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Characterizing affine \({\mathcal {C}}\)-semigroups

Abstract

Let \({\mathcal {C}} \subset {\mathbb {N}}^p\) be a finitely generated integer cone and \(S\subset {\mathcal {C}}\) be an affine semigroup such that the real cones generated by \({\mathcal {C}}\) and by S are equal. The semigroup S is called \({\mathcal {C}}\)-semigroup if \({\mathcal {C}}\setminus S\) is a finite set. In this paper, we characterize the \({\mathcal {C}}\)-semigroups from their minimal generating sets, and we give an algorithm to check if S is a \({\mathcal {C}}\)-semigroup and to compute its set of gaps. We also study the embedding dimension of \({\mathcal {C}}\)-semigroups obtaining a lower bound for it, and introduce some families of \({\mathcal {C}}\)-semigroups whose embedding dimension reaches our bound. In the last section, we present a method to obtain a decomposition of a \({\mathcal {C}}\)-semigroup into irreducible \({\mathcal {C}}\)-semigroups.

Introduction

An affine semigroup \(S\subset {\mathbb {N}}^p\) is called \({\mathcal {C}}_S\)-semigroup if \({\mathcal {C}}_S\setminus S\) is a finite set where \({\mathcal {C}}_S\subset {\mathbb {N}}^p\) is the minimal integer cone containing it. These semigroups are a natural generalization of numerical semigroups, and several of their invariants can be generalized. For a given numerical semigroup G, it is well-known that \({\mathbb {N}}\setminus G\) is finite; in fact, \(G\subset {\mathbb {N}}\) is a numerical semigroup if it is a submonoid of \({\mathbb {N}}\) and \({\mathbb {N}}\setminus G\) is finite (for topics related with numerical semigroups see [13] and the references therein). In general, it does not happen for affine semigroups.

\({\mathcal {C}}\)-semigroups are introduced in [8], where the authors study several properties about them (for example, an extended Wilf’s conjecture for \({\mathcal {C}}\)-semigroups is given). These semigroups appear in different contexts: when the integer points in an infinite family of some homothetic convex bodies in \({\mathbb {R}}^p_{\ge }\) are considered (see, for instance, [9, 10] and the references therein), or when the non-negative integer solutions of some modular Diophantine inequality are studied (see [5]), et cetera. In case the cone \({\mathcal {C}}\) is \({\mathbb {N}}^p\), \({\mathbb {N}}^p\)-semigroups are called generalized numerical semigroups and they were introduced in [6]. Recently, in [11] it is proved that the minimal free resolution of the associated algebra to any \({\mathcal {C}}\)-semigroup has maximal projective dimension possible.

In this context, \({\mathbb {N}}^p\)-semigroups are characterized in [3], but the general problem was opened, given any affine semigroup S, how to detect if Sis or not a \({\mathcal {C}}_S\)-semigroup? The primary goal of this work is to determine the conditions that any affine semigroup given by its minimal set of generators has to verify to be a \({\mathcal {C}}_S\)-semigroup. We solve this problem in Theorem 9, and in Algorithm 1 we provide a computational way to check it.

Another open problem is to compute the set of gaps of any \({\mathcal {C}}\)-semigroup defined by its minimal generating set. We solve this problem by means of setting a finite subset of \({\mathcal {C}}\) containing all the gaps of a given \({\mathcal {C}}\)-semigroup. Algorithm 2 computes the set of gaps of the given \({\mathcal {C}}\)-semigroup.

In this paper, we also go in-depth to study the embedding dimension of \({\mathcal {C}}\)-semigroups. In [8, Theorem 11], a lower bound of the embedding dimension of \({\mathbb {N}}^p\)-semigroups is provided, and some families of \({\mathbb {N}}^p\)-semigroups reaching this bound are given. Besides, in [8, Conjecture 12], it is proposed a conjecture about a lower bound for the embedding dimension of any \({\mathcal {C}}\)-semigroup. In Sect. 5, we introduce a lower bound of the embedding dimension of any \({\mathcal {C}}\)-semigroup, and some families of \({\mathcal {C}}\)-semigroups whose embedding dimension is equal to this new bound.

An important problem in Semigroup Theory is to determine some decomposition of a semigroup into irreducible semigroups (for example, see [13, Chapter 3] for numerical semigroups, or its generalization for \({\mathbb {N}}^p\)-semigroups in [2]). We propose an algorithm to compute a decomposition of any \({\mathcal {C}}\)-semigroup into irreducible \({\mathcal {C}}\)-semigroups.

The results of this work are illustrated with several examples. To this aim, we have used third-party software, such as Normaliz [4], and the libraries CharacterizingAffineCSemigroup and Irreducible [7] developed by the authors in Python [12].

The content of this work is organized as follows. Section 2 introduces the initial definitions and notations used throughout the paper, mainly related to finitely generated cones. In Sect. 3, a characterization of \({\mathcal {C}}\)-semigroups is provided, and an algorithm to check if an affine semigroup is a \({\mathcal {C}}\)-semigroup. Section 4 is devoted to give an algorithm to compute the set of gaps of a \({\mathcal {C}}\)-semigroup. Section 5 makes a study of the minimal generating sets of \({\mathcal {C}}\)-semigroups formulating explicitly a lower bound for their embedding dimensions. Finally, in Sect. 6 an algorithm for computing a decomposition of a \({\mathcal {C}}\)-semigroup into irreducible \({\mathcal {C}}\)-semigroups is presented.

Preliminaries

The sets of real numbers, rational numbers, integer numbers and the non-negative integer numbers are denoted by \({\mathbb {R}}\), \({\mathbb {Q}}\), \({\mathbb {Z}}\) and \({\mathbb {N}}\), respectively. Given a subset A of \({\mathbb {R}}\), \(A_\ge \) is the set of elements in A greater than or equal to zero. For any \(n\in {\mathbb {N}}\), [n] denotes the set \(\{1,\ldots n\}\). Given an element x in \({\mathbb {R}}^n\), \(||x||_1\) denotes the sum of the absolute value of its entries, that is, its 1-norm. In this paper we assume the set \(\{{\mathbf {e}}_1,\ldots , {\mathbf {e}}_p\}\) is the canonical basis of \({\mathbb {R}}^p\).

For a non empty subset B of \({\mathbb {R}}_\ge ^p\), we define the cone generated by B:

$$\begin{aligned} L(B){:}{=} \left\{ \sum _{i=1}^n \lambda _i {\mathbf {b}}_i \mid n\in {\mathbb {N}}, \{{\mathbf {b}}_1,\ldots ,{\mathbf {b}}_n\}\subset B,\text { and } \lambda _i \in {\mathbb {R}}_{\ge }, \forall i \in [n] \right\} . \end{aligned}$$

Given a real cone \({\mathcal {C}}\subset {\mathbb {R}}^p_\ge \), it is well-known that \({\mathcal {C}}\cap {\mathbb {N}}^p\) is finitely generated if and only if there exists a rational point in each extremal ray of \({\mathcal {C}}\). Moreover, any subsemigroup of \({\mathcal {C}}\) is finitely generated if and only if there exists an element in the semigroup in each extremal ray of \({\mathcal {C}}\). A good monograph about rational cones and affine monoids is [1]. From now on, we assume that the integer cones considered in this work are finitely generated.

Definition 1

Given an integer cone \({\mathcal {C}}\subset {\mathbb {N}}^p\), an affine semigroup \(S\subset {\mathcal {C}}\) is said to be a \({\mathcal {C}}\)-semigroup if \({\mathcal {C}}\setminus S\) is a finite set. If the cone \({\mathcal {C}}={\mathbb {N}}^p\), a \({\mathcal {C}}\)-semigroup is called \({\mathbb {N}}^p\)-semigroup.

