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Generalized Strongly Increasing Semigroups

1
Departamento de Matemáticas, Estadística e I.O. Sección de Matemáticas, Universidad de La Laguna, Apartado de Correos 456, 38200 La Laguna, Spain
2
Department of Mathematics/INDESS (Instituto Universitario para el Desarrollo Social Sostenible), University of Cadiz, 11510 Puerto Real, Spain
3
Department of Mathematics/INDESS (Instituto Universitario para el Desarrollo Social Sostenible), University of Cadiz, 11406 Jerez de la Frontera, Spain
*
Author to whom correspondence should be addressed.
Academic Editors: Alexei Kanel-Belov and Alexei Semenov
Mathematics 2021, 9(12), 1370; https://doi.org/10.3390/math9121370
Received: 3 May 2021 / Revised: 11 June 2021 / Accepted: 11 June 2021 / Published: 13 June 2021
(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic)

Abstract

In this work, we present a new class of numerical semigroups called GSI-semigroups. We see the relations between them and other families of semigroups and we explicitly give their set of gaps. Moreover, an algorithm to obtain all the GSI-semigroups up to a given Frobenius number is provided and the realization of positive integers as Frobenius numbers of GSI-semigroups is studied.
Keywords: generalized strongly increasing semigroup; strongly increasing semigroup; Frobenius number; Apéry set; singular analytic plane curve generalized strongly increasing semigroup; strongly increasing semigroup; Frobenius number; Apéry set; singular analytic plane curve

1. Introduction

Let N = { 0 , 1 , 2 , . . . } be the set of non-negative integers. A numerical semigroup is a subset S of N closed under addition, 0 S and N \ S , its gapset, is finite. The least non-zero element in S is called the multiplicity of S, we denote it by m ( S ) . Given a non-empty subset A = { a 1 , . . . , a n } of N , we denote by A the smallest submonoid of ( N , + ) containing A; the submonoid A is equal to the set N a 1 + + 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 the embedding dimension of S. It is well known (see Lemma 2.1 from [1]) that A is a numerical semigroup if and only if gcd ( A ) = 1 . The cardinality of N \ S is called the genus of S (denoted by g ( S ) ) and its maximum is known as the Frobenius number of S (denoted by F ( S ) ).
Numerical semigroups appear in several areas of mathematics and their theory is connected with Algebraic Geometry and Commutative Algebra (see [2,3]) as well as with Integer Optimization (see [4]) and Number Theory (see [5]). There are several studies on different families of numerical semigroups, for instance symmetric semigroups, irreducible semigroups and strongly increasing semigroups (see [6,7]) or on the study or characterization of invariants, for instance the Frobenius number, the Apéry set, the set of gaps, the genus, etc. (see [1,7,8,9,10]). It is an open problem to find a formula that gives the Frobenius number and algorithms to calculate it. Let S be a numerical semigroup, admitting as minimal system of generators the coprime positive integers a 1 , . . . , a n . It is well-known after [11] that the Frobenius number F ( S ) = a 1 a 2 a 1 a 2 for n = 2 . The computation of a similar formula when n 3 is more difficult and it remains open, in fact it is a hard problem from the computational point of view (see [12]). On the other hand, given m S \ { 0 } , the Apéry set of m in S is A p ( S , m ) = { s S : s m S } . Clearly, A p ( S , m ) = { w ( 0 ) = 0 , w ( 1 ) , . . . , w ( m 1 ) } , where w ( i ) is the least element of S congruent to i modulo m, for all i { 0 , . . . , m 1 } . Moreover, the set of gaps of S is the non-negative elements of A p ( S , m ) m , hence F ( S ) = max ( A p ( S , m ) ) m . As happens for the Frobenius number, given a numerical semigroup S, it is an open problem to determine explicit formulas of the Apéry set of an element m S , which can be expressed in terms of the minimal system of generators of S and of m itself. In fact, if there were, the problem of Frobenius would be solved in particular. It is still an open problem to give an explicit formula for the particular case of numerical semigroups, which are minimally generated by three elements. In this sense, we can find a positive response to some special families, as in [13] [Theorem 2.8], where the authors determined the explicit formula for the Apéry set of the multiplicity of numerical semigroups whose minimal generators are three pairwise relatively prime numbers.
Inspired in [6,8], and with the aim to study the sets of gaps of strongly increasing semigroups (shorted by SI-semigroups), we introduce the concept of generalized strongly increasing semigroup (shorted by GSI-semigroups). These numerical semigroups S ¯ = v 0 , . . . , v h , γ are the gluing of a semigroup S with N , we denote them by S d , γ N , where S = v 0 / d , . . . , v h / d , d = gcd ( v 0 , . . . , v h ) > 1 and γ N such that γ > max { d F ( S ) , v h } .
In [14], [Theorem 12] and [15] [Theorem 9.2, p. 125], we found the description of the Apéry set of an element in the gluing of two semigroups. Given the relationship between the Apéry set of an element of a numerical semigroup and the set of gaps of such a semigroup, the mentioned results in [14,15] could be used to describe the set of gaps of a gluing semigroup, but that description would not be explicit, since the description of the Apéry set in [14,15] are not, in the sense that they do not provide a list of the elements of the Apéry set and/or an explicit formula based on the minimal system of generators of the gluing semigroup, as it is done for example in [13] [Theorem 2.8].
Our main result is Theorem 2, where we describe the set of gaps of GSI-semigroups listing all its elements as explicit formulae based on the minimal system of generators of the semigroup. As a consequence, we determine the explicit list of elements of the Apéry set of the multiplicity of GSI-semigroups (see Corollary 4), following the spirit of [13] [Theorem 2.8] rather than that of [14] [Theorem 12] or [15] [Theorem 9.2, p. 125]. Since every SI-semigroup is a GSI-semigroup, our description of the gaps is also valid for SI-semigroups. Semigroups of values associated with plane branches are always SI-semigroups, and their sets of gaps describe topological invariants of the curves (see [6]), which are used to classify singular analytic plane curves. Due to the fact that the condition for being a GSI-semigroup is straightforward to check from a given system of generators, it is easy to construct subfamilies of GSI-semigroups and thus of SI-semigroups.
In this work we also compare the class of GSI-semigroups with other families of numerical semigroups obtained as gluing of numerical semigroups. These are the classes of telescopic, free and complete intersection numerical semigroups. In [8], the set of complete intersection numerical semigroups with given Frobenius number is constructed, and some special subfamilies as free and telescopic numerical semigroups, and numerical semigroups associated with irreducible singular plane curves are studied. As we pointed above, we prove that SI-semigroups are always GSI-semigroups. We also prove that GSI-semigroups are not included in the other three above-mentioned families.
Some algorithms for computing GSI-semigroups are provided in this work. One of them computes the set of GSI-semigroups up to a fixed Frobenius number. We prove that for any odd number f, there is at least one GSI-semigroup with Frobenius f, while this is not always the case for an even number f. Thus, GSI-semigroups with even Frobenius numbers are also studied, and we present an algorithm to check for a GSI-semigroup with a given even Frobenius number.
This work is organized as follows. In Section 2, we introduce the GSI-semigroups and some of their properties. We prove that SI-semigroups are GSI-semigroups (see Corollary 1). We also compare GSI-semigroups with other families such as free, telescopic and complete intersection numerical semigroups. In Section 3, the main result of this paper is presented (Theorem 2). This theorem gives us an explicit formula for the set of gaps of GSI-semigroups. In Section 4, we give an algorithm for computing the set of GSI-semigroups up to a fixed Frobenius number, and we show some properties of Frobenius numbers of GSI-semigroups. We also provide an algorithm to test whether there is a GSI-semigroup with given even Frobenius number. In the last section, Section 5, some conclusions and possible future research are set out. Appendix A presents the code of the functions in GAP using for computing the examples in this work.

