Generalized strongly increasing semigroups
E.R. García Barroso, J.I. García-García, A. Vigneron-Tenorio
aa r X i v : . [ m a t h . A C ] M a r Generalized strongly increasing semigroups
E.R. Garc´ıa Barroso, J.I. Garc´ıa-Garc´ıa,A. Vigneron-Tenorio ∗ Abstract
In this work we present a new class of numerical semigroups calledGSI-semigroups. We see the relations between them and others familiesof semigroups and we give explicitly their set of gaps. Moreover, analgorithm to obtain all the GSI-semigroups up to a given Frobeniusnumber is provided and the realization of positive integers as Frobeniusnumbers of GSI-semigroups is studied.2010
Mathematics Subject Classification:
Primary 20M14; Secondary14H20.
Key words:
Generalized strongly increasing semigroup, strongly in-creasing semigroup, Frobenius number, singular analytic plane curve.
Introduction
Let N = { , , , . . . } be the set of nonnegative integers. A numericalsemigroup is a subset S of N closed under addition, 0 ∈ S and N \ S ,its gapset, is finite. The least not zero element in S is called themultiplicity of S , we denote it by m ( S ). Given a nonempty subset A = { a , . . . , a n } of N we denote by h A i the smallest submonoid of ( N , +)containing A ; the submonoid h A i is equal to the set N a + · · · + N a n .The minimal system of generators of S is the smallest subset of S generating it, and its cardinality, denoted by e( S ), is known as theembedding dimension of S . It is well known (see Lemma 2.1 from [12])that h A i is a numerical semigroup if and only if gcd( A ) = 1. Thecardinality of N \ S is called the genus of S (denoted by g ( S )) and itsmaximum is known as the Frobenius number of S (denoted by F( S )).Numerical semigroups appear in several areas of mathematics andits theory is connected with Algebraic Geometry and CommutativeAlgebra (see [3], [6]) as well as with Integer Optimization (see [5]) andNumber Theory (see [2]). It is common the study of families of numer-ical semigroups, for instance symmetric semigroups, irreducible semi-groups and strongly increasing semigroups (see [9], [10]) or the studyor characterization of invariants, for instance the Frobenius number,the set of gaps, the genus, etc. (see [1], [8], [10] and [11]). ∗ The first-named author was partially supported by the Spanish Project MTM2016-80659-P. The second and third-named author were partially supported by the SpanishProject MTM2017-84890-P and Junta de Andaluc´ıa group FQM-366. nspired in [1] and [9], and with the aim of study the sets of gaps ofstrongly increasing semigroups (shorted by SI-semigroups), we intro-duce the concept of generalized strongly increasing semigroup (shortedby GSI-semigroups). These numerical semigroups ¯ S = h v , . . . , v h , γ i are the gluing of a semigroup S with N , we denote them by S ⊕ d,γ N ,where S = h v /d, . . . , v h /d i , d = gcd( v , . . . , v h ) > γ ∈ N suchthat γ > max { d F( S ) , v h } .Our main result is Theorem 6, where we describe the set of gapsof GSI-semigroups. Since every SI-semigroup is a GSI-semigroup, ourdescription of the gaps is also valid for SI-semigroups. Semigroups ofvalues associated with plane branches are always SI-semigroups andtheir sets of gaps describe topological invariants of the curves (see[9]) which are used to classify singular analytic plane curves. Due tothe fact that the condition for being GSI-semigroup is straightforwardto check from a given system of generators, it is easy to constructsubfamilies of GSI-semigroups and thus of SI-semigroups.In this work we also compare the class of GSI-semigroups with otherfamilies of numerical semigroups obtained as gluing of numerical semi-groups. These are the classes of telescopic, free and complete intersec-tion numerical semigroups. In [1], it is constructed the set of completeintersection numerical semigroups with given Frobenius number, andsome special subfamilies as free and telescopic numerical semigroups,and numerical semigroups associated with irreducible singular planecurves are studied. As we pointed above, we prove that SI-semigroupsare always GSI-semigroups. We also prove that GSI-semigroups arenot included in the other three above-mentioned families.Some algorithms for computing GSI-semigroups are provided in thiswork. One of them computes the set of GSI-semigroups up to a fixedFrobenius number. We prove that for any odd number f , there is atleast a GSI-semigroup which Frobenius number equal to f . For evennumbers, it does not always happen. Thus, GSI-semigroups with evenFrobenius numbers are also studied, and we present an algorithm tocheck whether a GSI-semigroup with a given even Frobenius number.This work is organized as follows. In Section 1, we introducethe GSI-semigroups and some of their properties. We prove that SI-semigroups are GSI-semigroups (see Corollary 5). We also compareGSI-semigroups with another families such as free, telescopic and com-plete intersection numerical semigroups. In Section 2, the main resultof this paper is presented (Theorem 6). This theorem gives us an ex-plicit formula for the set of gaps of GSI-semigroups. We finish thiswork with Section 3, where we give an algorithm for computing theset of GSI-semigroups up to a fixed Frobenius number and we showsome properties of Frobenius numbers of GSI-semigroups. In this lastsection, we also provide an algorithm to test whether there is a GSI-semigroup with given even Frobenius number. Generalized strongly increasing semigroups
The gluing of S = h v , . . . , v h i and N with respect to d and γ withgcd( d, γ ) = 1 (see [12, Chapter 8]) is the numerical semigroup N dv + · · · + N dv h + N γ . We denote it by S ⊕ d,γ N . Definition 1.
