Modular Frobenius pseudo-varieties

If $m \in \mathbb{N}$ and $A$ is a finite subset of $\bigcup_{k \in \mathbb{N} \setminus \{0,1\}} \{1,\ldots,m-1\}^k$, then we denote by \begin{align*} \mathscr{C}(m,A) = \left\{S\in \mathscr{S}_m \mid s_1+\cdots+s_k-m \in S \mbox{ if } (s_1,\ldots,s_k)\in S^k \mbox{ and }\right. \\ \left.(s_1 \bmod m, \ldots, s_k \bmod m)\in A \right\}. \end{align*} In this work we prove that $\mathscr{C}(m,A)$ is a Frobenius pseudo-variety. We also show algorithms that allows us to establish whether a numerical semigroup belongs to $\mathscr{C}(m,A)$ and to compute all the elements of $\mathscr{C}(m,A)$ with a fixed genus. Moreover, we introduce and study three families of numerical semigroups, called of second-level, thin and strong, and corresponding to $\mathscr{C}(m,A)$ when $A=\{1,\ldots,m-1\}^3$, $A=\{(1,1),\ldots,(m-1,m-1)\}$, and $A=\{1,\ldots,m-1\}^2 \setminus \{(1,1),\ldots,(m-1,m-1)\}$, respectively.


Introduction
Let N be the set of non-negative integers. A numerical semigroup is a subset S of N that is closed under addition, 0 ∈ S and N \ S is finite. The Frobenius number of S, denoted by F(S), is the greatest integer that does not belong to S. The cardinality of N \ S, denoted by g(S), is the genus of S.
Our main purpose in this work is to study the set C (m, A).
In Section 2 we show that C (m, A) is a Frobenius pseudo-variety with maximum element given by ∆(m) = {0, m, →} (where the symbol → means that every integer greater than m belongs to ∆(m)). Thus, we call the pseudovarieties that arise in this way modular Frobenius pseudo-varieties. Also, we give an algorithm that allows us to establish whether a numerical semigroup belongs to C (m, A). In addition, with the help of the results in [9], we can arrange C (m, A) in a rooted tree and we find an algorithm to compute all the elements of C (m, A) with a fixed genus.
It is well known (see Lemma 2.1 of [11]) that X is a numerical semigroup if and only if gcd(X) = 1. If M is a submonoid of (N, +) and M = X , then we say that X is a system of generators of M . Moreover, if M = Y for any subset Y X, then we say that X is a minimal system of generators of M . In Corollary 2.8 of [11] it is shown that each submonoid of (N, +) has a unique minimal system of generators and that such a system is finite. We denote by msg(M ) the minimal system of generators of M . The cardinality of msg(M ), denoted by e(M ), is the embedding dimension of M .
By applying Proposition 2.10 of [11], if S is a numerical semigroup, then we know that e(S) ≤ m(S). A numerical semigroup S has maximal embedding dimension if e(S) = m(S). This family of numerical semigroups has been extensively studied (for instance, see [2] and [11]). Let us denote by M m the set formed by the numerical semigroups that have maximal embedding dimension and with multiplicity m. It is easy to see that M m = C (m, {1, . . . , m − 1} 2 ).
In Sections 3,4, and 5 we study the family C (m, A) for respectively. Observe that, in a certain sense, these families are generalisation of M m . To finish this introduction, we are going to comment several ideas (see [3,4,14]) that motivate the study of modular Frobenius pseudo-varieties.
First of all, observe that modular Frobenius pseudo-varieties are related to the non-homogeneous patterns with positive coefficients that involve in their constant parameter the multiplicity of the numerical semigroup (see [4]).
The notion of non-homogeneous pattern was introduced in [4] as a generalisation of the notion of homogeneous pattern [3]. Thus, a linear pattern p(x 1 , . . . , x n ) is an expression of the form a 1 x 1 + · · · + a n x n + a 0 , with a 0 ∈ Z (as usual, Z is the set of integers numbers) and a 1 , . . . , a n ∈ Z \ {0}. In particular, the (linear) pattern p is homogeneous if a 0 = 0, and non-homogeneous if a 0 = 0.
