumerical semigroups It is clear thatS is a Frobenius varietyIf we take 119883 = 2 then the intersection of all elements of Scontaining 2 is exactly119872 = ⟨2⟩ which is not a numericalsemigroup

The next result will be key for our last purpose in thissection

Theorem 41 S119875119871

= S119875119871(119883) | 119883 is a nonempty finite

subset 119900119891 N 0

Proof By Proposition 39 we have that

SPL (119883) | 119883 is a nonempty finite subset ofN 0 sube SPL(8)

For the other inclusion it is enough to observe that if 119878 isin SPLthen (by Proposition 37) there exists a nonempty finite subset119883 of N 0 such that 119878 = SPL(119883)

Since SPL is a Frobenius variety by applying [3 Proposi-tion 24] we get the next result

Proposition 42 Let119872 be aS119875119871-monoid and let 119909 isin 119872Then

119872119909 is aS119875119871-monoid if and only if 119909 belongs to the minimal

S119875119871-system of generators of119872

As an immediate consequence of this propositionwe havethe following result

8 The Scientific World Journal

Corollary 43 Let 119883 be a nonempty subset of N 0 Thenthe minimal S

119875119871-system of generators of S

119875119871(119883) is 119909 isin 119883 |

S119875119871(119883) 119909 is a PL-semigroup

Example 44 By Proposition 21 119878 = ⟨3 7 11⟩ is a PL-semigroup By applying Proposition 14 we easily deduce that

119909 isin 3 7 11 | 119878 119909 is a PL-semigroup = 3 (9)

Therefore 119878 = SPL(3) and 3 is the minimalSPL-system ofgenerators of 119878

Now we want to describe SPL(119883) when 119883 is a fixednonempty finite set of positive integers Let us observe thatby Theorem 41 we know that every PL-semigroup can beobtained in this way

Let 1198991 119899

119901be positive integers We will denote by

119878(1198991 119899

119901) the set 120572

11198991+ sdot sdot sdot + 120572

119901119899119901+ 119903 | 119903 120572

1 120572

119901isin

N 119903 lt 1205721+ sdot sdot sdot + 120572

119901 cup 0 Our next purpose will be to show

that 119878(1198991 119899

119901) = SPL(1198991 119899119901)

Lemma 45 Let 119878 be a numerical semigroup let 1199041 119904

119905isin

119878 0 and let 1205721 120572

119905isin N Then ord(120572

11199041+ sdot sdot sdot + 120572

119905119904119905) ge

1205721+ sdot sdot sdot + 120572

119905

Proof Let 1198991 119899

119901 be the minimal system of generators

of 119878 Then for each 119894 isin 1 119905 there exist 1205731198941 120573

119894119901isin N

such that 119904119894= 12057311989411198991+ sdot sdot sdot + 120573

119894119901119899119901 Moreover since 119904

119894= 0 we

have that 1205731198941+ sdot sdot sdot + 120573

119894119901ge 1 Thus

12057211199041+ sdot sdot sdot + 120572

119905119904119905

= 1205721(120573111198991+ sdot sdot sdot + 120573

1119901119899119901) + sdot sdot sdot + 120572

119905(12057311990511198991+ sdot sdot sdot + 120573

119905119901119899119901)

= (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) 1198991+ sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901) 119899119901

(10)

Therefore

ord (12057211199041+ sdot sdot sdot + 120572

119905119904119905)

ge (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) + sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901)

= 1205721(12057311+ sdot sdot sdot + 120573

1119901) + sdot sdot sdot + 120572

119905(1205731199051+ sdot sdot sdot + 120573

119905119901)

ge 1205721+ sdot sdot sdot + 120572

119905

(11)

Theorem 46 If 1198991 119899

119901are positive integers then

119878(1198991 119899

119901) is the smallest (with respect to set inclusion)

119875119871-semigroup containing 1198991 119899

119901

Proof We divide the proof into five steps

(i) Let us see that if 119909 119910 isin 119878(1198991 119899

119901) 0 then 119909 +

119910 isin 119878(1198991 119899

119901) In effect we know that there exist

1205721 120572

119901 1205731 120573

119901 119903 1199031015840 nonnegative integers such

that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903 119910 = 120573

11198991+sdot sdot sdot+120573

119901119899119901+1199031015840

119903 lt 1205721+ sdot sdot sdot + 120572

119901 and 1199031015840 lt 120573

1+ + 120573

119901 Therefore

119909 + 119910 = (1205721+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840 with119903 + 1199031015840

lt (1205721+ 1205731) + sdot sdot sdot + (120572

119901+ 120573119899) Consequently

119909 + 119910 isin 119878(1198991 119899

119901)

(ii) Let us see that N 119878(1198991 119899

119901) is finite Since 119899

1=

1sdot1198991+0sdot1198992+sdot sdot sdot+0sdot119899

119901+0 and 2119899

1+1 = 2sdot119899

1+0sdot1198992+sdot sdot sdot+0sdot

119899119901+1 we have that 119899

1 21198991isin 119878(1198991 119899

119901) By applying

the first step we get that ⟨1198991 21198991⟩ sube 119878(119899

1 119899

119901)

Using the same reasoning as we did in the proof ofProposition 39 we have the result

(iii) From the previous steps we know that 119878(1198991 119899

119901)

is a numerical semigroup Let us see now that119878(1198991 119899

119901) is a PL-semigroup In order to do that

it is enough (by Theorem 5) to show that if 119909 119910 isin

119878(1198991 119899

119901) 0 then 119909 + 119910 + 1 isin 119878(119899

1 119899

119901)

Indeed arguing as in the first step we have that 119909 +119910 + 1 = (120572

1+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840

+ 1

with 119903 + 1199031015840 + 1 lt (1205721+1205731) + sdot sdot sdot + (120572

119901+120573119899) Therefore

119909 + 119910 + 1 isin 119878(1198991 119899

119901)

(iv) Following the proof of the second step it is clear that1198991 119899

119901 sube 119878(119899

1 119899

119901)

(v) Finally let us see that 119878(1198991 119899

119901) is the small-

est PL-semigroup that contains 1198991 119899

119901 In fact

we will show that if 119879 is PL-semigroup containing1198991 119899

119901 then 119878(119899

1 119899

119901) sube 119879 Thus let 119909 isin

119878(1198991 119899

119901) 0 Then there exist 120572

1 120572

119901 119903 isin N

such that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903with 119903 lt 120572

1+sdot sdot sdot+120572

119901

Since 1198991 119899

119901 sube 119879 by Proposition 9 we have that

12057211198991+sdot sdot sdot+120572

119901119899119901+0 ord(120572

11198991+sdot sdot sdot+120572

119901119899119901)minus1 sube 119879

By applying Lemma 45 we have that 119903 lt ord(12057211198991+

sdot sdot sdot + 120572119901119899119901) and therefore 119909 isin 119879

In this way we have proved the statement

The next result is an immediate consequence of theprevious theorem

Corollary 47 If 1198991 119899

119901are positive integers then

SPL(1198991 119899119901) = 119878(1198991 119899119901)

We finish this section with an example that illustrates itscontent

Example 48 It is clear that 119878(4 7) = 0 4 7 8 9 11 rarr =

⟨4 7 9⟩ Therefore SPL(4 7) = ⟨4 7 9⟩

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referee for hisher usefulcomments and suggestions that helped to improve this workBoth of the authors are supported by FQM-343 (Junta deAndalucıa) MTM2010-15595 (MICINN Spain) and FEDERfunds The second author is also partially supported by Juntade AndalucıaFeder Grant no FQM-5849

The Scientific World Journal 9

References

[1] M Lothaire Applied Combinatorics on Words vol 105 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press 2005

[2] J L Ramırez Alfonsın The Diophantine Frobenius ProblemOxford University Press 2005

[3] J C Rosales ldquoFamilies of numerical semigroups closed underfinite intersections and for the Frobenius numberrdquo HoustonJournal of Mathematics vol 34 no 2 pp 339ndash348 2008

[4] J C Rosales M B Branco and D Torrao ldquoBracelet monoidsand numerical semigroupsrdquo preprint

[5] J C Rosales and P A Garcıa-Sanchez Numerical Semigroupsvol 20 of Developments in Mathematics Springer New YorkNY USA 2009

[6] M Bras-Amoros P A Garcıa-Sanchez and A Vico-OtonldquoNonhomogeneous patterns on numerical semigroupsrdquo Inter-national Journal of Algebra and Computation vol 23 no 6 pp1469ndash1483 2013

[7] K Stokes and M Bras-Amoros ldquoLinear non-homogeneoussymmetric patterns and prime power generators in numeri-cal semigroups associated to combinatorial congurationsrdquo toappear in Semigroup Forum 2013

[8] L Bryant ldquoGoto numbers of a numerical semigroup ring andthe gorensteiness of associated graded ringsrdquo Communicationsin Algebra vol 38 no 6 pp 2092ndash2128 2010

[9] R Apery ldquoSur les branches superlineaires des courbesalgebriquesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol222 pp 1198ndash1200 1946

[10] J B Roberts ldquoNote on linear formsrdquo Proceedings of the Ameri-can Mathematical Society vol 7 pp 465ndash469 1956

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The Scientific World Journal 5

