Characterizing affine $\mathcal{C}$-semigroups

TL;DR

This paper characterizes affine -semigroups within finitely generated cones, providing algorithms for their identification, gap computation, and decomposition, along with bounds on their embedding dimension.

Contribution

It offers a characterization of -semigroups from minimal generators, an algorithm to verify and compute gaps, and a method to decompose them into irreducible components.

Findings

Characterization of -semigroups from minimal generating sets

Algorithm to check if a semigroup is a -semigroup and to compute gaps

Lower bounds for the embedding dimension of -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.

Equations

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

$$\begin{multlined}\Lambda_{S}=\{(2,0,0),(4,2,4),(0,1,0),(3,0,0),(6,3,6),(3,1,1),(4,1,1),\\ (3,1,2),(1,1,0),(3,2,3),(1,2,1)\}.\end{multlined}\Lambda_{S}=\{(2,0,0),(4,2,4),(0,1,0),(3,0,0),(6,3,6),(3,1,1),(4,1,1),\\ (3,1,2),(1,1,0),(3,2,3),(1,2,1)\}.$$

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

$${{\rm Y}}_1(\mathcal{A})=\{(0,0),(0,2),(1,1),(2,0),(2,2)\},$$

$${{\rm Y}}_2(\mathcal{A})=\{(0,0),(0,1),(1,0),(1,1),(2,0),(2,1)\},$$

$${{\rm Y}}_3(\mathcal{A})=\{(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)\}.$$

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

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

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

$$\begin{multlined}\cup_{\alpha\in\Upsilon_{2}(\mathcal{D})\setminus\{0\}}\{\mathbf{m}^{(2)}_{\alpha}\}=\{(0,1,0),(3,1,2),(1,1,0),(3,2,3),(2,0,0),\\ (2,1,0),(6,3,5),(3,1,1)\}\end{multlined}\cup_{\alpha\in\Upsilon_{2}(\mathcal{D})\setminus\{0\}}\{\mathbf{m}^{(2)}_{\alpha}\}=\{(0,1,0),(3,1,2),(1,1,0),(3,2,3),(2,0,0),\\ (2,1,0),(6,3,5),(3,1,1)\}$$

$$\begin{multlined}\cup_{\alpha\in\Upsilon_{3}(\mathcal{D})\setminus\{0\}}\{\mathbf{m}^{(3)}_{\alpha}\}=\{(1,1,0),(1,2,1),(2,0,0),(2,3,1),(2,4,2),\\ (3,1,1),(3,1,2),(3,2,3),(4,2,2),(4,3,3),(4,2,4),(5,3,4),(6,2,4)\}\end{multlined}\cup_{\alpha\in\Upsilon_{3}(\mathcal{D})\setminus\{0\}}\{\mathbf{m}^{(3)}_{\alpha}\}=\{(1,1,0),(1,2,1),(2,0,0),(2,3,1),(2,4,2),\\ (3,1,1),(3,1,2),(3,2,3),(4,2,2),(4,3,3),(4,2,4),(5,3,4),(6,2,4)\}$$

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

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

$${\Lambda }^{\prime \prime }\cup \bigcup _{i\in [q]}\{n_1^{(i)}{\mathbf{a}}_i,\dots ,n_{\mathrm{e}(S_i)}^{(i)}{\mathbf{a}}_i\}$$

$$\sum _{i\in [q]}(\mathrm{e}(S_i)+\mathrm{e}(T_i))+k-\sum _{i\in [k]}♯(M_i).$$

$${\Lambda }^{\prime \prime }\cup \bigcup _{i\in [q]}\{n_1^{(i)}{\mathbf{a}}_i,\dots ,n_{\mathrm{e}(S_i)}^{(i)}{\mathbf{a}}_i\}$$

$$\begin{multlined}\Lambda_{S}=\{(2,0,0),(4,2,4),(0,2,0),(3,0,0),(6,3,6),(0,3,0),(3,1,1),\\ (3,1,2),(1,1,0),(3,2,3),(1,2,1)\}.\end{multlined}\Lambda_{S}=\{(2,0,0),(4,2,4),(0,2,0),(3,0,0),(6,3,6),(0,3,0),(3,1,1),\\ (3,1,2),(1,1,0),(3,2,3),(1,2,1)\}.$$

