The Covariety of Saturated Numerical Semigroups with Fixed Frobenius Number

: In this work, we show that if F is a positive integer, then Sat ( F ) = { S | S is a saturated numerical semigroup with Frobenius number F } is a covariety. As a consequence, we present two algorithms: one that computes Sat ( F ) , and another which computes all the elements of Sat ( F ) with a fixed genus. If X ⊆ S \ ∆ ( F ) for some S ∈ Sat ( F ) , then we see that there exists the least element of Sat ( F ) containing X . This element is denoted by Sat ( F )[ X ] . If S ∈ Sat ( F ) , then we define the Sat ( F ) -rank of S as the minimum of { cardinality ( X ) | S = Sat ( F )[ X ] } . In this paper, we present an algorithm to compute all the elements of Sat ( F ) with a given Sat ( F ) -rank.


Introduction
Let N the set of nonnegative integers numbers.A numerical semigroup is a subset S of N which is closed by sum, 0 ∈ S and N\S is finite.The set N\S is known as the set of gaps of S and its cardinality, denoted by g(S), is the genus of S. The largest integer not belonging to S, is known as the Frobenius number of S and it will be denoted by F(S).
Let {n 1 < • • • < n p } ⊆ N with gcd(n 1 , . . ., n p ) = 1.Then n 1 , . . ., n p = { p i=1 λ i n i | {λ 1 , . . ., λ p } ⊆ N} is a numerical semigroup and every numerical semigroup has this form (see [19,Lemma 2.1]).The set n 1 < • • • < n p is called system of generators of S, and we write S = n 1 , . . ., n p .We say that a system of generators of a numerical semigroup is a minimal system of generators if none of its proper subsets generates the numerical semigroup.Every numerical semigroup has a unique minimal system of generators, which in addition is finite (see [19,Corollary 2.8]).The minimal system of generators of a numerical semigroup S is denoted by msg(S).Its cardinality is called the embedding dimension and will be denoted by e(S).Another invariant which we will use in this work is the multiplicity of S, denoted by m(S).It is defined as the minimum of S\{0}.
Given S a numerical semigroup the mulplicity, the genus and the Frobenius number of S are three invariants very important in the theory numerical semigroups (see for instance [16] and [2] and the reference given there) and they will play a very important role in this work.
The Frobenius problem (see [16]) for numerical semigroups, lies in finding formulas to obtain the Frobenius number and the genus of a numerical semigroup from its minimal system of generators.When the numerical semigroup has embedding dimension two, this problem was solved by J. J. Sylvester (see [22]).However, if the numerical semigroup has embedding dimension greater than or equal to three, the problem is still open.
Looking to find a solution to the Frobenius problem, in [12], we study the set A (F ) = {S | S is a numerical semigroup and F(S) = F }, being F ∈ N\{0}.The generalization of A (F ) as a family of numerical semigroups that verifies certain properties leads us introduce in [12], the concept of covariety.That is, a covariety is a nonempty family C of numerical semigroups that fulfills the following conditions: 1) C has a minimum, denoted by ∆(C ) = min(C ), with respect to set inclusion. 2 This concept has allowed us to study common properties of some families of numerical semigroups.For instance, in [13], we have studied the set of all numerical semigroups which have the Arf property (see for example, [2]) with a given Frobenius number, showing some algorithms to compute them.
In the semigroup literature one can find a long list of works dedicated to the study of one dimensional analytically irreducible domains via their value semigroup (see for instance [3], [5], [6], [11] and [23]).One of the properties studied for this kind of rings using this approach has been to be saturated.Saturated rings were introduced in three different ways by Zariski ([24]), ) and Campillo ([4]).Theses tree definitions coincide for algebraically closed fields of zero characteristic.The characterization of saturated ring in terms of their value semigroups gave rise to the notion of saturated numerical semigroup (see [ [8], [14]]).
If A ⊆ N and a ∈ A, then we denote by d If F ∈ N\{0}, we denote by Sat(F ) = {S | S is a saturated numerical semigroup and F(S) = S}.
In this work, we study the set of Sat(F ) by using the techniques of covarieties.
The structure of the paper is the following.Section 2 will be devoted to recall some concepts and result which will be used in this work.Also, we show how we can compute some of then with the help of the GAP [10] package numericalsgps [7].In Section 3 we will show that Sat(F ) is a covariety.This fact allow us to order the elements of Sat(F ) making a rooted tree, and consequently, to present an algorithm which computes all the elements of Sat(F ).
In Section 4, we will see who are the maximal elements of Sat(F ).We compute the set {g(S) | S ∈ Sat(F )} and we apply this result to give an algorithm which enables to calculate all the elements of Sat(F ) with a fixed genus.
A set X is called a Sat(F )-set, if it verifies the following conditions: 1) X ∩ {0, F + 1, →} = ∅, where the symbol → means that every integer greater than F + 1 belongs to the set.