⟨5 6 7 8 9⟩ ⟨4 6 7 9⟩ ⟨4 5 7⟩ ⟨4 5 6⟩ ⟨3 7 8⟩

⟨3 5 7⟩

⟨3 4 5⟩

⟨3 4⟩

⟨2 3⟩

⟨2 5⟩

⟨4 5 6 7⟩

⟨1⟩ = N

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

Figure 1

We finish this section with an illustrative example aboutthe above corollary

Example 19 Let 119878 be the numerical semigroupwithmsg(119878) =3 5 7 It is obvious that F(119878) = 4 By Proposition 17 weknow that 1198785 = ⟨3 7 8⟩ In addition 8minus7 = 1 notin 119878TherebyapplyingCorollary 18 we have thatmsg(1198785) = 3 7 8 Onthe other hand applying Proposition 17 again we have that119878 7 = ⟨3 5 10⟩ Finally since 10 minus 5 = 5 isin 119878 we concludethat msg(119878 7) = 3 5

4 PL-Semigroups with a Fixed Multiplicity

Let 119898 be a positive integer We will denote by Δ(119898) =

0119898 rarr It is clear that Δ(119898) is the greatest (with respectto set inclusion) PL-semigroup with multiplicity119898 Our firstaim in this section will be to show that there also exists thesmallest (with respect to set inclusion) PL-semigroup withmultiplicity119898

As an immediate consequence of item (4) inProposition 9 we have the next result

Lemma20 If 119878 is a119875119871-semigroup119898 isin 1198780 and 119896 isin N0then 119896119898 + 119894 isin 119878 for all 119894 isin 0 119896 minus 1

Proposition 21 Let 119898 isin N 0 Then the numericalsemigroup generated by (119894 + 1)119898 + 119894 | 119894 isin 0 119898 minus 1 isthe smallest (with respect to set inclusion) 119875119871-semigroup withmultiplicity119898

Proof Let 119878 = ⟨119898 2119898+1 1198982+(119898minus1)⟩ FromLemma 20we know that any PL-semigroup with multiplicity 119898 has tocontain 119878 In order to conclude the proof we will show that 119878is a PL-semigroup For this purpose since (119894 + 1)119898 + 119894 | 119894 isin

0 119898minus1 is a system of generators of 119878 it will be enoughto check item (2) of Proposition 9 that is if 119894 119895 isin 0 119898 minus

1 then (119894 + 1)119898+ 119894 + (119895 + 1)119898+ 119895+ 1 isin 119878 We distinguish twocases

(1) If 119894+ 119895+1 le 119898minus1 then (119894+1)119898+119894+ (119895+1)119898+119895+1 =(119894 + 119895 + 2)119898 + (119894 + 119895 + 1) isin 119878

(2) If 119894 + 119895 + 1 ge 119898 then (119894 + 1)119898 + 119894 + (119895 + 1)119898 + 119895 + 1 =(119894 + 119895 + 2)119898 + (119894 + 119895 + 1) = (119898+ 1)119898+ (119894 + 119895 minus119898+ 2)119898

+ (119894 + 119895 minus 119898 + 1) isin 119878

We will denote byΘ(119898) = ⟨119898 2119898 + 1 1198982

+ (119898 minus 1)⟩

and bySPL(119898) the set of all PL-semigroups with multiplicityequal to 119898 Let us recall that Δ(119898) = max(SPL(119898)) andΘ(119898) = min(SPL(119898))

As an application of the above comment we have the nextresult

Corollary 22 The set SPL(119898) is finite

Proof If 119878 isin SPL(119898) thenΘ(119898) sube 119878 sube Δ(119898) SinceΔ(119898) andΘ(119898) are numerical semigroups we have that Δ(119898) Θ(119898) isfinite Consequently SPL(119898) is also finite

Remark 23 The previous result can be considered a particu-lar case of [6 Theorem 66]

Now we are interested in computing the Frobeniusnumber and the genus ofΘ(119898) For that several concepts andresults are introduced

If 119878 is a numerical semigroup and 119898 isin 119878 0 then theApery set of119898 in 119878 (see [9]) is Ap(119878 119898) = 119904 isin 119878 | 119904 minus 119898 notin 119878It is clear (see for instance [5 Lemma 24]) that Ap(119878 119898) =120596(0) = 0 120596(1) 120596(119898 minus 1) where 120596(119894) is for each 119894 isin0 119898 minus 1 the least element of 119878 that is congruent with 119894modulo119898

The next result is [5 Proposition 212]

Lemma24 Let 119878 be a numerical semigroup and let119898 isin 1198780Then

(1) F(119878) = max(Ap(119878 119898)) minus 119898(2) g(119878) = (1119898)(sum

119908isinAp(119878119898) 119908) minus ((119898 minus 1)2)

If 119886 119887 are integers with 119887 = 0 we denote by 119886 mod 119887 theremainder of the division of 119886 by 119887 The following result is [5Proposition 35]

6 The Scientific World Journal

Lemma 25 Let 119898 isin N 0 and let 119883 = 120596(0) =

0 120596(1) 120596(119898minus1) sube N such that for each 119894 isin 1 119898minus1120596(119894) is congruent with 119894 modulo 119898 Let 119878 be the numericalsemigroup generated by 119883 cup 119898 The following conditions areequivalent

(1) Ap(119878 119898) = 119883(2) 120596(119894)+120596(119895) ge 120596((119894+119895) mod 119898) for all 119894 119895 isin 1 119898minus

1

Proposition 26 If119898 isin N 0 then

Ap (Θ (119898) 119898)

= 120596 (0) = 0 120596 (1) = 2119898 + 1 120596 (119898 minus 1) = 1198982

+ 119898 minus 1

(4)

Proof It is clear that 120596(119894) = (119894 + 1)119898 + 119894 is congruent with 119894modulo 119898 for all 119894 isin 1 119898 minus 1 Let us see now that if119894 119895 isin 1 119898 minus 1 then 120596(119894) + 120596(119895) ge 120596((119894 + 119895) mod 119898)Indeed 120596(119894) +120596(119895) = (119894 + 1)119898 + 119894 + (119895 + 1)119898 + 119895 gt (119894 + 119895 + 1)119898+ (119894 + 119895) ge ((119894 + 119895) mod 119898 + 1)119898 + (119894 + 119895) mod 119898 = 120596((119894 +119895) mod 119898) The proof follows from Lemma 25

Corollary 27 If119898 isin N 0 1 then

(1) F(Δ(119898)) = 119898 minus 1(2) g(Δ(119898)) = 119898 minus 1(3) F(Θ(119898)) = 1198982 minus 1(4) g(Θ(119898)) = (119898 minus 1)(119898 + 2)2

Proof Items (1) and (2) are trivial On the other hand items(3) and (4) are immediate consequences of Lemma 24 andProposition 26

Remark 28 Thenumerical semigroupΘ(119898) can be rewrittenas

Θ (119898)

= 119898119898+ (119898+ 1) 119898 +2 (119898 + 1) 119898 + (119898 minus 1) (119898 + 1)

(5)

Thus Θ(119898) is a numerical semigroup generated by an arith-metic sequencewith first term119898 and commondifference119898+1(see [2 10])

If 119878 is a numerical semigroup then the cardinality of theminimal system of generators of 119878 is called the embeddingdimension of 119878 and is denoted by e(119878) It is well known (see [5Proposition 212]) that e(119878) le m(119878) We say that a numericalsemigroup 119878 has maximal embedding dimension if e(119878) =m(119878) It is clear that 119898119898 + 1 2119898 minus 1 is the minimalsystem of generators of Δ(119898) Therefore Δ(119898) is a numericalsemigroupwithmaximal embedding dimensionNowwewillshow that Θ(119898) has also maximal embedding dimension

The next result is [5 Corollary 36]

Lemma 29 Let 119878 be a numerical semigroup with multiplicity119898 and assume thatAp(119878 119898) = 120596(0) = 0 120596(1) 120596(119898minus1)

Then 119878 has maximal embedding dimension if and only if 120596(119894)+120596(119895) gt 120596((119894 + 119895) mod 119898) for all 119894 119895 isin 1 119898 minus 1

Let us observe that in the proof of Proposition 26 wehave shown that 120596(119894) + 120596(119895) gt 120596((119894 + 119895) mod 119898) for all119894 119895 isin 1 119898 minus 1 Therefore by applying Lemma 29 weget the following result

Corollary 30 If 119898 isin N 0 then Θ(119898) is a numericalsemigroup with maximal embedding dimension

As a consequence of this corollary we have that 119898 2119898 +

1 1198982

+119898minus1 is theminimal systemof generators ofΘ(119898)Now we want to show orderly the elements of SPL(119898)

Thus we define the graph G(SPL(119898)) in the following way

(i) SPL(119898) is the set of vertices of G(SPL(119898))

(ii) (119878 1198781015840) isin SPL(119898) timesSPL(119898) is an edge of G(SPL(119898)) if1198781015840

= 119878 cup F(119878)

The next result is analogous to Theorem 13

Theorem 31 The graph G(S119875119871(119898)) is a tree with root equal

to Δ(119898) Moreover the sons of a vertex 119878 isin S119875119871(119898) are 119878

1199091 119878119909

119903 with 119909

1 119909

119903= 119909 isin msg(119878) | 119909 =119898 119909 gt

F(119878) 119886119899119889 119878 119909 isin S119875119871

By applyingTheorem 31 and Corollaries 15 and 18 we canget easily G(SPL(119898)) such as is shown in the next example

Example 32 We are going to depict G(SPL(3)) that is thetree of the PL-semigroups with multiplicity equal to 3

⟨3 7 8⟩

⟨3 5 7⟩ ⟨3 4⟩

⟨3 4 5⟩ = Δ(3)

⟨3 7 11⟩ = Θ(3)

If119879 = (119881 119864) is a tree then the height of119879 is themaximumof the lengths of the paths that connect each vertex with theroot Let us observe that the height of G(SPL(3)) is 3 Ingeneral the height of the tree G(SPL(119898)) is equal to

g (Θ (119898)) minus g (Δ (119898)) = (119898 minus 1) (119898 + 2)

