Some Properties of Affine ${\mathcal {C}}$-semigroups

Abstract

Numerical semigroups have been extensively studied throughout the literature, and many of their invariants have been characterized. In this work, we generalize some of the most important results about symmetry, pseudo-symmetry, or fundamental gaps, to affine ${\mathcal {C}}$-semigroups. In addition, we give algorithms to compute the tree of irreducible ${\mathcal {C}}$-semigroups and ${\mathcal {C}}$-semigroups with a given Frobenius vector.

1 Introduction

A ${\mathcal {C}}$-semigroup is a non-empty subset S of ${\mathbb {N}}^p$ (for some non-zero natural number p), containing 0 and closed under addition, such that ${\mathcal {C}}\setminus S$ is finite; ${\mathcal {C}}\subset {\mathbb {N}}^p$ denotes the positive integer cone generated by S. These semigroups are the natural generalization to higher dimensions of the numerical semigroups. Moreover, some objects related to numerical semigroups can be generalized to ${\mathcal {C}}$-semigroups. For example, the elements in ${\mathcal {C}}\setminus S$ are called gaps of S, and the cardinality of its gap set is called genus of S. We denote this set by ${\mathcal {H}}(S)$, and its cardinality by g(S).

There are other objects whose generalization needs to consider a total order on ${\mathbb {N}}^p$. For example, the Frobenius number of a numerical semigroup is the maximum integer that is not in it. Still, its generalization over the ${\mathcal {C}}$-semigroups is not unique if we do not fix a total order. So, fixed $\prec$ a total order on ${\mathbb {N}}^p$, the Frobenius element of S is $\max _{\prec } ({\mathcal {C}}\setminus S)$.

Even though ${\mathcal {C}}$-semigroups frequently appear in semigroup theory, it is not until the publication of [12] that they have become an object of study in their own right. This paper defines generalized numerical semigroups as the ${\mathcal {C}}$-semigroups where the cone ${\mathcal {C}}$ is ${\mathbb {N}}^p$. Since 2016, several works have been devoted to study different properties of ${\mathcal {C}}$-semigroups in general and generalized numerical semigroups in particular. For example, in [8], the authors show that any ${\mathbb {N}}^p$-semigroup has a unique minimal system of generators and provide an algorithm to compute its set of gaps from a set of generators of the ${\mathbb {N}}^p$-semigroup. In [14], an extension of Wilf’s conjecture for numerical semigroups is given to ${\mathcal {C}}$-semigroups, and in [5], another one is introduced for ${\mathbb {N}}^p$-semigroups. This paper also studies the irreducibleness of the ${\mathbb {N}}^p$-semigroups. More recent papers about ${\mathbb {N}}^p$-semigroups are [1, 6, 7, 9], and [19]. For any ${\mathcal {C}}$-semigroup, in [11], the authors mainly provide an algorithm to check if an affine semigroup given by a generating set is a ${\mathcal {C}}$-semigroup and to compute its gap set.

The main goal of this work is to generalize several results of numerical semigroups to ${\mathcal {C}}$-semigroups. A ${\mathcal {C}}$-semigroup is ${\mathcal {C}}$-reducible (simplifying reducible) when it can be expressed as an intersection of two ${\mathcal {C}}$-semigroups containing it properly (see [15]); S is ${\mathcal {C}}$-irreducible (simplifying irreducible) otherwise. In this work, we also characterize irreducible ${\mathcal {C}}$-semigroups from their genus and from their generalized Frobenius elements. We also study when a subset of a cone ${\mathcal {C}}$ is the gap set of a ${\mathcal {C}}$-semigroup or determines it. These results are complemented by some algorithms for checking the corresponding properties.

Moreover, some algorithms for computing some objects related to ${\mathcal {C}}$-semigroups are provided. In particular, it is defined as a tree whose vertex set is the set of all irreducible ${\mathcal {C}}$-semigroups with a fixed Frobenius vector. An algorithm to compute this tree is also introduced. For any integer cone ${\mathcal {C}}$ and any non-null element ${\textbf{f}}\in {\mathcal {C}}$, we give a procedure to obtain all ${\mathcal {C}}$-semigroups with Frobenius element equal to ${\textbf{f}}$.

The results of this work are illustrated with several examples. For this purpose, we have implemented all the algorithms shown in this work in our library CommutativeMonoids dedicated to the study of numerical and affine semigroups (see [13]) developed by the authors in Python [16] and C++. A notebook containing all the examples of this work can be found at the following link https://github.com/D-marina/CommutativeMonoids/blob/master/CClassCSemigroups/SomePropertiesCSemigroup.ipynb.

The content of this work is organized as follows: Sect. 1 is devoted to provide the reader with the necessary background for the correct understanding of the work. In Sect. 2, we introduce the concept of symmetric and pseudo-symmetric ${\mathcal {C}}$-semigroups, and some characterizations of these concepts are given. We turn our attention in Sect. 3 to the irreducible ${\mathcal {C}}$-semigroups, we prove that we can build a tree with all these semigroups with a fixed Frobenius vector, and we show an algorithm for computing them. Similarly, an algorithm for computing all the ${\mathcal {C}}$-semigroups with a fixed Frobenius vector is given in Sect. 5. Finally, in Sect. 4, we study the fundamental gaps of a ${\mathcal {C}}$-semigroup, and for any set $X\subset {\mathcal {C}}$, we give conditions to determine if ${\mathcal {C}}\setminus X$ is a ${\mathcal {C}}$-semigroup.

2 Preliminaries

In this work, ${\mathbb {Q}}$, ${\mathbb {Q}}_\ge$, and ${\mathbb {N}}$ denote the sets of rational numbers, non-negative rational numbers, and non-negative integer numbers, respectively. For any $n\in {\mathbb {N}}$, [n] denotes the set $\{1,\ldots , n\}$.

A non-degenerated rational cone in ${\mathbb {Q}}_\ge ^p$ is the convex hull of finitely many half lines in ${\mathbb {Q}}_\ge ^p$ emanating from the origin. These cones can also be determined from their supporting hyperplanes. We consider that the integer points of a rational cone form an integer cone in ${\mathbb {N}}^p$. It is well known that any integer cone ${\mathcal {C}}\subset {\mathbb {N}}^p$ is finitely generated if and only if a rational point exists in each of its extremal rays. Moreover, any subsemigroup of ${\mathcal {C}}$ is finitely generated if and only if there exists an element in the subsemigroup in each extremal ray of ${\mathcal {C}}$. Both results are proved in [4, Chapter 2], where an in-depth study on cones can also be found. We assume that any integer cone considered in this work is finitely generated.

Throughout this work, we use some particular gaps in ${\mathcal {H}}(S)$ whose definitions are the same for numerical semigroups [17]:

• ${\textbf{x}}\in {\mathcal {H}}(S)$ is a fundamental gap if $2{\textbf{x}},3{\textbf{x}}\in S$. The set of these elements is denoted by $\textrm{FG}(S)$.

• ${\textbf{x}}\in {\mathcal {H}}(S)$ is a pseudo-Frobenius element if ${\textbf{x}}+(S\setminus \{0\})\subset S$, the set of pseudo-Frobenius elements of S is denoted by $\textrm{PF}(S)$, and its cardinality is known as the type of S, t(S).