$$\begin{multlined}\Lambda_{S}=\{(2,0,0),(4,2,4),(0,2,0),(3,0,0),(6,3,6),(0,3,0),(3,1,1),\\ (4,1,2),(5,2,4),(2,1,0),(1,2,0),(3,2,3),(1,2,1)\}.\end{multlined}\Lambda_{S}=\{(2,0,0),(4,2,4),(0,2,0),(3,0,0),(6,3,6),(0,3,0),(3,1,1),\\ (4,1,2),(5,2,4),(2,1,0),(1,2,0),(3,2,3),(1,2,1)\}.$$

$$\begin{multlined}\Lambda_{S}=\{(2,0,0),(4,2,4),(0,1,0),(3,0,0),(6,3,6),(3,1,1),(4,1,1),\\ (3,1,2),(1,1,0),(3,2,3),(1,2,1)\},\end{multlined}\Lambda_{S}=\{(2,0,0),(4,2,4),(0,1,0),(3,0,0),(6,3,6),(3,1,1),(4,1,1),\\ (3,1,2),(1,1,0),(3,2,3),(1,2,1)\},$$

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

Full text

Keywords: affine semigroup, $\mathcal{C}$-semigroup, embedding dimension, gap of a semigroup, generalized numerical semigroup, irreducible semigroup.

2020 Mathematics Subject Classification: 20M14 (Primary), 68R05 (Secondary).

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 $\mathbb{R}^{p}_{\geq}$ 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 $S$ is or not a $\mathcal{C}_{S}$-semigroup? This work’s primary goal 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.

Other 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 $\mathcal{C}$-semigroups.

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 section 4, 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 $\mathcal{C}$-semigroups 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 1 introduces the initial definitions and notations used throughout the paper, mainly related to finitely generated cones. In Section 2, a characterization of $\mathcal{C}$-semigroups is provided, and an algorithm to check if an affine semigroup is a $\mathcal{C}$-semigroup. Section 3 is devoted to give an algorithm to compute the set of gaps of a $\mathcal{C}$-semigroup. Section 4 makes a study of the minimal generating sets of $\mathcal{C}$-semigroups formulating explicitly a lower bound for their embedding dimensions. Finally, in Section 5 an algorithm for computing a decomposition of a $\mathcal{C}$-semigroup into irreducible $\mathcal{C}$-semigroups is presented.

1 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 $A$ a subset of $\mathbb{R}$, $A_{\geq}$ 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 of $\mathbb{R}_{\geq}^{p}$, $B$, we define the cone

[TABLE]

Given a real cone $\mathcal{C}\subset\mathbb{R}^{p}_{\geq}$, 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 $L(S)=\{x\in\mathbb{R}_{\geq}^{d}\mid h_{1}(x)\geq 0,\ldots,h_{t}(x)\geq 0\}$. 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 $H_{i}:=\{h_{j_{1}}(x)=0,\ldots,h_{j_{p-1}}(x)=0\}$ where $J_{i}:=\{j_{1}<\cdots<j_{p-1}\}\subset[t]$ 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 $\tau_{i}=\{\lambda\mathbf{a}_{i}\mid\lambda\in\mathbb{R}_{\geq}\}$. 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 denotes by $\Upsilon_{i}(P)$ the element $(h_{j_{1}}(P),\ldots,h_{j_{p-1}}(P))\in\mathbb{N}^{p-1}$ with $J_{i}=\{j_{1}<\cdots<j_{p-1}\}$, $P\in\mathcal{C}_{S}$ 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.

2 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 $\tau_{i}\cap(\mathcal{C}_{S}\setminus S)$ respect the natural order. 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 $\tau_{i}\cap S$ to $\mathbf{f}_{i}$ 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 numerical semigroup $S_{i}$.

Proof.

Consider the isomorphism $\varphi:\tau_{i}\cap S\to S_{i}$ with $\varphi(\mathbf{w}):=\lambda$ such that $\mathbf{w}=\lambda\mathbf{a}_{i}$. ∎

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 $\tau$ 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 $\gcd(\{\lambda\mid\lambda\mathbf{a}\in\tau\cap\Lambda_{S}\})=n\neq 1$. 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, by Lemma 2, if $\gcd(\{\lambda\mid\lambda\mathbf{a}\in\tau\cap\Lambda_{S}\})=1$, $S_{i}$ is isomorphic to $\tau_{i}\cap S$. Therefore, $\tau\cap(\mathcal{C}_{S}\setminus S)$ is finite. ∎

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 $\mathcal{A}:=\{\sum_{i\in[q]}\lambda_{i}\mathbf{a}_{i}\mid 0\leq\lambda_{i}\leq 1\}\cap\mathbb{N}^{p}$ and $\mathcal{D}:=\{\sum_{i\in[q]}\lambda_{i}\mathbf{s}_{i}\mid 0\leq\lambda_{i}\leq 1\}\cap\mathbb{N}^{p}$.