2

minus (119898 minus 1)

=

(119898 minus 1)119898

2

(6)

Let us study now the possible values of the Frobeniusnumber and the genus for PL-semigroups with multiplicity119898

The Scientific World Journal 7

Proposition 33 If119898 isin N 0 then

(1) g(119878) | 119878 isin SPL(119898) = 119909 isin N | 119898 minus 1 le 119909 le

(119898 minus 1)(119898 + 2)2

(2) F(119878) | 119878 isin SPL(119898) = (Δ(119898) Θ(119898)) cup 119898 minus 1

Proof Let us assume that Δ(119898) Θ(119898) = 1199091gt 1199092gt sdot sdot sdot gt

119909120588(1) By Corollary 27 we know that

g (119878) | 119878 isin SPL (119898) sube 119898 minus 1

(119898 minus 1) (119898 + 2)

2

(7)

For the opposite inclusion it is enough to observe thatΘ(119898)cup119909119894 rarr isin SPL(119898) and that g(Θ(119898)cup119909119894 rarr ) = (119898minus1)(119898+

2)2 minus 119894(2) It is clear that Θ(119898) cup 119909

119894+ 1 rarr isin SPL(119898) with

Frobenius number equal to 119909119894 Thus (Δ(119898) Θ(119898)) cup 119898 minus

1 sube F(119878) | 119878 isin SPL(119898) For the other inclusion let ustake 119878 isin SPL(119898) such that 119878 = Δ(119898) Then F(119878) gt 119898 andthereby F(119878) isin Δ(119898) Since Θ(119898) sube 119878 we have F(119878) notin Θ(119878)Therefore we conclude that F(119878) isin Δ(119898) Θ(119878)

Example 34 By Proposition 33 g(119878) | 119878 isin SPL(3) =

2 3 4 5 Since Δ(3) = ⟨3 4 5⟩ and Θ(3) = ⟨3 7 11⟩ wehave that (Δ(3) Θ(3)) cup 2 = 2 4 5 8 Therefore byapplying Proposition 33 again we conclude that F(119878) | 119878 isinSPL(3) = 2 4 5 8

5 The Smallest PL-Semigroup That Contains aGiven Set of Positive Integers

Let us observe that in general the infinite intersection ofelements of SPL is not a numerical semigroup For instance⋂119899isinN0 119899 rarr = 0 On the other hand it is clear that the

(finite or infinite) intersection of numerical semigroups isalways a submonoid of (N +)

Let 119872 be a submonoid of (N +) We will say that 119872 isa SPL-monoid if it can be expressed like the intersection ofelements of SPL

The next lemma has an immediate proof

Lemma35 The intersection ofS119875119871-monoids is aS

119875119871-monoid

In view of this result we can give the following definition

Definition 36 Let 119883 be a subset of N The SPL-monoidgenerated by119883 (denoted bySPL(119883)) is the intersection of allSPL-monoids containing119883

If 119872 = SPL(119883) then we will say that 119883 is a SPL-systemof generators of119872 In addition if no proper subset of 119883 is aSPL-system of generators of 119872 then we will say that 119883 is aminimal SPL-system of generators of119872

Let us recall that by Proposition 12 we know that S119875119871

isa Frobenius variety Therefore by applying [3 Corollary 19]we have the next result

Proposition 37 Every S119875119871-monoid has a unique minimal

S119875119871-system of generators which in addition is finite

The proof of the following lemma is also immediate

Lemma 38 If 119883 sube N then S119875119871(119883) is the intersection of all

119875119871-semigroups that contain 119883

Proposition 39 If 119883 is a nonempty subset of N 0 thenS119875119871(119883) is a 119875119871-semigroup

Proof We know that SPL(119883) is a submonoid of (N +)Therefore in order to show that SPL(119883) is a numericalsemigroup it will be enough to see that N SPL(119883) is a finiteset

Let 119909 isin 119883 If 119878 is a PL-semigroup containing 119883 then (byTheorem 5) we know that 119909 2119909 + 1 sube 119878 and in this way⟨119909 2119909 + 1⟩ sube 119878 From Lemma 38 we have that ⟨119909 2119909 + 1⟩ subeSPL(119883) Since gcd119909 2119909 + 1 = 1 we get that ⟨119909 2119909 + 1⟩is a numerical semigroup and thus N ⟨119909 2119909 + 1⟩ is finiteConsequently N SPL(119883) is finite

Now let us see that SPL(119883) is a PL-semigroup Let 119909 119910 isinSPL(119883) 0 If 119878 is a PL-semigroup containing 119883 fromLemma 38 we deduce that 119909 119910 isin 1198780 and fromTheorem 5we have that 119909 + 119910 + 1 isin 119878 By applying again Lemma 38we have that 119909 + 119910 + 1 isin SPL(119883) Therefore by applyingTheorem 5 once more we can assert that SPL(119883) is a PL-semigroup

Remark 40 Let us observe that in general Proposition 39 isnot true for Frobenius varieties In fact let S be the set of allnumerical semigroups It is clear thatS is a Frobenius varietyIf we take 119883 = 2 then the intersection of all elements of Scontaining 2 is exactly119872 = ⟨2⟩ which is not a numericalsemigroup

The next result will be key for our last purpose in thissection

Theorem 41 S119875119871

= S119875119871(119883) | 119883 is a nonempty finite

subset 119900119891 N 0

Proof By Proposition 39 we have that

SPL (119883) | 119883 is a nonempty finite subset ofN 0 sube SPL(8)

For the other inclusion it is enough to observe that if 119878 isin SPLthen (by Proposition 37) there exists a nonempty finite subset119883 of N 0 such that 119878 = SPL(119883)

Since SPL is a Frobenius variety by applying [3 Proposi-tion 24] we get the next result

Proposition 42 Let119872 be aS119875119871-monoid and let 119909 isin 119872Then

119872119909 is aS119875119871-monoid if and only if 119909 belongs to the minimal

S119875119871-system of generators of119872

As an immediate consequence of this propositionwe havethe following result

8 The Scientific World Journal

Corollary 43 Let 119883 be a nonempty subset of N 0 Thenthe minimal S

119875119871-system of generators of S

119875119871(119883) is 119909 isin 119883 |

S119875119871(119883) 119909 is a PL-semigroup

Example 44 By Proposition 21 119878 = ⟨3 7 11⟩ is a PL-semigroup By applying Proposition 14 we easily deduce that

119909 isin 3 7 11 | 119878 119909 is a PL-semigroup = 3 (9)

Therefore 119878 = SPL(3) and 3 is the minimalSPL-system ofgenerators of 119878

Now we want to describe SPL(119883) when 119883 is a fixednonempty finite set of positive integers Let us observe thatby Theorem 41 we know that every PL-semigroup can beobtained in this way

Let 1198991 119899

119901be positive integers We will denote by

119878(1198991 119899

119901) the set 120572

11198991+ sdot sdot sdot + 120572

119901119899119901+ 119903 | 119903 120572

1 120572

119901isin

N 119903 lt 1205721+ sdot sdot sdot + 120572

119901 cup 0 Our next purpose will be to show

that 119878(1198991 119899

119901) = SPL(1198991 119899119901)

Lemma 45 Let 119878 be a numerical semigroup let 1199041 119904

119905isin

119878 0 and let 1205721 120572

119905isin N Then ord(120572

11199041+ sdot sdot sdot + 120572

119905119904119905) ge

1205721+ sdot sdot sdot + 120572

119905

Proof Let 1198991 119899

119901 be the minimal system of generators

of 119878 Then for each 119894 isin 1 119905 there exist 1205731198941 120573

119894119901isin N

such that 119904119894= 12057311989411198991+ sdot sdot sdot + 120573

119894119901119899119901 Moreover since 119904

119894= 0 we

have that 1205731198941+ sdot sdot sdot + 120573

119894119901ge 1 Thus

12057211199041+ sdot sdot sdot + 120572

119905119904119905

= 1205721(120573111198991+ sdot sdot sdot + 120573

1119901119899119901) + sdot sdot sdot + 120572

119905(12057311990511198991+ sdot sdot sdot + 120573

119905119901119899119901)

= (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) 1198991+ sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901) 119899119901

(10)

Therefore

ord (12057211199041+ sdot sdot sdot + 120572

119905119904119905)

ge (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) + sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901)

= 1205721(12057311+ sdot sdot sdot + 120573

1119901) + sdot sdot sdot + 120572

119905(1205731199051+ sdot sdot sdot + 120573

119905119901)

ge 1205721+ sdot sdot sdot + 120572

119905

(11)

Theorem 46 If 1198991 119899

119901are positive integers then

119878(1198991 119899

119901) is the smallest (with respect to set inclusion)

119875119871-semigroup containing 1198991 119899

119901

Proof We divide the proof into five steps

(i) Let us see that if 119909 119910 isin 119878(1198991 119899

119901) 0 then 119909 +

119910 isin 119878(1198991 119899

119901) In effect we know that there exist

1205721 120572

119901 1205731 120573

119901 119903 1199031015840 nonnegative integers such

that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903 119910 = 120573

11198991+sdot sdot sdot+120573

119901119899119901+1199031015840

119903 lt 1205721+ sdot sdot sdot + 120572

119901 and 1199031015840 lt 120573

1+ + 120573

119901 Therefore

119909 + 119910 = (1205721+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840 with119903 + 1199031015840

lt (1205721+ 1205731) + sdot sdot sdot + (120572

119901+ 120573119899) Consequently

119909 + 119910 isin 119878(1198991 119899

119901)