• ${\textbf{x}}\in {\mathcal {H}}(S)$ is a special gap of S if ${\textbf{x}}\in \textrm{PF}(S)$ and $2{\textbf{x}}\in S$. We denote by $\textrm{SG}(S)$ the set of special gaps of S.

In this work, we consider different orders on some sets. On a non-empty set $L\subset {\mathbb {N}}^p$ and ${\textbf{x}},{\textbf{y}}\in {\mathbb {N}}^p$, consider the partial order ${\textbf{x}}\le _L {\textbf{y}}$ if ${\textbf{y}}- {\textbf{x}}\in L$. Besides, we also fix $\preceq$ a total order on ${\mathbb {N}}^p$ determined by a monomial order. A monomial order is a total order on the set of all (monic) monomials in a given polynomial ring (see [10]). From the properties of a monomial order, the (induced) total order $\preceq$ on ${\mathbb {N}}^p$ satisfies:

• if ${\textbf{a}} \preceq {\textbf{b}}$ and ${\textbf{c}}\in {\mathbb {N}}^p$, then ${\textbf{a}}+{\textbf{c}}\preceq {\textbf{b}}+{\textbf{c}}$,

• if ${\textbf{c}}\in {\mathbb {N}}^p$, then $0\preceq {\textbf{c}}$.

Every monomial order can be represented via matrices. For a nonsingular integer ($p\times p$)-matrix M with rows $M_1,\ldots ,M_p,$ the M-ordering $\prec$ is defined by ${\textbf{a}}\prec {\textbf{b}}$ if and only if there exists an integer i belonging to $[p-1],$ such that $M_1{\textbf{a}}=M_1{\textbf{b}},\ldots ,M_i{\textbf{a}}=M_i{\textbf{b}}$ and $M_{i+1}{\textbf{a}}<M_{i+1}{\textbf{b}}.$

From the fixed total order on ${\mathbb {N}}^p$, the Frobenius vector of S, F(S), is the maximal element in ${\mathcal {H}}(S)$ respect to $\preceq$, and we set n(S) as the cardinality of ${\mathcal {N}}(S)=\{{\textbf{x}}\in S\mid {\textbf{x}}\preceq F(S) \}$.

The following lemma generalizes to ${\mathcal {C}}$-semigroups Proposition 2.26 in [17].

Lemma 1

Let S be a ${\mathcal {C}}$-semigroup. Then, $g(S)\le t(S)n(S)$.

Proof

Just as it occurs for numerical semigroups, for any ${\textbf{x}}\in {\mathcal {H}}(S)$, there exist $({\textbf{f}},{\textbf{s}})\in \textrm{PF}(S)\times S$ such that ${\textbf{f}}={\textbf{x}}+{\textbf{s}}$, and ${\textbf{f}}_{\textbf{x}}=\min _{\preceq }\{ {\textbf{f}}\in \textrm{PF}(S)\mid {\textbf{f}}-{\textbf{x}}\in S \}$. Hence, the map ${\mathcal {H}}(S)\rightarrow \textrm{PF}(S)\times {\mathcal {N}}(S)$, defined by $x\mapsto ({\textbf{f}}_{\textbf{x}},{\textbf{f}}_{\textbf{x}}-{\textbf{x}})$ is injective, and thus $g(S)\le t(S)n(S)$. $\square$

3 Symmetric and Pseudo-symmetric ${\mathcal {C}}$-semigroups

Fix $S\subset {\mathbb {N}}^p$ a ${\mathcal {C}}$-semigroup with genus g. In this section, we characterize the symmetric and pseudo-symmetric ${\mathcal {C}}$-semigroups using their genus. We say that S is ${\mathcal {C}}$-irreducible when $\textrm{PF}(S)$ is equal to $\{F(S)\}$ or $\{F(S),F(S)/2\}$ (see [15]). If $\textrm{PF}(S)=\{F(S)\}$, we say that S is symmetric, and pseudo-symmetric when $\textrm{PF}(S)=\{F(S),F(S)/2\}$.

For any element ${\textbf{n}}$ in ${\mathcal {C}}$, let $I_S({\textbf{n}})$ be the set $\{ {\textbf{s}}\in S\mid {\textbf{s}}\le _{\mathcal {C}}{\textbf{n}}\}$.

Remark 2

Note that, for any ${\textbf{s}}\in S$, ${\textbf{s}}\in I_S(F(S))$ if and only if $F(S)-{\textbf{s}}\in {\mathcal {H}}(S)$. Thus, $g\ge \sharp I_S(F(S))$.

We have the following characterizations of symmetric and pseudo-symmetric ${\mathcal {C}}$-semigroups.

Proposition 3

Let S be a ${\mathcal {C}}$-semigroup with genus g. Then, S is symmetric if and only if $g=\sharp I_S(F(S))$.

Proof

Assume that S is symmetric. Thus, F(S) is the unique pseudo-Frobenius element of S. Furthermore, for any ${\textbf{x}}\in {\mathcal {H}}(S)$, there exists ${\textbf{s}}\in S$ such that ${\textbf{x}}+{\textbf{s}}=F(S)$, that is ${\textbf{s}}\in I_S(F(S))$, and then $\sharp I_S(F(S))\ge g$. Since $g\ge \sharp I_S(F(S))$, we conclude that $g=\sharp I_S(F(S))$.

Conversely, note that $I_S(F(S))=\{{\textbf{s}}\in S\mid F(S)-{\textbf{s}}\in {\mathcal {H}}(S)\}$, and suppose that $g= \sharp I_S(F(S))$. Hence, every ${\textbf{x}}\in {\mathcal {H}}(S){\setminus } \{F(S)\}$ satisfies $F(S)-{\textbf{x}}\in S$, and then ${\textbf{x}}$ is not a pseudo-Frobenius element of S. $\square$

Proposition 4

Let S be a ${\mathcal {C}}$-semigroup with genus g. Then, S is pseudo-symmetric if and only if $g=1+\sharp I_S(F(S))$ and $F(S)/2\in {\mathbb {N}}^p$.

Proof

Assume that S is pseudo-symmetric, thus $\textrm{PF}(S)=\{F(S),F(S)/2\}$, and $g>\sharp I_S(F(S))$. For all ${\textbf{x}}\in {\mathcal {H}}(S){\setminus }\{F(S)/2\}$, there exists some ${\textbf{s}}\in S$ such that ${\textbf{x}}+{\textbf{s}}= F(S)$, or ${\textbf{x}}+{\textbf{s}}= F(S)/2$. If the first option is satisfied, ${\textbf{s}}\in I_S(F(S))$. In other case, ${\textbf{x}}+{\textbf{s}}+F(S)/2= F(S)$ and then ${\textbf{s}}+F(S)/2$ also belongs to $I_S(F(S))$. Besides, $F(S)/2+{\textbf{s}}\ne F(S)$ for every ${\textbf{s}}\in S$. Hence, $\sharp I_S(F(S))\ge g-1$.

