# Some properties of affine $\mathcal{C}$-semigroups

J. I. García-García
D. Marín-Aragón
A. Sánchez-Loureiro
A. Vigneron-Tenorio

###### 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.

_Keywords:_ $\mathcal{C}$-semigroup, Frobenius element, fundamental gap,
irreducible semigroup, pseudo-Frobenius element, pseudo-symmetric semigroup,
special gap, symmetric semigroup.

_2020 Mathematics Subject Classification:_ 20M14 (Primary), 68R05 (Secondary).

## Introduction

A $\mathcal{C}$-semigroup is a non-empty subset 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 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 [10] 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 [6], 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 [12], an extension of Wilf’s conjecture for
numerical semigroups is given to $\mathcal{C}$-semigroups, and in [3], 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], [4], [5], [7], and [17]. For any
$\mathcal{C}$-semigroup, in [9], 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
[13]); $S$ is $\mathcal{C}$-irreducible (simplifying irreducible) in another
case. In this work, we also characterize irreducible $\mathcal{C}$-semigroups
from their genus and from their generalized Frobenius numbers. 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 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
$\mathbf{f}\in\mathcal{C}$, we give a procedure to obtain all
$\mathcal{C}$-semigroups with Frobenius element equal to $\mathbf{f}$.

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 [11]) developed by the authors in Python [14] 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: Section 1 is devoted to
provide the reader with the necessary background for the correct understanding
of the work. In Section 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 Section 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 Section 5.
Finally, in Section 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.

##  1 Preliminaries

In this work, $\mathbb{Q}$, $\mathbb{Q}_{\geq}$, 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 $\mathbb{Q}_{\geq}^{p}$ is the convex hull
of finitely many half lines in $\mathbb{Q}_{\geq}^{p}$ 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 [2, 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 [15]:

  * •

$\mathbf{x}\in\mathcal{H}(S)$ is a _fundamental gap_ if
$2\mathbf{x},3\mathbf{x}\in S$. The set of these elements is denoted by
$\mathrm{FG}(S)$.

  * •

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

  * •

$\mathbf{x}\in\mathcal{H}(S)$ is a _special gap_ of $S$ if
$\mathbf{x}\in\mathrm{PF}(S)$ and $2\mathbf{x}\in S$. We denote by
$\mathrm{SG}(S)$ 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 $\mathbf{x},\mathbf{y}\in\mathbb{N}^{p}$,
consider the partial order $\mathbf{x}\leq_{L}\mathbf{y}$ if
$\mathbf{y}-\mathbf{x}\in L$. 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
[8]). From the properties of a monomial order, the (induced) total order
$\preceq$ on $\mathbb{N}^{p}$ satisfies:

  * •

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

  * •

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

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 $\mathbf{a}\prec\mathbf{b}$ if and only if
there exists an integer $i$ belonging to $[p-1],$ such that
$M_{1}\mathbf{a}=M_{1}\mathbf{b},\ldots,M_{i}\mathbf{a}=M_{i}\mathbf{b}$ and
$M_{i+1}\mathbf{a}<M_{i+1}\mathbf{b}.$

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 $\mathcal{N}(S)=\\{\mathbf{x}\in
S\mid\mathbf{x}\preceq F(S)\\}$.

The following lemma generalizes to $\mathcal{C}$-semigroups Proposition 2.26
in [15].

######  Lemma 1.

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

###### Proof.

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

##  2 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 $\mathrm{PF}(S)$ is equal to $\\{F(S)\\}$ or
$\\{F(S),F(S)/2\\}$ (see [13]). If $\mathrm{PF}(S)=\\{F(S)\\}$, we say that
$S$ is symmetric, and pseudo-symmetric when
$\mathrm{PF}(S)=\\{F(S),F(S)/2\\}$.

For any element $\mathbf{n}$ in $\mathcal{C}$, let $I_{S}(\mathbf{n})$ be the
set $\\{\mathbf{s}\in S\mid\mathbf{s}\leq_{\mathcal{C}}\mathbf{n}\\}$.