(ii) Let us see that N 119878(1198991 119899

119901) is finite Since 119899

1=

1sdot1198991+0sdot1198992+sdot sdot sdot+0sdot119899

119901+0 and 2119899

1+1 = 2sdot119899

1+0sdot1198992+sdot sdot sdot+0sdot

119899119901+1 we have that 119899

1 21198991isin 119878(1198991 119899

119901) By applying

the first step we get that ⟨1198991 21198991⟩ sube 119878(119899

1 119899

119901)

Using the same reasoning as we did in the proof ofProposition 39 we have the result

(iii) From the previous steps we know that 119878(1198991 119899

119901)

is a numerical semigroup Let us see now that119878(1198991 119899

119901) is a PL-semigroup In order to do that

it is enough (by Theorem 5) to show that if 119909 119910 isin

119878(1198991 119899

119901) 0 then 119909 + 119910 + 1 isin 119878(119899

1 119899

119901)

Indeed arguing as in the first step we have that 119909 +119910 + 1 = (120572

1+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840

+ 1

with 119903 + 1199031015840 + 1 lt (1205721+1205731) + sdot sdot sdot + (120572

119901+120573119899) Therefore

119909 + 119910 + 1 isin 119878(1198991 119899

119901)

(iv) Following the proof of the second step it is clear that1198991 119899

119901 sube 119878(119899

1 119899

119901)

(v) Finally let us see that 119878(1198991 119899

119901) is the small-

est PL-semigroup that contains 1198991 119899

119901 In fact

we will show that if 119879 is PL-semigroup containing1198991 119899

119901 then 119878(119899

1 119899

119901) sube 119879 Thus let 119909 isin

119878(1198991 119899

119901) 0 Then there exist 120572

1 120572

119901 119903 isin N

such that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903with 119903 lt 120572

1+sdot sdot sdot+120572

119901

Since 1198991 119899

119901 sube 119879 by Proposition 9 we have that

12057211198991+sdot sdot sdot+120572

119901119899119901+0 ord(120572

11198991+sdot sdot sdot+120572

119901119899119901)minus1 sube 119879

By applying Lemma 45 we have that 119903 lt ord(12057211198991+

sdot sdot sdot + 120572119901119899119901) and therefore 119909 isin 119879

In this way we have proved the statement

The next result is an immediate consequence of theprevious theorem

Corollary 47 If 1198991 119899

119901are positive integers then

SPL(1198991 119899119901) = 119878(1198991 119899119901)

We finish this section with an example that illustrates itscontent

Example 48 It is clear that 119878(4 7) = 0 4 7 8 9 11 rarr =

⟨4 7 9⟩ Therefore SPL(4 7) = ⟨4 7 9⟩

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referee for hisher usefulcomments and suggestions that helped to improve this workBoth of the authors are supported by FQM-343 (Junta deAndalucıa) MTM2010-15595 (MICINN Spain) and FEDERfunds The second author is also partially supported by Juntade AndalucıaFeder Grant no FQM-5849

The Scientific World Journal 9

References

[1] M Lothaire Applied Combinatorics on Words vol 105 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press 2005

[2] J L Ramırez Alfonsın The Diophantine Frobenius ProblemOxford University Press 2005

[3] J C Rosales ldquoFamilies of numerical semigroups closed underfinite intersections and for the Frobenius numberrdquo HoustonJournal of Mathematics vol 34 no 2 pp 339ndash348 2008

[4] J C Rosales M B Branco and D Torrao ldquoBracelet monoidsand numerical semigroupsrdquo preprint

[5] J C Rosales and P A Garcıa-Sanchez Numerical Semigroupsvol 20 of Developments in Mathematics Springer New YorkNY USA 2009

[6] M Bras-Amoros P A Garcıa-Sanchez and A Vico-OtonldquoNonhomogeneous patterns on numerical semigroupsrdquo Inter-national Journal of Algebra and Computation vol 23 no 6 pp1469ndash1483 2013

[7] K Stokes and M Bras-Amoros ldquoLinear non-homogeneoussymmetric patterns and prime power generators in numeri-cal semigroups associated to combinatorial congurationsrdquo toappear in Semigroup Forum 2013

[8] L Bryant ldquoGoto numbers of a numerical semigroup ring andthe gorensteiness of associated graded ringsrdquo Communicationsin Algebra vol 38 no 6 pp 2092ndash2128 2010

[9] R Apery ldquoSur les branches superlineaires des courbesalgebriquesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol222 pp 1198ndash1200 1946

[10] J B Roberts ldquoNote on linear formsrdquo Proceedings of the Ameri-can Mathematical Society vol 7 pp 465ndash469 1956

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6 The Scientific World Journal

Lemma 25 Let 119898 isin N 0 and let 119883 = 120596(0) =

0 120596(1) 120596(119898minus1) sube N such that for each 119894 isin 1 119898minus1120596(119894) is congruent with 119894 modulo 119898 Let 119878 be the numericalsemigroup generated by 119883 cup 119898 The following conditions areequivalent

(1) Ap(119878 119898) = 119883(2) 120596(119894)+120596(119895) ge 120596((119894+119895) mod 119898) for all 119894 119895 isin 1 119898minus

1

Proposition 26 If119898 isin N 0 then

Ap (Θ (119898) 119898)

= 120596 (0) = 0 120596 (1) = 2119898 + 1 120596 (119898 minus 1) = 1198982

+ 119898 minus 1

(4)

Proof It is clear that 120596(119894) = (119894 + 1)119898 + 119894 is congruent with 119894modulo 119898 for all 119894 isin 1 119898 minus 1 Let us see now that if119894 119895 isin 1 119898 minus 1 then 120596(119894) + 120596(119895) ge 120596((119894 + 119895) mod 119898)Indeed 120596(119894) +120596(119895) = (119894 + 1)119898 + 119894 + (119895 + 1)119898 + 119895 gt (119894 + 119895 + 1)119898+ (119894 + 119895) ge ((119894 + 119895) mod 119898 + 1)119898 + (119894 + 119895) mod 119898 = 120596((119894 +119895) mod 119898) The proof follows from Lemma 25

Corollary 27 If119898 isin N 0 1 then

(1) F(Δ(119898)) = 119898 minus 1(2) g(Δ(119898)) = 119898 minus 1(3) F(Θ(119898)) = 1198982 minus 1(4) g(Θ(119898)) = (119898 minus 1)(119898 + 2)2

Proof Items (1) and (2) are trivial On the other hand items(3) and (4) are immediate consequences of Lemma 24 andProposition 26

Remark 28 Thenumerical semigroupΘ(119898) can be rewrittenas

Θ (119898)

= 119898119898+ (119898+ 1) 119898 +2 (119898 + 1) 119898 + (119898 minus 1) (119898 + 1)

(5)

Thus Θ(119898) is a numerical semigroup generated by an arith-metic sequencewith first term119898 and commondifference119898+1(see [2 10])

If 119878 is a numerical semigroup then the cardinality of theminimal system of generators of 119878 is called the embeddingdimension of 119878 and is denoted by e(119878) It is well known (see [5Proposition 212]) that e(119878) le m(119878) We say that a numericalsemigroup 119878 has maximal embedding dimension if e(119878) =m(119878) It is clear that 119898119898 + 1 2119898 minus 1 is the minimalsystem of generators of Δ(119898) Therefore Δ(119898) is a numericalsemigroupwithmaximal embedding dimensionNowwewillshow that Θ(119898) has also maximal embedding dimension

The next result is [5 Corollary 36]

Lemma 29 Let 119878 be a numerical semigroup with multiplicity119898 and assume thatAp(119878 119898) = 120596(0) = 0 120596(1) 120596(119898minus1)

Then 119878 has maximal embedding dimension if and only if 120596(119894)+120596(119895) gt 120596((119894 + 119895) mod 119898) for all 119894 119895 isin 1 119898 minus 1

Let us observe that in the proof of Proposition 26 wehave shown that 120596(119894) + 120596(119895) gt 120596((119894 + 119895) mod 119898) for all119894 119895 isin 1 119898 minus 1 Therefore by applying Lemma 29 weget the following result

Corollary 30 If 119898 isin N 0 then Θ(119898) is a numericalsemigroup with maximal embedding dimension

As a consequence of this corollary we have that 119898 2119898 +

1 1198982

+119898minus1 is theminimal systemof generators ofΘ(119898)Now we want to show orderly the elements of SPL(119898)

Thus we define the graph G(SPL(119898)) in the following way

(i) SPL(119898) is the set of vertices of G(SPL(119898))

(ii) (119878 1198781015840) isin SPL(119898) timesSPL(119898) is an edge of G(SPL(119898)) if1198781015840

= 119878 cup F(119878)

The next result is analogous to Theorem 13

Theorem 31 The graph G(S119875119871(119898)) is a tree with root equal

to Δ(119898) Moreover the sons of a vertex 119878 isin S119875119871(119898) are 119878

1199091 119878119909

119903 with 119909

1 119909

119903= 119909 isin msg(119878) | 119909 =119898 119909 gt

F(119878) 119886119899119889 119878 119909 isin S119875119871

By applyingTheorem 31 and Corollaries 15 and 18 we canget easily G(SPL(119898)) such as is shown in the next example

Example 32 We are going to depict G(SPL(3)) that is thetree of the PL-semigroups with multiplicity equal to 3

⟨3 7 8⟩

⟨3 5 7⟩ ⟨3 4⟩

⟨3 4 5⟩ = Δ(3)

⟨3 7 11⟩ = Θ(3)