Fix a finitely generated semigroup \(S\subset {\mathbb {N}}^p\), we denote by \({\mathcal {C}}_S\) the integer cone \(L(S)\cap {\mathbb {N}}^p\). Note that, if S is a \({\mathcal {C}}\)-semigroup, the cone \({\mathcal {C}}\) is \({\mathcal {C}}_S\). Obviously, a unique cone corresponds to infinite different semigroups.

The cone L(S) is a polyhedron and we denote by \(\{h_1(x)=0,\ldots , h_t(x)=0\}\) the set of its supported hyperplanes. We suppose \(L(S)=\{ x\in {\mathbb {R}}_\ge ^d \mid h_1(x)\ge 0, \ldots , h_t(x)\ge 0\}\). Unless otherwise stated, the considered coefficients of each \(h_i(x)\) are integers and relatively primes.

Assume L(S) has q extremal rays denoted by \(\tau _1,\ldots ,\tau _q\). Then, each \(\tau _i\) is determined by the set of linear equations \(H_i{:}{=}\{h_{j^{(i)}_1}(x)=0,\ldots , h_{j^{(i)}_{p-1}}(x)=0\}\) where \(J_i{:}{=}\{j^{(i)}_1<\cdots <j^{(i)}_{p-1}\}\subset [t]\) is the index set of the supported hyperplanes containing \(\tau _i\). So, for each \(i\in [q]\), there exists the minimal non-negative integer vector \({\mathbf {a}}_i\) such that \(\tau _i=\{\lambda {\mathbf {a}}_i\mid \lambda \in {\mathbb {R}}_\ge \}\). The set \(\{{\mathbf {a}}_1,\ldots , {\mathbf {a}}_q\}\) is a generating set of L(S).

Note that a necessary condition for S to be a \({\mathcal {C}}_S\)-semigroup is the set \(\tau _i \cap ({\mathcal {C}}_S\setminus S)\) is finite for all \(i\in [q]\).

From each extremal ray \(\tau _i\) of L(S), we define \(\upsilon _{i}(\alpha )\) as the parallel line to \(\tau _i\) given by the solutions of the linear equations \(\bigcup _{j\in J_i} \{h_j(x)=\alpha _j\}\) where \(\alpha =(\alpha _{j_1},\ldots ,\alpha _{j_{p-1}})\in {\mathbb {Z}}^{p-1}\). For every integer point \(P\in {\mathbb {Z}}^p\) and \(i\in [q]\), there exists \(\alpha \in {\mathbb {Z}}^{p-1}\) such that P belongs to \(\upsilon _{i}(\alpha )\); if \(P\in {\mathcal {C}}_S\), \(\alpha \in {\mathbb {N}}^{p-1}\). We denote by \(\Upsilon _i(P)\) the element \((h_{j^{(i)}_1}(P),\ldots , h_{j^{(i)}_{p-1}}(P))\in {\mathbb {N}}^{p-1}\) with \(J_i=\{j^{(i)}_1<\cdots <j^{(i)}_{p-1}\}\), \(P\in {\mathcal {C}}_S\) and \(i\in [q]\). Note that for any \(P\in {\mathcal {C}}_S\), \(P\in \upsilon _i(\alpha )\) if and only if \(\alpha =\Upsilon _i(P)\).

Since all the semigroups appearing in this work are finitely generated, from now on, we omit the term affine when affine semigroups are considered.

An algorithm to detect if a semigroup is a \({\mathcal {C}}\)-semigroup

In this section, we study the conditions that a semigroup has to satisfy to be a \({\mathcal {C}}\)-semigroup. This characterization depends on the minimal set of generators of the given semigroup.

Let \(S\subset {\mathbb {N}}^p\) be the affine semigroup minimally generated by \(\Lambda _S=\{{\mathbf {s}}_1,\ldots , {\mathbf {s}}_q,{\mathbf {s}}_{q+1}, \ldots , {\mathbf {s}}_n\}\) and \(\tau _1,\ldots ,\tau _q\) be the extremal rays of L(S). Assume that for every \(i\in [q]\), \(\tau _i \cap ({\mathcal {C}}_S\setminus S)\) is finite and \({\mathbf {s}}_i\) is the minimum (respect to the natural order) element in \(\Lambda _S\) belonging to \(\tau _i\). We denote by \({\mathbf {f}}_i\) the maximal element in \(\tau _i \cap ({\mathcal {C}}_S\setminus S)\) with respect to the natural order in \({\mathbb {N}}^p\). Recall that \({\mathbf {a}}_i\) is the minimal non-negative integer vector defining \(\tau _i\), and let \({\mathbf {c}}_i\in S\) be the element \({\mathbf {f}}_i+{\mathbf {a}}_i\). In case \(\tau _i \cap ({\mathcal {C}}_S\setminus S)=\emptyset \), we fix \({\mathbf {f}}_i=-{\mathbf {a}}_i\). The elements \({\mathbf {f}}_i\) and \({\mathbf {c}}_i\) are a generalization on the semigroup \(\tau _i \cap S\) of the concepts Frobenius number and conductor of a numerical semigroup; for numerical semigroups, the Frobenius number is the maximal natural number that is not in the semigroup, and the conductor is Frobenius number plus one (see [13, Chapter 1]). Hence, we call Frobenius element and conductor of the semigroup \(\tau _i\cap S\) the elements \({\mathbf {f}}_i\) and \({\mathbf {c}}_i\), respectively. One easy but important property of S is for every \(P\in S\), \(P+{\mathbf {c}}_i+\lambda {\mathbf {a}}_i\in S\) for any \(i\in [q]\) and \(\lambda \in {\mathbb {N}}\).

Note that \(\tau _i\cap {\mathbb {N}}^p\) is equal to \(\{\lambda {\mathbf {a}}_i\mid \lambda \in {\mathbb {N}}\}\). So, there exists \(S_i\subset {\mathbb {N}}\) such that \(\tau _i\cap S=\{\lambda {\mathbf {a}}_i\mid \lambda \in S_i \}\). If we assume that \(\tau _i \cap ({\mathcal {C}}_S\setminus S)\) is finite, it is easy to prove that \(S_i\) is a numerical semigroup.

Lemma 2

The \(\tau _i\)-semigroup \(\tau _i \cap S\) is isomorphic to the semigroup \(S_i=\{\lambda \in {\mathbb {N}}\mid \lambda {\mathbf {a}}_i\in S\}\). Moreover \(\tau _i\cap ({\mathcal {C}}_S\setminus S)\) is finite if and only if \(S_i\) is a numerical semigroup.

Proof

Consider the isomorphism \(\varphi :\tau _i \cap S\rightarrow S_i\) with \(\varphi ({\mathbf {w}}){:}{=}\lambda \) such that \({\mathbf {w}}=\lambda {\mathbf {a}}_i\). The second statement holds since \(\tau _i\cap ({\mathcal {C}}_S\setminus S)=\{\lambda {\mathbf {a}}_i\in {\mathbb {N}}\mid \lambda {\mathbf {a}}_i\not \in S,\lambda \in {\mathbb {N}}\}\). \(\square \)

Corollary 3

Given the semigroup \(\tau _i \cap S\), \({\mathbf {f}}_i\) is equal to \(f{\mathbf {a}}_i\) and \({\mathbf {c}}_i=c\,{\mathbf {a}}_i\) where f and c are the Frobenius number and the conductor of the numerical semigroup \(S_i\), respectively.