Conversely, suppose that $g= \sharp I_S(F(S))+1$ and $F(S)/2\in {\mathbb {N}}^p$. If there exist ${\textbf{x}},{\textbf{y}}\in \textrm{PF}(S){\setminus } \{F(S)\}$ with ${\textbf{x}}\ne {\textbf{y}}$, then $g\ge \sharp I_S(F(S))+2$. Hence, $\textrm{PF}(S)=\{F(S),{\textbf{x}}\}$. If ${\textbf{x}}\ne F(S)/2$, then there is ${\textbf{s}}\in S$ such that $F(S)/2+{\textbf{s}}=F(S)$, and $F(S)/2\in S$, but it is not possible. So, ${\textbf{x}}= F(S)/2$. $\square$

Consider the Apéry set of a ${\mathcal {C}}$-semigroup S relative to ${\textbf{b}} \in S \setminus \{0\}$ as $\textrm{Ap}(S,{\textbf{b}}) = \{{\textbf{a}} \in S \mid {\textbf{a}}-{\textbf{b}} \in {\mathcal {H}}(S)\}$. The following proposition shows the relationship between the pseudo-Frobenius elements of S and its Apéry set.

Proposition 5

[15, Proposition 16]. Let S be a ${\mathcal {C}}$-semigroup and ${\textbf{b}} \in S {\setminus } \{0\}$. Then,

$$\begin{aligned} \textrm{PF}(S) = \{{\textbf{a}} - {\textbf{b}} \mid {\textbf{a}} \in \textrm{maximals}_{\le _S} \textrm{Ap}(S,{\textbf{b}}) \}. \end{aligned}$$

From this result, we can generalize the corollaries 4.12 and 4.19 in [17].

Corollary 6

Let S be a ${\mathcal {C}}$-semigroup and ${\textbf{b}} \in S {\setminus } \{0\}$. The semigroup S is symmetric if and only if $\textrm{maximals}_{\le _S} \textrm{Ap}(S,{\textbf{b}})=\{F(S)+{\textbf{b}}\}$.

Corollary 7

Let S be a ${\mathcal {C}}$-semigroup and ${\textbf{b}} \in S {\setminus } \{0\}$. The semigroup S is pseudo-symmetric if and only if $\textrm{maximals}_{\le _S} \textrm{Ap}(S,{\textbf{b}})=\{F(S)+{\textbf{b}},F(S)/2+{\textbf{b}}\}$.

The Frobenius number of a numerical semigroup is the maximum non-negative integer that is not an element of the semigroup. We define the (generalized) Frobenius number of a ${\mathcal {C}}$-semigroup S as ${\mathcal {F}}(S)=\sharp I_S(F(S))+g(S)$. We can easily rewrite the previous propositions 3 and 4 from this definition.

Corollary 8

Let S be a ${\mathcal {C}}$-semigroup with genus g. Then, S is symmetric if and only if $2g={\mathcal {F}}(S)$.

Corollary 9

Let S be a ${\mathcal {C}}$-semigroup with genus g. Then, S is pseudo-symmetric if and only if $2g=1+{\mathcal {F}}(S)$ and $F(S)/2\in {\mathbb {N}}^p$.

These corollaries specialized to numerical semigroups or ${\mathbb {N}}^p$-semigroups are equivalent to Corollary 4.5 in [17], and to theorems 5.6 and 5.7 in [7], respectively. Some results appearing in this section have been proved independently by O. P. Bhardwaj, K. Goel, and I. Sengupta (see [2, Section 3]).

We illustrate the previous results with one easy example.

Example 10

Let ${\mathcal {C}}\subset {\mathbb {N}}^2$ be the cone with extremal rays $\overrightarrow{(7,3)}$ and $\overrightarrow{(15,1)}$.

The ${\mathcal {C}}$-semigroup $S_1$ minimally generated by

$$\begin{aligned} \Lambda _{S_1}= & {} \{ (3, 1), (4, 1), (5, 1), (6, 1), (7, 1), (7, 3), (8, 1), (8, 3), (9, 1), (10, 1),\\{} & {} (11, 1), (12, 1), (12, 5), (13, 1), (14, 1), (15, 1) \} \end{aligned}$$

is symmetric, while the $S_2$ minimally generated by

$$\begin{aligned} \Lambda _{S_2}= & {} \{ (3, 1), (5, 2), (6, 1), (7, 1), (7, 2), (7, 3), (8, 1), (9, 1), (10,1) \\{} & {} (11, 1), (12, 1), (13, 1), (14, 1), (15, 1) \} \end{aligned}$$

is pseudo-symmetric.

Note that $\textrm{PF}(S_1)=\textrm{SG}(S_1)={\mathcal {H}}(S_1)=\{(5,2)\}$, but ${\mathcal {H}}(S_2)=\{(4, 1),$ $(5, 1), (8, 2)\}$, $\textrm{PF}(S_2)=\{(4, 1), (8, 2)\}$, and $\textrm{SG}(S_2)=\{(8, 2)\}$.

4 Trees of Irreducible ${\mathcal {C}}$-semigroups

This section describes a tree whose vertex set is the set of all irreducible ${\mathcal {C}}$-semigroups with a fixed Frobenius vector.

Again, consider ${\mathcal {C}}\subset {\mathbb {N}}^p$ an integer cone and ${\textbf{f}}\in {\mathcal {C}}\setminus \{0\}$. Consider a monomial order $\preceq$ on ${\mathbb {N}}^p$ and decompose the set $I_{\mathcal {C}}({\textbf{f}})$ as $I_{\mathcal {C}}({\textbf{f}})=\{{\textbf{0}}\}\sqcup I_1({\textbf{f}})\sqcup I_2({\textbf{f}})$ with $I_1({\textbf{f}})=\{{\textbf{x}}\in I_{\mathcal {C}}({\textbf{f}})\mid {\textbf{0}}\ne {\textbf{x}}\preceq {\textbf{f}}/2\}$ and $I_2({\textbf{f}})=\{{\textbf{x}}\in I_{\mathcal {C}}({\textbf{f}})\mid {\textbf{x}}\succ {\textbf{f}}/2\}$ (when ${\textbf{f}}/2\notin {\mathbb {N}}^p$, consider $\preceq$ as the monomial order extended to ${\mathbb {Q}}_{\ge }^p$). We define the ${\mathcal {C}}$-semigroup $S({\textbf{f}})$ as $\big ( {\mathcal {C}}{\setminus }\{{\textbf{f}}\}\big ){\setminus } I_1({\textbf{f}})$. This semigroup will be the root of our tree of irreducible ${\mathcal {C}}$-semigroups; this root depends on the fixed monomial order, as the following example shows.