If119879 = (119881 119864) is a tree then the height of119879 is themaximumof the lengths of the paths that connect each vertex with theroot Let us observe that the height of G(SPL(3)) is 3 Ingeneral the height of the tree G(SPL(119898)) is equal to

g (Θ (119898)) minus g (Δ (119898)) = (119898 minus 1) (119898 + 2)

2

minus (119898 minus 1)

=

(119898 minus 1)119898

2

(6)

Let us study now the possible values of the Frobeniusnumber and the genus for PL-semigroups with multiplicity119898

The Scientific World Journal 7

Proposition 33 If119898 isin N 0 then

(1) g(119878) | 119878 isin SPL(119898) = 119909 isin N | 119898 minus 1 le 119909 le

(119898 minus 1)(119898 + 2)2

(2) F(119878) | 119878 isin SPL(119898) = (Δ(119898) Θ(119898)) cup 119898 minus 1

Proof Let us assume that Δ(119898) Θ(119898) = 1199091gt 1199092gt sdot sdot sdot gt

119909120588(1) By Corollary 27 we know that

g (119878) | 119878 isin SPL (119898) sube 119898 minus 1

(119898 minus 1) (119898 + 2)

2

(7)

For the opposite inclusion it is enough to observe thatΘ(119898)cup119909119894 rarr isin SPL(119898) and that g(Θ(119898)cup119909119894 rarr ) = (119898minus1)(119898+

2)2 minus 119894(2) It is clear that Θ(119898) cup 119909

119894+ 1 rarr isin SPL(119898) with

Frobenius number equal to 119909119894 Thus (Δ(119898) Θ(119898)) cup 119898 minus

1 sube F(119878) | 119878 isin SPL(119898) For the other inclusion let ustake 119878 isin SPL(119898) such that 119878 = Δ(119898) Then F(119878) gt 119898 andthereby F(119878) isin Δ(119898) Since Θ(119898) sube 119878 we have F(119878) notin Θ(119878)Therefore we conclude that F(119878) isin Δ(119898) Θ(119878)

Example 34 By Proposition 33 g(119878) | 119878 isin SPL(3) =

2 3 4 5 Since Δ(3) = ⟨3 4 5⟩ and Θ(3) = ⟨3 7 11⟩ wehave that (Δ(3) Θ(3)) cup 2 = 2 4 5 8 Therefore byapplying Proposition 33 again we conclude that F(119878) | 119878 isinSPL(3) = 2 4 5 8

5 The Smallest PL-Semigroup That Contains aGiven Set of Positive Integers

Let us observe that in general the infinite intersection ofelements of SPL is not a numerical semigroup For instance⋂119899isinN0 119899 rarr = 0 On the other hand it is clear that the

(finite or infinite) intersection of numerical semigroups isalways a submonoid of (N +)

Let 119872 be a submonoid of (N +) We will say that 119872 isa SPL-monoid if it can be expressed like the intersection ofelements of SPL

The next lemma has an immediate proof

Lemma35 The intersection ofS119875119871-monoids is aS

119875119871-monoid

In view of this result we can give the following definition

Definition 36 Let 119883 be a subset of N The SPL-monoidgenerated by119883 (denoted bySPL(119883)) is the intersection of allSPL-monoids containing119883

If 119872 = SPL(119883) then we will say that 119883 is a SPL-systemof generators of119872 In addition if no proper subset of 119883 is aSPL-system of generators of 119872 then we will say that 119883 is aminimal SPL-system of generators of119872

Let us recall that by Proposition 12 we know that S119875119871

isa Frobenius variety Therefore by applying [3 Corollary 19]we have the next result

Proposition 37 Every S119875119871-monoid has a unique minimal

S119875119871-system of generators which in addition is finite

The proof of the following lemma is also immediate

Lemma 38 If 119883 sube N then S119875119871(119883) is the intersection of all

119875119871-semigroups that contain 119883

Proposition 39 If 119883 is a nonempty subset of N 0 thenS119875119871(119883) is a 119875119871-semigroup

Proof We know that SPL(119883) is a submonoid of (N +)Therefore in order to show that SPL(119883) is a numericalsemigroup it will be enough to see that N SPL(119883) is a finiteset

Let 119909 isin 119883 If 119878 is a PL-semigroup containing 119883 then (byTheorem 5) we know that 119909 2119909 + 1 sube 119878 and in this way⟨119909 2119909 + 1⟩ sube 119878 From Lemma 38 we have that ⟨119909 2119909 + 1⟩ subeSPL(119883) Since gcd119909 2119909 + 1 = 1 we get that ⟨119909 2119909 + 1⟩is a numerical semigroup and thus N ⟨119909 2119909 + 1⟩ is finiteConsequently N SPL(119883) is finite

Now let us see that SPL(119883) is a PL-semigroup Let 119909 119910 isinSPL(119883) 0 If 119878 is a PL-semigroup containing 119883 fromLemma 38 we deduce that 119909 119910 isin 1198780 and fromTheorem 5we have that 119909 + 119910 + 1 isin 119878 By applying again Lemma 38we have that 119909 + 119910 + 1 isin SPL(119883) Therefore by applyingTheorem 5 once more we can assert that SPL(119883) is a PL-semigroup

Remark 40 Let us observe that in general Proposition 39 isnot true for Frobenius varieties In fact let S be the set of allnumerical semigroups It is clear thatS is a Frobenius varietyIf we take 119883 = 2 then the intersection of all elements of Scontaining 2 is exactly119872 = ⟨2⟩ which is not a numericalsemigroup

The next result will be key for our last purpose in thissection

Theorem 41 S119875119871

= S119875119871(119883) | 119883 is a nonempty finite

subset 119900119891 N 0

Proof By Proposition 39 we have that

SPL (119883) | 119883 is a nonempty finite subset ofN 0 sube SPL(8)

For the other inclusion it is enough to observe that if 119878 isin SPLthen (by Proposition 37) there exists a nonempty finite subset119883 of N 0 such that 119878 = SPL(119883)

Since SPL is a Frobenius variety by applying [3 Proposi-tion 24] we get the next result

Proposition 42 Let119872 be aS119875119871-monoid and let 119909 isin 119872Then

119872119909 is aS119875119871-monoid if and only if 119909 belongs to the minimal

S119875119871-system of generators of119872

As an immediate consequence of this propositionwe havethe following result

8 The Scientific World Journal

Corollary 43 Let 119883 be a nonempty subset of N 0 Thenthe minimal S

119875119871-system of generators of S

119875119871(119883) is 119909 isin 119883 |

S119875119871(119883) 119909 is a PL-semigroup

Example 44 By Proposition 21 119878 = ⟨3 7 11⟩ is a PL-semigroup By applying Proposition 14 we easily deduce that

119909 isin 3 7 11 | 119878 119909 is a PL-semigroup = 3 (9)

Therefore 119878 = SPL(3) and 3 is the minimalSPL-system ofgenerators of 119878

Now we want to describe SPL(119883) when 119883 is a fixednonempty finite set of positive integers Let us observe thatby Theorem 41 we know that every PL-semigroup can beobtained in this way

Let 1198991 119899

119901be positive integers We will denote by

119878(1198991 119899

119901) the set 120572

11198991+ sdot sdot sdot + 120572

119901119899119901+ 119903 | 119903 120572

1 120572

119901isin

N 119903 lt 1205721+ sdot sdot sdot + 120572

119901 cup 0 Our next purpose will be to show

that 119878(1198991 119899

119901) = SPL(1198991 119899119901)

Lemma 45 Let 119878 be a numerical semigroup let 1199041 119904

119905isin

119878 0 and let 1205721 120572

119905isin N Then ord(120572

11199041+ sdot sdot sdot + 120572

119905119904119905) ge

1205721+ sdot sdot sdot + 120572

119905

Proof Let 1198991 119899

119901 be the minimal system of generators

of 119878 Then for each 119894 isin 1 119905 there exist 1205731198941 120573

119894119901isin N

such that 119904119894= 12057311989411198991+ sdot sdot sdot + 120573

119894119901119899119901 Moreover since 119904

119894= 0 we

have that 1205731198941+ sdot sdot sdot + 120573

119894119901ge 1 Thus

12057211199041+ sdot sdot sdot + 120572

119905119904119905

= 1205721(120573111198991+ sdot sdot sdot + 120573

1119901119899119901) + sdot sdot sdot + 120572

119905(12057311990511198991+ sdot sdot sdot + 120573

119905119901119899119901)

= (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) 1198991+ sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901) 119899119901

(10)

Therefore

ord (12057211199041+ sdot sdot sdot + 120572

119905119904119905)

ge (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) + sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901)

= 1205721(12057311+ sdot sdot sdot + 120573

1119901) + sdot sdot sdot + 120572

119905(1205731199051+ sdot sdot sdot + 120573

119905119901)

ge 1205721+ sdot sdot sdot + 120572

119905

(11)

Theorem 46 If 1198991 119899

119901are positive integers then

119878(1198991 119899

119901) is the smallest (with respect to set inclusion)

119875119871-semigroup containing 1198991 119899

119901

Proof We divide the proof into five steps

(i) Let us see that if 119909 119910 isin 119878(1198991 119899

119901) 0 then 119909 +

119910 isin 119878(1198991 119899

119901) In effect we know that there exist

1205721 120572

119901 1205731 120573

119901 119903 1199031015840 nonnegative integers such

that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903 119910 = 120573

11198991+sdot sdot sdot+120573

119901119899119901+1199031015840

119903 lt 1205721+ sdot sdot sdot + 120572

119901 and 1199031015840 lt 120573

1+ + 120573

119901 Therefore

119909 + 119910 = (1205721+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840 with119903 + 1199031015840

lt (1205721+ 1205731) + sdot sdot sdot + (120572

119901+ 120573119899) Consequently

119909 + 119910 isin 119878(1198991 119899

119901)