To test whether \(\tau _i \cap ({\mathcal {C}}_S\setminus S)\) is finite, the following result can be used.

Lemma 4

Let \(S\subset {\mathbb {N}}^p\) be a semigroup and \(\tau \) be an extremal ray of L(S) satisfying \(\tau \cap {\mathbb {N}}^p=\{\lambda {\mathbf {a}}\mid \lambda \in {\mathbb {N}}\}\) with \({\mathbf {a}}\in {\mathbb {N}}^p\). Then, \(\tau \cap ({\mathcal {C}}_S\setminus S)\) is finite if and only if \(\gcd (\{\lambda \mid \lambda {\mathbf {a}}\in \tau \cap \Lambda _S \})=1\).

Proof

Assume that \(\tau \cap ({\mathcal {C}}_S\setminus S)\) is finite and suppose that \(\gcd (\{\lambda \mid \lambda {\mathbf {a}}\in \tau \cap \Lambda _S \})=n\ne 1\). Hence, every element \(\lambda {\mathbf {a}}\) with \(\gcd (n,\lambda )=1\) does not belong to S, and then \(\tau \cap ({\mathcal {C}}_S\setminus S)\) is not finite.

Conversely if \(\gcd (\{\lambda \mid \lambda {\mathbf {a}}\in \tau \cap \Lambda _S \})=1\), then the semigroup \(S'=\{\lambda \in {\mathbb {N}}\mid \lambda {\mathbf {a}}\in S\}\) is a numerical semigroup. From the proof of Lemma 2, \(S'\) is isomorphic to \(\tau \cap S\). Therefore, \(\tau \cap ({\mathcal {C}}_S\setminus S)\) is finite. \(\square \)

To introduce the announced characterization, we need to define some subsets of L(S) and prove some of their properties. Associated to the integer cone \({\mathcal {C}}_S\), consider the sets \({\mathcal {A}}{:}{=}\{ \sum _{i\in [q]} \lambda _i{\mathbf {a}}_i \mid 0\le \lambda _i\le 1\}\cap {\mathbb {N}}^p\) and \({\mathcal {D}}{:}{=}\{ \sum _{i\in [q]} \lambda _i{\mathbf {s}}_i \mid 0\le \lambda _i\le 1\}\cap {\mathbb {N}}^p\).

Lemma 5

Given \(P\in {\mathcal {C}}_S\), there exist \(Q\in {\mathcal {A}}\) and \(\beta \in {\mathbb {N}}^q\) such that \(P=Q+\sum _{i\in [q]} \beta _i{\mathbf {a}}_i\). Moreover, \(\Upsilon _j(P)=\Upsilon _j(Q)+\sum _{i\in [q]}\beta _i \Upsilon _j(\mathbf {a_i})\) for every \(j\in [q]\).

Proof

Since \(P\in {\mathcal {C}}_S\), \(P=\sum _{i\in [q]}\mu _i{\mathbf {a}}_i\) with \(\mu _i\in {\mathbb {Q}}_\ge \). For each \(\mu _i\) there exists \(\lambda _i\in [0,1)\) satisfying \(\mu _i=\lfloor \mu _i\rfloor +\lambda _i\). Hence, \(P=Q +\sum _{i\in [q]}\lfloor \mu _i\rfloor {\mathbf {a}}_i\) where \(Q=\sum _{i\in [q]}\lambda _i{\mathbf {a}}_i=P-\sum _{i\in [q]}\lfloor \mu _i\rfloor {\mathbf {a}}_i\in {\mathcal {A}}\). Trivially, \(\Upsilon _j(P)\) is equal to \(\Upsilon _j(Q)+\sum _{i\in [q]}\beta _i \Upsilon _j({\mathbf {a}}_i)\) for every \(j\in [q]\). \(\square \)

For every \(i\in [q]\), consider \( T_i\subset {\mathbb {N}}^{p-1}\) the semigroup generated by the finite set \(\{\Upsilon _i(Q)\mid Q\in {\mathcal {A}}\}\) and let \(\Gamma _i\) be its minimal generating set. Note that the sets \({\mathcal {A}}\), \( T_i\) and \(\Gamma _i\) only depend on the cone \({\mathcal {C}}_S\), and \(0\in T_i\), since \({\mathbf {a}}_i\in {\mathcal {A}}\). The relationships between the elements in \({\mathcal {C}}_S\) and S, and the elements belonging to \(T_i\) and \(\Gamma _i\) are explicitly determined in the following results for each \(i\in [q]\).

Lemma 6

Let P be an element in \({\mathcal {C}}_S\) such that \(P\in \upsilon _i(\alpha )\) for some \(\alpha \in {\mathbb {N}}^{p-1}\), then \(\alpha \in T_i\).

Proof

By definition, \(P\in \upsilon _i(\alpha )\) means that \(\alpha =\Upsilon _i(P)\). Using Lemma 5, \(P=Q+\sum _{j\in [q]} \beta _j{\mathbf {a}}_j\) with \(Q,{\mathbf {a}}_1,\ldots ,{\mathbf {a}}_q\in {\mathcal {A}}\) and \(\beta _1,\ldots ,\beta _q\in {\mathbb {N}}\). Therefore, \(\Upsilon _i (P)=\Upsilon _i(Q)+\sum _{j\in [q]}\beta _j \Upsilon _i({\mathbf {a}}_j)\in T_i\). \(\square \)

Corollary 7

For every \(\alpha \in T_i\), \({\mathcal {C}}_S\cap \upsilon _i(\alpha )\ne \emptyset \) if and only if \({\mathcal {C}}_S\cap \upsilon _i(\beta )\ne \emptyset \) for all \(\beta \in \Gamma _i\).

Proof

Since \(\Gamma _i\subset T_i\), if for all \(\alpha \in T_i\), \({\mathcal {C}}_S\cap \upsilon _i(\alpha )\ne \emptyset \) then \({\mathcal {C}}_S\cap \upsilon _i(\beta )\ne \emptyset \) for all \(\beta \in \Gamma _i\).

Assume that \({\mathcal {C}}_S\cap \upsilon _i(\beta )\ne \emptyset \) for all \(\beta \in \Gamma _i\) and let \(\alpha \) be an element in \( T_i\). Then, there exist \(\beta _1,\ldots ,\beta _k\in \Gamma _i\), \(\mu _1,\ldots ,\mu _k\in {\mathbb {N}}\) and \(Q_1,\ldots ,Q_k\in {\mathcal {A}}\) such that \(\alpha =\sum _{j\in [k]}\mu _j\beta _j\) and \(\Upsilon _i(Q_j)=\beta _j\) for \(j\in [k]\). Note that \(P=\sum _{j\in [k]}\mu _jQ_j\in {\mathcal {C}}_S\) belongs to \(\upsilon _i(\alpha )\). \(\square \)

Corollary 8

For every \(\alpha \in T_i\), \(S\cap \upsilon _i(\alpha )\ne \emptyset \) if and only if \(S\cap \upsilon _i(\beta )\ne \emptyset \) for all \(\beta \in \Gamma _i\).

Proof

Since \(S\cap \upsilon _i(\beta )\ne \emptyset \) for all \(\beta \), then there exists \(Q_1,\ldots ,Q_k\in {\mathcal {A}}\) such that \(\Upsilon _i(Q_j)=\beta _j\) for \(j\in [k]\). Thus, the proof of this corollary is analogous to the proof of Corollary 7. \(\square \)

Note that if \(P\in S\cap \upsilon _i(\alpha )\) for some \(\alpha \in {\mathbb {N}}^{p-1}\) and \(i\in [q]\), then \(P+{\mathbf {c}}_i+\lambda {\mathbf {a}}_i\in S\) and \(\Upsilon _i(P+{\mathbf {c}}_i+\lambda {\mathbf {a}}_i)=\alpha \) for all \(\lambda \in {\mathbb {N}}\).