2. Generalized Strongly Increasing Semigroups

In this section, we introduce the family of GSI-semigroups and give some of their properties. We also compare this new family with other families of numerical semigroups already known.
The gluing of S = v 0 , . . . , v h and N with respect to d and γ , with gcd ( d , γ ) = 1 and d , γ > 1 (see [15] [Chapter 8]), is the numerical semigroup N d v 0 + + N d v h + N γ . We denote it by S d , γ N . In general, given S 1 and S 2 , two numerical semigroups minimally generated by { v 0 , . . . , v r 1 } and { v r , . . . , v t } , respectively, f S 1 \ { v 0 , . . . , v r 1 } and d S 2 \ { v r , . . . , v t } such that gcd ( f , d ) = 1 , the numerical semigroup minimally generated by { d v 0 , . . . , d v r 1 , f v r , . . . , f v t } is called a gluing of S 1 and S 2 .
Definition 1.
A numerical semigroup S ¯ is a generalized strongly increasing semigroup whenever S ¯ is the gluing of a numerical semigroup S = v 0 , , v h with respect to d and γ (that is, S ¯ = S d , γ N ), where d N \ { 0 , 1 } and γ N with γ > max { d F ( S ) , d max { v 0 , . . . , v h } } (note that d and γ are coprimes).
Since the gluing does not depend on the order of the generators of the semigroup, the definition of GSI-semigroup is also independent of this order.
The first example of GSI-semigroups are numerical semigroups generated by two positive integers S ¯ = a , b with a < b . For these semigroups, set S = N = 1 , d = a , and γ = b . Since F ( S ) = 1 , γ = b > max { d F ( S ) , d } = { a · ( 1 ) , a } = a . By a result of Sylvester (see [11]), we know that the Frobenius numbers of these semigroups are given by the formula a · b a b . Hence, every odd natural number is realizable as a Frobenius number of a GSI-semigroup.
Our definition of GSI-semigroup is motivated by the already established notion of a SI-semigroup. We remind you how they are defined (see [16] for further details).
A sequence of positive integers ( v 0 , . . . , v h ) is called a characteristic sequence if it satisfies the following two properties:
(CS1)
Let e k = gcd ( v 0 , . . . , v k ) for 0 k h . Then e k < e k 1 for 1 k h and e h = 1 .
(CS2)
e k 1 v k < e k v k + 1 for 1 k h 1 .
We put n k = e k 1 e k for 1 k h . Therefore, n k > 1 for 1 k h and n h = e h 1 . If h = 0 , the only characteristic sequence is ( v 0 ) = ( 1 ) . If h = 1 , the sequence ( v 0 , v 1 ) is a characteristic sequence if and only if gcd ( v 0 , v 1 ) = 1 . Property (CS2) plays a role if and only if h 2 .
Lemma 1.
(Ref. [16] [Lemma 1.1]) Let ( v 0 , , v h ) , h 2 be a characteristic sequence. Then,
(i)
v 1 < < v h and v 0 < v 2 .
(ii)
Let v 1 < v 0 . If v 0 0 (mod v 1 ) then ( v 1 , v 0 , v 2 , . . . , v h ) is a characteristic sequence. If v 0 0 (mod v 1 ) then ( v 1 , v 2 , . . . , v h ) is a characteristic sequence.
We denote by v 0 , . . . , v h the semigroup generated by the characteristic sequence ( v 0 , . . . , v h ) . Observe that v 0 , , v h is a numerical semigroup. A semigroup S N is Strongly Increasing (SI-semigroup) if S { 0 } and it is generated by a characteristic sequence. Note that by Lemma 1, we can assume that v 0 < < v h .
Theorem 1.
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 conditions holds:
1.
h = 1 , S ¯ = N d , γ N = d , γ , where d and γ are two coprime integers.
2.
h > 1 , S ¯ = S d , γ N , where S = v 0 , . . . , v h 1 is a strongly increasing semigroup with embedding dimension h and γ , d > 1 are two coprime integer numbers such that γ > d gcd ( v 0 , . . . , v h 2 ) v h 1 .
Proof. 
The case h = 1 is trivial by definition of characteristic sequences.
Assume h > 1 and that S ¯ = v ¯ 0 , , v ¯ h is a strongly increasing numerical semigroup with embedding dimension strictly greater than 2 . Let e ¯ i = gcd ( v ¯ 0 , , v ¯ i ) for 0 i h . Put v i = v ¯ i e ¯ h 1 for 0 i h 1 . Then, ( v 0 , . . . , v h 1 ) is a characteristic sequence. Let S = v 0 , . . . , v h 1 . Since e ( S ¯ ) = h + 1 , then e ( S ) = h . Set γ = v ¯ h and d = e ¯ h 1 , we get S ¯ = S d , γ N . We have that γ = v ¯ h > v ¯ h 1 = d v h 1 , and since S ¯ is a SI-semigroup,
γ = v ¯ h > e ¯ h 2 e ¯ h 1 v ¯ h 1 = gcd ( v ¯ 0 , , v ¯ h 2 ) e ¯ h 1 v ¯ h 1 = gcd ( e ¯ h 1 v 0 , , e ¯ h 1 v h 2 ) e ¯ h 1 e ¯ h 1 v h 1 = e ¯ h 1 gcd ( v 0 , . . . , v h 2 ) v h 1 = d gcd ( v 0 , . . . , v h 2 ) v h 1 .