A numerical semigroup ¯ S is a generalized strongly in-creasing semigroup whenever ¯ S is the gluing of a numerical semigroup S = h v , . . . , v h i with respect to d and γ (that is, ¯ S = S ⊕ d,γ N ), where d ∈ N \ { , } and γ ∈ N with γ > max { d F( S ) , dv h } (note that d and γ are coprimes). The first example of GSI-semigroups are numerical semigroups gen-erated by two positive integers ¯ S = h a, b i with a < b . For these semi-groups, set S = N = h i , d = a , and γ = b . Since F( S ) = − γ = b > max { d F( S ) , d } = { a · ( − , a } = a . From Sylvester (see [13]),we know that the Frobenius numbers of these semigroups are given bythe formula a · b − a − b . Hence, every odd natural number is realizableas Frobenius number of a GSI-semigroup.Our definition of GSI-semigroups is inspired on the one of SI-semigroups. We remind you how they are defined (see [4] for furtherdetails).A sequence of positive integers ( v , . . . , v h ) is called a characteristicsequence if it satisfies the following two properties:(CS1) Put e k = gcd( v , . . . , v k ) for 0 ≤ k ≤ h . Then e k < e k − for1 ≤ k ≤ h and e h = 1.(CS2) e k − v k < e k v k +1 for 1 ≤ k ≤ h − n k = e k − e k for 1 ≤ k ≤ h . Therefore n k > ≤ k ≤ h and n h = e h − . If h = 0, the only characteristic sequence is ( v ) = (1). If h = 1, the sequence ( v , v ) is a characteristic sequence if and only ifgcd( v , v ) = 1. Property (CS2) plays a role if and only if h ≥ Lemma 2. ([4, Lemma 1.1]) Let ( v , . . . , v h ) , h ≥ be a characteristicsequence. Then, (i) v < · · · < v h and v < v . (ii) Let v < v . If v v ) then ( v , v , v , . . . , v h ) is acharacteristic sequence. If v ≡ v ) then ( v , v , . . . , v h ) is a characteristic sequence. We denote by h v , . . . , v h i the semigroup generated by the charac-teristic sequence ( v , . . . , v h ). Observe that h v , . . . , v h i is a numericalsemigroup. A semigroup S ⊆ N is Strongly Increasing (SI-semigroup)if S = { } and it is generated by a characteristic sequence. Note thatby Lemma 2, we can assume that v < · · · < v h . Theorem 3.
Let ¯ S be a numerical semigroup with e( ¯ S ) = h + 1 . Then, ¯ S is strongly increasing if and only if one of the two next conditionsholds:1. h = 1 , ¯ S = N ⊕ d,γ N = h d, γ i , where d and γ are two coprimeintegers. . h > , ¯ S = S ⊕ d,γ N , where S = h v , . . . , v h − i is a strongly in-creasing semigroup with embedding dimension h and γ, d > aretwo coprime integer numbers such that γ > d gcd( v , . . . , v h − ) v h − . Proof.