2) There is S ∈ Sat(F ) such that X ⊆ S.
In Section 5, we will see that if X is a Sat(F )-set, then there is the least element of Sat(F ) containing a X.This element will denote by Sat(F ) [X].
If S = Sat(F )[X], then we will say that X is a Sat(F )-system of generators of S. Also, we will show that every element of Sat(F ) admits a unique minimal Sat(F )-sytem of generators.
The Sat(F )-rank of an element of Sat(F ) is the cardinal of its minimal Sat(F )-sytem of generators.In Section 6, we presente an algorithmic procedure to compute all the elements of Sat(F ) with a given Sat(F )-rank.

Preliminaires
In this section we present some concepts and results which are necessary for the understant of the work.The following result appears in [19,Proposition 3.10].
A numerical semigroup S is said to have maximal embedding dimension (from now on MED-semigroup) if e(S) = m(S).
By applying the results of [19, Chapter 3], we have the following result.
Following the notation introduced in [17], an integer z is a pseudo-Frobenius number of a numerical semigroup S if z / ∈ S and z + s ∈ S for all s ∈ S\{0}.We denote by PF(S) the set formed by the pseudo-Frobenius numbers of S. The cardinality of PF(S) is an important invariant of S (see [9] and [2]) called the type of S, denoted by t(S).
Proposition 2.3.Let S be a numerical semigroup and x ∈ N\S.Then x ∈ SG(S) if and only if S ∪ {x} is a numerical semigroup.
From Proposition 3.1 of [19], we can deduce the following result.The next lemma has an immediate proof.
The proof of the following result is very simple.
Proposition 2.8.If S is a numerical semigroup and S = N, then Remark 2.9.Observe that as a consequence from Propositions 2.6, 2.7 and 2.8, if S is a numerical semigroup and we know the set Ap(S, n) for some n ∈ S\{0}, then we easily can calculate the set SG(S).
The following result is well known as well as it is very easy to prove.
Proposition 2.10.Let S and T be numerical semigroups and x ∈ S. Then the following hold: 1) S ∩ T is a numerical semigroup and F(S ∩ T ) = max{F(S), F(T )}.
2) S\{x} is a numerical semigroup if and only if x ∈ msg(S).
The following result is Lemma 2.14 of [19].
3 The tree associated to Sat(F ) Our first aim in this section will be to prove that if F is a positive integer, then the set Sat(F ) = {S | S is a saturated numerical semigroup and F(S) = F } is a covariety.
The following result can be consulted in [20, Proposition 5].
Lemma 3.1.If S and T are saturated numerical semigroups, then S ∩ T is also a saturated numerical semigroup.
The following result has an immediate proof.
Lemma 3.2.Let F be a positive integer.Then the following properties are verified: 2) ∆(F + 1) is the minimun of Sat(F ).
3) If S is a saturated numerical semigroup, then S\{m(S)} is also a saturated numerical semigroup.
By applying Proposition 2.10; and Lemmas 3.1 and 3.2, we can easily deduce the following result.
The elements of V and E are called vertices and edges, respectively.A path, of length n, connecting the vertices x and y of G is a sequence of different edges of the form A graph G is a tree if there exists a vertex r (known as the root of G) such that for any other vertex x of G, there exists a unique path connecting x and r.If (u, v) is an edge of the tree G, we say that u is a child of v.
Define the graph G(F ) as follows: As a consequence from [12, Proposition 2.6] and Proposition 3.3, we have the following result.Proposition 3.4.Let F be a positive integer.Then G(F ) is a tree with root ∆(F + 1).
A tree can be built recurrently starting from the root and connecting, through an edge, the vertices already built with their children.Hence, it is very interesting to characterize the children of an arbitrary vertex of G(F ).For this reason, we will introduce some concepts and results.
The following result is deduced from Proposition 3. Proposition 3.8.Let S be a numerical semigroup and x ∈ SG(S) such that x < m(S) and S ∪{x} is a MED-semigroup.Then the following conditions hold.
Remark 3.9.Note that as a consequence of Propositions 2.2, 3.5 and 3.8, if S ∈ Sat(F ) and we know the set msg(S), then we can easily compute msg(T ) for every child T of S in the tree G(F ).
Input: A positive integer F. Output: Sat(F ).
(3) If  (5) For all S ∈ C compute msg(S), by using Proposition 3.8.Next we illustrate this algorithm with an example.
• The algorithm returns 4 The elements of Sat(F ) with fixed genus Given F and g positve integers, denote by From Proposition 2.11 it is deduced the following result.
Let S be a numerical semigroup, then the associated sequence to S is recurrently defined as follows: Let S be a numerical semigroup.We say that an element s of S is an small elemnent of S if s < F(S).Denote by N(S) the set of small elements of S. The cardinality of N(S) will be denoted by n(S).