Example 11

Let ${\mathcal {C}}\subset {\mathbb {N}}^2$ be the integer cone with extremal rays $\overrightarrow{(1,0)}$ and $\overrightarrow{(1,2)}$, and ${\textbf{f}}=(4,2)$. Then, ${\textbf{f}}/2=(2,1)$ and

$$\begin{aligned} I_{\mathcal {C}}({\textbf{f}})=\{(0,0),(1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2), (3, 0), (3, 1), (3, 2), (4, 2)\}. \end{aligned}$$

Let $\prec _1$ and $\prec _2$ be the orders defined by the matrices $\left( \begin{array}{cc} 1 &{}\quad 1\\ 1 &{}\quad 0 \end{array} \right)$ and $\left( \begin{array}{cc} 1 &{}\quad 1\\ 0 &{}\quad 1 \end{array} \right)$, respectively. In the first case, $I_1({\textbf{f}})_{\prec _1}=\{(1, 0), (1, 1), (1, 2), (2, 0), (2, 1)\}$ and

$$\begin{aligned} S({\textbf{f}})_{\prec _1}= & {} \langle (3, 0), (4, 0), (5, 0), (3, 1), (4, 1),\\{} & {} (5, 1), (2, 2), (3, 2), (2, 3), (3, 3), (4, 3), (2, 4), (3, 4), (3, 5), (3, 6) \rangle . \end{aligned}$$

In the other one, $I_1({\textbf{f}})_{\prec _2}=\{(1, 0), (1, 1), (2, 0), (2, 1), (3, 0)\}$ and

$$\begin{aligned} S({\textbf{f}})_{\prec _2}= & {} \langle (4, 0), (5, 0), (6, 0), (7, 0), (3, 1),\\{} & {} (4, 1), (5, 1), (6, 1), (1, 2), (2, 2), (3, 2), (2, 3), (3, 3) \rangle . \end{aligned}$$

The set $S({\textbf{f}})$ satisfies interesting properties collected in the following lemma.

Lemma 12

The ${\mathcal {C}}$-semigroup $S({\textbf{f}})$ is irreducible. Moreover, ${\textbf{f}}$ is the Frobenius vector of $S({\textbf{f}})$ for any monomial order, and $S({\textbf{f}})$ is the unique irreducible ${\mathcal {C}}$-semigroup satisfying all its gaps belong to $I_1({\textbf{f}})\cup \{{\textbf{f}}\}$.

Proof

By definition of $S({\textbf{f}})$, ${\textbf{f}}$ is the unique maximum in ${\mathcal {H}}(S({\textbf{f}}))$ respect to $\le _{\mathcal {C}}$. So, it is also the unique maximum in ${\mathcal {H}}(S({\textbf{f}}))$ respect to $\le _{{\mathbb {N}}^p}$. This fact implies that ${\textbf{f}}$ is the Frobenius vector of $S({\textbf{f}})$ for any monomial order.

Note that the set of gaps of $S({\textbf{f}})$ is the set ${\mathcal {H}}(S({\textbf{f}}))=I_1({\textbf{f}})\cup \{{\textbf{f}}\}$, and $I_{S({\textbf{f}})}({\textbf{f}})=I_2({\textbf{f}})\setminus \{{\textbf{f}}\}$. Besides, for any ${\textbf{x}}\in {\mathcal {H}}(S({\textbf{f}}))$, the element ${\textbf{f}}-{\textbf{x}}$ belongs to $I_{S({\textbf{f}})}({\textbf{f}})$. Otherwise, ${\textbf{f}} = {\textbf{f}} - {\textbf{x}} + {\textbf{x}} \prec {\textbf{f}}/2+{\textbf{f}}/2={\textbf{f}}$. Furthermore, since ${\textbf{x}}\in I_{S({\textbf{f}})}({\textbf{f}})$ if and only if ${\textbf{f}}-{\textbf{x}}\in {\mathcal {H}}(S({\textbf{f}}))$, we have that the cardinality of ${\mathcal {H}}(S({\textbf{f}}))$ is equal to $1+\sharp I_{S({\textbf{f}})}({\textbf{f}})$ when ${\textbf{f}}\in 2{\mathbb {N}}^p$, or equal to $\sharp I_{S({\textbf{f}})}({\textbf{f}})$ in the other case. By Proposition 4 or Proposition 3 (respectively), $S({\textbf{f}})$ is an irreducible ${\mathcal {C}}$-semigroup.

The uniqueness of $S({\textbf{f}})$ is given by its definition. $\square$

The following proposition gives us an irreducible ${\mathcal {C}}$-semigroups from an existing one, such that both have the same Frobenius vector. This result generalizes [3, Proposition 2.5].

Proposition 13

Let S be an irreducible ${\mathcal {C}}$-semigroup with Frobenius vector ${\textbf{f}}$, and ${\textbf{x}}\in I_S({\textbf{f}})$ be one of its minimal generators such that:

• 1. $2{\textbf{x}}-{\textbf{f}}\notin S$.

• 2. $3{\textbf{x}}\ne 2{\textbf{f}}$.

• 3. $4{\textbf{x}}\ne 3{\textbf{f}}$.

Then, $S'= (S\setminus \{{\textbf{x}}\})\cup \{{\textbf{f}}-{\textbf{x}}\}$ is an irreducible ${\mathcal {C}}$-semigroup with Frobenius vector ${\textbf{f}}$.

Proof

Note $F(S')= {\textbf{f}}$. We prove that $S'$ is closed under addition. Since ${\textbf{x}}=({\textbf{f}}-{\textbf{x}})+ (2{\textbf{x}}-{\textbf{f}})$, $2{\textbf{x}}-{\textbf{f}}$ can not belong to S, that is, the second condition is necessary.

Trivially, given two elements in $S\setminus \{{\textbf{x}}\}$, their addition belongs to the same set. Besides, ${\textbf{f}}-{\textbf{x}}+{\textbf{s}}\in S\setminus \{{\textbf{x}}\}$ for any ${\textbf{s}}\in S\setminus \{{\textbf{x}}\}$. Otherwise, ${\textbf{f}}-{\textbf{x}}+{\textbf{s}}= {\textbf{x}}$ or ${\textbf{f}}-{\textbf{x}}+{\textbf{s}}\in {\mathcal {H}}(S)$, for some ${\textbf{s}}\in S\setminus \{{\textbf{x}}\}$. If ${\textbf{f}}-{\textbf{x}}+{\textbf{s}}= {\textbf{x}}$, then ${\textbf{s}}=2{\textbf{x}}-{\textbf{f}}\notin S$. If ${\textbf{f}}-{\textbf{x}}+{\textbf{s}}\in {\mathcal {H}}(S)$, then there exists ${\textbf{s}}'\in S$ such that ${\textbf{f}}-{\textbf{x}}+{\textbf{s}}+{\textbf{s}}'\in \textrm{PF}(S)$. When ${\textbf{f}}-{\textbf{x}}+{\textbf{s}}+{\textbf{s}}'={\textbf{f}}/2$, we have $2({\textbf{s}}+{\textbf{s}}')=2{\textbf{x}}-{\textbf{f}}\notin S$, and when ${\textbf{f}}-{\textbf{x}}+{\textbf{s}}+{\textbf{s}}'={\textbf{f}}$, ${\textbf{x}}={\textbf{s}}+{\textbf{s}}'$. Both conclusions are not possible.