The case h = 1 is trivial by definition of characteristic se-quences.Assume h > S = h ¯ v , . . . , ¯ v h i is a strongly increasingnumerical semigroup with embedding dimension strictly greater than2 . Let ¯ e i = gcd(¯ v , . . . , ¯ v i ) for 0 ≤ i ≤ h . Put v i = ¯ v i ¯ e h − for 0 ≤ i ≤ h −
1. Then, ( v , . . . , v h − ) is a characteristic sequence. Let S = h v , . . . , v h − i . Since e( ¯ S ) = h + 1, then e( S ) = h . Set γ = ¯ v h and d = ¯ e h − , we get ¯ S = S ⊕ d,γ N . We have that γ = ¯ v h > ¯ v h − = dv h − ,and since ¯ S is a SI-semigroup, γ = ¯ v h > ¯ e h − ¯ e h − ¯ v h − = gcd(¯ v , . . . , ¯ v h − )¯ e h − ¯ v h − = gcd(¯ e h − v , . . . , ¯ e h − v h − )¯ e h − ¯ e h − v h − = ¯ e h − gcd( v , . . . , v h − ) v h − = d gcd( v , . . . , v h − ) v h − . Conversely, let S = h v , . . . , v h − i be a strongly increasing semi-group with embedding dimension h, and γ, d > γ > d gcd( v , . . . , v h − ) v h − . Denote e i =gcd( v , . . . , v i ) for i = 0 , . . . , h −
1. Take ¯ S = h ¯ v , . . . , ¯ v h = γ i the gluingsemigroup S ⊕ d,γ N and define ¯ e i = gcd(¯ v , . . . , ¯ v i ) for i = 0 , . . . , h . Wehave that ¯ e = de > · · · > ¯ e h − = de h − = d > ¯ e h = gcd( γ, d ) = 1.Since e i − v i < e i v i +1 , for 1 ≤ i ≤ h − e i − v i d < e i v i +1 d and therefore ¯ e i − ¯ v i < ¯ e i ¯ v i +1 . By hypothesis de h − v h − < γ , hence d e h − v h − < dγ and therefore ¯ e h − ¯ v h − < ¯ e h − γ . We conclude that¯ S is a SI-semigroup.The following result give us a formula for the conductor (the Frobe-nius number plus 1) of a SI-semigroup. Proposition 4. ([9, Proposition 2.3 (4)], [4, Proposition 1.2]) Let S = h v , . . . , v h i be the semigroup generated by the characteristic sequence ( v , . . . , v h ) . The conductor of the semigroup S is c ( S ) = h X i =1 ( n i − v i − v + 1 . Moreover, the conductor of S is an even number and its genus is g ( S ) = c ( S )2 . y Proposition 4, we getF( S ) = h X i =1 ( n i − v i − v = h X i =1 n i v i − h X i =1 v i − v = ( n v − v ) + ( n v − v ) + · · · +( n h − v h − − v h ) + n h v h − v − v ≤ − ( h −
1) + n h v h − v − v = e h − v h − v − v − h + 1 < e h − v h − v − v . Assume that ¯ S = h ¯ v , . . . , ¯ v h i is a SI-semigroup satisfying that¯ v < · · · < ¯ v h = γ . Set d = ¯ e h − = gcd(¯ v , . . . , ¯ v h − ), γ = ¯ v h and S = h ¯ v /d, . . . , ¯ v h − /d i . We have that dγ = ¯ e h − ¯ v h > ¯ e h − ¯ v h − ¯ v − ¯ v > F( ¯ S ) = P hi =1 (¯ n i − v i − ¯ v = d F( S )+ (¯ e h − − v h = d F( S )+ ( d − γ .Thus γ > d F( S ). Since ¯ S is a SI-semigroup, the property (CS2) iffulfilled. Using that the generators are ordered, we obtain that ¯ v h < ¯ e h − ¯ v h = d ¯ v h < γ .So we can state the following result. Corollary 5.
Every SI-semigroup is a GSI-semigroup.
There are semigroups with similar definitions to SI and GSI semi-groups. For example, telescopic, free and complete intersection.Let S = h v , . . . , v h i . For k ∈ { , . . . , h } , set e k = gcd( v , . . . , v k − )( e = v ). We say that S is free whenever it is equal to N or it is thegluing of a free with N . The semigroup S is telescopic if it is free forthe rearrangement v < · · · < v h . A semigroup is complete intersectionif it is the gluing of two complete intersection numerical semigroups.The above three definitions are from [1].It is easy to check that SI-semigroups are telescopic, telescopic arefree semigroups and free semigroups are complete intersection. In gen-eral, GSI-semigroups are neither strongly increasing nor telescopic norfree nor complete intersection. Clearly, h , , , i = h , , i⊕ , N and 23 > max { h , , i ) , · } . Thus this is a GSI-semigroup. Wedefine the functions IsSIncreasingNumericalSemigroup and
IsGSI to check is a numerical semigroup is a SI-semigroup and a GSI-semigroup,respectively (the code of these functions is showed in Table 1).Applying our functions and the functions
IsFreeNumericalSemigroup , IsTelescopicNumericalSemigroup and
IsCompleteIntersection of[7] to the semigroup h , , , i , we obtain the following outputs: gap> IsFreeNumericalSemigroup(NumericalSemigroup(6,14,22,23));falsegap> IsTelescopicNumericalSemigroup(NumericalSemigroup(6,14,22,23));falsegap> IsCompleteIntersection(NumericalSemigroup(6,14,22,23));falsegap> IsSIncreasing(NumericalSemigroup(6,14,22,23)); alsegap> IsGSI(NumericalSemigroup(6,14,22,23));true From the results of the above computations, we conclude that theclass of GSI-semigroups contains the class of SI-semigroups, but it isdifferent to the classes of free, telescopic and complete intersectionsemigroups.