It is clear that the disjoint union of the sets N(S) and N\S is the set {0, . . ., F(S)}.Therefore, we have the following result.If S is a numerical semigroup and {S n } n∈N is its associated sequence, then the set Cad(S) = {S 0 , S 1 , . . ., S n(S)−1 } is called the associated chain to S. Observe that S 0 = S and S n(S)−1 = ∆(F(S) + 1).
Observe that, from Proposition 3.3, we know that if S ∈ Sat(F ), then Cad(S) ⊆ Sat(F ).Therefore, we can enounce the following result.For integers a and b, we say that a divides b if there exists an integer c such that b = ca, and we denote this by a | b.Otherwise, a does not divide b, and we denote this by a ∤ b.
The nex lemma is [21,Lemma 2.3] and it shows a characterization of saturated numerical semigroups.Lemma 4.4.Let S be a numerical semigroup.Then S is a saturated numerical semigroup if and only if there are positive integers a 1 , b 1 , • • • , a n , b n verifying the following properties: The next Lemma is an immediate consequence of Lemma 4.4.
The following result is a consequence of Lemmas 4.4 and 4.5.
Theorem 4.6.With the previous notation, S is a maximal element of Sat(F ) if and only if S = T(x, F + 1) for some x ∈ B(F ).
In the following Example, we illustrate how the previous theorem works.By using this corollary, in the following example we calculate the minimum genus of the elements belonging to Sat (7), as well as the minimum genus of the elementos of Sat (6).We have now all the ingredients needed to present the announced algorithm.Algorithm 4.10.
Input: Two positive integers F and g such that F +1 2 ≤ g ≤ F. Output: Sat(F, g).
(1) Compute the smallest positive integer p such that p ∤ F.
Lemma 5.3.Let S ∈ Sat(F ) and s ∈ S such that 0 < s < F and d S (s) = d S (s ′ ) for all s ′ ∈ S being 0 < s ′ < s.If X is a Sat(F )-system of generators of S, then s ∈ X.
Proof.By Lemma 5.2, we know that S\{s} is an element of Sat(F ).If s / ∈ X, then X ⊆ S\{s} and, by applying Proposition 5.1, we have that S = Sat(F )[X] ⊆ S\{s}, which is absurd.
The following result can be consulted in [20,Theorem 4].
Lemma 5.4.Let A ⊆ N such that 0 ∈ A and gcd(A) = 1.Then the following conditions are equivalent.1) A is a saturated numerical semigroup.
2) a + d A (a) ∈ A for all a ∈ A. Proof.Let T = Sat(F )[X].As X ⊆ S, then by applying Proposition 5.1, we have that T ⊆ S. Now we will see the reverse inclusion, that is, S ⊆ T. Assume that X = {x 1 , . . ., x n }, s ∈ S\{0} and Then d S (s) = d S (x k ) = d T (x k ) and s = x k + a for some a ∈ N. We deduce then d S (x k ) | a and so s = x k + t • d S (x k ) for some t ∈ N. Consequently, by applying Lemma 5.4, As a consequence of Lemmas 5.3 y 5.5, we can assert that the minimal Sat(F )-system of generators is unique.This is the content of the following proposition.
Proposition 5.6.If S ∈ Sat(F ), then the unique minimal Sat(F )-system of generators of S is the set {x ∈ S\{0} | x < F and d S (x) = d S (y) for all y ∈ S such that y < x}.
If S ∈ Sat(F ), then we denote by Sat(F )msg(S) the minimal Sat(F )-system of generators of S. The cardinality of Sat(F )msg(S) is called the Sat(F )-rank of S and it will denote by Sat(F )-rank (S).Let us illustrate these two concepts with an example.
The following result is a direct consequence of Proposition 5.6.2) Sat(F )-rank (S) = 0 if and only if S = ∆(F + 1).
As a consequence of Proposition 5.12 and Lemma 5.8, we have the following result.
Corollary 5.13.Under the standing notation, the following conditions are equivalent.
2) There is m ∈ N such that 2 ≤ m < F, m ∤ F and S = T (m, F + 1).
6 Sat(F )-sequences As a consequence of Theorem 6.1 and Corollary 6.2, if we want to compute all the elements belonging to Sat(F ) with Sat(F )-rank equal to p, it will be enough to do the following steps:

Proposition 2 . 5 .
Let S be a numerical semigroup.Then S is a MED-semigroup if and only if msg(S) = (Ap(S, m(S))\{0}) ∪ {m(S)}.Let S be a numerical semigroup.Over Z we define the following order relation: a ≤ S b if b − a ∈ S. The following result is Lemma 10 from [17].Proposition 2.6.If S is a numerical semigroup and n ∈ S\{0}.Then PF(S) = {w − n | w ∈ Maximals ≤S Ap(S, n)}.
Our next aim is to determine the minimum element of the set {g(S) | S ∈ Sat(F )}.For this purpose we introduce the following notation.Let {a, b} ⊆ N, then we denote by T(a, b) = a ∪ {x ∈ N | x ≥ b}.