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 $\mu_{i}\in\mathbb{Q}_{\geq}$. 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 $j\in[q]$. ∎

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 $\Gamma_{i}$ its minimal generating set. Note that the sets $\mathcal{A}$, $T_{i}$ and $\Gamma_{i}$ only depend on the cone $\mathcal{C}_{S}$, and since $\mathbf{a}_{i}\in\mathcal{A}$, $0\in T_{i}$. 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, $\Upsilon_{i}(P)=\Upsilon_{i}(Q)+\sum_{j\in[q]}\beta_{j}\Upsilon_{i}(\mathbf{a}_{j})\in T_{i}$. ∎

Corollary 7**.**

For every $\alpha\in T_{i}$, $\mathcal{C}_{S}\cap\upsilon_{i}(\alpha)\neq\emptyset$ if and only if $\mathcal{C}_{S}\cap\upsilon_{i}(\beta)\neq\emptyset$ for all $\beta\in\Gamma_{i}$.

Proof.

Since $\Gamma_{i}\subset T_{i}$, if for all $\alpha\in T_{i}$, $\mathcal{C}_{S}\cap\upsilon_{i}(\alpha)\neq\emptyset$ then $\mathcal{C}_{S}\cap\upsilon_{i}(\beta)\neq\emptyset$ for all $\beta\in\Gamma_{i}$.

Assume that $\mathcal{C}_{S}\cap\upsilon_{i}(\beta)\neq\emptyset$ 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 $Q_{1},\ldots,Q_{k}\in\mathcal{D}$ such that $\alpha=\sum_{j\in[k]}\mu_{j}\beta_{j}$ and $\Upsilon_{i}(Q_{j})=\beta_{j}$ for $j\in[k]$. Note that $P=\sum_{j\in[k]}\mu_{j}Q_{j}\in\mathcal{C}_{S}$ belongs to $\upsilon_{i}(\alpha)$. ∎

Corollary 8**.**

For every $\alpha\in T_{i}$, $S\cap\upsilon_{i}(\alpha)\neq\emptyset$ if and only if $S\cap\upsilon_{i}(\beta)\neq\emptyset$ for all $\beta\in\Gamma_{i}$.

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:

$\tau_{i}\cap(\mathcal{C}_{S}\setminus S)$* is finite for all $i\in[q]$.* 2. 2.

$\Lambda_{S}\cap\upsilon_{i}(\alpha)\neq\emptyset$* 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 $i\in[q]$. Assume that $\Lambda_{S}\cap\upsilon_{i}(\alpha)=\emptyset$ for some $\alpha\in\Gamma_{i}$ and some $i\in[q]$. 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 element in $T_{i}$, $\sum_{j\in[n]}\mu_{j}+\sum_{j\in[n]}\sum_{k\in[q]\setminus\{i\}}\mu_{j}\beta_{jk}=1$. Hence, there exists $\mathbf{s}\in\Lambda_{S}$ such that $\Upsilon_{i}(\mathbf{s})=\alpha$ and then $\Lambda_{S}\cap\upsilon_{i}(\alpha)\neq\emptyset$.

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 $\Lambda_{S}\cap\upsilon_{i}(\alpha)\neq\emptyset$ (recall that $\mathbf{c}_{i}=\mathbf{f}_{i}+\mathbf{a}_{i}$). The second condition implies that for $\beta=\Upsilon_{i}(Q)$ with $Q\in\mathcal{D}$, each line $\upsilon_{i}(\beta)$ included an 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 (respect to the 1-norm) element 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 $\mathcal{D}_{i}\cap S=\mathcal{D}_{i}\cap\mathcal{C}_{S}=\mathcal{D}_{i}$, so $\mathbf{x}_{i}+\mathcal{C}_{S}\subset S$. We define the finite set $\mathcal{X}:=\{\sum_{i\in[q]}\lambda_{i}\mathbf{x}_{i}\mid 0\leq\lambda_{i}\leq 1\}$. Since $\mathbf{x}_{i}+\mathcal{C}_{S}\subset S$ for every $i\in[q]$, $\mathcal{C}_{S}\setminus S\subset\mathcal{X}$. Therefore, $S$ is a $\mathcal{C}_{S}$-semigroup. ∎

Example 10 illustrates Theorem 9 and Algorithm 1.