Now, we introduce a characterization of \({\mathcal {C}}\)-semigroups. This characterization depends on the minimal generating set of the given semigroup. Besides, from its proof, we provide an algorithm for checking if a semigroup is a \({\mathcal {C}}\)-semigroup (Algorithm 1). Note that most of the parts of Algorithm 1 can be parallelized at least in q stand-alone processes.

Theorem 9

A semigroup S minimally generated by \(\Lambda _S=\{{\mathbf {s}}_1,\ldots ,{\mathbf {s}}_n\}\) is a \({\mathcal {C}}_S\)-semigroup if and only if:

  1. 1.

    \(\tau _i \cap ({\mathcal {C}}_S\setminus S)\) is finite for all \(i\in [q]\).

  2. 2.

    \(\Lambda _S\cap \upsilon _i(\alpha )\ne \emptyset \) for all \(\alpha \in \Gamma _i\) and \(i\in [q]\).

Proof

Let S be a \({\mathcal {C}}_S\)-semigroup. Trivially, \(\tau _i \cap ({\mathcal {C}}_S\setminus S)\) is finite for all \(i\in [q]\). Now let \(i\in [q]\) and \(\alpha \in \Gamma _i\), we probe that \(\Lambda _S\cap \upsilon _i(\alpha )\ne \emptyset \). Since \(\alpha \in \Gamma _i\), there exists \(Q\in {\mathcal {A}}\) such that \(\alpha =\Upsilon _i(Q)\). Besides, \(Q+\lambda {\mathbf {a}}_i\in {\mathcal {C}}_S\) and \(\Upsilon _i(Q+\lambda {\mathbf {a}}_i)=\alpha \) for all \(\lambda \in {\mathbb {N}}\). For some \(\lambda \in {\mathbb {N}}\), \(Q+\lambda {\mathbf {a}}_i\) has to be in S (S is \({\mathcal {C}}_S\)-semigroup), that is to say, \(Q+\lambda {\mathbf {a}}_i=\sum _{j\in [n]} \mu _j {\mathbf {s}}_j\) with \(\mu _1,\ldots ,\mu _n\in {\mathbb {N}}\). Therefore, \(\alpha =\Upsilon _i (Q+\lambda {\mathbf {a}}_i)= \sum _{j\in [n]} \mu _j \Upsilon _i({\mathbf {s}}_j)\). By Lemma 5, for all \(j\in [n]\), \({\mathbf {s}}_j=Q_j+\sum _{k\in [q]} \beta _{jk}{\mathbf {a}}_k\) for some \(Q_j\in {\mathcal {A}}\) and \(\beta _{j1},\ldots ,\beta _{jq}\in {\mathbb {N}}\). So, \(\alpha = \sum _{j\in [n]} \mu _j \Upsilon _i(Q_j+\sum _{k\in [q]} \beta _{jk}{\mathbf {a}}_k)= \sum _{j\in [n]} \mu _j \Upsilon _i(Q_j) +\sum _{j\in [n]} \sum _{k\in [q]} \mu _j \beta _{jk}\Upsilon _i( {\mathbf {a}}_k)\). Since \(\alpha \) is a minimal generator of \( T_i\), \(\sum _{j\in [n]} \mu _j +\sum _{j\in [n]} \sum _{k\in [q]\setminus \{i\}} \mu _j \beta _{jk}=1\). So \(\beta _{jk}=0\) for all \(j\in [n]\) and for all \(k\in [q]\setminus \{i\}\), and there exists \(l\in [n]\) such that \(\mu _l=1\) and \(\mu _j=0\) for all \(j\in [n]\setminus \{l\}\). Hence, there exists \({\mathbf {s}}\in \Lambda _S\) such that \(\Upsilon _i({\mathbf {s}})=\alpha \) and then \(\Lambda _S\cap \upsilon _i(\alpha )\ne \emptyset \).

Conversely, we assume that \(\forall i\in [q]\) and \(\forall \alpha \in \Gamma _i\), \(\tau _i \cap ({\mathcal {C}}_S\setminus S)\) is finite and \(\Lambda _S\cap \upsilon _i(\alpha )\ne \emptyset \) (recall that \({\mathbf {c}}_i={\mathbf {f}}_i+{\mathbf {a}}_i\)). Let Q be an element in \({\mathcal {D}}\). By Lemmas 5 and 6, \(Q\in \alpha _i(\Upsilon _i(Q))\) and \(\Upsilon _i(Q) \in T_i\). If \(Q\in \tau _i\) for some \(i\in [q]\), then \(v_i(\Upsilon _i(Q))=\tau _i\) and, by the first condition, \(S\cap v_i(\Upsilon _i(Q))\ne \emptyset \). If Q is not in any ray, by the second condition and Corollary 8, \(S\cap v_i(\Upsilon _i(Q))\ne \emptyset \). Therefore, for every \(Q\in {\mathcal {D}}\), the line \(\upsilon _i(\Upsilon _i(Q))\) includes a unique non zero minimum (respect 1-norm) point belonging to S. Denote by \(\{{\mathbf {m}}_{i1},\ldots , {\mathbf {m}}_{id_i}\}\) the set obtained from the union of above points for the different elements in \({\mathcal {D}}\) (some of these elements belong to \(\Lambda _S\)). Note that \({\mathbf {m}}_{ij}+{\mathbf {c}}_i+\lambda {\mathbf {a}}_i\in S\) for all \(j\in [d_i]\) and \(\lambda \in {\mathbb {N}}\). Consider \(n_i{:}{=}\max \{||{\mathbf {m}}_{i1}+{\mathbf {c}}_i||_1,\ldots , ||{\mathbf {m}}_{id_i}+{\mathbf {c}}_i ||_1\}\), and \({\mathbf {x}}_i\) the minimum element (respect to the 1-norm) in \(\tau _i\cap S\) such that \(||{\mathbf {x}}_i||_1\) is greater than or equal to \(n_i\). The set \({\mathcal {D}}_i{:}{=}{\mathcal {D}}+{\mathbf {x}}_i\) satisfies that \({\mathcal {D}}_i \cap S={\mathcal {D}}_i \cap {\mathcal {C}}_S={\mathcal {D}}_i\). Consider \({\mathbf {a}}\in {\mathbf {x}}_i+{\mathcal {C}}_S\), proceeding as in the proof of Lemma 5, \({\mathbf {a}} = {\mathbf {x}}_i + P + \sum _{j\in [q]} \beta _j {\mathbf {s}}_j\) for some \(P\in {\mathcal {D}}\) and \(\beta _1,\ldots ,\beta _q\in {\mathbb {N}}\), and hence, \({\mathbf {x}}_i+{\mathcal {C}}_S\subset S\). We define the bounded set \({\mathcal {X}}{:}{=} \{\sum _{i\in [q]}\lambda _i{\mathbf {x}}_i \mid 0\le \lambda _i\le 1 \} \). Since \({\mathbf {x}}_i+{\mathcal {C}}_S\subset S\) for every \(i\in [q]\) and \(L({\mathcal {C}}_S)=\{ \sum _{i\in [q]} \lambda _i {\mathbf {x}}_i | \lambda _i\in {\mathbb {R}}_\ge \}\), \({\mathcal {C}}_S\setminus S\subset {\mathcal {X}}\). Therefore, S is a \({\mathcal {C}}_S\)-semigroup. \(\square \)

figure a

Example 10 illustrates Theorem 9 and Algorithm 1.