Conversely, let S = v 0 , . . . , v h 1 be a strongly increasing semigroup with embedding dimension h , and γ , d > 1 be two coprime integer numbers such that γ > d gcd ( v 0 , , v h 2 ) v h 1 . Denote e i = gcd ( v 0 , , v i ) for i = 0 , . . . , h 1 . Take S ¯ = v ¯ 0 , , v ¯ h = γ the gluing semigroup S d , γ N and define e ¯ i = gcd ( v ¯ 0 , , v ¯ i ) for i = 0 , . . . , h . We have that e ¯ 0 = d e 0 > > e ¯ h 1 = d e h 1 = d > e ¯ h = gcd ( γ , d ) = 1 . Since e i 1 v i < e i v i + 1 , for 1 i h 1 then e i 1 v i d 2 < e i v i + 1 d 2 and therefore e ¯ i 1 v ¯ i < e ¯ i v ¯ i + 1 . By hypothesis d e h 2 v h 1 < γ , hence d 2 e h 2 v h 1 < d γ and therefore e ¯ h 2 v ¯ h 1 < e ¯ h 1 γ . We conclude that S ¯ is a SI-semigroup. □
The following result gives us a formula for the conductor (the Frobenius number plus 1) of a SI-semigroup.
Proposition 1.
(Ref. [6] [Proposition 2.3 (4)], [16] [Proposition 1.2]) Let S = v 0 , , v h be the semigroup generated by the characteristic sequence ( v 0 , , v h ) . The conductor of the semigroup S is
c ( S ) = i = 1 h ( n i 1 ) v i v 0 + 1 .
Moreover, the conductor of S is an even number and the genus of S is g ( S ) = c ( S ) 2 .
By Proposition 1, we get
F ( S ) = i = 1 h ( n i 1 ) v i v 0 = i = 1 h n i v i i = 1 h v i v 0 = ( n 1 v 1 v 2 ) + ( n 2 v 2 v 3 ) + + ( n h 1 v h 1 v h ) + n h v h v 1 v 0 ( h 1 ) + n h v h v 1 v 0 = e h 1 v h v 0 v 1 h + 1 < e h 1 v h v 0 v 1 .
Assume S ¯ = v ¯ 0 , . . . , v ¯ h is a SI-semigroup satisfying that v ¯ 0 < . . . < v ¯ h = γ . Set d = e ¯ h 1 = gcd ( v ¯ 0 , . . . , v ¯ h 1 ) , γ = v ¯ h and S = v ¯ 0 / d , . . . , v ¯ h 1 / d . We have that d γ = e ¯ h 1 v ¯ h > e ¯ h 1 v ¯ h v ¯ 0 v ¯ 1 > F ( S ¯ ) = i = 1 h ( n ¯ i 1 ) v ¯ i v ¯ 0 = d F ( S ) + ( e ¯ h 1 1 ) v ¯ h = d F ( S ) + ( d 1 ) γ . Thus, γ > d F ( S ) . Since S ¯ is a SI-semigroup, the property (CS2) is fulfilled. Using the ordered generators, we obtain v ¯ h < e ¯ h 1 v ¯ h = d v ¯ h < γ .
So we can state the following result.
Corollary 1.
Every SI-semigroup is a GSI-semigroup.
There are some semigroups with similar definitions to SI and GSI semigroups; for example, telescopic, free and complete intersection semigroups. All of them are defined using the gluings of different types of numerical semigroups.
Let S be a numerical semigroup and let { v 0 , . . . , v h } be its minimal set of generators. Put e 0 = v 0 and for k { 1 , . . . , h } , set e k = gcd ( v 0 , . . . , v k ) and n k = e k 1 e k . The semigroup S is free for the arrangement ( v 0 , . . . , v h ) if for all k { 1 , . . . , h } , n k > 1 and n k v k v 0 , . . . , v k 1 (see [5] [page 30]). In terms of gluings, a semigroup S is free whenever it is equal to N or it is the gluing of a free semigroup with N (see [15] [Chapter 8]). The semigroup S is telescopic if it is free for the rearrangement v 0 < < v h . A semigroup S is complete intersection if its associated algebra is complete intersection (see [17]). In terms of gluings, a semigroup is complete intersection if it is equal to N or it is the gluing of two complete intersection numerical semigroups (see [18] [Proposition 9]).
It is well known that SI-semigroups are telescopic, telescopic are free semigroups and free semigroups are complete intersection (see [19] [Figure 1]). In general, GSI-semigroups are neither strongly increasing nor telescopic nor free nor complete intersection. Clearly, 6 , 14 , 22 , 23 = 3 , 7 , 11 2 , 23 N and 23 > max { 2 F ( 3 , 7 , 11 ) , 2 · 11 } . Thus, this is a GSI-semigroup. We define the functions IsSIncreasingNumericalSemigroup and IsGSI to check if a numerical semigroup is a SI-semigroup and a GSI-semigroup, respectively, (the code of these functions is showed in Appendix A).
Applying our functions and the functions IsFreeNumericalSemigroup, IsTelescopicNumericalSemigroup and IsCompleteIntersection of [20] to the semigroup 6 , 14 , 22 , 23 , we obtain the following outputs:
  • gap>   IsFreeNumericalSemigroup (
  •                  NumericalSemigroup (6,14,22,23));
  • false
  • gap>   IsTelescopicNumericalSemigroup (
  •                  NumericalSemigroup (6,14,22,23));
  • false
  • gap>   IsCompleteIntersection (
  •                  NumericalSemigroup (6,14,22,23));
  • false
  • gap>   IsSIncreasing (NumericalSemigroup (6,14,22,23));
  • false
  • gap>   IsGSI(NumericalSemigroup (6,14,22,23));
  • true
From the results of the above computations, we conclude that the class of GSI-semigroups contains the class of SI-semigroups, but it is different to the classes of free, telescopic and complete intersection semigroups.

