Numerical semigroups bounded by the translation of a plane monoid

Let N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {N}$$\end{document} be the set of nonnegative integer numbers. A plane monoid is a submonoid of (N2,+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbb {N}^2,+)$$\end{document}. Let M be a plane monoid and p,q∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p,q\in \mathbb {N}$$\end{document}. We will say that an integer number n is M(p, q)-bounded if there is (a,b)∈M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a,b)\in M$$\end{document} such that a+p≤n≤b-q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a+p\le n \le b-q$$\end{document}. We will denote by A(M(p,q))={n∈N∣nisM(p,q)-bounded}.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm A}(M(p,q))=\{n\in \mathbb {N}\mid n \text { is } M(p,q)\text {-bounded}\}.$$\end{document} An A(p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}(p,q)$$\end{document}-semigroup is a numerical semigroup S such that S=A(M(p,q))∪{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S= {\mathrm A}(M(p,q))\cup \{0\}$$\end{document} for some plane monoid M. In this work we will study these kinds of numerical semigroups.


Introduction
Let Z be the set of integer numbers and N = {z ∈ Z | z ≥ 0}. Let k ∈ N\{0}, a submonoid of (N k , +) is a subset M of N k which is closed under addition and contains the element (0, . . . , 0). A plane monoid is a submonoid of (N 2 , +). A numerical semigroup is a submonoid S of (N, +) such that N\S = {x ∈ N | x / ∈ S} is finite. Let M be a plane monoid. We will say that an integer number n is bounded by M if there is (a, b) ∈ M such that a < n < b. We will denote it by A(M ) = {n ∈ N | n is bounded by M }. In [4] it is proven that if A(M ) = ∅ then A(M ) ∪ {0} is a numerical semigroup. An A-semigroup is a numerical semigroup S such that S = A(M ) ∪ {0} for some plane monoid M . In [4,Theorem 2.4] it is shown that a numerical semigroup S is an A-semigroup if and only if x + y − 1, x + y + 1 ∈ S for all x, y ∈ S\{0}.
Let M be a plane monoid and p, q ∈ N. We will say that an integer number n is M (p, q)-bounded if there is (a, b) ∈ M such that a + p ≤ n ≤ b − q. We will denote it by A(M (p, q)) = {n ∈ N | n is M (p, q)-bounded}. In [5] it is proven that A(M (p, q)) ∪ {0} is a submonoid of (N, +). An A(p, q)-semigroup is a numerical semigroup S such that S = A(M (p, q)) ∪ {0} for some plane monoid M.
If S is a numerical semigroup, then m(S) = min(S\{0}), F(S) = max{z ∈ Z | z / ∈ S} and g(S) = (N\S) (where (X) denotes the cardinality of a set X), are three invariants of S which are called multiplicity, Frobenius number and genus of S, respectively.
Following the notation introduced in [3], a Frobenius pseudo-variety is a nonempty family P of numerical semigroups such that the following conditions hold: 1) P has a maximum element, max(P) (with respect to the inclusion order).

3) If S ∈ P and S = max(P), then S ∪ {F(S)} ∈ P.
We will denote it by A(p, q)[m] = {S | S is an A(p, q)-semigroup and m(S) = m}. In Sect. 3, we will see that A(p, q)[m] is a Frobenius pseudo-variety. This fact allows us to order the elements of A(p, q)[m] making a tree with root. Consequently, we present an algorithmic procedure to calculate all the elements of A(p, q) [m].
In Sect. 4, we will see that if X is a nonempty subset of N\{0, 1} and p ≤ min(X), then there exists a unique smallest (with respect to set inclusion) A(p, q)-semigroup containing X and it will be denoted by A(p, q)(X). If m = min(X), then we will see that A(p, q)(X) is the intersection of all elements of A(p, q)[m] containing X. Moreover, we will present an algorithm which calculates Following the terminology introduced in [2] an AC-semigroup is a numerical semigroup S such that S = A(M )∪{0} for some plane and cyclic monoid M . Note that if M = {(ka, kb) | k ∈ N} then A(M ) = {n ∈ N | ka < n < kb for some k ∈ N}.
We say that a numerical semigroup S is an AC(p, q)-semigroup if S = A(M (p, q)) ∪ {0} for some plane and cyclic monoid M . Note that if M = {(ka, kb) | k ∈ N} then A(M (p, q)) = {n ∈ N | ka+p ≤ n ≤ kb−q for some k ∈ N}. Also, notice that the concepts of AC-semigroup and AC(1, 1)-semigroup are the same. Our main aim in Sect. 5 is to give formulas to compute, in terms of a, b, p and q, the multiplicity, the Frobenius number and the genus of A(M (p, q)) ∪ {0}.

A(p, q)-semigroups
Let M be a plane monoid and p, q ∈ N. Our first aim in this section will be to study the conditions that M , p and q have to fulfil to make A(M (p, q)) ∪ {0} a numerical semigroup.
The following result is a consequence of [8, Lemma 1.1].