On the other hand, it is said that a numerical semigroup S admits the homogeneous pattern p if p(s 1 , . . . , s n ) ∈ S for every non-increasing sequence (s 1 , . . . , s n ) ∈ S n . The corresponding definition for non-homogeneous patterns is a bit different: a numerical semigroup S admits the non-homogeneous pattern p if p(s 1 , . . . , s n ) ∈ S for every non-increasing sequence (s 1 , . . . , s n ) ∈ (S \ {0}) n .
Having in mind that M (S) = S \ {0} is an ideal of the numerical semigroup S (in fact, M (S) is the maximal ideal of S), in [14] the concepts of the above paragraph were extended in the following way: an ideal I of a numerical semigroups S admits the pattern p if p(s 1 , . . . , s n ) ∈ S for every non-increasing sequence (s 1 , . . . , s n ) ∈ I n .
At this point we have the main difference between the above-mentioned papers and our proposal in this work: we do not keep the non-increasing condition (on the sequences in which we evaluate the pattern) in mind. In addition, we take sequences in sets A without structure (that is, A does not have to be an ideal of a numerical semigroup). Now, let us denote by S m (p) the family of numerical semigroups with multiplicity m that admit the pattern p. If we take the patterns p 1 = 2x 1 + x 2 − m and p 2 = x 1 + 2x 2 − m, then S m (p 1 , p 2 ) = S m (p 1 ) ∩ S m (p 2 ) (that is, the family of numerical semigroups which admit p 1 and p 2 simultaneously) is equal to C (m, A) for being this one an easy example of the connection between modular Frobenius pseudo-varieties and families of numerical semigroups defined by nonhomogeneous patterns (as considered in [4]).
A first question studied in [4,14] is the next one: if p = a 1 x 1 +· · ·+a n x n +a 0 is a non-homogeneous pattern, for which values of a 0 we have that the family S (p), of numerical semigroups that admit the pattern p, is non-empty? If the answer is affirmative, then it is said that p is an admissible pattern. In our case, we have imposed that a 0 = −m, where m will be the multiplicity of all the numerical semigroups, in order to get a relevant advantage: we want to build in a (more or less) easy and explicit way the Apéry set of the numerical semigroups belonging to C (m, A). As a consequence of this fact, we will be able to obtain extra information about such families of numerical semigroups.
Of course, in addition to −m, it is possible to consider other values that maintain the pseudo-variety structure. Thus, for p = x 1 + · · · + x n + a 0 , we have that, independently of the chosen set A, every numerical semigroup S, such that a 0 ∈ S, admits the pattern p.
A second question that appears in [3,14] is about the equivalence of patterns. Briefly, the pattern p 1 induces the pattern p 2 if S (p 2 ) ⊆ S (p 1 ) and, moreover, p 1 and p 2 are equivalent patterns if they induce each other. Maybe the first result of this type is the equivalence between the homogeneous patterns 2x 1 −x 2 and x 1 + x 2 − x 3 , which correspond to the family of Arf numerical semigroups (see [5]). Since the set A has an important role (not to say the main role) in the families C (m, A), the large number of possibilities in the choice of A leads us to believe that this is an issue that deserves a new work.
By the way, in Section 4 we study the family of thin numerical semigroups, denoted by T m , and in Section 5 we consider the family of strong numerical semigroups, denoted by R m . These families are associated with the patterns 2x − m and x + y − m, respectively. However, the set A is different in each case and, as a consequence, we have that there is not a inclusion relation between both of them (see Examples 4.6 and 5.6). This fact may give an idea about the difficulty of obtaining similar results to those seen in [3,14].
In any case, we can show simple results about the equivalence (or, at least, the inclusions) of the families C (m, A). Remark 1.1. All the patterns p = x 1 +· · ·+x n −m are equivalent (independently of the chosen set A) if n ≥ m, in which case we have that C (m, A) = S m . Indeed, applying the pigeonhole principle, if s 1 , . . . , s n are elements of S ∈ S m such that s i ≡ 0 (mod m), 1 ≤ i ≤ n, then there exist i, j ∈ {1, . . . , n}, with i < j, such that s i + · · · + s j = km for some k ∈ N \ {0}. Consequently, S ∈ C (m, A). . In order to verify this inclusion, we take a 1 , a 2 , a 3 ∈ S ∈ C (m, A).
More generally, let us suppose that, for each b ∈ B, there exists a ∈ A such that b is obtained by adding coordinates to a. Then C (m, A) ⊆ C (m, B).