3. Set of Gaps of a GSI-Semigroup

In this section, we state and prove the main theorem of this article. We have seen that GSI-semigroups are easy to obtain from any numerical semigroup just by gluing it with N with appropriate elements d and γ . Hence, these semigroups form a large family within the set of numerical semigroups. Now, we deepen their study by explicitly determining their set of gaps.
Hereafter, the notation [ a mod n ] for an integer a and a natural number n means the remainder of the division of a by n, and [ a ] n denotes the co-set 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 real interval [ a , b ] . Let a be the integral part of the real number a.
Theorem 2.
Let S = v 0 , , v h be a numerical semigroup with v 0 < < v h , d 2 and v h + 1 as two natural co-prime numbers such that v h + 1 > max { d F ( S ) , d v h } . Then, the gaps of the GSI-semigroup S ¯ = S d , v h + 1 N are
N \ S ¯ = 1 , , d v 0 1 x ( d v 0 , v h + 1 ) N : x d S A d = 1 d 2 B d ,
where
B d , = v h + 1 + [ v h + 1 mod d ] + k d : 0 k v h + 1 d 1
and
A d = k = 1 d 1 d ( N \ S ) + k v h + 1 ( A d = when S = N ) .
Moreover, the sets on the right-hand side of (1) are pairwise disjoint.
Proof .
Let
H : = { 1 , . . . , d v 0 1 } { x ( d v 0 , v h + 1 ) N x d S } A d = 1 d 2 B d , .
First, we will prove the inclusion H N \ S ¯ . It is clear that { 1 , . . . , d v 0 1 } is included in N \ S ¯ .
Consider x ( d v 0 , v h + 1 ) N such that x d S , and suppose that x S ¯ . Since x < v h + 1 , then there are λ i with 0 i h such that x = λ 0 d v 0 + + λ h d v h = d ( λ 0 v 0 + + λ h v h ) d S , which is a contradiction. Hence, we conclude that { x ( d v 0 , v h + 1 ) N : x d S } N \ S ¯ .
Suppose that S N . Fix 1 k d 1 and let x d ( N \ S ) + k v h + 1 . We get x = d α + k v h + 1 , for some α N \ S . Suppose that x S ¯ . So, there exist α 1 , . . . , α h , β N such that x = d α + k v h + 1 = d α 1 v 1 + + d α h v h + β v h + 1 and then ( k β ) v h + 1 = d ( α 1 v 1 + + α h v h α ) . If k = β , the element α has to belong to S, which is not possible. Moreover, since d and v h + 1 are co-prime, d divides k β when k β 0 . If β > k , d α = d ( α 1 v 1 + + α h v h ) + ( β k ) v h + 1 with v h + 1 S , that is, α S . Again, it is not possible. We may therefore assume k > β , then k β d and k d . In any case, the set d ( N \ S ) + k v h + 1 is included in N \ S ¯ for any integer k in [ 1 , d 1 ] N .
Let us prove now that B d , N \ S ¯ . Suppose that x = v h + 1 + [ v h + 1 mod d ] + k d S ¯ for some 1 d 2 and 0 k v h + 1 d 1 . Let α 0 , α 1 , . . . , α h + 1 N such that x = v h + 1 + [ v h + 1 mod d ] + k d = α 0 d v 0 + + α h d v h + α h + 1 v h + 1 . Hence, ( α h + 1 1 ) v h + 1 [ v h + 1 mod d ] = d ( k α 0 v 0 α h v h ) and [ ( α h + 1 1 ) v h + 1 ] d = [ 0 ] d . Since d and v h + 1 are co-prime, then d divides α h + 1 1 . However, max B d , = ( + 1 ) v h + 1 d so we have α h + 1 { 0 , 1 , , } , hence 1 α h + 1 1 1 or equivalently 1 + α h + 1 + 1 + 1 and α h + 1 + 1 + is a multiple of d, which is a contradiction since < d 1 .
Taking into account the reasoning done so far we have
H = { 1 , . . . , d v 0 1 } { x ( d v 0 , v h + 1 ) N x d S } A d = 1 d 2 B d , N \ S ¯ .
Now, let us prove that the sets on the right-hand side of (1) are pairwise disjoint.
When A d is a non-empty set, let A d , k = d ( N \ S ) + k v h + 1 for any fixed 1 k d 1 . In this case, if B d , is non-empty we have
max B d , < min A d , + 1 for 1 d 2 .
Observe that
[ x ] d = [ k v h + 1 ] d for any x A d , k
and
[ y ] d = [ ( + 1 ) v h + 1 ] d for any y B d , .
Since 1 k , < d we get that any two sets A d , k and A d , k are disjoint for k k and any two sets B d , and B d , are also disjoint for . Moreover, A d and B d , are also disjoint for any 1 d 2 . Indeed, let x A d B d , for some 1 d 2 . Hence, there is k { 1 , . . . , d 1 } such that x A d , k and by (3) and (4), [ x ] d = [ k v h + 1 ] d = [ ( + 1 ) v h + 1 ] d . Given that d and v h + 1 are co-prime and 1 k , < d , we get k = + 1 . So, x A d , + 1 B d , , which is a contradiction by inequality (2).
In order to finish the proof, we will show that there is no gap of S ¯ outside H .
First at all, observe that if x N \ S ¯ and x < v h + 1 , x { 1 , , d v 0 1 } { x ( d v 0 , v h + 1 ) N x d S } .
Claim 1: if x N \ S ¯ and v h + 1 < x , then [ x ] d = [ k v h + 1 ] d , for some k { 1 , . . . , d 1 } .