Example 10

Let \(S\subset {\mathbb {N}}^3\) be the semigroup minimally generated by

$$\begin{aligned} \Lambda _S=\{ (2, 0, 0), (4, 2, 4), (0, 1, 0), (3, 0, 0), (6, 3, 6), (3, 1, 1), (4, 1, 1),\\ (3, 1, 2), (1, 1, 0), (3, 2, 3), (1, 2, 1) \}. \end{aligned}$$

The cone L(S) is \(\langle (1,0,0),(2,1,2),(0,1,0) \rangle _{{\mathbb {R}}_\ge }\) and its supported hyperplanes are \(h_1(x,y,z)\equiv 2y-z =0\), \(h_2(x,y,z)\equiv x-z =0\) and \(h_3(x,y,z)\equiv z=0\). Recall \({\mathcal {C}}_S=L(S)\cap {\mathbb {N}}^3\). By \({\mathbf {a}}_1\), \({\mathbf {a}}_2\) and \({\mathbf {a}}_3\) we denote the vectors (1, 0, 0), (2, 1, 2) and (0, 1, 0) respectively, and \(\tau _1\), \(\tau _2\) and \(\tau _3\) are the extremal rays with sets of defining equations \(\{h_1(x,y,z)=0,h_3(x,y,z)=0\}\), \(\{h_1(x,y,z)=0,h_2(x,y,z)=0\}\) and \(\{h_2(x,y,z)=0,h_3(x,y,z)=0\}\), respectively. Hence, \(S_1=(\tau _1\setminus \{(1,0,0)\})\cap {\mathbb {N}}^3\), \(S_2=\tau _2\setminus \{(2,1,2)\}\cap {\mathbb {N}}^3\) and \(S_3=\tau _3\cap {\mathbb {N}}^3\), and the first condition in Theorem 9 holds.

The set \({\mathcal {A}}\) is equal to

$$\begin{aligned} \{(0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (2, 1, 1), (2,1, 2),\nonumber \\ (2, 2, 2), (3, 1, 2), (3, 2, 2)\}, \end{aligned}$$
(1)

and

$$\begin{aligned} \Upsilon _1({\mathcal {A}})&=\{(0, 0), (0, 2), (1, 1), (2, 0), (2, 2)\}, \\ \Upsilon _2({\mathcal {A}})&=\{(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1)\}, \\ \Upsilon _3({\mathcal {A}})&=\{(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2) \}. \end{aligned}$$

Therefore, \(\Gamma _1=\{(0,2),(1,1),(2,0)\}\) and \(\Gamma _2=\Gamma _3=\{(0,1),(1,0)\}\).

Since \(\Upsilon _1(\{(3,1,1),(3,1,2),(1,1,0)\})=\Gamma _1\), \(\Upsilon _2(\{(3,1,2),(3,2,3)\})=\Gamma _2\), and \(\Upsilon _3(\{(1,1,0),(1,2,1)\})=\Gamma _3\), S satisfies the second condition in Theorem 9. Hence, S is a \({\mathcal {C}}_S\)-semigroup.

By using our implementation of Algorithm 1, we can confirm that S is a \({\mathcal {C}}_S\)-semigroup,

figure b

To finish this section, it should be pointed out that there exist some special cases of semigroups where Theorem 9 can be simplified: \({\mathbb {N}}^p\)-semigroups and two-dimensional case.

Note that, if the integer cone \({\mathcal {C}}_S\) is \({\mathbb {N}}^p\), its supported hyperplanes are \(\{x_1=0,\ldots ,x_p=0\}\). Moreover, since its extremal rays are the axes, \(\tau _i\equiv \{\lambda {\mathbf {e}}_i\mid \lambda \in {\mathbb {Q}}_\ge \}\) is determined by the equations \(\cup _{j\in [p]\setminus \{i\}}\{x_{j}=0\}\), and for any canonical generator \({\mathbf {e}}\) of \({\mathbb {N}}^{p-1}\), there exists P in \({\mathbb {N}}^p\) such that \(\Upsilon _i(P)={\mathbf {e}}\). Furthermore, \(\cup _{j\in [p]\setminus \{i\}} \{\Upsilon _i({\mathbf {e}}_j)\}\) is the canonical basis of \({\mathbb {N}}^{p-1}\). Hence, \(\Gamma _1=\cdots =\Gamma _p\) is the canonical basis of \({\mathbb {N}}^{p-1}\). From previous considerations, the same characterization of \({\mathbb {N}}^p\)-semigroups in [3, Theorem 2.8] is obtained from Theorem 9.

Corollary 11

A semigroup S minimally generated by \(\Lambda _S\) is an \({\mathbb {N}}^p\)-semigroup if and only if:

  1. 1.

    for all \(i\in [p]\), the non null entries of the elements in \(\tau _i \cap \Lambda _S\) are coprime, or \({\mathbf {s}}_i={\mathbf {e}}_i\).

  2. 2.

    for all \(i,j\in [p]\) with \(i\ne j\), \({\mathbf {e}}_i+\lambda _j{\mathbf {e}}_j\in \Lambda _S\) for some \(\lambda _j\in {\mathbb {N}}\).

Focus on two dimensional case, note that the extremal rays and the supported hyperplanes of a cone are equal. Since for each extremal ray the coefficients of its defining linear equation are relatively primes, the linear equations \(h_1(x,y)=1\) and \(h_2(x,y)=1\) always have non-negative integer solutions. So, any semigroup \(S\subset {\mathbb {N}}^2\) is a \({\mathcal {C}}_S\)-semigroup if and only if \(\tau _i \cap ({\mathcal {C}}_S\setminus S)\) is finite for \(i=1,2\), and both sets \(\Lambda _S\cap \{h_1(x,y)=1\}\) and \(\Lambda _S\cap \{h_2(x,y)=1\}\) are non empty.

Set of gaps of \({\mathcal {C}}\)-semigroups

This section gives an algorithm to compute the set of gaps of a \({\mathcal {C}}\)-semigroup, i.e. the set \({\mathcal {H}}(S)={\mathcal {C}}_S\setminus S\). This algorithm is obtained from Theorem 9. To introduce such an algorithm, let us start by redefining some objects used to prove that theorem.

Given S a \({\mathcal {C}}_S\)-semigroup with q extremal rays, for any \(i\in [q]\), let \({\mathbf {c}}_i\) be the conductor of the semigroup \(\tau _i\cap S\). By Corollary 8, for any \(\alpha \in \Upsilon _i({\mathcal {D}})\) the intersection \(\upsilon _i(\alpha )\cap S\) is not empty. Hence, set \({\mathbf {m}}^{(i)}_\alpha \) the element in \(\upsilon _i(\alpha )\cap S\) with minimal 1-norm and \(\alpha \in \Upsilon _i({\mathcal {D}})\setminus \{0\}\). Note that \({\mathbf {m}}^{(i)}_\alpha + {\mathbf {c}}_i+\lambda {\mathbf {a}}_i\in S\) for all \(\lambda \in {\mathbb {N}}\). Let \(n_i{:}{=}||{\mathbf {c}}_i ||_1+ \max \big (\{||{\mathbf {m}}^{(i)}_\alpha ||_1\mid \alpha \in \Upsilon _i({\mathcal {D}}) \setminus \{0\} \}\big )\), and \({\mathbf {x}}_i\) the minimal element in \(\tau _i\cap S\) such that \(||{\mathbf {x}}_i||_1\) is greater than or equal to \(n_i\). The vector \({\mathbf {x}}_i\) can be computed as follows: let Q be the non-negative rational solution of the systems of linear equations \(\{x_1+\cdots +x_p=n_i, h_{j^{(i)}_1}(x)=0,\ldots , h_{j^{(i)}_{p-1}}(x)=0\}\) (recall that \(h_{j^{(i)}_1}(x)=0,\ldots , h_{j^{(i)}_{p-1}}(x)=0\) are the equations defining \(\tau _i\)), then \({\mathbf {x}}_i= \Big \lceil \frac{||Q||_1}{||{\mathbf {a}}_i||_1}\Big \rceil {\mathbf {a}}_i\).