We have seen that GSI-semigroups are easy to obtain from any numer-ical semigroup just gluing it with N with appropiate elements d and γ .Hence these semigroups form a large family within the set of numericalsemigroups. In this section, we deepen into their study by explicitlydetermining their set of gaps.Hereafter the notation [ a mod n ] for an integer a and a naturalnumber n means the remainder of the division of a by n , and [ a ] n denotes the coset of a modulo n . For any two real numbers a ≤ b we denote by [ a, b ] N the set of natural numbers belonging to the realinterval [ a, b ]. Put ⌊ a ⌋ the integral part of the real number a . Theorem 6.
Let S = h v , . . . , v h i be a numerical semigroup with v < · · · < v h , d ≥ and v h +1 be two natural coprime numbers such that v h +1 > max { d F( S ) , dv h } . Then the gaps of the GSI-semigroup ¯ S = S ⊕ d,v h +1 N are N \ ¯ S = { , . . . , dv − } ∪ { x ∈ ( dv , v h +1 ) ∩ N : x / ∈ dS } ∪A d ∪ S d − ℓ =1 B d,ℓ , (1) where B d,ℓ = (cid:26) v h +1 + [ ℓv h +1 mod d ] + kd : 0 ≤ k ≤ (cid:22) ℓv h +1 d (cid:23) − (cid:27) and A d = d − [ k =1 (cid:0) d ( N \ S ) + kv h +1 (cid:1) ( A d = ∅ when S = N ) . Moreover (1) is a partition of the gapset of ¯ S (we do not write A d or B d,l if they are empty).Proof. It is clear that { , . . . , dv − } is included in N \ ¯ S .Consider x ∈ ( dv , v h +1 ) ∩ N such that x / ∈ dS , and suppose that x ∈ ¯ S . Since x < v h +1 then there are λ i with 0 ≤ i ≤ h such that x = λ dv + · · · + λ h dv h = d ( λ v + · · · + λ h v h ) ∈ dS , which is acontradiction. Hence we conclude that { x ∈ ( dv , v h +1 ) ∩ N : x / ∈ dS } ⊆ N \ ¯ S .Suppose that S = N . Fix 1 ≤ k ≤ d − x ∈ d ( N \ S )+ kv h +1 .We get x = dα + kv h +1 , for some α ∈ N \ S . Suppose that x ∈ ¯ S .So, there exist α , . . . , α h , β ∈ N such that x = dα + kv h +1 = dα v + · · + dα h v h + βv h +1 and then ( k − β ) v h +1 = d ( α v + · · · + α h v h − α ).If k = β , the element α have to belong to S which it is not possible.Moreover, since d and v h +1 are coprime, d divides k − β . If β > k , dα = d ( α v + · · · + α h v h ) + ( β − k ) v h +1 with v h +1 ∈ S , that is, α ∈ S .Again, it is not possible. If we assume k > β then k − β ≥ d and k ≥ d .In any case, the set d ( N \ S ) + kv h +1 is included in N \ ¯ S for any integer k in [1 , d − N .Let us prove now that B d,ℓ ⊆ N \ ¯ S . Suppose that x = v h +1 +[ ℓv h +1 mod d ] + kd ∈ ¯ S for some 1 ≤ ℓ ≤ d − ≤ k ≤ j ℓv h +1 d k −
1. Let α , α , . . . , α h +1 ∈ N such that x = v h +1 + [ ℓv h +1 mod d ] + kd = α dv + · · · + α h dv h + α h +1 v h +1 . Hence, ( α h +1 − v h +1 − [ ℓv h +1 mod d ] = d ( k − α v − · · · − α h v h ) and [( α h +1 − − ℓ ) v h +1 ] d =[0] d . Since d and v h +1 are coprime then d divides α h +1 − − ℓ . Butmax B d,ℓ = ( ℓ + 1) v h +1 − d so we have α h +1 ∈ { , , . . . , ℓ } , hence − − ℓ ≤ α h +1 − − ℓ ≤ − ℓ ≥ − α h +1 + 1 + ℓ ≥ − α h +1 + 1 + ℓ is a multiple of d which is a contradiction since ℓ < d − H = { , . . . , dv − }∪{ x ∈ ( dv , v h +1 ) ∩ N | x / ∈ dS }∪A d ∪ d − [ ℓ =1 B d,ℓ ⊆ N \ ¯ S. Let us prove that H is a partition (we do not write A d or B d,ℓ whenthey are the emptyset).When A d is a nonempty set, let A d,k = d ( N \ S ) + kv h +1 for a fix1 ≤ k ≤ d −
1. In this case, if B d,ℓ is nonempty we havemax B d,ℓ < min A d,ℓ +1 for 1 ≤ ℓ ≤ d − . (2)Observe that [ x ] d = [ kv h +1 ] d for any x ∈ A d,k (3)and [ y ] d = [( ℓ + 1) v h +1 ] d for any y ∈ B d,ℓ . (4)Since 1 ≤ k, ℓ < d we get that any two sets A d,k and A d,k ′ aredisjoint for k = k ′ and any two sets B d,ℓ and B d,ℓ ′ are also disjointfor ℓ = ℓ ′ . Moreover A d and B d,ℓ are also disjoint for any 1 ≤ ℓ ≤ d −
2. Indeed, let x ∈ A d ∩ B d,ℓ for some 1 ≤ ℓ ≤ d −