Indeed, if we suppose that x = λ d for some λ N , by hypothesis we get d F ( S ) < v h + 1 < x = λ d , in particular λ > F ( S ) , so x d S S ¯ . Since [ x ] d [ 0 ] d and gcd ( d , v h + 1 ) = 1 we get [ x ] d { [ 1 ] d , , [ d 1 ] d } = { [ k v h + 1 ] d : 1 k d 1 } , that is, any x N \ S ¯ with v h + 1 < x is congruent with k v h + 1 modulo d for some integer k { 1 , . . . , d 1 } .
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 k v h + 1 modulo d is max A d , k , for 1 k d 1 .
Let x N \ S ¯ with [ x ] d = [ k v h + 1 ] d and x > max A d , k , then x = d F ( S ) + k v h + 1 + λ d for some non-zero natural number λ . So, x = d ( F ( S ) + λ ) + k v h + 1 S ¯ since F ( S ) + λ S .
Claim 3: There are no gaps of S ¯ congruent with ( + 1 ) v h + 1 modulo d between max B d , and min A d , + 1 .
Remember that [ max B d , ] d = [ min A d , + 1 ] d = [ ( + 1 ) v h + 1 ] d . Suppose that x N \ S ¯ with max B d , < x < min A d , + 1 and [ x ] d = [ ( + 1 ) v h + 1 ] d . Since max B d , = ( + 1 ) v h + 1 d and min A d , + 1 = ( + 1 ) v h + 1 + d , the only possibility for x is ( + 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 0 { 1 , , d 1 } such that [ x ] d = [ k 0 v h + 1 ] d and v h + 1 < x max A d , k 0 . In particular, there is an integer λ such that x = k 0 v h + 1 + λ d . Hence, if x [ min A d , k 0 , max A d , k 0 ] N then x A d , k 0 . Indeed, in this case λ N and λ S , otherwise x S ¯ .
By Claim 3, we can conclude that if v h + 1 < x < min A d , k 0 then v h + 1 < x max B d , k 0 1 = k 0 v h + 1 d .
Claim 4: The set of all the integers in ( v h + 1 , max B d , k 0 1 ] congruent with k 0 v h + 1 modulo d is B d , k 0 1 .
By (4) we have [ max B d , k 0 1 ] d = [ k 0 v h + 1 ] d . Moreover
{ max B d , k 0 1 , max B d , k 0 1 d , max B d , k 0 1 2 d , . . . , min B d , k 0 1 } = B d , k 0 1
and min B d , k 0 1 d = v h + 1 + [ ( k 0 1 ) v h + 1 mod d ] d < v h + 1 .
Hence, x has to belong to B d , k 0 1 and we are done with the proof for the case A d .
Suppose now that A d = , that is S = N and S ¯ is generated by d and v 1 ( h = 0 ).
Claim 5: If A d = , then max B d , is the greatest gap of S ¯ , which is congruent with ( + 1 ) v h + 1 modulo d, for 1 d 2 .
Observe that max B d , = ( + 1 ) v 1 d . For any natural number x > max B d , with [ x ] d = [ ( + 1 ) v 1 ] d , there is α N \ { 0 } such that x = ( + 1 ) v 1 d + α d = ( + 1 ) v 1 + ( α 1 ) d S ¯ . □
The above result provides us an explicit formula for the gaps, except the elements of A d . We now give some examples of GSI-semigroups where the set A d is easily known.
Example 1.
Let S = 2 , 7 . We have N \ S = { 1 , 3 , 5 } and F ( S ) = 5 . Take now d = 2 and γ = 15 . Since γ > max { 2 · 5 , 2 · 7 } , the semigroup S 2 , 15 N is a GSI-semigroup. The set A 2 is equal to 2 { 1 , 3 , 5 } + 1 · 15 = { 17 , 21 , 25 } and therefore F ( S 2 , 15 N ) = 25 .
Example 2.
Consider now the semigroup S = 5 , 6 , 7 , 8 , 9 , d = 3 and γ = 31 . We have N \ S = { 1 , 2 , 3 , 4 } and F ( S ) = 4 . Since γ > max { 3 · 4 , 3 · 9 } , the semigroup S 3 , 31 N is a GSI-semigroup and A 3 = ( 3 { 1 , 2 , 3 , 4 } + 1 · 31 ) ( 3 { 1 , 2 , 3 , 4 } + 2 · 31 ) = { 34 , 37 , 40 , 43 , 65 , 68 , 71 , 74 } . Thus, F ( S 3 , 31 N ) = 74 .
Corollary 2.
Let S = v 0 , , v h be a semigroup, and d , v h + 1 N two natural numbers such that S ¯ = d v 0 , . . . , d v h , v h + 1 is a GSI-semigroup. Then,
F ( S ¯ ) = max A d if A d max B d , d 2 otherwise ,
where A d and B d , d 2 are from (1).
Proof. 
If A d , then, by inequality (2), F ( S ¯ ) = max A d = d F ( S ) + ( d 1 ) v h + 1 . Otherwise, S = N and S ¯ is generated by d and v 1 ( h = 0 ). So, F ( S ¯ ) = ( d 1 ) ( v 1 1 ) 1 = max B d , d 2 .  □
Corollary 3.
The Frobenius number of a GSI-semigroup S d , γ N is
F ( S d , γ N ) = d F ( S ) + ( d 1 ) γ .
Proof. 
It is a consequence of the proof of Theorem 2. □
From the proof of Theorem 2, we obtain the Frobenius number of a GSI-semigroup S d , γ N , which is equal to
F ( S d , γ N ) = d F ( S ) + ( d 1 ) γ .
Corollary 4.
Let S = v 0 , , v h be a numerical semigroup with v 0 < < v h and d 2 and v h + 1 be two natural coprime numbers such that v h + 1 > max { d F ( S ) , d v h } . Then the Apéry set of v 0 in the GSI-semigroup S ¯ = S d , v h + 1 N is
A p ( S ¯ , v 0 ) = ( ( P + v 0 ) { 0 } ) S ¯ ,
where P is the partition (1).
Proof .
It is a consequence of the equality
A p ( T , b ) = T ( [ 0 , b ) ( ( N \ T ) + b ) ) ,
for any numerical semigroup T and any b T . □