By the proof of Theorem 9, \({\mathcal {C}}_S\setminus S\subset {\mathcal {X}}\), with \({\mathcal {X}}= \{\sum _{i\in [q]}\lambda _i{\mathbf {x}}_i \mid 0\le \lambda _i\le 1 \}\). Algorithm 2 shows the process to computed the set of gaps of S. Note that several of its steps can be computed in a parallel way.

figure c

We illustrate Algorithm 2 in the following example. Besides, we confirm our handmade computations by using our free software [7].

Example 12

Consider the \({\mathcal {C}}_S\)-semigroup S defined in example 10. So, \({\mathbf {s}}_1={\mathbf {c}}_1= (2,0,0)\), \({\mathbf {s}}_2={\mathbf {c}}_2= (4,2,4)\), \({\mathbf {s}}_3= (0,1,0)\) and \({\mathbf {c}}_3= (0,0,0)\). The set \({\mathcal {D}}\) is

$$\begin{aligned} \{ (0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (2, 0, 0), (2, 1, 0), (2, 1, 1),\\ (2, 1, 2), (2, 2, 2), (3, 1, 1), (3, 1, 2), (3, 2, 2), (3, 2, 3), (4, 1, 2), (4, 2, 2),\\ (4, 2, 3), (4, 2, 4), (4, 3, 4), (5, 2, 3), (5, 2, 4), (5, 3, 4), (6, 2, 4), (6, 3, 4) \}. \end{aligned}$$

For example, for the extremal ray \(\tau _1\), \(\Upsilon _1({\mathcal {D}})\) is the set

$$\begin{aligned} \{(0, 0), (0, 2), (0, 4), (1, 1), (1, 3), (2, 0), (2, 2), (2, 4)\}, \end{aligned}$$

and \(\cup _{\alpha \in \Upsilon _1({\mathcal {D}}) \setminus \{0\}} \{{\mathbf {m}}^{(1)}_\alpha \}\) is

$$\begin{aligned} \{ (0, 1, 0), (3, 1, 1), (3, 1, 2), (3, 2, 2), (3, 2, 3), (4, 2, 4), (4, 3, 4) \} \end{aligned}$$

For \(\tau _2\) and \(\tau _3\),

$$\begin{aligned} \cup _{\alpha \in \Upsilon _2({\mathcal {D}})\setminus \{0\}} \{{\mathbf {m}}^{(2)}_\alpha \}= \{(0, 1, 0), (3, 1, 2), (1, 1, 0), (3, 2, 3), (2, 0, 0),\\ (2, 1, 0), (6, 3, 5), (3, 1, 1)\} \\ \cup _{\alpha \in \Upsilon _3({\mathcal {D}})\setminus \{0\}} \{{\mathbf {m}}^{(3)}_\alpha \}= \{ (1, 1, 0), (1, 2, 1), (2, 0, 0), (2, 3, 1), (2, 4, 2),\\ (3, 1, 1), (3, 1, 2), (3, 2, 3), (4, 2, 2), (4, 3, 3), (4, 2, 4), (5, 3, 4), (6, 2, 4) \} \end{aligned}$$

Then \(n_1=13\), \(n_2=24\) and \(n_3=12\), and \({\mathbf {x}}_1= (14,0,0)\), \({\mathbf {x}}_2= (10,5,10)\) and \({\mathbf {x}}_3= (0,13,0)\). Therefore, the set of gaps of S is,

$$\begin{aligned} \{ (1,0,0), (1,1,1), (2,1,1), (2,1,2), (2,2,1), (2,2,2), (2,3,2),\\ (4,1,2), (4,2,3), (5,2,4), (5,3,5), (8,4,7) \}. \end{aligned}$$

By using our implementation of Algorithm 2, we obtain the same gaps:

figure d

Embedding dimension of \({\mathcal {C}}\)-semigroups

In [8], it is proved that the embedding dimension of an \({\mathbb {N}}^p\)-semigroup is greater than or equal to 2p, and this bound holds. Furthermore, a conjecture about a lower bound of embedding dimension of any \({\mathcal {C}}\)-semigroup is proposed. In this section, we determine a lower bound of the embedding dimension e(S) of a given \({\mathcal {C}}\)-semigroup S by studying its elements belonging to \({\mathcal {A}}\).

As in previous sections, let \({\mathcal {C}}\subset {\mathbb {N}}^p\) be a finitely generated cone and \(\tau _1,\ldots ,\tau _q\) its extremal rays. For any \(i\in [q]\), \({\mathbf {a}}_i\) is the generator of \(\tau _i\cap {\mathbb {N}}^p\), \({\mathcal {A}}\) is the finite set \(\{ \sum _{i\in [q]} \lambda _i{\mathbf {a}}_i \mid 0\le \lambda _i\le 1\}\cap {\mathbb {N}}^p\) and \(\Gamma _i\) denotes the minimal generating set of the semigroup \(T_i\subset {\mathbb {N}}^{p-1}\) generated by \(\Upsilon _i({\mathcal {A}})\). Given a \({\mathcal {C}}\)-semigroup S, consider \(\Lambda '_S{:}{=}\{{\mathbf {s}}_{t_1},\ldots ,{\mathbf {s}}_{t_k}\}\) the set of minimal generators of S belonging to \({\mathcal {A}}{\setminus \cup _{i\in [q]}\tau _i}\), and \(M_l{:}{=}\{i\in [q] \mid \Upsilon _{i}({\mathbf {s}}_{t_l})\in \Gamma _i\}\) for \(l\in [k]\).

The following result provides us with a lower bound for the embedding dimension of any \({\mathcal {C}}\)-semigroup.

Proposition 13

Given a \({\mathcal {C}}\)-semigroup \(S\subset {\mathbb {N}}^p\), then

$$\begin{aligned} \mathrm {e}(S)\ge \sum _{i\in [q]} (\mathrm {e}(S_i)+\mathrm {e}(T_i))+k-\sum _{i\in [k]}\sharp (M_i). \end{aligned}$$
(2)