2. Hence,there is k ∈ { , . . . , d − } such that x ∈ A d,k and by (3) and (4),[ x ] d = [ kv h +1 ] d = [( ℓ + 1) v h +1 ] d . Given that d and v h +1 are coprimeand 1 ≤ k, ℓ < d we get k = ℓ + 1. So x ∈ A d,ℓ +1 ∩ B d,ℓ , which is acontradiction by inequality (2).In order to finish the proof we will show that there is not a gap of¯ S outside H .First at all, observe that if x ∈ N \ ¯ S and x < v h +1 , x ∈ { , . . . , dv − } S { x ∈ ( dv , v h +1 ) ∩ N | x / ∈ dS } .Claim 1: if x ∈ N \ ¯ S and v h +1 < x then [ x ] d = [ kv h +1 ] d , for some k ∈ { , . . . , d − } . ndeed, if we suppose that x = λd for some λ ∈ N , by hypoth-esis we get d F( S ) < v h +1 < x = λd , in particular λ > F( S ), so x ∈ dS ⊂ ¯ S . Since [ x ] d = [0] d and gcd( d, v h +1 ) = 1 we get [ x ] d ∈{ [1] d , . . . , [ d − d } = { [ kv h +1 ] d : 1 ≤ k ≤ d − } , that is, any x ∈ N \ ¯ S with v h +1 < x is congruent with kv h +1 module d for some integer k ∈ { , . . . , d − } .We distinguish two cases, depending on A d . First, we suppose that A d = ∅ .Claim 2: The greatest gap of ¯ S which is congruent with kv h +1 modulo d is max A d,k , for 1 ≤ k ≤ d − x ∈ N \ ¯ S with [ x ] d = [ kv h +1 ] d and x > max A d,k then x = d F( S ) + kv h +1 + λd for some non zero natural number λ . So x = d (F( S ) + λ ) + kv h +1 ∈ ¯ S since F( S ) + λ ∈ S .Claim 3: There are not gaps of ¯ S congruent with ( ℓ + 1) v h +1 mod-ulo d , between max B d,ℓ and min A d,ℓ +1 .Remember that [max B d,ℓ ] d = [min A d,ℓ +1 ] d = [( ℓ + 1) v h +1 ]. Supposethat x ∈ N \ ¯ S with max B d,ℓ < x < min A d,ℓ +1 and [ x ] d = [( ℓ +1) v h +1 ] d .Since max B d,ℓ = ( ℓ + 1) v h +1 − d and min A d,ℓ +1 = ( ℓ + 1) v h +1 + d theonly possibility for x is ( ℓ + 1) v h +1 which is an element of ¯ S .By Claims 1 and 2 we deduce that for any x ∈ N \ ¯ S with v h +1 < x there exists an integer k ∈ { , . . . , d − } such that [ x ] d = [ k v h +1 ] d and v h +1 < x ≤ max A d,k . In particular there is an integer number λ such that x = k v h +1 + λd . Hence if x ∈ [min A d,k , max A d,k ] N then x ∈ A d,k . Indeed, in this case λ ∈ N and λ S , otherwise x ∈ ¯ S .By Claim 3, we can assume that if v h +1 < x < min A d,k then v h +1 < x ≤ max B d,k − = k v h +1 − d .Claim 4: The set of all the integers in ( v h +1 , max B d,k − ] congruentwith k v h +1 module d is B d,k − .By (4) we have [max B d,k − ] d = [ k v h +1 ] d . Moreover { max B d,k − , max B d,k − − d, max B d,k − − d . . . , min B d,k − } = B d,k − and min B d,k − − d = v h +1 + [( k − v h +1 mod d ] − d < v h +1 .Hence x has to belong to B d,k − and we finish the proof for thecase A d = ∅ .Suppose now that A d = ∅ , that is S = N and ¯ S is generated by d and v ( h = 0).Claim 5: If A d = ∅ then max B d,ℓ is the greatest gap of ¯ S which iscongruent with ( ℓ + 1) v h +1 modulo d , for 1 ≤ ℓ ≤ d − B d,ℓ = ( ℓ + 1) v − d . For any natural number x > max B d,ℓ with [ x ] d = [( ℓ + 1) v ] d , there is α ∈ N \ { } such that x =( ℓ + 1) v − d + αd = ( ℓ + 1) v + ( α − d ∈ ¯ S . he above result provides us an explicit formula for the gaps exceptthe elements of A d . We now give some examples of GSI-semigroupswhere the set A d is easily known. Example . Let S = h , i . We have N \ S = { , , } and F( S ) = 5.Take now d = 2 and γ = 15. Since γ > max { · , · } , the semigroup S ⊕ , N is a GSI-semigroup. The set A is equal to 2 { , , } +1 ·
15 = { , , } and therefore F( S ⊕ , N ) = 25. Example . Consider now the semigroup S = h , , , , i , d = 3 and γ = 31. We have N \ S = { , , , } and F( S ) = 4. Since γ > max { · , · } , the semigroup S ⊕ , N is a GSI-semigroup and A =(3 { , , , } +1 · ∪ (3 { , , , } +2 ·
31) = { , , , , , , , } .Thus, F( S ⊕ , N ) = 74. Corollary 9.