As a final recommendation, it is worth mentioning that in [4, Section 2] and in [14,Introduction] there are several motivating examples, and references, as to why it is interesting to study (non-homogeneous) patterns of numerical semigroups.

The pseudo-variety C (m, A)
In this section, m is an integer greater than or equal to 2 and A is a finite subset where x mod m denotes the remainder after division of x by m.
If S is a numerical semigroup and x ∈ S \ {0}, then the Apéry set of x in S (see [1] Observe that an integer s belongs to S if and only if there exists t ∈ N such that s = w(s mod x) + tx. Then the following conditions are equivalent.
Then there exist t 1 , . . . , t k ∈ N such that s j = w(i j ) + t j m, 1 ≤ j ≤ k, and thus, By using the function AperyListOfNumericalSemigroupWRTElement(S,m) of [6], we can compute Ap(S, m) from a system of generators of S. Thereby, we have the following algorithm to decide if a numerical semigroup S belongs or not to C (m, A).

Algorithm 2.2.
INPUT: A finite subset G of positive integers.
Let us illustrate the working of the previous algorithm through an example. • min(G) = 5.
Let m ∈ N\{0, 1}. Then it is easy to show that S∩T ∈ S m for all S, T ∈ S m . Moreover, ∆(m) is the maximum of S m and, if S ∈ S m and S = ∆(m), then S ∪ {F(S)} ∈ S m . From all this, we conclude the following result. Proof. From Lemmas 2.4 and 2.5, we have that ∆(m) is the maximum of C (m, A). It is also easy to see that, if S, T ∈ C (m, A), then S ∩ T ∈ C (m, A). In order to finish the proof, let us see that, if S ∈ C (m, A) and S = ∆(m), then Our next purpose in this section is to show an algorithm that allows us to build all the elements of C (m, A) that have a fixed genus. To do that, we use the concept of rooted tree.
A graph G is a pair (V, E) where V is a non-empty set (whose elements are called vertices of G) and E is a subset of {(v, w) ∈ V × V | v = w} (whose elements are called edges of G). A path (of length n) connecting the vertices x and y of G is a sequence of different edges We say that a graph G is a (rooted) 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 there exists a path connecting x and y, then we say that x is a descendant of y. In particular, if (x, y) is an edge of the tree, then we say that x is a child of y. (See [12]. ) We define the graph G C (m, A) in the following way: C (m, A) is the set of vertices and (S, The following result is a consequence of Lemma 4.2 and Theorem 4.3 of [9]. Let S be a numerical semigroup and let x ∈ S. Then it is clear that S \ {x} is a numerical semigroup if and only if x ∈ msg(S). Moreover, let us observe that, if x ∈ msg(S) and m ∈ S \ {0, x}, then In the following result we characterise the children of S ∈ C (m, A). (Necessity.) If w(i 1 ) + · · · + w(i k ) = m + x, then m + x / ∈ {w(i 1 ), . . . , w(i k )} and, therefore, w(i 1 ) = w ′ (i 1 ), . . . , w(i k ) = w ′ (i k ). Moreover, w(i 1 ) + · · · + w(i k ) − m = x / ∈ S \ {x} and, consequently, S \ {x} / ∈ C (m, A). (Sufficiency.) In order to prove that S \ {x} ∈ C (m, A), by Proposition 2.1, it is enough to see that, if (i 1 , . . . , i k ) ∈ A, then w ′ (i 1 )+· · ·+w ′ (i k )−m ∈ S\{x}.
Since S ∈ C (m, A), we easily deduce that w ′ (i 1 ) Let us observe that a tree can be built in a recurrent way starting from its root and connecting each vertex with its children. Let us also observe that the elements of C (m, A) with genus equal to g + 1 are precisely the children of the elements of C (m, A) with genus equal to g.
We are ready to show the above announced algorithm. (1) If g < m − 1, return ∅.

Second-level numerical semigroups
We say that a numerical semigroup S is of second-level if x + y + z − m(S) ∈ S for all (x, y, z) ∈ (S \ {0}) 3 . We denote by L 2,m the set of all the second-level numerical semigroups with multiplicity equal to m.
Proposition 3.1. Let S be a numerical semigroup with minimal system of generators given by {m = n 1 < n 2 < · · · < n e }. Then the following two conditions are equivalents.
Proof. (1. ⇒ 2.) It is evident from the definition of second-level numerical semigroup.