4. Algorithms for GSI-Semigroups

In this section, we propose some algorithms for computing GSI-semigroups. These algorithms focus on computing the GSI-semigroups up to a given Frobenius 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, in this section we dedicate a special study to GSI-semigroups with even Frobenius number.
Algorithm 1 computes the set of GSI-semigroups with Frobenius number less than or equal to a fixed non-negative integer. Note that in step 5 of the algorithm, we use F ( S ¯ ) = d F ( S ) + ( d 1 ) γ and γ > d F ( S ) , which implies that F ( S ¯ ) d 2 F ( S ) where S ¯ = S d , γ N .
Denote by M ( S ) the largest element of the minimal system of generators of a numerical semigroup S.
Algorithm 1: Computation of the set of GSI-semigroups with Frobenius number
less than or equal to f.
Mathematics 09 01370 i001
Remark 1.
If A is a minimal system of generators of a numerical semigroup S and d N \ { 0 , 1 } , then d A is a minimal system of generators of d S = { d s s S } d N . Furthermore, if γ N \ { 1 } and gcd ( d , γ ) = 1 , then γ d N \ { 0 } . Thus, γ d S and d A { γ } is a minimal system of generators of d A { γ } .
We give in Table 1 all the GSI-semigroups with Frobenius number less than or equal to 15.
Remember that every numerical semigroup generated by two elements is a GSI-semigroup. Hence, for any odd natural number there exists at least one GSI-semigroup with such Frobenius number.
From Table 1, one might think that there are no GSI-semigroups with even Frobenius number. This is not so, and we can check that 9 , 12 , 15 , 16 = 3 , 4 , 5 3 , 16 N is a GSI-semigroup and its Frobenius number is 38,
  • gap>   FrobeniusNumber (NumericalSemigroup (9, 12, 15, 16));
  • 38
  • gap>   IsGSI(NumericalSemigroup (9, 12, 15, 16));
  • true
This is the first even integer that is realizable as the Frobenius number of a GSI-semigroup. We explain this fact: we want to obtain an even number f from the Formula (6), f = F ( S d , γ N ) = d F ( S ) + ( d 1 ) γ . Since gcd ( d , γ ) = 1 , then d has to be odd and F ( S ) even. Thus, the lowest number f is obtained for the numerical semigroup S with the smallest even Frobenius number, the smallest odd number d 3 and the smaller feasible integer γ , that is, S = 3 , 4 , 5 , d = 3 and γ = 16 . Thus, the GSI-semigroup with the minimum even Frobenius number is 3 , 4 , 5 3 , 16 N .
Note that not every even number is obtained as the Frobenius number of a GSI-semigroup 3 , 4 , 5 3 , γ N for some γ 16 with gcd ( d , γ ) = 1 . In this way, we only obtain the values of the form 36 + 2 k with k N and k 0 mod 3 (if γ = 16 + k for k N , F ( 3 , 4 , 5 3 , γ N ) = 38 + 2 k ). The numbers of the form 42 + 6 k , with k N , are not obtained (see Table 2).
We now look for GSI-semigroups with Frobenius number of the form 42 + 6 k . Reasoning as above, we use again the semigroup 3 , 4 , 5 and set now d = 5 . In this case, the smallest Frobenius number is 114, and it is given by the semigroup 3 , 4 , 5 5 , 26 N . In general, for the semigroups 3 , 4 , 5 5 , γ N , the formula of their Frobenius numbers is 5 · 2 + 4 γ with γ 26 and γ 0 mod 5 (see Table 3). For S = 3 , 4 , 5 , we fill all the even Frobenius number f 114 , excepting if f is of the form f = 10 + 4 k with k = 15 k . That is, f cannot be a number of the form f = 10 + 60 k with k N \ { 0 , 1 } , for instance 130 and 190.
The above procedures are useful to construct GSI-semigroups with even Frobenius numbers, but with them we cannot determine if a given even positive integer is realizable as the Frobenius number of a GSI-semigroup.
Given an even number f, we are interested in providing an algorithm to check if there exists at least one GSI-semigroup S d , γ N such that F ( S d , γ N ) = f .
As γ has to be greater than or equal to d F ( S ) + 1 3 F ( S ) + 1 (recall that γ > max { d F ( S ) , d M ( S ) } and d 3 ) and formula (6), we obtain that if F ( S d , γ N ) = f , then 2 F ( S ) f 2 9 .
Let t 2 , f 2 9 be the Frobenius number of S. Hence, f = d t + ( d 1 ) γ d 2 t + d 1 and d 3 , 1 + 4 f t + 4 t + 1 2 t .
The next lemma follows from the previous considerations.
Lemma 2.
Given an even number f, S d , γ N is a GSI-semigroup with Frobenius number f if and only if F ( S ) is an even number belonging to 2 , f 2 9 , d is an odd number verifying
d 3 , 1 + 4 f F ( S ) + 4 F ( S ) + 1 2 F ( S ) ,
and γ = f d F ( S ) d 1 is an integer number such that gcd ( γ , d ) = 1 and γ > max { d F ( S ) , d M ( S ) } .
We present a family formed by semigroups S of even Frobenius number with F ( S ) 10 and such that M ( S ) F ( S ) .
Proposition 2.