Let S = h v , . . . , v h i be a semigroup, and d, v h +1 ∈ N twonatural numbers such that ¯ S = h dv , . . . , dv h , v h +1 i is a GSI-semigroup.Then F( ¯ S ) = (cid:26) max A d if A d = ∅ max B d,d − otherwise, where A d and B d,d − are from (1).Proof. If A d = ∅ , then, by inequality (2) , F( ¯ S ) = max A d = d F( S ) +( d − v h +1 . Otherwise, S = N and ¯ S is generated by d and v ( h = 0).So F( ¯ S ) = ( d − v − − B d,d − . From the proof of Theorem 6, we obtain the Frobenius number ofa GSI-semigroup S ⊕ d,γ N , which is equal toF( S ⊕ d,γ N ) = d F( S ) + ( d − γ. (5) We finish this work with some algorithms for computing GSI-semigroups.These algorithms focus on computing the GSI-semigroups up to a givenFrobenius number, and on checking whether there is at least one GSI-semigroup with a given even Frobenius number. For any odd number,there is a GSI-semigroup with this number as its Frobenius number,however, this does not happen for a given even number. Thus, inthis section we dedicate a special study to GSI-semigroups with evenFrobenius number.Algorithm 1 computes the set of GSI-semigroups with Frobeniusnumber least than or equal to a fixed nonnegative integer. Note thatin step 5 of the algorithm we use that F( ¯ S ) = d F( S ) + ( d − γ and γ > d F( S ) implies that F( ¯ S ) ≥ d F( S ) where ¯ S = S ⊕ d,γ N .Denote by M ( S ) the largest element of the minimal system of gen-erators of a numerical semigroup S . Remark . If A is a minimal system of generators of a numericalsemigroup S and d ∈ N \ { , } , then dA is a minimal system of gen-erators of dS = { ds | s ∈ S } ⊂ d N . Furthermore, if γ ∈ N \ { } andgcd( d, γ ) = 1, then γ d N \ { } . Thus, γ dS and dA ∪ { γ } is aminimal system of generators of h dA ∪ { γ }i . lgorithm 1: Computation of the set of GSI-semigroups with Frobe-nius number least than or equal to f . Data: f ∈ N \ { } . Result:
The set { ¯ S | ¯ S is a GSI-semigroup with F( ¯ S ) ≤ f } . A = ∅ ; forall k ∈ {− } ∪ { , , . . . , f } do B = { S | F( S ) = k } ; forall S ∈ B do D S = { d ∈ N \ { , } | d F( S ) ≤ f } ; G d,S = { ( d, γ ) ∈ D S × N | gcd( γ, d ) = 1 , γ > max { d F( S ) , dM ( S ) } , d F( S ) + ( d − γ ≤ f } ; A = A ∪ { S ⊕ d,γ N | ( d, γ ) ∈ G d,S } ; return A ; We give in Table 2 all the GSI-semigroups with Frobenius numberleast than or equal to 15.
Frobenius number Set of GSI-semigroups1 {h , i} ∅ {h , i} ∅ {h , i , h , i} ∅ {h , i , h , i} ∅ {h , i , h , , i} ∅ {h , i , h , i , h , i , h , , i} ∅ {h , i , h , i , h , , i} ∅ {h , i , h , , i , h , , , i} Table 2: Sets of GSI-semigroups with Frobenius number up to 15.
