Generalized strongly increasing semigroups

In this work we present a new class of numerical semigroups called GSI-semigroups. We see the relations between them and others families of semigroups and we give explicitly 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.


Introduction
Let N = {0, 1, 2, . . .} be the set of nonnegative integers. A numerical semigroup 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 the multiplicity of S, we denote it by m(S). Given a nonempty 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 Na 1 + · · · + Na 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 [12]) 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 its theory is connected with Algebraic Geometry and Commutative Algebra (see [3], [6]) as well as with Integer Optimization (see [5]) and Number Theory (see [2]). It is common the study of families of numerical semigroups, for instance symmetric semigroups, irreducible semigroups and strongly increasing semigroups (see [9], [10]) or the study or characterization of invariants, for instance the Frobenius number, the set of gaps, the genus, etc. (see [1], [8], [10] and [11]).
The first example of GSI-semigroups are numerical semigroups generated by two positive integersS = a, b with a < b. For these semigroups, set S = N = 1 , d = a, and γ = b. Since F(S) = −1, γ = b > max{dF(S), d} = {a · (−1), a} = a. From Sylvester (see [13]), 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 Frobenius number of a GSI-semigroup.
Our definition of GSI-semigroups is inspired on the one of SIsemigroups. We remind you how they are defined (see [4] for further details).
A sequence of positive integers (v 0 , . . . , v h ) is called a characteristic sequence if it satisfies the following two properties: 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.
We denote by v 0 , . . . , v h the semigroup generated by the charac- and it is generated by a characteristic sequence. Note that by Lemma 2, we can assume that v 0 < · · · < v h . Theorem 3. LetS 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: Proof. The case h = 1 is trivial by definition of characteristic sequences.
Assume h > 1 and thatS = v 0 , . . . ,v h is a strongly increasing numerical semigroup with embedding dimension strictly greater than 2.
Since e(S) = h + 1, then e(S) = h. Set γ =v h and d =ē h−1 , we getS = S ⊕ d,γ N. We have that γ =v h >v h−1 = dv h−1 , and sinceS is a SI-semigroup, 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 The following result give us a formula for the conductor (the Frobenius number plus 1) of a SI-semigroup.

Moreover, the conductor of S is an even number and its genus is
By Proposition 4, we get SinceS is a SI-semigroup, the property (CS2) if fulfilled. Using that the generators are ordered, we obtain thatv h < e h−1vh = dv 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 semigroups. For example, telescopic, free and complete intersection.
. We say that S is free whenever it is equal to N or it is the gluing of a free with N. The semigroup S is telescopic if it is free for the rearrangement v 0 < · · · < v h . A semigroup is complete intersection if 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 are free semigroups and free semigroups are complete intersection. 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{2F( 3, 7, 11 ), 2 · 11}. Thus this is a GSI-semigroup. We define 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 6, 14, 22, 23 , we obtain the following outputs: 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.

Set of gaps of a GSI-semigroup
We have seen that GSI-semigroups are easy to obtain from any numerical semigroup just gluing it with N with appropiate elements d and γ. Hence these semigroups form a large family within the set of numerical semigroups. In this section, we deepen into 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 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 real interval [a, b]. Put ⌊a⌋ the integral part of the real number a. where

Moreover (1) is a partition of the gapset ofS (we do not write
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 Taking into account the reasoning done so far we have Let us prove that H is a partition (we do not write A d or B d,ℓ when they are the emptyset).
When A d is a nonempty set, let A d,k = d(N \ S) + kv h+1 for a fix 1 ≤ k ≤ d − 1. In this case, if B d,ℓ is nonempty we have Observe that and Since 1 ≤ k, ℓ < 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,ℓ and B d,ℓ ′ are also disjoint for ℓ = ℓ ′ . Moreover A d and B d,ℓ are also disjoint for any (3) and (4), Given that d and v h+1 are coprime and 1 ≤ k, ℓ < d we get k = ℓ + 1. So x ∈ A d,ℓ+1 ∩ B d,ℓ , which is a contradiction 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 Indeed, if we suppose that x = λd for some λ ∈ N, by hypothesis we get dF(S) < v h+1 < x = λd, in particular λ > F(S), so We distinguish two cases, depending on A d . First, we suppose that A d = ∅.

Claim 2: The greatest gap ofS which is congruent with
Since max B d,ℓ = (ℓ + 1)v h+1 − d and min A d,ℓ+1 = (ℓ + 1)v h+1 + d the only possibility for x is (ℓ + 1)v h+1 which is an element ofS. k0 . In particular there is an integer number λ such that x = k 0 v h+1 + λd. Hence if x ∈ [min A d,k0 , max A d,k0 ] N then x ∈ A d,k0 . Indeed, in this case λ ∈ N and λ ∈ S, otherwise x ∈S.

Claim 4: The set of all the integers in
Hence x has to belong to B d,k0−1 and we finish the proof for the case A d = ∅.
Suppose now that A d = ∅, that is S = N andS is generated by d and v 1 (h = 0).
From the proof of Theorem 6, we obtain the Frobenius number of a GSI-semigroup S ⊕ d,γ N, which is equal to

Algorithms for GSI-semigroups
We finish this work with 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 GSIsemigroup 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 least than or equal to a fixed nonnegative integer. Note that in step 5 of the algorithm we use that F(S) = dF(S) + (d − 1)γ and γ > dF(S) implies that F(S) ≥ d 2 F(S) whereS = S ⊕ d,γ N.
Denote by M (S) the largest element of the minimal system of generators of a numerical semigroup S. Remark 10. If A is a minimal system of generators of a numerical semigroup S and d ∈ N \ {0, 1}, then dA is a minimal system of generators of dS = {ds | s ∈ S} ⊂ dN. Furthermore, if γ ∈ N \ {1} and gcd(d, γ) = 1, then γ ∈ dN \ {0}. Thus, γ ∈ dS and dA ∪ {γ} is a minimal system of generators of dA ∪ {γ} . Algorithm 1: Computation of the set of GSI-semigroups with Frobenius number least than or equal to f .
We give in Table 2 all the GSI-semigroups with Frobenius number least than or equal to 15.

Frobenius number
Set of GSI-semigroups 1  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 2, one might think that there are no GSI-semigrups 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 (5) 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.
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 GSIsemigroup.
Fixed 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 .
Using that γ has to be greater than or equal to dF(S)+1 ≥ 3F(S)+1 (recall that γ > max{dF(S), dM (S)} and d ≥ 3) and from formula (5), be the Frobenius number of S.
. Therefore, for , γ equals f −dt d−1 . The next lemma follows from the previous considerations.
Hence, there are no GSI-semigroups with Frobenius number 42.
Example 14. Consider f = 4620. Using the code in Table 5, we check that there are no GSI-semigroups of the form S ⊕ d,γ N, with