For every even number f 10 , the numerical semigroup S f minimally generated by A = { f / 2 1 , f 1 } ( [ f / 2 + 2 , f 3 ] N ) has a Frobenius number that equals f.
Proof. 
Since 2 ( f / 2 1 ) = f 2 A , ( f / 2 1 ) + ( f / 2 + 2 ) = f + 1 and gcd ( A ) = 1 , the set A is a minimal system of generators of S f and f S f .
The elements f + 1 = ( f / 2 1 ) + ( f / 2 + 2 ) , f + 2 = ( f / 2 1 ) + ( f / 2 + 3 ) , . . . , f + ( f / 2 4 ) = ( f / 2 1 ) + ( f 3 ) , f + ( f / 2 3 ) = ( f / 2 1 ) + ( f / 2 1 ) + ( f / 2 1 ) , f + ( f / 2 2 ) = ( f / 2 1 ) + ( f 1 ) , f + ( f / 2 1 ) = ( f / 2 + 2 ) + ( f 3 ) are f / 2 1 consecutive elements in S f . Hence, F ( S f ) = f . □
The numerical semigroups with Frobenius numbers 2, 4, 6 and 8 are the following:
{ 3 , 4 , 5 } ,
{ 3 , 5 , 7 , 5 , 6 , 7 , 8 , 9 } ,
{ 4 , 5 , 7 , 4 , 7 , 9 , 10 , 5 , 7 , 8 , 9 , 11 , 7 , 8 , 9 , 10 , 11 , 12 , 13 } ,
and
{ 3 , 7 , 11 , 3 , 10 , 11 , 5 , 6 , 7 , 9 , 5 , 6 , 9 , 13 , 5 , 7 , 9 , 11 , 13 , 5 , 9 , 11 , 12 , 13 , 6 , 7 , 9 , 10 , 11 , 6 , 9 , 10 , 11 , 13 , 14 , 7 , 9 , 10 , 11 , 12 , 13 , 15 , 9 , . . . , 17 } ,
respectively.
The semigroups of the sets (7)–(10) and the families of Proposition 2 are the seeds to determine the even natural numbers that are realizable as Frobenius numbers of GSI-semigroups. More precisely, fixed with an even natural number f, these seeds allow us to check if there exist GSI-semigroups with Frobenius number f, and in this case to construct one of them. This is done with Algorithm 2.
Note that several steps of Algorithm 2 can be computed in a parallel way. We now illustrate it with a couple of examples.
Example 3.
Let f = 42 , since 42 2 9 = 4 , by Algorithm 2, only the numerical semigroups with Frobenius number 2 and 4 must be considered.
If F ( S ) = 2 , then d { 3 , 5 } , since the odd numbers of the set 3 , 1 + 505 4 N are 3 and 5. For d = 3 , we have that γ = 42 3 · 2 3 1 = 18 , but gcd ( d , γ ) = 1 so we do not obtain any GSI-semigroup with Frobenius number 42 from S with F ( S ) = 2 and d = 3 . For d = 5 , γ = 42 5 · 2 5 1 = 8 8 = d F ( S ) = 5 · 2 = 10 , obtaining again no GSI-semigroups.
If F ( S ) = 4 , then d = 3 , since [ 3 , 3 ] = { 3 } , which is odd. We obtain that γ = 42 3 · 4 3 1 = 15 . By (8), for F ( S ) = 4 , we have M ( S ) 7 . In this case, 15 max { d F ( S ) , d M ( S ) } = max { 3 · 4 , 3 · 7 } = 21 .
Hence, there are no GSI-semigroups with Frobenius number 42.
Example 4.
Consider f = 4620 . Using the code in Appendix A, we check that there are no GSI-semigroups of the form S d , γ N , with F ( S ) { 2 , 4 , 6 , 8 } . Nevertheless, the number 4620 is realizable as the Frobenius number of a GSI-semigroup: the Frobenius number of S 12 13 , 372 N , S 12 17 , 276 N and S 12 19 , 244 N is 4620.
With the code below, we also obtain other examples of Frobenius numbers of GSI-semigroups that cannot be constructed from semigroups S with F ( S ) { 2 , 4 , 6 , 8 } .
  • gap> t:=30000; # Bound of the Frobenius numbers.
  • gap> s1:=Difference([2..(t-2)],
  •        Union(ListOfFrobeniusD(2,t/2,t),
  •          Union(ListOfFrobeniusD(4,t/2,t),
  •            Union(ListOfFrobeniusD(6,t/2,t),
  •              ListOfFrobeniusD(8,t/2,t))))
  •        );
  • gap> s2:=ListOfFrobeniusD(12,t/2,t);
  • gap> Print(Intersection(s1,s2));
  • [ 4620, 7980, 26460 ]
The new Frobenius numbers are 7980 and 26460. Some GSI-semigroups with these Frobenius numbers are: S 12 13 , 652 N and S 12 17 , 486 N for 7980, and S 12 13 , 2192 N and S 12 17 , 1641 N for 26460.
Algorithm 2: Computation of a GSI-semigroup with even Frobenius number f
(if possible).
Mathematics 09 01370 i002

5. Conclusions

The present paper introduces a new family of numerical semigroups called generalized strongly increasing semigroups, using the technique of gluing of numerical semigroups. Our definition of GSI-semigroup is motivated by the already established notion of SI-semigroups, which are the semigroups associated with singular plane branches, and that it is indeed a generalization: every SI-semigroup is a GSI-semigroup. The SI-semigroups determine the topological classification of singular plane branches, and the knowledge of their gaps allows us to study the analytical classification of these curves.
Our main result is an accurate description of the set of gaps of a GSI-semigroup. As an application of this result, some algorithms for finding GSI-semigroups with certain Frobenius numbers are formulated.
For future research, it would be interesting to deepen in the study of this new family of semigroups, determining new properties and invariants, and the geometric varieties associated with.