Remember that every numerical semigroup generated by two ele-ments is a GSI-semigroup. Hence, for any odd natural number thereexists at least one GSI-semigroup with such Frobenius number.From Table 2, one might think that there are no GSI-semigrupswith even Frobenius number. This is not so and we can check that h , , , i = h , , i ⊕ , N is a GSI-semigroup and its Frobenius umber is 38, gap> FrobeniusNumber(NumericalSemigroup(9,12,15,16));38gap> IsGSI(NumericalSemigroup(9,12,15,16));true This is the first even integer that is realizable as the Frobenius numberof a GSI-semigroup. We explain this fact: we want to obtain an evennumber f from the formula (5), f = F( S ⊕ d,γ N ) = d F( S ) + ( d − γ .Since gcd( d, γ ) = 1, then d has to be odd and F( S ) even. Thus, thelowest number f is obtained for the numerical semigroup S with thesmallest even Frobenius number, the smallest odd number d ≥ γ , that is, S = h , , i , d = 3 and γ = 16.Thus, the GSI-semigroup with the minimum even Frobenius numberis h , , i ⊕ , N .Note that not every even number is obtained as the Frobeniusnumber of a GSI-semigroup h , , i ⊕ ,γ N for some γ ≥
16 withgcd( d, γ ) = 1. In this way, we only obtain the values of the form36 + 2 k with k ∈ N and k γ = 16 + k for k ∈ N ,F( h , , i ⊕ ,γ N ) = 38 + 2 k ). The numbers of the form 42 + 6 k , with k ∈ N , are not obtained (see Table 3). γ
16 17 18 19 20 21 22 . . . F( h , , i ⊕ ,γ N ) 38 40 * 44 46 * 50 . . . Table 3: Values of γ such that gcd(3 , γ ) = 1 are marked with *. We now look for GSI-semigroups with Frobenius number of theform 42 + 6 k . Reasoning as above, we use again the semigroup h , , i and set now d = 5. In this case, the smallest Frobenius number is 114,and it is given by the semigroup h , , i ⊕ , N . In general, for thesemigroups h , , i ⊕ ,γ N , the formula of their Frobenius numbers is5 · γ with γ ≥
26 and γ S = h , , i ,we fill all the even Frobenius number f ≥ f is of theform f = 10 + 4 k with k = 15 k ′ . That is, f cannot be a number of theform f = 10 + 60 k ′ with k ′ ∈ N \ { , } , for instance 130 and 190. γ
26 27 28 29 30 31 32 . . . F( h , , i ⊕ ,γ N ) 114 118 122 126 * 134 138 . . . Table 4: Values γ such that gcd(5 , γ ) = 1 are marked with *. The above procedures are useful to construct GSI-semigroups witheven Frobenius numbers, but with them we cannot determine if a giveneven positive integer is realizable as the Frobenius number of a GSI-semigroup.Fixed an even number f , we are interested in providing an algo-rithm to check if there exists at least one GSI-semigroup S ⊕ d,γ N suchthat F( S ⊕ d,γ N ) = f . sing that γ has to be greater than or equal to d F( S )+1 ≥ S )+1(recall that γ > max { d F( S ) , dM ( S ) } and d ≥
3) and from formula (5),we obtain that if F( S ⊕ d,γ N ) = f , then 2 ≤ F( S ) ≤ ⌊ f − ⌋ .Let t ∈ h , f − i be the Frobenius number of S . Hence, f = dt +( d − γ ≥ d t + d − d ∈ h , j − √ ft +4 t +12 t ki . Therefore, for t ∈ h , f − i and d ∈ h , j − √ ft +4 t +12 t ki , γ equals f − dtd − .The next lemma follows from the previous considerations. Lemma 11.
Given an even number f , S ⊕ d,γ N is a GSI-semigroupwith Frobenius number f if and only if F( S ) is an even number belong-ing to h , f − i , d is an odd number verifying d ∈ " , $ − p f F( S ) + 4F( S ) + 12F( S ) % , and γ = f − d F( S ) d − is an integer number such that gcd( γ, d ) = 1 and γ > max { d F( S ) , dM ( S ) } . We present a family formed by semigroups S of even Frobeniusnumber with F( S ) ≥
10 and such that M ( S ) ≤ F( S ). Proposition 12.