(ii) Let us see that N 119878(1198991 119899

119901) is finite Since 119899

1=

1sdot1198991+0sdot1198992+sdot sdot sdot+0sdot119899

119901+0 and 2119899

1+1 = 2sdot119899

1+0sdot1198992+sdot sdot sdot+0sdot

119899119901+1 we have that 119899

1 21198991isin 119878(1198991 119899

119901) By applying

the first step we get that ⟨1198991 21198991⟩ sube 119878(119899

1 119899

119901)

Using the same reasoning as we did in the proof ofProposition 39 we have the result

(iii) From the previous steps we know that 119878(1198991 119899

119901)

is a numerical semigroup Let us see now that119878(1198991 119899

119901) is a PL-semigroup In order to do that

it is enough (by Theorem 5) to show that if 119909 119910 isin

119878(1198991 119899

119901) 0 then 119909 + 119910 + 1 isin 119878(119899

1 119899

119901)

Indeed arguing as in the first step we have that 119909 +119910 + 1 = (120572

1+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840

+ 1

with 119903 + 1199031015840 + 1 lt (1205721+1205731) + sdot sdot sdot + (120572

119901+120573119899) Therefore

119909 + 119910 + 1 isin 119878(1198991 119899

119901)

(iv) Following the proof of the second step it is clear that1198991 119899

119901 sube 119878(119899

1 119899

119901)

(v) Finally let us see that 119878(1198991 119899

119901) is the small-

est PL-semigroup that contains 1198991 119899

119901 In fact

we will show that if 119879 is PL-semigroup containing1198991 119899

119901 then 119878(119899

1 119899

119901) sube 119879 Thus let 119909 isin

119878(1198991 119899

119901) 0 Then there exist 120572

1 120572

119901 119903 isin N

such that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903with 119903 lt 120572

1+sdot sdot sdot+120572

119901

Since 1198991 119899

119901 sube 119879 by Proposition 9 we have that

12057211198991+sdot sdot sdot+120572

119901119899119901+0 ord(120572

11198991+sdot sdot sdot+120572

119901119899119901)minus1 sube 119879

By applying Lemma 45 we have that 119903 lt ord(12057211198991+

sdot sdot sdot + 120572119901119899119901) and therefore 119909 isin 119879

In this way we have proved the statement

The next result is an immediate consequence of theprevious theorem

Corollary 47 If 1198991 119899

119901are positive integers then

SPL(1198991 119899119901) = 119878(1198991 119899119901)

We finish this section with an example that illustrates itscontent

Example 48 It is clear that 119878(4 7) = 0 4 7 8 9 11 rarr =

⟨4 7 9⟩ Therefore SPL(4 7) = ⟨4 7 9⟩

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referee for hisher usefulcomments and suggestions that helped to improve this workBoth of the authors are supported by FQM-343 (Junta deAndalucıa) MTM2010-15595 (MICINN Spain) and FEDERfunds The second author is also partially supported by Juntade AndalucıaFeder Grant no FQM-5849

The Scientific World Journal 9

References

[1] M Lothaire Applied Combinatorics on Words vol 105 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press 2005

[2] J L Ramırez Alfonsın The Diophantine Frobenius ProblemOxford University Press 2005

[3] J C Rosales ldquoFamilies of numerical semigroups closed underfinite intersections and for the Frobenius numberrdquo HoustonJournal of Mathematics vol 34 no 2 pp 339ndash348 2008

[4] J C Rosales M B Branco and D Torrao ldquoBracelet monoidsand numerical semigroupsrdquo preprint

[5] J C Rosales and P A Garcıa-Sanchez Numerical Semigroupsvol 20 of Developments in Mathematics Springer New YorkNY USA 2009

[6] M Bras-Amoros P A Garcıa-Sanchez and A Vico-OtonldquoNonhomogeneous patterns on numerical semigroupsrdquo Inter-national Journal of Algebra and Computation vol 23 no 6 pp1469ndash1483 2013

[7] K Stokes and M Bras-Amoros ldquoLinear non-homogeneoussymmetric patterns and prime power generators in numeri-cal semigroups associated to combinatorial congurationsrdquo toappear in Semigroup Forum 2013

[8] L Bryant ldquoGoto numbers of a numerical semigroup ring andthe gorensteiness of associated graded ringsrdquo Communicationsin Algebra vol 38 no 6 pp 2092ndash2128 2010

[9] R Apery ldquoSur les branches superlineaires des courbesalgebriquesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol222 pp 1198ndash1200 1946

[10] J B Roberts ldquoNote on linear formsrdquo Proceedings of the Ameri-can Mathematical Society vol 7 pp 465ndash469 1956

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 7

Proposition 33 If119898 isin N 0 then

(1) g(119878) | 119878 isin SPL(119898) = 119909 isin N | 119898 minus 1 le 119909 le

(119898 minus 1)(119898 + 2)2

(2) F(119878) | 119878 isin SPL(119898) = (Δ(119898) Θ(119898)) cup 119898 minus 1

Proof Let us assume that Δ(119898) Θ(119898) = 1199091gt 1199092gt sdot sdot sdot gt

119909120588(1) By Corollary 27 we know that

g (119878) | 119878 isin SPL (119898) sube 119898 minus 1

(119898 minus 1) (119898 + 2)

2

(7)

For the opposite inclusion it is enough to observe thatΘ(119898)cup119909119894 rarr isin SPL(119898) and that g(Θ(119898)cup119909119894 rarr ) = (119898minus1)(119898+

2)2 minus 119894(2) It is clear that Θ(119898) cup 119909

119894+ 1 rarr isin SPL(119898) with

Frobenius number equal to 119909119894 Thus (Δ(119898) Θ(119898)) cup 119898 minus

1 sube F(119878) | 119878 isin SPL(119898) For the other inclusion let ustake 119878 isin SPL(119898) such that 119878 = Δ(119898) Then F(119878) gt 119898 andthereby F(119878) isin Δ(119898) Since Θ(119898) sube 119878 we have F(119878) notin Θ(119878)Therefore we conclude that F(119878) isin Δ(119898) Θ(119878)

Example 34 By Proposition 33 g(119878) | 119878 isin SPL(3) =

2 3 4 5 Since Δ(3) = ⟨3 4 5⟩ and Θ(3) = ⟨3 7 11⟩ wehave that (Δ(3) Θ(3)) cup 2 = 2 4 5 8 Therefore byapplying Proposition 33 again we conclude that F(119878) | 119878 isinSPL(3) = 2 4 5 8

5 The Smallest PL-Semigroup That Contains aGiven Set of Positive Integers

Let us observe that in general the infinite intersection ofelements of SPL is not a numerical semigroup For instance⋂119899isinN0 119899 rarr = 0 On the other hand it is clear that the

(finite or infinite) intersection of numerical semigroups isalways a submonoid of (N +)

Let 119872 be a submonoid of (N +) We will say that 119872 isa SPL-monoid if it can be expressed like the intersection ofelements of SPL

The next lemma has an immediate proof

Lemma35 The intersection ofS119875119871-monoids is aS

119875119871-monoid

In view of this result we can give the following definition

Definition 36 Let 119883 be a subset of N The SPL-monoidgenerated by119883 (denoted bySPL(119883)) is the intersection of allSPL-monoids containing119883

If 119872 = SPL(119883) then we will say that 119883 is a SPL-systemof generators of119872 In addition if no proper subset of 119883 is aSPL-system of generators of 119872 then we will say that 119883 is aminimal SPL-system of generators of119872

Let us recall that by Proposition 12 we know that S119875119871

isa Frobenius variety Therefore by applying [3 Corollary 19]we have the next result

Proposition 37 Every S119875119871-monoid has a unique minimal

S119875119871-system of generators which in addition is finite

The proof of the following lemma is also immediate

Lemma 38 If 119883 sube N then S119875119871(119883) is the intersection of all

119875119871-semigroups that contain 119883

Proposition 39 If 119883 is a nonempty subset of N 0 thenS119875119871(119883) is a 119875119871-semigroup

Proof We know that SPL(119883) is a submonoid of (N +)Therefore in order to show that SPL(119883) is a numericalsemigroup it will be enough to see that N SPL(119883) is a finiteset

Let 119909 isin 119883 If 119878 is a PL-semigroup containing 119883 then (byTheorem 5) we know that 119909 2119909 + 1 sube 119878 and in this way⟨119909 2119909 + 1⟩ sube 119878 From Lemma 38 we have that ⟨119909 2119909 + 1⟩ subeSPL(119883) Since gcd119909 2119909 + 1 = 1 we get that ⟨119909 2119909 + 1⟩is a numerical semigroup and thus N ⟨119909 2119909 + 1⟩ is finiteConsequently N SPL(119883) is finite

Now let us see that SPL(119883) is a PL-semigroup Let 119909 119910 isinSPL(119883) 0 If 119878 is a PL-semigroup containing 119883 fromLemma 38 we deduce that 119909 119910 isin 1198780 and fromTheorem 5we have that 119909 + 119910 + 1 isin 119878 By applying again Lemma 38we have that 119909 + 119910 + 1 isin SPL(119883) Therefore by applyingTheorem 5 once more we can assert that SPL(119883) is a PL-semigroup

Remark 40 Let us observe that in general Proposition 39 isnot true for Frobenius varieties In fact let S be the set of allnumerical semigroups It is clear thatS is a Frobenius varietyIf we take 119883 = 2 then the intersection of all elements of Scontaining 2 is exactly119872 = ⟨2⟩ which is not a numericalsemigroup