To finish this proof, we show that $2({\textbf{f}}-{\textbf{x}})\in S{\setminus }\{{\textbf{x}}\}$. Assume that $2({\textbf{f}}-{\textbf{x}})\notin S\setminus \{{\textbf{x}}\}$, so $2({\textbf{f}}-{\textbf{x}})= {\textbf{x}}$, or $2({\textbf{f}}-{\textbf{x}})+{\textbf{s}}\in \textrm{PF}(S)$ for some ${\textbf{s}}\in S$. Since $3{\textbf{x}}\ne 2{\textbf{f}}$, $2({\textbf{f}}-{\textbf{x}})\ne {\textbf{x}}$. The semigroup S to be irreducible implies that $2({\textbf{f}}-{\textbf{x}})+{\textbf{s}}\in \{{\textbf{f}},{\textbf{f}}/2\}$, but $2({\textbf{f}}-{\textbf{x}})+{\textbf{s}}$ is not equal to ${\textbf{f}}$ because of $2{\textbf{x}}-{\textbf{f}}\notin S$. Hence, $2({\textbf{f}}-{\textbf{x}})+{\textbf{s}}={\textbf{f}}/2$. Since $4{\textbf{x}}\ne 3{\textbf{f}}$, ${\textbf{s}}\ne 0$, and from $2{\textbf{x}}-{\textbf{f}}\notin S$, we obtain $2({\textbf{f}}-{\textbf{x}})+{\textbf{s}}\ne {\textbf{f}}/2$. We conclude $2({\textbf{f}}-{\textbf{x}})\in S\setminus \{{\textbf{x}}\}$.

Since $\sharp I_S({\textbf{f}})=\sharp I_{S'}({\textbf{f}})$ and $\sharp {\mathcal {H}}(S)= \sharp {\mathcal {H}}(S')$, $S'$ is irreducible by the Propositions 3 and 4. $\square$

If $S\ne S({\textbf{f}})$, m(S) denotes the minimum element with respect to the monomial order $\preceq$ in $I_1({\textbf{f}})\cap S$. If $I_1({\textbf{f}})\cap S=\emptyset$, then $S= S({\textbf{f}})$.

We denote by ${\mathfrak {I}}({\textbf{f}})$ the set of the irreducible ${\mathcal {C}}$-semigroups with Frobenius vector ${\textbf{f}}$. Given $S\in {\mathfrak {I}}({\textbf{f}})$, consider $S_0=S,$ and $S_n=(S_{n-1}\setminus \{m(S_{n-1})\})\cup \{{\textbf{f}}-m(S_{n-1})\}$ when $m(S_{n-1})\in I_1({\textbf{f}})$, or $S_n=S_{n-1}$ in other case, for $n>1$. Note that $S_n=S_{n-1}$ if $S_n=S({\textbf{f}})$. Since $I_1({\textbf{f}})$ is a finite set, the set $\{S_0,S_1,\ldots \}$ is also finite. Let $G=(V,E)$ be the digraph given by the set of vertices $V={\mathfrak {I}}({\textbf{f}})$, and edge set $E=\big \{(A,B)\in V\times V\mid m(A)\prec {\textbf{f}}/2 \text { and } B=(A{\setminus }\{m(A)\})\cup \{{\textbf{f}}-m(A)\}\big \}$.

Theorem 14

Let $\preceq$ be a monomial order on ${\mathbb {N}}^p$, ${\mathcal {C}}\subset {\mathbb {N}}^p$ be an integer cone, and ${\textbf{f}}\in {\mathcal {C}}$ be a non zero element. The digraph G is a rooted tree with root $S(\mathbf {f)}$.

Proof

Let S be an element belonging to ${\mathfrak {I}}({\textbf{f}})$. If $S\ne S({\textbf{f}})$, then $m(S)\prec {\textbf{f}}/2$, that is, $2\,m(S)-{\textbf{f}}\notin S$, $3\,m(S)\ne 2{\textbf{f}}$, and $4\,m(S)\ne 3{\textbf{f}}$. By Proposition 13, $S_1=(S\setminus \{m(S)\})\cup \{{\textbf{f}}-m(S)\}$ is irreducible. That means $(S,S_1)\in E$. Following this construction, G is a tree whose root is $S(\mathbf {f)}$. $\square$

We obtain an algorithm from previous construction and results to compute a tree of all irreducible ${\mathcal {C}}$-semigroups with a given Frobenius vector and a fixed monomial order (Algorithm 1). The symbol $\Lambda _S$ denotes the minimal generating set of a ${\mathcal {C}}$-semigroup S.

Algorithm 1

Computing a tree of irreducible ${\mathcal {C}}$-semigroups with a given Frobenius vector.

The following example shows how to apply Algorithm 1 using the semigroups of Example 11.

Example 15

Let $S({{\textbf {f}}})_{\prec _1}$ be the semigroup spanned by

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

Applying Algorithm 1, we have that $I_{2}({{\textbf {f}}})_{\prec _1}=\{(2, 2), (3, 0), (3, 1), (3, 2), (4, 2)\}$. Hence, $S({{\textbf {f}}})_{\prec _1}$ has three children:

• $\langle (4, 0), (5, 0), (6, 0),(7, 0), (3, 1), (4, 1), (5, 1),$ (6, 1), (1, 2), (2, 2), (3, 2),  $(2, 3), (3, 3)\rangle$,

• $\langle (3, 0), (4, 0),$ $(5, 0), (1, 1), (3, 2), (2, 3), (2, 4), (3, 6) \rangle$,

• $\langle (2, 0), (3, 0), (3, 1), (4, 1), (3, 2),$ (2, 3),  (3, 3), (2, 4), (3, 4), (3, 5), (4, 5),  $(3, 6)\rangle$.

After repeating this procedure, the tree in Fig. 1 is obtained. Since the definition of $S({{\textbf {f}}}$) depends on the monomial order, we get a new tree if we change it. For example, when we use the order $\prec _2$, Fig. 2 appears.

Fig. 1

Tree of irreducible ${\mathcal {C}}$-semigroups with $\prec _1$

Fig. 2

Tree of irreducible ${\mathcal {C}}$-semigroups with $\prec _2$

5 Fundamental Gaps of ${\mathcal {C}}$-semigroups

In this section, we generalize to ${\mathcal {C}}$-semigroups several results related to the fundamental gaps of a numerical semigroup (see [17, Chapter 4]). The first results allow us to check when ${\mathcal {C}}\setminus X$ is a ${\mathcal {C}}$-semigroup for any finite subset $X\subset {\mathcal {C}}$. Denote by D(X) the set $\{{\textbf{a}}\in {\mathcal {C}}\mid n{\textbf{a}}\in X \text { for some }n\in {\mathbb {N}}\}$.

Proposition 16

Let ${\mathcal {C}}\subset {\mathbb {N}}^p$ be an integer cone and X be a finite subset of ${\mathcal {C}}\setminus \{0\}$. Then, ${\mathcal {C}}\setminus X$ is a ${\mathcal {C}}$-semigroup if and only if ${\textbf{x}}-{\textbf{s}}\in X$ for every $({\textbf{x}},{\textbf{s}})\in X\times ({\mathcal {C}}\setminus X)$ with ${\textbf{s}}\le _{\mathcal {C}}{\textbf{x}}$.

Proof

Let S be the set ${\mathcal {C}}\setminus X$, and assume that S is a ${\mathcal {C}}$-semigroup. Set $({\textbf{x}},{\textbf{s}})\in X\times S$ with ${\textbf{s}}\le _{\mathcal {C}}{\textbf{x}}$. Since ${\textbf{s}}\le _{\mathcal {C}}{\textbf{x}}$, we have that ${\textbf{x}}-{\textbf{s}}\in {\mathcal {C}}$. If ${\textbf{x}}-{\textbf{s}}\notin X$, then ${\textbf{x}}={\textbf{s}}+{\textbf{s}}'$ for some ${\textbf{s}}'\in S$, and S is not a semigroup. So, ${\textbf{x}}-{\textbf{s}}\in X$ for any $({\textbf{x}},{\textbf{s}})\in X\times ( {\mathcal {C}}\setminus X)$ with ${\textbf{s}}\le _{\mathcal {C}}{\textbf{x}}$.