Example 10*.*

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

[TABLE]

The cone $L(S)$ is $\langle(1,0,0),(2,1,2),(0,1,0)\rangle_{\mathbb{R}_{\geq}}$ 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

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

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,

In [1]: IsCsemigroup([[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]]) Out[1]: True

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, $\tau_{i}\equiv\{\lambda\mathbf{e}_{i}\mid\lambda\in\mathbb{Q}_{\geq}\}$ 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, a characterization of $\mathbb{N}^{p}$-semigroups equivalent to [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:

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 $i\neq j$, $\mathbf{e}_{i}+\lambda_{j}\mathbf{e}_{j}\in\Lambda_{S}$ 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.

3 Set of gaps of $\mathcal{C}$-semigroups

This section gives an algorithm to compute the set of gaps of a $\mathcal{C}$-semigroup. 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 $\{x_{1}+\cdots+x_{p}=n_{i},h_{j_{1}}(x)=0,\ldots,h_{j_{p-1}}(x)=0\}$ (recall that $h_{j_{1}}(x)=0,\ldots,h_{j_{p-1}}(x)=0$ 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 $\mathcal{X}=\{\sum_{i\in[q]}\lambda_{i}\mathbf{x}_{i}\mid 0\leq\lambda_{i}\leq 1\}$. 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.

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

[TABLE]

For example, for the extremal ray $\tau_{1}$, $\Upsilon_{1}(\mathcal{D})$ is the set

[TABLE]

and $\cup_{\alpha\in\Upsilon_{1}(\mathcal{D})\setminus\{0\}}\{\mathbf{m}^{(1)}_{\alpha}\}$ is

[TABLE]

For $\tau_{2}$ and $\tau_{3}$,

[TABLE]

[TABLE]

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,

[TABLE]

By using our implementation of Algorithm 2, we obtain those gaps,

In [1]: ComputeGaps([[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]]) Out[1]: [[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]]

4 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 of a given $\mathcal{C}$-semigroup 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 $\{\sum_{i\in[q]}\lambda_{i}\mathbf{a}_{i}\mid 0\leq\lambda_{i}\leq 1\}\cap\mathbb{N}^{p}$ and $\Gamma_{i}=\{\alpha^{(i)}_{1},\ldots,\alpha^{(i)}_{m_{i}}\}$ 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 $\Lambda^{\prime}:=\{\mathbf{s}_{t_{1}},\ldots,\mathbf{s}_{t_{k}}\}$ the set of minimal elements of $S$ in $\mathcal{A}$, and $M_{l}:=\{i\in[q]\mid\Upsilon_{i}(\mathbf{s}_{t_{l}})\in\Gamma_{i}\cup\{0\}\}$ for $l\in[k]$.

The following result provides a lower bound for the embedding dimension of any $\mathcal{C}$-semigroup such that $\Lambda^{\prime}$ is the set of its minimal elements in $\mathcal{A}$.

Proposition 13**.**

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

[TABLE]

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 $\upsilon_{i}(\gamma)$. But, it is possible that one element in $\Lambda_{S}\cap\mathcal{A}$ belongs to two (or more) different lines $\upsilon_{i}(\gamma)$ and $\upsilon_{j}(\gamma^{\prime})$ with $\gamma\in\Gamma_{i}\cup\{0\}$ and $\gamma^{\prime}\in\Gamma_{j}\cup\{0\}$ (in that case, $\upsilon_{i}(\gamma)\cap\upsilon_{j}(\gamma^{\prime})$ is this minimal generator). Since each one of these points in $\Lambda_{S}\cap\mathcal{A}$ can be the only minimal generator of $S$ in those lines, $\sharp(M_{l})=n>1$ means that one minimal generator can be the only minimal generator for $n$ different elements in $\cup_{i\in[q]}\Gamma_{i}\cup\{0\}$. So, counting the minimal amount of elements needed to have almost one minimal generator in each line $\upsilon_{i}(\gamma)$ for each $\gamma\in\Gamma_{i}\cup\{0\}$ and $i\in[q]$, we have that the embedding dimension of $S$ is greater than or equal to $\sum_{i\in[q]}(\mathrm{e}(S_{i})+\mathrm{e}(T_{i}))+k-\sum_{i\in[k]}\sharp(M_{i}).$ ∎