The next result will be key for our last purpose in thissection

Theorem 41 S119875119871

= S119875119871(119883) | 119883 is a nonempty finite

subset 119900119891 N 0

Proof By Proposition 39 we have that

SPL (119883) | 119883 is a nonempty finite subset ofN 0 sube SPL(8)

For the other inclusion it is enough to observe that if 119878 isin SPLthen (by Proposition 37) there exists a nonempty finite subset119883 of N 0 such that 119878 = SPL(119883)

Since SPL is a Frobenius variety by applying [3 Proposi-tion 24] we get the next result

Proposition 42 Let119872 be aS119875119871-monoid and let 119909 isin 119872Then

119872119909 is aS119875119871-monoid if and only if 119909 belongs to the minimal

S119875119871-system of generators of119872

As an immediate consequence of this propositionwe havethe following result

8 The Scientific World Journal

Corollary 43 Let 119883 be a nonempty subset of N 0 Thenthe minimal S

119875119871-system of generators of S

119875119871(119883) is 119909 isin 119883 |

S119875119871(119883) 119909 is a PL-semigroup

Example 44 By Proposition 21 119878 = ⟨3 7 11⟩ is a PL-semigroup By applying Proposition 14 we easily deduce that

119909 isin 3 7 11 | 119878 119909 is a PL-semigroup = 3 (9)

Therefore 119878 = SPL(3) and 3 is the minimalSPL-system ofgenerators of 119878

Now we want to describe SPL(119883) when 119883 is a fixednonempty finite set of positive integers Let us observe thatby Theorem 41 we know that every PL-semigroup can beobtained in this way

Let 1198991 119899

119901be positive integers We will denote by

119878(1198991 119899

119901) the set 120572

11198991+ sdot sdot sdot + 120572

119901119899119901+ 119903 | 119903 120572

1 120572

119901isin

N 119903 lt 1205721+ sdot sdot sdot + 120572

119901 cup 0 Our next purpose will be to show

that 119878(1198991 119899

119901) = SPL(1198991 119899119901)

Lemma 45 Let 119878 be a numerical semigroup let 1199041 119904

119905isin

119878 0 and let 1205721 120572

119905isin N Then ord(120572

11199041+ sdot sdot sdot + 120572

119905119904119905) ge

1205721+ sdot sdot sdot + 120572

119905

Proof Let 1198991 119899

119901 be the minimal system of generators

of 119878 Then for each 119894 isin 1 119905 there exist 1205731198941 120573

119894119901isin N

such that 119904119894= 12057311989411198991+ sdot sdot sdot + 120573

119894119901119899119901 Moreover since 119904

119894= 0 we

have that 1205731198941+ sdot sdot sdot + 120573

119894119901ge 1 Thus

12057211199041+ sdot sdot sdot + 120572

119905119904119905

= 1205721(120573111198991+ sdot sdot sdot + 120573

1119901119899119901) + sdot sdot sdot + 120572

119905(12057311990511198991+ sdot sdot sdot + 120573

119905119901119899119901)

= (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) 1198991+ sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901) 119899119901

(10)

Therefore

ord (12057211199041+ sdot sdot sdot + 120572

119905119904119905)

ge (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) + sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901)

= 1205721(12057311+ sdot sdot sdot + 120573

1119901) + sdot sdot sdot + 120572

119905(1205731199051+ sdot sdot sdot + 120573

119905119901)

ge 1205721+ sdot sdot sdot + 120572

119905

(11)

Theorem 46 If 1198991 119899

119901are positive integers then

119878(1198991 119899

119901) is the smallest (with respect to set inclusion)

119875119871-semigroup containing 1198991 119899

119901

Proof We divide the proof into five steps

(i) Let us see that if 119909 119910 isin 119878(1198991 119899

119901) 0 then 119909 +

119910 isin 119878(1198991 119899

119901) In effect we know that there exist

1205721 120572

119901 1205731 120573

119901 119903 1199031015840 nonnegative integers such

that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903 119910 = 120573

11198991+sdot sdot sdot+120573

119901119899119901+1199031015840

119903 lt 1205721+ sdot sdot sdot + 120572

119901 and 1199031015840 lt 120573

1+ + 120573

119901 Therefore

119909 + 119910 = (1205721+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840 with119903 + 1199031015840

lt (1205721+ 1205731) + sdot sdot sdot + (120572

119901+ 120573119899) Consequently

119909 + 119910 isin 119878(1198991 119899

119901)

(ii) Let us see that N 119878(1198991 119899

119901) is finite Since 119899

1=

1sdot1198991+0sdot1198992+sdot sdot sdot+0sdot119899

119901+0 and 2119899

1+1 = 2sdot119899

1+0sdot1198992+sdot sdot sdot+0sdot

119899119901+1 we have that 119899

1 21198991isin 119878(1198991 119899

119901) By applying

the first step we get that ⟨1198991 21198991⟩ sube 119878(119899

1 119899

119901)

Using the same reasoning as we did in the proof ofProposition 39 we have the result

(iii) From the previous steps we know that 119878(1198991 119899

119901)

is a numerical semigroup Let us see now that119878(1198991 119899

119901) is a PL-semigroup In order to do that

it is enough (by Theorem 5) to show that if 119909 119910 isin

119878(1198991 119899

119901) 0 then 119909 + 119910 + 1 isin 119878(119899

1 119899

119901)

Indeed arguing as in the first step we have that 119909 +119910 + 1 = (120572

1+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840

+ 1

with 119903 + 1199031015840 + 1 lt (1205721+1205731) + sdot sdot sdot + (120572

119901+120573119899) Therefore

119909 + 119910 + 1 isin 119878(1198991 119899

119901)

(iv) Following the proof of the second step it is clear that1198991 119899

119901 sube 119878(119899

1 119899

119901)

(v) Finally let us see that 119878(1198991 119899

119901) is the small-

est PL-semigroup that contains 1198991 119899

119901 In fact

we will show that if 119879 is PL-semigroup containing1198991 119899

119901 then 119878(119899

1 119899

119901) sube 119879 Thus let 119909 isin

119878(1198991 119899

119901) 0 Then there exist 120572

1 120572

119901 119903 isin N

such that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903with 119903 lt 120572

1+sdot sdot sdot+120572

119901

Since 1198991 119899

119901 sube 119879 by Proposition 9 we have that

12057211198991+sdot sdot sdot+120572

119901119899119901+0 ord(120572

11198991+sdot sdot sdot+120572

119901119899119901)minus1 sube 119879

By applying Lemma 45 we have that 119903 lt ord(12057211198991+

sdot sdot sdot + 120572119901119899119901) and therefore 119909 isin 119879

In this way we have proved the statement

The next result is an immediate consequence of theprevious theorem

Corollary 47 If 1198991 119899

119901are positive integers then

SPL(1198991 119899119901) = 119878(1198991 119899119901)

We finish this section with an example that illustrates itscontent

Example 48 It is clear that 119878(4 7) = 0 4 7 8 9 11 rarr =

⟨4 7 9⟩ Therefore SPL(4 7) = ⟨4 7 9⟩

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referee for hisher usefulcomments and suggestions that helped to improve this workBoth of the authors are supported by FQM-343 (Junta deAndalucıa) MTM2010-15595 (MICINN Spain) and FEDERfunds The second author is also partially supported by Juntade AndalucıaFeder Grant no FQM-5849

The Scientific World Journal 9

References

[1] M Lothaire Applied Combinatorics on Words vol 105 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press 2005

[2] J L Ramırez Alfonsın The Diophantine Frobenius ProblemOxford University Press 2005

[3] J C Rosales ldquoFamilies of numerical semigroups closed underfinite intersections and for the Frobenius numberrdquo HoustonJournal of Mathematics vol 34 no 2 pp 339ndash348 2008

[4] J C Rosales M B Branco and D Torrao ldquoBracelet monoidsand numerical semigroupsrdquo preprint

[5] J C Rosales and P A Garcıa-Sanchez Numerical Semigroupsvol 20 of Developments in Mathematics Springer New YorkNY USA 2009

[6] M Bras-Amoros P A Garcıa-Sanchez and A Vico-OtonldquoNonhomogeneous patterns on numerical semigroupsrdquo Inter-national Journal of Algebra and Computation vol 23 no 6 pp1469ndash1483 2013

[7] K Stokes and M Bras-Amoros ldquoLinear non-homogeneoussymmetric patterns and prime power generators in numeri-cal semigroups associated to combinatorial congurationsrdquo toappear in Semigroup Forum 2013

[8] L Bryant ldquoGoto numbers of a numerical semigroup ring andthe gorensteiness of associated graded ringsrdquo Communicationsin Algebra vol 38 no 6 pp 2092ndash2128 2010

[9] R Apery ldquoSur les branches superlineaires des courbesalgebriquesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol222 pp 1198ndash1200 1946

[10] J B Roberts ldquoNote on linear formsrdquo Proceedings of the Ameri-can Mathematical Society vol 7 pp 465ndash469 1956

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 The Scientific World Journal

Corollary 43 Let 119883 be a nonempty subset of N 0 Thenthe minimal S

119875119871-system of generators of S

119875119871(119883) is 119909 isin 119883 |

S119875119871(119883) 119909 is a PL-semigroup

Example 44 By Proposition 21 119878 = ⟨3 7 11⟩ is a PL-semigroup By applying Proposition 14 we easily deduce that

119909 isin 3 7 11 | 119878 119909 is a PL-semigroup = 3 (9)