For every even number f ≥ , the numerical semi-group S f minimally generated by A = { f / − , f / , f / , . . ., f / − − , f / −
1) + 1 } has Frobenius number equal to f .Proof. Since 2( f / −
1) = f − A , ( f / −
1) + ( f / f + 1and gcd( A ) = 1, the set A is a minimal system of generators of S f and f S f .The elements f + 1 = ( f / −
1) + ( f / f + 2 = ( f / −
1) +( f / . . . , f +( f / −
4) = ( f / − f / − − f +( f / −
3) =( f / −
1) + ( f / −
1) + ( f / − f + ( f / −
2) = ( f / −
1) + ( f − f + ( f / −
1) = ( f / f / − −
1) are f / − S f . Hence F( S f ) = f .The numerical semigroups with Frobenius numbers 2, 4, 6 and 8are the following: {h , , i} , (6) {h , , i , h , , , , i} , (7) {h , , i , h , , , i , h , , , , i , h , , , , , , i} , (8)and {h , , i , h , , i , h , , , i , h , , , i , h , , , , i , h , , , , i , h , , , , i , h , , , , , i , h , , , , , , i , h , . . ., i} , (9)respectively.The semigroups of the sets (6), (7), (8) and (9) and the families ofProposition 12 are the seeds to construct the integers that are realizable s Frobenius numbers of GSI-semigroups. We propose Algorithm 2to check if there exist GSI-semigroups with a given even Frobeniusnumber. Algorithm 2:
Computation of a GSI-semigroup with even Frobeniusnumber f (if possible). Data: f an even number. Result:
If there exists, a GSI-semigroup with Frobenius number f . if f < then return ∅ . S = { S numerical semigroup | F( S ) ∈ N ∩ [2 , min { , ⌊ f − ⌋} ] } ; forall S ∈ S , d ∈ (cid:20) , (cid:22) − √ f F( S )+4F( S )+12F( S ) (cid:23)(cid:21) odd, and γ = f − d F( S ) d − ∈ N do if (cid:0) ( γ > max { d F( S ) , dM ( S ) } ) ∧ (gcd( d, γ ) = 1) (cid:1) then return S ⊕ d,γ N ; A = n t ∈ [10 , ⌊ f − ⌋ ] | t even o ; while A 6 = ∅ do t = First( A ); A = A \ { t } ; B = { d ∈ h , j − √ ft +4 t +12 t ki | d odd } ; while B 6 = ∅ do d = First( B ); B = B \ { d } ; γ = f − dtd − ; if (cid:0) ( γ ∈ N ) ∧ ( γ > dt ) ∧ (gcd( d, γ ) = 1) (cid:1) then return S t ⊕ d,γ N ; return A ;Note that several steps of Algorithm 2 can be computed in parallelway. We now illustrate it with a couple of examples. Example . Let f = 42, since ⌊ − ⌋ = 4, by Algorithm 2, onlythe numerical semigroups with Frobenius number 2 and 4 must beconsidered.If F( S ) = 2, then d ∈ { , } , since the odd numbers of the set h , ⌊ − √ ⌋ i N are 3 and 5. For d = 3, we have that γ = − · − =18, but gcd( d, γ ) = 1 so we do not obtain any GSI-semigroup withFrobenius number 42 from S with F( S ) = 2 and d = 3. For d = 5, γ = − · − = 8 > d F( S ) = 5 · S ) = 4, then d = 3, since [3 ,
3] = { } , which is odd. We obtainthat γ = − · − = 15. By (7), for F( S ) = 4, we have M ( S ) ≥
7. Inthis case 15 > max { d F( S ) , dM ( S ) } = max { · , · } = 21.Hence, there are no GSI-semigroups with Frobenius number 42. Example . Consider f = 4620. Using the code in Table 5, wecheck that there are no GSI-semigroups of the form S ⊕ d,γ N , with ( S ) ∈ { , , , } . Nevertheless, the number 4620 is realizable asthe Frobenius number of a GSI-semigroup: the Frobenius number of S ⊕ , N , S ⊕ , N and S ⊕ , N is 4620.With the code below, we also obtain other examples of Frobeniusnumbers of GSI-semigroups that cannot be constructed from semi-groups S with F( S ) ∈ { , , , } . gap> t:=30000; The new Frobenius numbers are 7980 and 26460. Some GSI-semigroupswith these Frobenius numbers are: S ⊕ , N and S ⊕ , N for7980, and S ⊕ , N and S ⊕ , N for 26460. References [1] A. Assi and P. A. Garc´ıa-S´anchez. Constructing the set of com-plete intersection numerical semigroups with a given Frobeniusnumber.
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Finitely Generated Com-mutative Monoids . Nova Science Pub Inc, 1999.[13] J. J. Sylvester, Mathematical questions with their solutions, Edu-cational Times 41(1884), 21.Evelia Rosa Garc´ıa BarrosoDepartamento de Matem´aticas, Estad´ıstica e I.O.Secci´on de Matem´aticas, Universidad de La LagunaApartado de Correos 45638200 La Laguna, Tenerife, Spaine-mail: [email protected] Ignacio Garc´ıa-Garc´ıaDepartamento de Matem´aticasUniversidad de C´adizE-11510 Puerto Real, C´adiz, Spaine-mail: [email protected] Vigneron-TenorioDepartamento de Matem´aticasUniversidad de C´adizE-11406 Jerez de la Frontera, C´adiz, Spaine-mail: [email protected] The input is a NumericalSemigroup S
Table 1: GAP code of functions to check if a numerical semigroup is SIand/or GSI. 16 istOfFrobenius:=function(fS,d,bound)local f,listF,gamma,lowerBound;listF:=[];f:=0;lowerBound:=d*fS;if(fS=2) then lowerBound:=fS*5; fi;if(fS=4) then lowerBound:=fS*7; fi;if(fS=6) then lowerBound:=fS*7; fi;if(fS=8) then lowerBound:=fS*9; fi;for gamma in [(lowerBound+1)..(bound-1)] doif(GcdInt(gamma,d)=1) thenf:=d*fS+(d-1)*gamma;if(f