Conversely, since ${\textbf{x}}-{\textbf{s}}$ belongs to X for every $({\textbf{x}},{\textbf{s}})\in X\times S$ with ${\textbf{s}}\le _{\mathcal {C}}{\textbf{x}}$, S is an additive submonoid of ${\mathbb {N}}^p$ with finite complement in ${\mathcal {C}}$, that is, S is a ${\mathcal {C}}$-semigroup. $\square$

From the above proposition, ${\mathcal {C}}\setminus X$ to be a ${\mathcal {C}}$-semigroup implies that $X=D(X)$; for example, if we consider ${\mathcal {C}}$ the cone generated by $\{(1,0), (1,1), (1,2)\}$ and $X=\{(2,0), (2,1)\}$, ${\mathcal {C}}\setminus X$ is not a semigroup because of D(X) $=\{(2,0), (2,1),(1,0)\}$. We now provide an algorithm to determine if ${\mathcal {C}}\setminus X$ is a ${\mathcal {C}}$-semigroup (Algorithm 2).

Algorithm 2

Checking if ${\mathcal {C}}\setminus X$ is a ${\mathcal {C}}$-semigroup.

Since, for each ${\textbf{x}}\in X$, the set $\{{\textbf{s}}\in {\mathcal {C}}{\setminus } X\mid {\textbf{s}}\le _{\mathcal {C}}{\textbf{x}}\}$ can be very very large, the condition ${\textbf{x}}-{\textbf{s}}\notin X$ has to be checked many, many times in Algorithm 2, and many iterations are required for the worst cases. To improve the computational resolution of this problem, we provide an alternative algorithm (Algorithm 3) obtained from the following lemma and [14, Lemma 3].

Lemma 17

Fix a total order $\preceq$ on ${\mathbb {N}}^p$, and let $X=\{{\textbf{x}}_1\preceq {\textbf{x}}_2\preceq \cdots \preceq {\textbf{x}}_t\}$ be a subset of an integer cone $S_0={\mathcal {C}}\subset {\mathbb {N}}^p$. Assume that $S_t={\mathcal {C}}{\setminus } X$ is a ${\mathcal {C}}$-semigroup. Then, $S_i=S_{i-1}\setminus \{{\textbf{x}}_i\}$ is a ${\mathcal {C}}$-semigroup, and ${\textbf{x}}_i$ is a minimal generator of $S_{i-1}$, for every $i\in [t]$.

Proof

Note that ${\textbf{x}}_i$ is the Frobenius vector of $S_{i}$ respect to $\preceq$. Hence, $S_{i-1}=S_i\cup \{{\textbf{x}}_i\}$ is a ${\mathcal {C}}$-semigroup and ${\textbf{x}}_i$ is a minimal generator of $S_{i-1}$, for every $i\in [t]$. $\square$

Algorithm 3

Checking if ${\mathcal {C}}\setminus X$ is a ${\mathcal {C}}$-semigroup.

We illustrate this algorithm with the following example.

Example 18

Let ${\mathcal {C}}$ be the cone generated by $\Lambda _{{\mathcal {C}}}=\{(1,0), (1,1), (1,2)\}$ and $X=\{(1,0),(1,1),$ $(1,2),(2,0),(2,1),(2,2),(2,3),(2,4)\}$. Since $X\not \subset \Lambda _{{\mathcal {C}}}$ and $X=D(X)$, if we apply Algorithm 3, we obtain that:

• $t=0$, $\Lambda = \{(2, 0), (3, 0), (1, 1), (2, 1), (1, 2)\}$,

• $t=1$, $\Lambda = \{(2, 0), (3, 0), (2, 1), (3, 1), (1, 2), (2, 2), (2, 3)\}$,

• $t=2$, $\Lambda = \{(2, 0), (3, 0), (2, 1), (3, 1), (2, 2), (3, 2), (2, 3), (3, 3), (2, 4),$ (3, 4),  $(3, 5), (3, 6)\}$.

Therefore,

$$\begin{aligned} {\mathcal {C}}\setminus X= & {} \big \langle (3, 0), (4, 0), (5, 0), (3, 1), (4, 1), (5, 1), (3, 2), (4, 2), (5, 2),(3, 3),\\{} & {} (4, 3), (5, 3), (3, 4), (4, 4), (5, 4), (3, 5), (4, 5), (5, 5), (3, 6), (4, 6),\\{} & {} (5, 6), (4, 7),(5, 7), (4, 8), (5, 8), (5, 9), (5, 10) \big \rangle . \end{aligned}$$

Fix $S\subset {\mathbb {N}}^p$ a ${\mathcal {C}}$-semigroup minimally generated by $\Lambda =$ $\{{\textbf{s}}_1,\ldots , {\textbf{s}}_q,{\textbf{s}}_{q+1},\ldots ,{\textbf{s}}_t\}$, and consider $\Lambda _{\mathcal {C}}=\{{\textbf{a}}_1,\ldots , {\textbf{a}}_q,{\textbf{a}}_{q+1},\ldots ,{\textbf{a}}_m\}$ the minimal generating set of ${\mathcal {C}}$, with ${\textbf{s}}_i,{\textbf{a}}_i\in \tau _i$ for $i=1,\ldots , q$ (we assume that the integer cone ${\mathcal {C}}$ has q extremal rays $\{\tau _1,\ldots ,\tau _q\}$).

Note that, the elements ${\textbf{x}}$ of $\textrm{SG}(S)$ are those elements in ${\mathcal {H}}(S)$ such that $S\cup \{{\textbf{x}}\}$ is again a ${\mathcal {C}}$-semigroup. These gaps play an important role in decomposing a ${\mathcal {C}}$-semigroup into irreducible ${\mathcal {C}}$-semigroups ([11]).

Similarly to numerical semigroups, given two ${\mathcal {C}}$-semigroups S and T with $S\subsetneq T$, any ${\textbf{x}}\in \max _{\le _{\mathcal {C}}} (T{\setminus } S)$ belongs to $\textrm{SG}(S)$, that is $S\cup \{{\textbf{x}}\}$ is a ${\mathcal {C}}$-semigroup. Note that if ${\textbf{x}}\in \max _{\le _{\mathcal {C}}} (T{\setminus } S)$, then $2{\textbf{x}}\in S$. From this fact, we can prove the following proposition.

Proposition 19

Let S be a ${\mathcal {C}}$-semigroup and G be a subset of ${\mathcal {H}}(S)$. Then, $S\in \max _{\subseteq } \{ T\text { is a }{\mathcal {C}}\text {-semigroup}\mid G\subseteq {\mathcal {H}}(T)\}$ if and only if $\textrm{SG}(S)\subseteq G$.

Proof

We know that ${\textbf{x}}\in \textrm{SG}(S)$ if and only if $S\cup \{{\textbf{x}}\}$ is a ${\mathcal {C}}$-semigroup. So, if $S\in \max _{\subseteq } \{ T\text { is a }{\mathcal {C}}\text {-semigroup}\mid G\subseteq {\mathcal {H}}(T)\}$, then ${\textbf{x}}\in G$. Otherwise, $S\subsetneq S\cup \{{\textbf{x}}\}$ and S is not maximal.