As an immediate consequence of Propositions 2.6 and 3.3, we have the following result. Our next step is to build the tree associated with the pseudo-variety L 2,m .
To do this, we should characterise the possible children of each element in L 2,m .
Let S be a numerical semigroup with msg(S) = {n 1 , . . . , n e }. If s ∈ S, then we denote by L S (s) = max{a 1 + · · · + a e | (a 1 , . . . , a e ) ∈ N e and a 1 n 1 + · · · + a e n e = s}. Proof. (Necessity.) Let us suppose that L S\{x} (x + m) ≥ 3. Then there exists (a, b, c) ∈ (S\{0, x}) 3 such that x+m = a+b+c. Thus, a+b+c−m = x / ∈ S\{x} and, therefore, By applying Theorem 2.7, Propositions 3.3 and 3.5, and Lemma 2.10, we can easily build the tree G(L 2,m ).   4, 9, 10, 11 4, 7, 10, 13 4, 7, 9 4, 6, 11, 13 4, 6, 9 The Frobenius problem (see [7]) consists in finding formulas that allow us to compute the Frobenius number and the genus of a numerical semigroup in terms of the minimal system of generators of such a numerical semigroup. This problem was solved in [15] for numerical semigroups with embedding dimension two. At present, the Frobenius problem is open for embedding dimension greater than or equal to 3. However, if we know the Apéry set Ap(S, x) for some x ∈ S \ {0}, then we have solved the Frobenius problem for S because we have the following result from [13]. Now our purpose is to show that, if S ∈ L 2,m , then is rather easy to compute Ap(S, m). We need the following easy result. As an immediate consequence of the above lemma we can formulate the following result.  .
Following the notation introduced in [10], we say that x ∈ Z \ S is a pseudo-Frobenius number of S if x + s ∈ S for all s ∈ S \ {0}. We denote by PF(S) the set of all the pseudo-Frobenius numbers of S. The cardinality of PF(S) is an important invariant of S (see [2]) that is the so-called type of S and it is denoted by t(S).
Let S be a numerical semigroup. Then we define the following binary relation over Z: a ≤ S b if b − a ∈ S. In [11] it is shown that ≤ S is a partial order (that is, reflexive, transitive, and antisymmetric). Moreover, Proposition 2.20 of [11] is the next result. Let us observe that, if w, w ′ ∈ Ap(S, x), then w ′ − w ∈ S if and only if w ′ − w ∈ Ap(S, x). Therefore, Maximals ≤S (Ap(S, x)) is the set We finish this section with an example that illustrates the above results. Remark 3.13. The definition of second-level numerical semigroup can be easily generalised to greater levels. Thus, we say that a numerical semigroup is of nth-level if x 1 + · · · + x n+1 − m(S) ∈ S for all (x 1 , . . . , x n+1 ) ∈ (S \ {0}) n+1 and denote by L n,m the set of all the nth-level numerical semigroups with multiplicity equal to m.
It is clear that for nth-level we obtain similar results to those of second-level. In particular, L n,m = C (m, {1, . . . , m−1} n+1 ) and Proposition 3.5 remains true taking L S\{x} (x + m) ≤ n.
On the other hand, having in mind that L 1,m is the family of numerical semigroups with maximal embedding dimension and that (by Remark 1.1) L m−1,m = S m , we can observe that where all the inclusions are strict, as we can deduce from the following example.
Remark 3.15. Let us observe that the chain obtained in Remark 3.13 is reminiscent, in some sense, of the chain associated with subtraction patterns (see [3,Section 6]).

Thin numerical semigroups
We say that a numerical semigroup S is thin if 2x− m(S) ∈ S for all x ∈ S \ {0}.
We denote by T m the set of all the thin numerical semigroups with multiplicity equal to m.
Proposition 4.1. Let S be a numerical semigroup with minimal system of generators given by {m = n 1 < n 2 < · · · < n e }. Then the following two conditions are equivalents.
The above proposition allows us to easily decide whether a numerical semigroup is thin or not.   In order to build the tree associated with the pseudo-variety S ∈ T m , we are going to characterise the possible children of each S ∈ T m .   Remark 4.9. Let us observe that we can generalise the concept of thin numerical semigroups in the following way: setting n ∈ N \ {0, 1}, we say that a numerical semigroup S is n-thin if nx − m(S) ∈ S for all x ∈ S \ {0}, and denote by T n,m the set of all k-thin numerical semigroups with multiplicity equal to m. It is clear that T m = T 2,m . Again, T n,m is a modular pseudo-variety, Proposition 4.5 is valid for the condition x+m n ∈ S, and we can build the chain Note that Example 3.14 also gives us the construction that ensures the strict inclusions in this case.