Author Contributions

Conceptualization, E.R.G.B., J.I.G.-G. and A.V.-T.; methodology, E.R.G.B., J.I.G.-G. and A.V.-T.; software, E.R.G.B., J.I.G.-G. and A.V.-T.; validation, E.R.G.B., J.I.G.-G. and A.V.-T.; investigation, E.R.G.B., J.I.G.-G. and A.V.-T.; resources, E.R.G.B., J.I.G.-G. and A.V.-T.; writing—original draft preparation, E.R.G.B., J.I.G.-G. and A.V.-T.; writing—review and editing, E.R.G.B., J.I.G.-G. and A.V.-T.; project administration, E.R.G.B., J.I.G.-G. and A.V.-T.; funding acquisition, E.R.G.B., J.I.G.-G. and A.V.-T. All authors have read and agreed to the published version of the manuscript.

Funding

The first-named author was partially supported by the Spanish Projects MTM2016-80659-P and PID2019-105896GB-I00. The second and third-named authors were partially supported by the Spanish Project MTM2017-84890-P and Junta de Andalucía groups FQM-343 and FQM-366.

Acknowledgments

The authors thank the referees for their helpful observations.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A. GAP Code

  • #   The   inputs   is   a   NumericalSemigroup   S
  • #   The   function   returns   true   if   S   is   a   SI-semigroup
  • IsStronglyIncreasing:= function (S)
  •       local   k,    lEs,   i,   smg;
  •       smg:= MinimalGeneratingSystemOfNumericalSemigroup (S);
  •       for   i   in   [2..Length(smg)]   do
  •               if (Gcd(smg{[1..i-1]})<=Gcd(smg{[1..i]}))   then
  •                              return   false;
  •               fi;
  •       od;
  •       for   k   in   [1..(Length(smg)-2)]   do
  •               if   (Gcd(smg{[1..k]})∗smg[k+1]>=
  •                                      Gcd(smg{[1..(k+1)]})∗smg[k+2])   then
  •                              return   false;
  •              fi;
  •       od;
  •       return   true;
  • end;
  • #   The   input   is   a   NumericalSemigroup   S
  • #   The   function   returns   true   if   S is   a   GSI-semigroup
  • IsGeneralizedStronglyIncreasing:= function (S)
  •        local smg, d, gamma, aux, S1, fn1;
  •        smg:=MinimalGeneratingSystemOfNumericalSemigroup (S);
  •        gamma:=smg[Length(smg)];
  •        d:=Gcd(smg{[1..Length(smg)-1]});
  •        aux:=(1/d)∗smg{[1..Length(smg)-1]};
  •        S1:=NumericalSemigroup(aux);
  •        fn1:=FrobeniusNumber(S1);
  •        if (gamma<=d∗fn1)   then
  •                 return   false;
  •        fi;
  •           if (gamma<=d∗aux[Length(aux)])   then
  •                 return false;
  •        fi;
  •        return   true;
  • end;
  • #   The   inputs   are:
  • #   fS   (value   of   F(S)),   d   (value   of   d),
  • #   and   b   (maximum   of   the   Frobenius   of     list F)
  • #   The   function   returns   list F:
  • #   the   list   of   Frobenius   numbers   lower   than   b
  • #   that   are   Frobenius   number   of   at   least   a   GSI-semigroup
  • #   for   a   given   value   of   d
  • # obtained   using   a   semigroup   with   Frobenius   equal   to   fS
  • ListOfFrobenius:=function (fS,d,b)
  •        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)..(b-1)]   do
  •                if (GcdInt(gamma,d)=1)   then
  •                        f:=d∗fS+(d-1)∗gamma;
  •                        if (f<b)   then   Append(listF,[f]);
  •                        fi;
  •                fi;
  •        od;
  •        return   listF;
  • end;
  • #   The   inputs   are:   fS   (value   of   F(S)),
  • #   boundD   (maximum   value   of   d   we   want   to   use),
  • #   and   b   (maximum   of   the   Frobenius   of   listF)
  • #   The   function   returns   listF:
  • #   the   list   of   Frobenius   numbers   lower   than   b
  • #   that   are   Frobenius   number   of   at   least   a   GSI-semigroup
  • #   obtained   using   a   semigroup   with   Frobenius   equal   to   fS
  • ListOfFrobeniusD:=function (fS,boundD,b)
  •         local   listF,d;
  •         d:=3;   listF:=[];
  •         for   d   in   List([1..Int((boundD-1)/2)],k->2∗k+1)   do
  •                 listF:=Union(listF,ListOfFrobenius(fS,d,b));
  •         od;
  •         return   listF;
  • end;

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Table 1. Sets of GSI-semigroups with Frobenius number up to 15.
Table 1. Sets of GSI-semigroups with Frobenius number up to 15.
Frobenius NumberSet of GSI-Semigroups
1 { 2 , 3 }
2
3 { 2 , 5 }
4
5 { 2 , 7 , 3 , 4 }
6
7 { 2 , 9 , 3 , 5 }
8
9 { 2 , 11 , 4 , 6 , 7 }
10
11 { 2 , 13 , 3 , 7 , 4 , 5 , 4 , 6 , 9 }
12
13 { 2 , 15 , 3 , 8 , 4 , 6 , 11 }
14
15 { 2 , 17 , 4 , 6 , 13 , 6 , 8 , 10 , 11 }
Table 2. Values of γ such that gcd ( 3 , γ ) 1 are marked with *.
Table 2. Values of γ such that gcd ( 3 , γ ) 1 are marked with *.
γ 16171819202122
F ( 3 , 4 , 5 3 , γ N ) 3840*4446*50 . . .
Table 3. Values γ such that gcd ( 5 , γ ) 1 are marked with *.
Table 3. Values γ such that gcd ( 5 , γ ) 1 are marked with *.
γ 26272829303132
F ( 3 , 4 , 5 5 , γ N ) 114118122126*134138 . . .
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