Assume that S is not maximal but $\textrm{SG}(S)\subseteq G$, so there exists T a ${\mathcal {C}}$-semigroup such that $S\subsetneq T$ and $G\subseteq {\mathcal {H}}(T)$. Let ${\textbf{x}}\in \max _{\le _{\mathcal {C}}} (T{\setminus } S)$, thus ${\textbf{x}}\in \textrm{SG}(S)\cap T$, but it is not possible ($\textrm{SG}(S)\subseteq G\subseteq T$). $\square$

There is another interesting subset related to the set of gaps of S. A subset X of ${\mathcal {H}}(S)$ is said to determine ${\mathcal {H}}(S)$ if $S= \max _{\subseteq } \{ T\text { is a }{\mathcal {C}}\text {-semigroup}\mid X\subseteq {\mathcal {H}}(T)\}$. These subsets were introduced in [18] for numerical semigroups.

Proposition 20

Let X be a finite subset of an integer cone ${\mathcal {C}}\subset {\mathbb {N}}^p$. Then, X determines the set of gaps of a ${\mathcal {C}}$-semigroup if and only if ${\mathcal {C}}\setminus D(X)$ is a ${\mathcal {C}}$-semigroup.

Proof

Fix $\Lambda _{\mathcal {C}}=\{{\textbf{a}}_1,\ldots , {\textbf{a}}_q,{\textbf{a}}_{q+1},\ldots ,{\textbf{a}}_m\}$ the minimal generating set of ${\mathcal {C}}\subset {\mathbb {N}}^p$.

Assume that X determines ${\mathcal {H}}(S)$ for a ${\mathcal {C}}$-semigroup S, so $X\subset D(X)\subset {\mathcal {H}}(S)$ and $S\subset {\mathcal {C}}{\setminus } D(X)$. Let $S'$ be the non-empty set

$$\begin{aligned} \{0\}\cup \bigcup _{i=1}^q\Big \{h_i{\textbf{a}}_i+{\mathcal {C}}\mid h_i=\min _{n\in {\mathbb {N}}}\{(n{\textbf{a}}_i+{\mathcal {C}})\cap X=\emptyset \}\Big \}. \end{aligned}$$

Note that $S'$ is a ${\mathcal {C}}$-semigroup. Let ${\textbf{a}}$ and ${\textbf{b}}$ be two elements in $S'$, so ${\textbf{a}}= h_i {\textbf{a}}_i + \sum _{k=1}^m \alpha _k {\textbf{a}}_k$, and ${\textbf{b}}= h_j {\textbf{a}}_j + \sum _{k=1}^m \beta _k {\textbf{a}}_k$ for some $h_i,h_j,j,i,\alpha _k,\beta _k\in {\mathbb {N}}$ with $i,j\in [q]$ and $k\in [m]$. Hence, ${\textbf{a}} + {\textbf{b}} = h_i {\textbf{a}}_i + (h_j {\textbf{a}}_j + \sum _{k=1}^m (\alpha _k +\beta _k) {\textbf{a}}_k)\in h{\textbf{a}}_i+{\mathcal {C}}$. Furthermore, ${\mathcal {C}}{\setminus } S'$ is finite. Get any ${\textbf{a}} \in \Lambda _{\mathcal {C}}$, then ${\textbf{a}} = \sum _{i=1}^q \alpha _i {\textbf{a}}_i$ for some $\alpha _1,\ldots ,\alpha _q\in {\mathbb {Q}}_\ge$, and hence $k {\textbf{a}} = \sum _{i=1}^q \beta _i {\textbf{a}}_i$ for some $\beta _1,\ldots ,\beta _q,k\in {\mathbb {N}}$. We can assume that $\beta _i\ge h_i$. In that case, ${\mathcal {C}}\setminus S'$ is a subset of the finite set $\{\sum _{i=1}^q \gamma _i {\textbf{a}}_i\mid 0\le \gamma _i\le \beta _i \}$. We obtain that $S'$ is a finitely generated ${\mathcal {C}}$-semigroup; let $\Lambda _{S'}$ its minimal generating set. The set X is a subset of ${\mathcal {H}}(S')$ by construction.

For every ${\textbf{a}}\in {\mathcal {C}}\setminus D(X)$, we can define $S_{\textbf{a}}$ as the semigroup generated by $\{{\textbf{a}}\}\cup \Lambda _{S'}$. Since $X\subset {\mathcal {H}}(S_{\textbf{a}})$, and X determines ${\mathcal {H}}(S)$, we have that $S_{\textbf{a}}\subset S$. Hence, ${\mathcal {C}}\setminus D(X) \subset S$ and then ${\mathcal {C}}\setminus D(X)$ is a ${\mathcal {C}}$-semigroup.

Conversely, any ${\mathcal {C}}$-semigroup T such that $X\subset {\mathcal {H}}(T)$ satisfies that $D(X)\subset {\mathcal {H}}(T)$. Thus, X determines the set of gaps of the ${\mathcal {C}}$-semigroup ${\mathcal {C}}\setminus D(X)$. $\square$

The sets determining the set of gaps of a ${\mathcal {C}}$-semigroup are related to its set of fundamental gaps.

Lemma 21

Let S be a ${\mathcal {C}}$-semigroup and X be a subset of ${\mathcal {H}}(S)$. Then, X determines ${\mathcal {H}}(S)$ if and only if $\textrm{FG}(S)\subseteq X$.

Proof

By Proposition 20, if X determines ${\mathcal {H}}(S)$, then ${\mathcal {H}}(S)=D(X)$. Thus, for all ${\textbf{x}}\in {\mathcal {H}}(S)$, $h{\textbf{x}}\in X$ for some $h\in {\mathbb {N}}$. In particular, for every fundamental gap of S, the integer h has to be one. Hence, if ${\textbf{x}}\in \textrm{FG}(S)$ then ${\textbf{x}}\in X$.

Conversely, since $X\subset {\mathcal {H}}(S)$, we know that $D(X)\subseteq {\mathcal {H}}(S)$. Let ${\textbf{x}}\in {\mathcal {H}}(S)$ and consider $h=\max \{k\in {\mathbb {N}}\mid k{\textbf{x}}\in {\mathcal {H}}(S)\}$. In that case, $h{\textbf{x}}\in {\mathcal {H}}(S)$, and $2\,h{\textbf{x}},3\,h{\textbf{x}} \in S$. Therefore, $h{\textbf{x}}\in \textrm{FG}(S)\subseteq X$, ${\textbf{x}}\in D(X)$, and ${\mathcal {H}}(S)\subseteq D(X)$. $\square$

Analogously to the case of numerical semigroups, it happens that $\textrm{FG}(S)$ is the smallest subset of ${\mathcal {H}}(S)$ determining ${\mathcal {H}}(S)$. Also, the relationship between the special and fundamental gaps of a ${\mathcal {C}}$-semigroup is equivalent to their relationship for numerical semigroups.

Lemma 22

Let S be a ${\mathcal {C}}$-semigroup. Then, $\textrm{SG}(S)=\max _{\le _S} \textrm{FG}(S)$.