Strong numerical semigroups
We say that a numerical semigroup S is strong if x + y − m(S) ∈ S for all (x, y) ∈ (S \ {0}) 2 such that x ≡ y (mod m(S)). We denote by R m the set of all the strong numerical semigroups with multiplicity equal to m.
Proposition 5.1. Let S be a numerical semigroup with minimal system of generators given by {m = n 1 < n 2 < · · · < n e }. Then the following two conditions are equivalents.
The above proposition allows us to easily decide whether a numerical semigroup is strong or not. To see the other inclusion, let S ∈ C (m, A) and (x, y) ∈ (S \ {0}) 2 . On the one hand, if 0 ∈ {x mod m, y mod m}, then it is clear that x + y − m ∈ S. On the other hand, if 0 / ∈ {x mod m, y mod m}, then there exist (i, j) ∈ A and (p, q) ∈ N 2 such that x = w(i) + pm and y = w(j) + qm. Therefore, x + y − m = (w(i) + w(i) − m) + (p + q)m ∈ S and, consequently, S ∈ R m . From Propositions 2.6 and 5.3, we get the following result. We are now interested in the description of the tree associated with the pseudo-variety S ∈ R m . In order to do that, we are going to characterise the children of an arbitrary S ∈ R m . Proof. (Necessity.) If a, b ∈ msg(S)\{m, x} and a = b, then (a, b) ∈ (S \{0, x}) 2 and a ≡ b (mod m). Since S \ {x} ∈ R m , we have that a + b − m ∈ S \ {x} and, therefore, a + b − x = x. Thus, x + m / ∈ {a + b | a, b ∈ msg(S) \ {m, x}, a = b}. On the other hand, if a ∈ msg(S) \ {m, x}, then (a, 2a) ∈ (S \ {0, x}) 2 and a ≡ 2a (mod m). Once again, since S \{x} ∈ R m , we have that 3a−m ∈ S \{x} and, therefore, a + b − x = x. Thus, x + m / ∈ {3a | a ∈ msg(S) \ {m, x}}. (Sufficiency.) Let a, b ∈ S \ {0, x} such that a = b. Since S ∈ R m , we have that a + b − m ∈ S and 3a − m ∈ S. Now, by Lemma 2.10, we know that msg(S) \ {x} ⊆ msg(S \ {x}) ⊆ (msg(S) \ {x}) ∪ {x + m}. Thus, from this fact and the hypothesis, it easily follows that a + b − m = x and 3a − m = x, that is, a + b − m, 3a − m ∈ S \ {x}. By applying Proposition 5.1, we conclude that S \ {x} ∈ R m . From Theorem 2.7, Propositions 5.3 and 5.5, and Lemma 2.10, we can build the tree G(R m ). Let us see an example.
Example 5.6. In the next figure we have the first four levels of G(R 4 ). 4, 9, 10, 11 4, 7, 10, 13 4, 7, 9 4, 6, 11, 13 We finish this section describing the Apéry set for S ∈ R m . It is interesting to observe that, under the hypotheses of Proposition 5.7, n i ∈ Maximals ≤S (Ap(S, m)) if and only if 2n i / ∈ Ap(S, m). In addition, 2n i ∈ Ap(S, m) if and only if 2n i ∈ Maximals ≤S (Ap(S, m)). Therefore, from Propositions 3.11 and 5.7, we get the next result. It is well known that, if S is a numerical semigroup, then e(S) ≤ m(S) and t(S) ≤ m(S) − 1 (see Proposition 2.10 and Corollary 2.23 in [11]). Combining these facts with Corollaries 5.8 and 5.9, we have the next result. Remark 5.11. Contrary to what happens in Sections 3 and 4, the concept of strong numerical semigroup does not have a natural generalisation. In fact, we have, at least, two possibilities.
2. x 1 + · · · x n − m(S) ∈ S for all (x 1 , . . . , x n ) ∈ B n , where It is clear that, if n ≥ 3, then A n B n and, in consequence, C (m, B n ) ⊆ C (m, A n ). In addition, it is guessed that, if n < m, then the second inclusion will be strict.