Proof

From Theorem 9, for any \(i\in [q]\), there exist \(\mathrm {e}(S_i)\) minimal generators of S in \(\tau _i\). Moreover, for each element \(\gamma \in \Gamma _i\), there is at least an element of \(\Lambda _S\) in \(\upsilon _i(\gamma )\). Note that, for every \({\mathbf {s}}\in \Lambda _S\setminus {\mathcal {A}}\), there is no \(\gamma \in \Gamma _i\) and \(\gamma '\in \Gamma _j\) such that \({\mathbf {s}}\in \upsilon _i(\gamma ) \cap \upsilon _j(\gamma ')\), since for any \(i,j\in [q]\), \(\gamma \in \Gamma _i\) and \(\gamma '\in \Gamma _j\), the intersection \(\upsilon _i(\gamma ) \cap \upsilon _j(\gamma ')\) is empty or belongs to \({\mathcal {A}}\). However, if \({\mathbf {s}}\in \Lambda '_S\), then it is possible that \({\mathbf {s}}\) belongs to two (or more) different lines \(\upsilon _i(\gamma )\) and \(\upsilon _j(\gamma ')\) with \(\gamma \in \Gamma _i\) and \(\gamma '\in \Gamma _j\) (in that case, \(\upsilon _i(\gamma )\cap \upsilon _j(\gamma ')=\{{\mathbf {s}}\}\)). Thus, the value of \(\sharp (M_l)\) indicates the number of different lines \(v_i(\gamma _i)\) with \(\gamma _i\in \Gamma _i\) to which \({\mathbf {s}}_{t_l}\in \Lambda _S'\) belongs. So, counting the minimal amount of elements needed to have at least one minimal generator in each line \(\upsilon _i(\gamma )\) for each \(\gamma \in \Gamma _i\) and \(i\in [q]\), we have that the embedding dimension of S is greater than or equal to \(\sum _{i\in [q]} (\mathrm {e}(S_i)+\mathrm {e}(T_i))+k-\sum _{i\in [k]}\sharp (M_i).\) \(\square \)

Example 14

Consider the \({\mathcal {C}}_S\)-semigroup S given in example 10. In that case, \(\Lambda '_S=\{(3,1,2),(1,1,0)\}\), \(\sharp (M_1)=2\) (i.e. \(\Upsilon _i(3,1,2)\in \Gamma _i\) for \(i=1,2\)), and \(\sharp (M_2)=2\) (\(\Upsilon _1 (1,1,0) \in \Gamma _1\) and \(\Upsilon _2 (1,1,0) \in \Gamma _3\)). So, \(\sum _{i\in [q]} (\mathrm {e}(S_i)+\mathrm {e}(T_i))+k-\sum _{i\in [k]}\sharp (M_i)= 5 + 7 +{2} - 2-2= {10}\) that is smaller than \(\mathrm {e}(S)=11\).

Given any bound, the first interesting question about it is if the bound is reached for some \({\mathcal {C}}\)-semigroup. The answer is affirmative for (2), and this fact is formulated as follows.

Lemma 15

Let \({\mathcal {C}}\subset {\mathbb {N}}^p\) be an integer cone generated by \(\{{\mathbf {a}}_1,\ldots ,{\mathbf {a}}_q\}\) and let \(S_1,\ldots ,S_q\) be the non proper numerical semigroups minimally generated by \(\{n_1^{(i)},\ldots ,n_{\mathrm {e}(S_i)}^{(i)}\}\) for each \(i\in [q]\). Let \(\Lambda ''\subset {\mathcal {C}}\setminus \cup _{i\in [q]}\tau _i\) be a finite set satisfying:

  • for every \(\gamma \in \Gamma _i\) and \(i\in [q]\), there exists a unique \({\mathbf {d}}\in \Lambda ''\) such that \(\Upsilon _i({\mathbf {d}})=\gamma \),

  • for every \({\mathbf {d}}\in \Lambda ''\), \(\Upsilon _i({\mathbf {d}})\in \Gamma _i\) for some \(i\in [q]\).

Then, the embedding dimension of the \({\mathcal {C}}\)-semigroup S generated by

$$\begin{aligned} \Lambda '' \cup \bigcup _{i\in [q]} \{n_1^{(i)}{\mathbf {a}}_i,\ldots ,n_{\mathrm {e}(S_i)}^{(i)}{\mathbf {a}}_i\} \end{aligned}$$

is

$$\begin{aligned} \sum _{i\in [q]} (\mathrm {e}(S_i)+\mathrm {e}(T_i))+k-\sum _{i\in [k]}\sharp (M_i), \end{aligned}$$

where k is the cardinality of \(\Lambda '_S=\{{\mathbf {s}}_{1},\ldots ,{\mathbf {s}}_{k}\}\), the set of minimal generators of S belonging to \({\mathcal {A}}\setminus \cup _{i\in [q]}\tau _i\), and \(M_l=\{i\in [q] \mid \Upsilon _{i}({\mathbf {s}}_{l})\in \Gamma _i \}\) for \(l\in [k]\).

Proof

By the hypothesis, there are exactly \(\sum _{i\in [q]} \mathrm {e}(T_i)+k-\sum _{i\in [k]}\sharp (M_i)\) minimal generators of S outside its extremal rays, and \(\sum _{i\in [q]} \mathrm {e}(S_i)\) belonging to its extremal rays. \(\square \)

Example 16

Let \(S\subset {\mathbb {N}}^3\) be the semigroup minimally generated by

$$\begin{aligned} \Lambda _S=\{ (2, 0, 0), (4, 2, 4), (0, 2, 0), (3, 0, 0), (6, 3, 6), (0, 3, 0), (3, 1, 1),\\ (3, 1, 2), (1, 1, 0), (3, 2, 3), (1, 2, 1) \}. \end{aligned}$$

Note that the cone \({\mathcal {C}}_S\) is the same as the cone in example 10. So, \({\mathcal {A}}\), \(\Gamma _1\), \(\Gamma _2\) and \(\Gamma _3\) are the sets given in that example. For the semigroup S, \(\Upsilon _1(\{(3,1,1),(3,1,2),(1,1,0)\})=\Gamma _1\), \(\Upsilon _2(\{(3,1,2),(3,2,3)\})=\Gamma _2\) and \(\Upsilon _3(\{(1,1,0),(1,2,1)\})=\Gamma _3\). Since \((1,1,0),(3,1,2)\in {\mathcal {A}}\), \(\mathrm {e}(S)= 11= 6+7+2-2-2 = \sum _{i\in [3]} (\mathrm {e}(S_i)+\mathrm {e}(T_i))+2-\sum _{i\in [2]}\sharp (M_i)\).

Fix a cone \({\mathcal {C}}\), studying the different possibilities to select sets of points \(K\subset {\mathcal {C}}\) such that \(\cup _{i\in [q]}\Gamma _i\) is the union of the minimal generating set of the semigroup given by \(\cup _{Q\in K}\Upsilon _i(Q)\) (for i from 1 to q), we can state results like the following:

Corollary 17

Let \(S_1,\ldots ,S_q\) be the non proper numerical semigroups minimally generated by \(\{n_1^{(i)},\ldots ,n_{\mathrm {e}(S_i)}^{(i)}\}\) for each \(i\in [q]\), and \(\Lambda ''\subset {\mathcal {C}}\) satisfying the hypothesis of Lemma 15. Thus, if \(\Lambda ''\subset {\mathcal {C}}\setminus {\mathcal {A}}\), then the embedding dimension of the \({\mathcal {C}}\)-semigroup generated by \(\Lambda '' \cup \bigcup _{i\in [q]} \{n_1^{(i)}{\mathbf {a}}_i,\ldots ,n_{\mathrm {e}(S_i)}^{(i)}{\mathbf {a}}_i\}\) is \(\sum _{i\in [q]} (\mathrm {e}(S_i) + \mathrm {e}(T_i))\).

Finally, we illustrate the above result with an example.