Lemma 2.
Let M be a submonoid of (N, +). Then M is a numerical semigroup if and only if gcd(M ) = 1.
We can now give the result announced at the beginning of this section. (1) There is (m, n) ∈ M such that m < n.
(2) There is a numerical semigroup S such that {(s, s) | s ∈ S} ⊆ M and p = q = 0.
Proof. (Necessity) We suppose that 1) does not hold. Let s ∈ A(M (p, q)), then there is (a, b) ∈ M such that a + p ≤ s ≤ b − q. As b ≤ a, we deduce p = q = 0 and a = b = s. Therefore, Hence 2) holds.
Our next aim will be to prove Theorem 8. For this purpose, we need to introduce some notions and results.
Let k ∈ N\{0} and let X be a nonempty set of N k . We denote by X the submonoid of (N k , +) generated by X, that is, X = {λ 1 x 1 + · · · + λ n x n | n ∈ N\{0}, {x 1 , . . . , x n } ⊆ X and {λ 1 , . . . , λ n } ⊆ N}. It is clear that this submonoid is the smallest (with respect to set inclusion) submonoid of (N k , +) containing X and it is the intersection of all submonoids of (N k , +) containing X.
If M = X , we will say that X is a system of generators There are submonoids of (N k , +) which are not finitely generated for all k ≥ 2 (see Exercise 2 from [7, Chapter 3]). The following result is Corollary 2.8 from [8].
Lemma 5. Every submonoid of (N, +) has a unique minimal system of generators. Moreover, this system is finite.
From the proof of this result we obtain the following consequence.
Corollary 9. Let p, q ∈ N and let S be a numerical semigroup such that p ≤ m(S). Then the following conditions are equivalent: (2) There is a plane and finitely generated monoid We finish this section by giving a very useful criterion to check whether or not a numerical semigroup is an A(p, q)-semigroup.
Corollary 10. Let p, q ∈ N and let S be a numerical semigroup such that p ≤ m(S) and msg(S) = {n 1 , . . . n e }. Then the following conditions are equivalent.

The Frobenius pseudo-variety A(p, q)[m]
By using Theorem 8, we can easily deduce that the concepts of A(0, 0)semigroup and numerical semigroup are the same. Therefore, for the rest of this section we suppose that p, q ∈ N and p + q = 0. The following result can be deduced from Lemma 6 and Theorem 8. Proof. Notice that T = S ∪ X for some X ⊆ N\S. To complete the proof it is enough to recall that N\S is finite.  T = m, 2m−p, . . . , 2m+q , as gcd{m, 2m−p, . . . , 2m+q} = 1, T is a numerical semigroup. Then A(p, q)[m] ⊆ {S | S is a numerical semigroup and T ⊆ S}. According to Lemma 13 this last set is finite; therefore A(p, q)[m] is finite.
In the following result, we prove that if A(p, q)[m] = ∅, then A(p, q)[m] is in fact a Frobenius pseudo-variety. We define the directed graph G (A(p, q)

[m]) as follows: A(p, q)[m] is its set of vertices and (S, T ) ∈ A(p, q)[m] × A(p, q)[m] is an edge if S ∪ {F(S)} = T.
As a consequence of Lemma 11 from [3], we have the following result.

Then G(A(p, q)[m]) is a tree with root Δ(m).
A tree can be built recursively starting from the root and connecting, through an edge, the already built vertices with their children. Hence, it is very interesting to know how the children of arbitrary vertices in a tree are.
The following result is a consequence of Lemma 12 and Theorem 3 of [3].

Then the set formed by the children of a vertex S in the tree G(A(p, q)[m]) is {S\{x} | x ∈ msg(S), x > F(S) and S\{x} ∈ A(p, q)[m]} .
The following result is Lemma 1.7 from [6].  Example 20. Now we are going to recursively build the tree G (A(1, 0) [5]), starting in Δ(5) = 5, 6, 7, 8, 9 . Observe that by Propositions 17 and 19, the set formed by the children of S in this tree is Besides, note that the number which appears on the edge in the following figure represents the element that we have removed from the father vertex to obtain its corresponding child. It is clear that this number matches the Frobenius number of the child as well.
Theorem 23. Let p, q ∈ N such that p + q = 0, ∅ = X ⊆ N\{0, 1} and m = min(X) . Then there is an A(p, q)-semigroup containing X if and only if p ≤  m. Moreover, in this case A(p, q) A(p, q)[m] containing X.

(X) is the intersection of all elements of
Proof. Let S be an A(p, q)-semigroup such that X ⊆ S. Then m(S) ≤ m. By applying Lemma 12, we deduce that p ≤ m(S) or p > m(S) and S = N. If p ≤ m(S), then p ≤ m. If p > m(S) and S = N, then we deduce that p = 2. Therefore, p = 2 ≤ m.
Conversely, by applying Theorem 8, we easily deduce that {0, m, →} is an A(p, q)-semigroup containing X. Let Corollary 24. Let p, q ∈ N such that p+q = 0, ∅ = X ⊆ N\{0, 1}, m = min(X) and p ≤ m. Then A(p, q)(X) is an A(p, q)-semigroup. A(p, q)[m] containing X. By Proposition 14, the set A(p, q)[m] is finite. Therefore, A(p, q)(X) is the intersection of a finite number of elements belonging to A(p, q) [m]. By applying now Theorem 15 we can conclude that A(p, q)(X) is an A(p, q)-semigroup.

Proof. By Theorem 23, A(p, q)(X) is the intersection of all the elements of
The following algorithm computes A(p, q)(X); Corollary 10 validates the operation.
Output: The minimal system of generators of A(p, q)(X).  Our next goal will be to present an algorithm that computes all the elements of A(p, q)[m] with a fixed genus. In order to do so we first need to introduce some concepts and results.

AC(p, q)-semigroups
Our main aim in this section will be to present formulas in terms of a, b, c y d to compute the Frobenius number and the genus of S(a, b, c, d Proof. Conditions 1) and 2) are trivial and condition 3) is deduced from [2,Corollary 8].
If q ∈ Q, we use the notation q = max{z ∈ Z | z ≤ q} and q = min{z ∈ Z | q ≤ z}. The following result is Proposition 9 from [2].