Example 14*.*

Consider the $\mathcal{C}_{S}$-semigroup $S$ given in example 10. In that case, $\Lambda^{\prime}=\{(3,1,2),(0,1,0),(1,1,0)\}$, $\sharp(M_{1})=2$ (i.e. $\Upsilon_{i}(3,1,2)\in\Gamma_{i}$ for $i=1,2$), $\sharp(M_{2})=2$ ($\Upsilon_{1}(0,1,0)\in\Gamma_{1}$ and $\Upsilon_{3}(0,1,0)=(0,0,0)$), and $\sharp(M_{3})=2$ ($\Upsilon_{1}(1,1,0)\in\Gamma_{1}$ and $\Upsilon_{3}(1,1,0)\in\Gamma_{3}$). So, $\sum_{i\in[q]}(\mathrm{e}(S_{i})+\mathrm{e}(T_{i}))+k-\sum_{i\in[k]}\sharp(M_{i})=5+7+3-2-2-2=9$ 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, and this fact is formulated as follows.

Lemma 15**.**

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 $\Lambda^{\prime\prime}\subset\mathcal{C}\setminus\{\mathbf{a}_{1},\ldots,\mathbf{a}_{q}\}$ be a set satisfying

• •

for every $\gamma\in\Gamma_{i}$ and $i\in[q]$, there exists an unique $\mathbf{d}\in\Lambda^{\prime\prime}$ such that $\Upsilon_{i}(\mathbf{d})=\gamma$,

• •

if there exist $i,j\in[q]$ and $\mathbf{d},\mathbf{d}^{\prime}\in\Lambda^{\prime\prime}$ such that $\Upsilon_{i}(\mathbf{d})=\Upsilon_{j}(\mathbf{d}^{\prime})$, then $\mathbf{d}=\mathbf{d}^{\prime}$.

Then, the embedding dimension of the $\mathcal{C}$-semigroup generated by

[TABLE]

is

[TABLE]

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 in the $\mathcal{C}$-semigroup generated by

[TABLE]

outside its extremal rays, and $\sum_{i\in[q]}\mathrm{e}(S_{i})$ belonging to its extremal rays.

∎

Example 16*.*

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

[TABLE]

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 $\Lambda^{\prime\prime}\subset\mathcal{C}\setminus\mathcal{A}$ be a set satisfying that for every $\gamma\in\Gamma_{i}$ and $i\in[q]$, there exists an unique $\mathbf{d}\in\Lambda^{\prime\prime}$ such that $\Upsilon_{i}(\mathbf{d})=\gamma$. Then, the embedding dimension of the $\mathcal{C}$-semigroup generated by $\Lambda^{\prime\prime}\cup\bigcup_{i\in[q]}\{n_{1}^{(i)}\mathbf{a}_{i},\ldots,n_{\mathrm{e}(S_{i})}^{(i)}\mathbf{a}_{i}\}$ 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

[TABLE]

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 $\Lambda_{S}\subset\mathcal{C}_{S}\setminus\mathcal{A}$. As previous corollary asserts, $\mathrm{e}(S)=13=6+7=\sum_{i\in[3]}(\mathrm{e}(S_{i})+\mathrm{e}(T_{i}))$.

5 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 $\textrm{PF}(S)=\{\mathbf{a}\in\mathcal{H}(S)\mid\mathbf{a}+(S\setminus\{0\})\subset S\}$, and the set of special gaps of $S$ as $\textrm{SG}(S)=\{\mathbf{a}\in\textrm{PF}(S)\mid 2\mathbf{a}\in S\}$. Note that the elements $\mathbf{a}$ of $\textrm{SG}(S)$ 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 $|\textrm{SG}(S)|\leq 1$. This definition generalizes the definitions of irreducible numerical semigroup (see [13]) and irreducible $\mathbb{N}^{p}$-semigroup (see [2]).

Our decomposition method into irreducible is based on adding to a $\mathcal{C}$-semigroup elements of $\textrm{SG}(S)$. 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 $\textrm{SG}(S)$ is obtained from $\textrm{PF}(S)$. If $S$ is determined by its minimal generating set, then $\textrm{PF}(S)$ 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].

Example 19*.*

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

[TABLE]

with

[TABLE]

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

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

Csemigroup([[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]]).DecomposeIrreducible()

Acknowledgements

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).