Example 18

Let \(S\subset {\mathbb {N}}^3\) be the semigroup minimally generated by

$$\begin{aligned} \Lambda _S=\{(2, 0, 0), (4, 2, 4), (0, 2, 0), (3, 0, 0), (6, 3, 6), (0, 3, 0), (3, 1, 1),\\ (4, 1, 2), (5, 2, 4), (2, 1, 0), (1, 2, 0), (3, 2, 3), (1, 2, 1) \}. \end{aligned}$$

Again, the cone \({\mathcal {C}}_S\) is the cone appearing in example 10. Note that the elements (2, 0, 0) and (3, 0, 0) are in \(S_1\), (4, 2, 4) and (6, 3, 6) belong to \(S_2\), and (0, 2, 0) and (0, 3, 0) are in \(S_3\). Moreover, \(\Upsilon _1(\{ (3, 1, 1), (4, 1, 2), (2, 1, 0) \})=\Gamma _1\), \(\Upsilon _2(\{(5, 2, 4), (3, 2, 3)\})=\Gamma _2\), \(\Upsilon _3(\{(1, 2, 0),(1, 2, 1)\})=\Gamma _3\), and \(\Lambda _S\setminus { \cup _{i\in [q]} \tau _i}\subset {\mathcal {C}}_S\setminus {\mathcal {A}}\). As previous corollary asserts, \(\mathrm {e}(S)=13 = 6 + 7 = \sum _{i\in [3]} (\mathrm {e}(S_i) + \mathrm {e}(T_i))\).

On the decomposition of a \({\mathcal {C}}\)-semigroup in terms of irreducible \({\mathcal {C}}\)-semigroups

We define the set of pseudo-Frobenius of a \({\mathcal {C}}\)-semigroup S as \(\text {PF}(S)=\{{\mathbf {a}}\in {\mathcal {H}}(S)\mid {\mathbf {a}}+(S\setminus \{0\})\subset S\}\) (recall that \({\mathcal {H}}(S)={\mathcal {C}}\setminus S\)), and the set of special gaps of S as \(\text {SG}(S)=\{{\mathbf {a}}\in \text {PF}(S)\mid 2{\mathbf {a}}\in S\}\). Note that the elements \({\mathbf {a}}\) of \(\text {SG}(S)\) are those elements in \({\mathcal {C}}\setminus S\) such that \(S\cup \{{\mathbf {a}}\}\) is again a \({\mathcal {C}}\)-semigroup.

A \({\mathcal {C}}\)-semigroup is \({\mathcal {C}}\)-reducible (simplifying reducible) if it can be expressed as an intersection of two \({\mathcal {C}}\)-semigroups containing it properly (see [11]). Equivalently, S is \({\mathcal {C}}\)-irreducible (simplifying irreducible) if and only if \(|\text {SG}(S)|\le 1\). A decomposition of a \({\mathcal {C}}\)-semigroup S in terms of irreducible \({\mathcal {C}}\)-semigroups is to express S as intersection of irreducible \({\mathcal {C}}\)-semigroups. This definition generalizes the definitions of irreducible numerical semigroups (see [13]) and irreducible \({\mathbb {N}}^p\)-semigroups (see [2]).

Our decomposition method into irreducible is based on adding to a \({\mathcal {C}}\)-semigroup elements of \(\text {SG}(S)\). If we repeat this operation, we always reach an irreducible \({\mathcal {C}}\)-semigroup or the cone \({\mathcal {C}}\). Since the set of gaps \({\mathcal {H}}(S)\) is finite, this process can be performed only a finite number of times. This allows us to state the following algorithm inspired by [13, Algorithm 4.49].

By definition, the set \(\text {SG}(S)\) is obtained from \(\text {PF}(S)\). If S is determined by its minimal generating set, then \(\text {PF}(S)\) can be computed from the set \({\mathcal {H}}(S)\) obtained with Algorithm 2, or using the two different ways given in [11, Corollary 9 and Example 10].

figure e

Example 19

Consider the \({\mathcal {C}}\)-semigroup S given in examples 10 and 12. It is minimally generated by

$$\begin{aligned} \Lambda _S=\{ (2, 0, 0), (4, 2, 4), (0, 1, 0), (3, 0, 0), (6, 3, 6), (3, 1, 1), (4, 1, 1),\\ (3, 1, 2), (1, 1, 0), (3, 2, 3), (1, 2, 1) \}, \end{aligned}$$

with

$$\begin{aligned} {\mathcal {H}}(S)=\{ (1,0,0), (1,1,1), (2,1,1), (2,1,2), (2,2,1), (2,2,2), (2,3,2),\\ (4,1,2), (4,2,3), (5,2,4), (5,3,5), (8,4,7) \}. \end{aligned}$$

Hence, \(\text {PF}(S)=\{(2, 2, 1), (2, 3, 2), (4, 1, 2), (8, 4, 7)\}\), and \(\text {SG}(S)\) is equal to \(\text {PF}(S)\).

Applying Algorithm 3 to S, we obtain the decomposition into six irreducible \({\mathcal {C}}\)-semigroups, \(S=S_1\cap \cdots \cap S_{6}\) where

  • \(S_1=\langle (3, 0, 0), (2, 0, 0), (1, 1, 0), (0, 1, 0), (4, 1, 1), (3, 1, 1), (3, 1, 2),(4, 1, 2)\), \((1, 2, 1), (2, 2, 1), (2, 2, 2), (3, 2, 3), (4, 2, 4), (6, 3, 6) \rangle \);

  • \(S_2=\langle (3, 0, 0), (2, 0, 0), (1, 1, 0), (0, 1, 0), (4, 1, 1), (3, 1, 1), (2, 1, 2),(3, 1, 2)\), \( (1, 2, 1), (2, 2, 1), (3, 2, 3) \rangle \);

  • \(S_3=\langle (1, 0, 0), (0, 1, 0), (2, 1, 1), (3, 1, 2), (1, 2, 1), (3, 2, 3), (4, 2, 4),(5, 3, 5)\), \((6, 3, 6) \rangle \);

  • \(S_4=\langle (3, 0, 0), (2, 0, 0), (1, 1, 0), (0, 1, 0), (2, 1, 1), (1, 1, 1), (3, 1, 2),(4, 1, 2)\), \( (3, 2, 3), (4, 2, 4), (6, 3, 6) \rangle \);

  • \(S_5=\langle (3, 0, 0), (2, 0, 0), (1, 1, 0), (0, 1, 0), (2, 1, 1), (1, 1, 1), (3, 1, 2),(3, 2, 3)\), \( (4, 2, 4), (5, 2, 4), (6, 3, 6) \rangle \);

  • \(S_6=\langle (3, 0, 0), (2, 0, 0), (1, 1, 0), (0, 1, 0), (4, 1, 1), (3, 1, 1), (2, 1, 2),(3, 1, 2)\), \((1, 2, 1), (3, 2, 3), (4, 2, 3) \rangle \);

To get these semigroups we have used our implementation in [7] by typing the following

figure f

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Acknowledgements

The authors thank the referees for their helpful observations. The authors were partially supported by Junta de Andalucía research group FQM-366. The first author was supported by the Programa Operativo de Empleo Juvenil 2014-2020, financed by the European Social Fund within the Youth Guarantee initiative. The second, third and fourth authors were partially supported by the project MTM2017-84890-P (MINECO/FEDER, UE), and the fourth author was partially supported by the project MTM2015-65764-C3-1-P (MINECO/FEDER, UE).

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Correspondence to J. I. García-García.

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Díaz-Ramírez, J.D., García-García, J.I., Marín-Aragón, D. et al. Characterizing affine \({\mathcal {C}}\)-semigroups. Ricerche mat 71, 283–296 (2022). https://doi.org/10.1007/s11587-022-00693-6

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  • DOI: https://doi.org/10.1007/s11587-022-00693-6

Keywords

  • Affine semigroup
  • \({\mathcal {C}}\)-semigroup
  • Embedding dimension
  • Gap of a semigroup
  • Generalized numerical semigroup
  • Irreducible semigroup

Mathematics Subject Classification

  • 20M14 Primary
  • 68R05 Secondary