Proof

Trivially, for any ${\textbf{x}}\in \textrm{SG}(S)$, $2{\textbf{x}},3{\textbf{x}}\in S$, and then $\textrm{SG}(S)\subseteq \textrm{FG}(S)$. Assume that, for an element ${\textbf{x}}\in \textrm{SG}(S)$, there exists some ${\textbf{y}}\in \textrm{FG}(S)$ with ${\textbf{x}}\le _{S} {\textbf{y}}$. So, ${\textbf{x}}+{\textbf{s}}= {\textbf{y}}$ for some ${\textbf{s}}\in S$. Since ${\textbf{x}}$ is a pseudo-Frobenius element of S, ${\textbf{y}}\in S$. It is not possible, then ${\textbf{x}}\in \max _{\le _S} \textrm{FG}(S)$. $\square$

A ${\mathcal {C}}$-irreducible semigroup can also be characterized from its fundamental gaps using the above lemma.

Corollary 23

S is a ${\mathcal {C}}$-irreducible semigroup if and only if the cardinality of $\max _{\le _S} \textrm{FG}(S)$ is equal to one.

Table 1 All ${\mathcal {C}}$-semigroups with ${\mathcal {C}}=\langle (1,0),(1,1),(1,2) \rangle$, and Frobenius vector equal to (2, 1); $\circ \equiv \text {gap}$, ${\text {blacksquare}} \equiv \text {minimal generator}$, ${\text {red bullet}} \equiv \text {element in }S$

The next example illustrates many results appearing in this section.

Example 24

Let ${\mathcal {C}}$ be the cone with extremal rays $\tau _1=\langle (1,0) \rangle$ and $\tau _2=\langle (1,1) \rangle$ and $X=\{ (1, 1), (3, 0), (3, 1), (3, 2), (5, 1), (5, 2) \}$. Since $D(X)=\{(1,0),$ $(1, 1), (3, 0), (3, 1), (3, 2), (5, 1), (5, 2) \}$, we have that

$$\begin{aligned}{} & {} \{(x,s)\in D(X)\times ({\mathcal {C}}\setminus D(X))\mid s\le _{{\mathcal {C}}} x\}\\{} & {} \quad =\{((1, 1),(0,0)),\\{} & {} \qquad ((3, 0),(0,0)), ((3, 1),(0,0)), ((3, 2),(0,0)),((5, 1),(0,0)),((5, 2),(0,0)),\\{} & {} \qquad ((3,0),(2,0)), ((3,1),(2,0)), ((5,1),(2,0)),((5,2),(2,0)),((3,2),(2,1)),\\{} & {} \qquad ((5,1),(2,1)), ((5,2),(2,1)),((3,2),(2,2)),((5,2),(2,2)), ((5,1),(4,0)), \\{} & {} \qquad ((1, 0), (0, 0)), ((3, 1), (2, 1)),((5, 1), (4, 1)),((5, 2), (4, 1)),((5, 2), (4, 2))\} \end{aligned}$$

Therefore, by Proposition 16, ${\mathcal {C}}\setminus D(X)$ is a ${\mathcal {C}}$-semigroup and, by Proposition 20, X determines the set of gaps of a ${\mathcal {C}}$-semigroup. If we call this semigroup S, we have that ${\mathcal {H}}(S)=D(X)$. It is not difficult to check that $S=\langle (2, 0), (5, 0), (2, 1), (2, 2), (3, 3) \rangle$ and that, in this case, $\textrm{FG}(S)=X$. Moreover, we can compute the set of pseudo-Frobenius elements of S, and we get $\textrm{PF}(S)=\{(5,1), (5,2)\}$, so $\textrm{SG}(S)=\{(5,1), (5,2)\}$. On the other hand, $\textrm{FG}(S)=\{(1, 1), (3, 0), (3, 1), (3, 2), (5, 1), (5, 2)\}$ and $\max _{\le _S}\textrm{FG}(S) = \{(5,1), (5,2)\}$, as we know by Lemma 22.

6 Computing All the ${\mathcal {C}}$-semigroups with a Given Frobenius Vector

Let ${\mathcal {C}}\subset {\mathbb {N}}^p$ be an integer cone, $\preceq$ be a monomial order on ${\mathbb {N}}^p$, and S be a ${\mathcal {C}}$-semigroup with Frobenius vector $F(S)\in C\setminus \{0\}$. Note that F(S) is a minimal generator of $S\cup \{F(S)\}$.

Conversely to Lemma 17, we can consider the following sequence of ${\mathcal {C}}$-semigroups for some $t\in {\mathbb {N}}$: $S_t=S$, $S_{i-1}= S_i\cup {F(S_i)}$ for all $i=1,\dots ,t$, and $S_0={\mathcal {C}}$. Such a sequence can be constructed for any ${\mathcal {C}}$-semigroup with Frobenius vector F(S). So, from a minimal system of generators of ${\mathcal {C}}$, we obtain new ${\mathcal {C}}$-semigroups just by removing a minimal generator ${\textbf{s}}$ fulfilling that ${\textbf{s}}\preceq F(S)$. Performing this process as many times as possible, we obtain all the ${\mathcal {C}}$-semigroups with Frobenius vector F(S). Note that this process is finite due to the finiteness of the set $\{ {\textbf{s}} \in {\mathcal {C}}\mid {\textbf{s}} \preceq F(S)\}$. This idea allows us to provide an algorithm for computing all the ${\mathcal {C}}$-semigroups with a fixed Frobenius vector (Algorithm 4). Moreover, this algorithm can be modified to obtain all the ${\mathcal {C}}$-semigroups with the Frobenius vector less than or equal to a fixed Frobenius vector. For any set of ordered pairs, A, $\pi _1(A)$ denotes the set of the first projection of its elements.

Algorithm 4

Computing ${\mathcal {C}}$-semigroups with a given Frobenius vector.

Example 25

Let ${\mathcal {C}}$ be the cone generated by $\{(1,0),(1,1),(1,2)\}$ and $F=(2,1)$. Then, applying Algorithm 1 with the graded lexicographic order, we get the following ${\mathcal {C}}$-semigroups:

• $S_1=\langle (2, 0), (3, 0), (1, 1), (1, 2)\rangle$,

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

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

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

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

• $S_6=\langle (3, 0), (4, 0), (5, 0), (3, 1), (4, 1), (5, 1), (2, 2), (3, 2), (4, 2), (2, 3),$(3, 3), (4, 3),  $(2, 4), (3, 4), (3, 5), (3, 6)\rangle$,

• $S_7=\langle (1, 1), (2, 3), (2, 4), (3, 0), (3, 1), (3, 2), (3, 6), (4, 0), (5, 0)\rangle$,

• $S_8=\langle (1, 0), (2, 2), (2, 3), (2, 4), (3, 1), (3, 5),$ $(3, 6)\rangle$,

• $S_9=\langle (2, 0), (2, 2), (2, 3), (2, 4), (3, 0), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5),$ (3, 6),  $(4, 1)\rangle$,

• $S_{10}=\langle (1, 1), (2, 0),$ $(2, 3), (2, 4), (3, 0), (3, 2), (3, 6)\rangle$

These semigroups are shown in Table 1.