Therefore 119878 = SPL(3) and 3 is the minimalSPL-system ofgenerators of 119878

Now we want to describe SPL(119883) when 119883 is a fixednonempty finite set of positive integers Let us observe thatby Theorem 41 we know that every PL-semigroup can beobtained in this way

Let 1198991 119899

119901be positive integers We will denote by

119878(1198991 119899

119901) the set 120572

11198991+ sdot sdot sdot + 120572

119901119899119901+ 119903 | 119903 120572

1 120572

119901isin

N 119903 lt 1205721+ sdot sdot sdot + 120572

119901 cup 0 Our next purpose will be to show

that 119878(1198991 119899

119901) = SPL(1198991 119899119901)

Lemma 45 Let 119878 be a numerical semigroup let 1199041 119904

119905isin

119878 0 and let 1205721 120572

119905isin N Then ord(120572

11199041+ sdot sdot sdot + 120572

119905119904119905) ge

1205721+ sdot sdot sdot + 120572

119905

Proof Let 1198991 119899

119901 be the minimal system of generators

of 119878 Then for each 119894 isin 1 119905 there exist 1205731198941 120573

119894119901isin N

such that 119904119894= 12057311989411198991+ sdot sdot sdot + 120573

119894119901119899119901 Moreover since 119904

119894= 0 we

have that 1205731198941+ sdot sdot sdot + 120573

119894119901ge 1 Thus

12057211199041+ sdot sdot sdot + 120572

119905119904119905

= 1205721(120573111198991+ sdot sdot sdot + 120573

1119901119899119901) + sdot sdot sdot + 120572

119905(12057311990511198991+ sdot sdot sdot + 120573

119905119901119899119901)

= (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) 1198991+ sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901) 119899119901

(10)

Therefore

ord (12057211199041+ sdot sdot sdot + 120572

119905119904119905)

ge (120572112057311+ sdot sdot sdot + 120572

1199051205731199051) + sdot sdot sdot + (120572

11205731119901+ sdot sdot sdot + 120572

119905120573119905119901)

= 1205721(12057311+ sdot sdot sdot + 120573

1119901) + sdot sdot sdot + 120572

119905(1205731199051+ sdot sdot sdot + 120573

119905119901)

ge 1205721+ sdot sdot sdot + 120572

119905

(11)

Theorem 46 If 1198991 119899

119901are positive integers then

119878(1198991 119899

119901) is the smallest (with respect to set inclusion)

119875119871-semigroup containing 1198991 119899

119901

Proof We divide the proof into five steps

(i) Let us see that if 119909 119910 isin 119878(1198991 119899

119901) 0 then 119909 +

119910 isin 119878(1198991 119899

119901) In effect we know that there exist

1205721 120572

119901 1205731 120573

119901 119903 1199031015840 nonnegative integers such

that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903 119910 = 120573

11198991+sdot sdot sdot+120573

119901119899119901+1199031015840

119903 lt 1205721+ sdot sdot sdot + 120572

119901 and 1199031015840 lt 120573

1+ + 120573

119901 Therefore

119909 + 119910 = (1205721+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840 with119903 + 1199031015840

lt (1205721+ 1205731) + sdot sdot sdot + (120572

119901+ 120573119899) Consequently

119909 + 119910 isin 119878(1198991 119899

119901)

(ii) Let us see that N 119878(1198991 119899

119901) is finite Since 119899

1=

1sdot1198991+0sdot1198992+sdot sdot sdot+0sdot119899

119901+0 and 2119899

1+1 = 2sdot119899

1+0sdot1198992+sdot sdot sdot+0sdot

119899119901+1 we have that 119899

1 21198991isin 119878(1198991 119899

119901) By applying

the first step we get that ⟨1198991 21198991⟩ sube 119878(119899

1 119899

119901)

Using the same reasoning as we did in the proof ofProposition 39 we have the result

(iii) From the previous steps we know that 119878(1198991 119899

119901)

is a numerical semigroup Let us see now that119878(1198991 119899

119901) is a PL-semigroup In order to do that

it is enough (by Theorem 5) to show that if 119909 119910 isin

119878(1198991 119899

119901) 0 then 119909 + 119910 + 1 isin 119878(119899

1 119899

119901)

Indeed arguing as in the first step we have that 119909 +119910 + 1 = (120572

1+ 1205731)1198991+ sdot sdot sdot + (120572

119901+ 120573119899)119899119901+ 119903 + 119903

1015840

+ 1

with 119903 + 1199031015840 + 1 lt (1205721+1205731) + sdot sdot sdot + (120572

119901+120573119899) Therefore

119909 + 119910 + 1 isin 119878(1198991 119899

119901)

(iv) Following the proof of the second step it is clear that1198991 119899

119901 sube 119878(119899

1 119899

119901)

(v) Finally let us see that 119878(1198991 119899

119901) is the small-

est PL-semigroup that contains 1198991 119899

119901 In fact

we will show that if 119879 is PL-semigroup containing1198991 119899

119901 then 119878(119899

1 119899

119901) sube 119879 Thus let 119909 isin

119878(1198991 119899

119901) 0 Then there exist 120572

1 120572

119901 119903 isin N

such that 119909 = 12057211198991+sdot sdot sdot+120572

119901119899119901+119903with 119903 lt 120572

1+sdot sdot sdot+120572

119901

Since 1198991 119899

119901 sube 119879 by Proposition 9 we have that

12057211198991+sdot sdot sdot+120572

119901119899119901+0 ord(120572

11198991+sdot sdot sdot+120572

119901119899119901)minus1 sube 119879

By applying Lemma 45 we have that 119903 lt ord(12057211198991+

sdot sdot sdot + 120572119901119899119901) and therefore 119909 isin 119879

In this way we have proved the statement

The next result is an immediate consequence of theprevious theorem

Corollary 47 If 1198991 119899

119901are positive integers then

SPL(1198991 119899119901) = 119878(1198991 119899119901)

We finish this section with an example that illustrates itscontent

Example 48 It is clear that 119878(4 7) = 0 4 7 8 9 11 rarr =

⟨4 7 9⟩ Therefore SPL(4 7) = ⟨4 7 9⟩

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referee for hisher usefulcomments and suggestions that helped to improve this workBoth of the authors are supported by FQM-343 (Junta deAndalucıa) MTM2010-15595 (MICINN Spain) and FEDERfunds The second author is also partially supported by Juntade AndalucıaFeder Grant no FQM-5849

The Scientific World Journal 9

References

[1] M Lothaire Applied Combinatorics on Words vol 105 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press 2005

[2] J L Ramırez Alfonsın The Diophantine Frobenius ProblemOxford University Press 2005

[3] J C Rosales ldquoFamilies of numerical semigroups closed underfinite intersections and for the Frobenius numberrdquo HoustonJournal of Mathematics vol 34 no 2 pp 339ndash348 2008

[4] J C Rosales M B Branco and D Torrao ldquoBracelet monoidsand numerical semigroupsrdquo preprint

[5] J C Rosales and P A Garcıa-Sanchez Numerical Semigroupsvol 20 of Developments in Mathematics Springer New YorkNY USA 2009

[6] M Bras-Amoros P A Garcıa-Sanchez and A Vico-OtonldquoNonhomogeneous patterns on numerical semigroupsrdquo Inter-national Journal of Algebra and Computation vol 23 no 6 pp1469ndash1483 2013

[7] K Stokes and M Bras-Amoros ldquoLinear non-homogeneoussymmetric patterns and prime power generators in numeri-cal semigroups associated to combinatorial congurationsrdquo toappear in Semigroup Forum 2013

[8] L Bryant ldquoGoto numbers of a numerical semigroup ring andthe gorensteiness of associated graded ringsrdquo Communicationsin Algebra vol 38 no 6 pp 2092ndash2128 2010

[9] R Apery ldquoSur les branches superlineaires des courbesalgebriquesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol222 pp 1198ndash1200 1946

[10] J B Roberts ldquoNote on linear formsrdquo Proceedings of the Ameri-can Mathematical Society vol 7 pp 465ndash469 1956

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 9

References

[1] M Lothaire Applied Combinatorics on Words vol 105 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press 2005

[2] J L Ramırez Alfonsın The Diophantine Frobenius ProblemOxford University Press 2005

[3] J C Rosales ldquoFamilies of numerical semigroups closed underfinite intersections and for the Frobenius numberrdquo HoustonJournal of Mathematics vol 34 no 2 pp 339ndash348 2008

[4] J C Rosales M B Branco and D Torrao ldquoBracelet monoidsand numerical semigroupsrdquo preprint

[5] J C Rosales and P A Garcıa-Sanchez Numerical Semigroupsvol 20 of Developments in Mathematics Springer New YorkNY USA 2009

[6] M Bras-Amoros P A Garcıa-Sanchez and A Vico-OtonldquoNonhomogeneous patterns on numerical semigroupsrdquo Inter-national Journal of Algebra and Computation vol 23 no 6 pp1469ndash1483 2013

[7] K Stokes and M Bras-Amoros ldquoLinear non-homogeneoussymmetric patterns and prime power generators in numeri-cal semigroups associated to combinatorial congurationsrdquo toappear in Semigroup Forum 2013

[8] L Bryant ldquoGoto numbers of a numerical semigroup ring andthe gorensteiness of associated graded ringsrdquo Communicationsin Algebra vol 38 no 6 pp 2092ndash2128 2010

[9] R Apery ldquoSur les branches superlineaires des courbesalgebriquesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol222 pp 1198ndash1200 1946

[10] J B Roberts ldquoNote on linear formsrdquo Proceedings of the Ameri-can Mathematical Society vol 7 pp 465ndash469 1956

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of