######  Remark 2.

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

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 $\mathbf{x}\in\mathcal{H}(S)$, there
exists $\mathbf{s}\in S$ such that $\mathbf{x}+\mathbf{s}=F(S)$, that is
$\mathbf{s}\in I_{S}(F(S))$, and then $\sharp I_{S}(F(S))\geq g$. Since
$g\geq\sharp I_{S}(F(S))$, we conclude that $g=\sharp I_{S}(F(S))$.

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

######  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 $\mathrm{PF}(S)=\\{F(S),F(S)/2\\}$,
and $g>\sharp I_{S}(F(S))$. For all
$\mathbf{x}\in\mathcal{H}(S)\setminus\\{F(S)/2\\}$, there exists some
$\mathbf{s}\in S$ such that $\mathbf{x}+\mathbf{s}=F(S)$, or
$\mathbf{x}+\mathbf{s}=F(S)/2$. If the first option is satisfied,
$\mathbf{s}\in I_{S}(F(S))$. In other case,
$\mathbf{x}+\mathbf{s}+F(S)/2=F(S)$ and then $\mathbf{s}+F(S)/2$ also belongs
to $I_{S}(F(S))$. Besides, $F(S)/2+\mathbf{s}\neq F(S)$ for every
$\mathbf{s}\in S$. Hence, $\sharp I_{S}(F(S))\geq g-1$.

Conversely, suppose that $g=\sharp I_{S}(F(S))+1$ and
$F(S)/2\in\mathbb{N}^{p}$. Hence, there exists only one
$\mathbf{x}\in\mathcal{H}(S)\setminus\\{F(S)\\}$ with
$\mathbf{x}+\mathbf{s}\neq F(S)$ for all $\mathbf{s}\in S$. Hence,
$\mathrm{PF}(S)=\\{F(S),\mathbf{x}\\}$. If $\mathbf{x}\neq F(S)/2$, then there
is $\mathbf{s}\in S$ such that $F(S)/2+\mathbf{s}=F(S)$, and $F(S)/2\in S$,
but it is not possible. So, $\mathbf{x}=F(S)/2$. ∎

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

######  Proposition 5.

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

$\mathrm{PF}(S)=\\{\mathbf{a}-\mathbf{b}\mid\mathbf{a}\in\mathrm{maximals}_{\leq_{S}}\mathrm{Ap}(S,\mathbf{b})\\}.$

From this result, we can generalize the corollaries 4.12 and 4.19 in [15].

######  Corollary 6.

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

######  Corollary 7.

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

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 [15], and the
theorems 5.6 and 5.7 in [5], respectively.

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

$\begin{multlined}\Lambda_{S_{1}}=\\{(3,1),(4,1),(5,1),(6,1),(7,1),(7,3),(8,1),(8,3),(9,1),(10,1),\\\
(11,1),(12,1),(12,5),(13,1),(14,1),(15,1)\\}\end{multlined}\Lambda_{S_{1}}=\\{(3,1),(4,1),(5,1),(6,1),(7,1),(7,3),(8,1),(8,3),(9,1),(10,1),\\\
(11,1),(12,1),(12,5),(13,1),(14,1),(15,1)\\}$

is symmetric, while the $S_{2}$ minimally generated by

$\begin{multlined}\Lambda_{S_{2}}=\\{(3,1),(5,2),(6,1),(7,1),(7,2),(7,3),(8,1),(9,1),(10,1)\\\
(11,1),(12,1),(13,1),(14,1),(15,1)\\}\end{multlined}\Lambda_{S_{2}}=\\{(3,1),(5,2),(6,1),(7,1),(7,2),(7,3),(8,1),(9,1),(10,1)\\\
(11,1),(12,1),(13,1),(14,1),(15,1)\\}$

is pseudo-symmetric.

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

##  3 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
$\mathbf{f}\in\mathcal{C}\setminus\\{0\\}$. Consider a monomial order
$\preceq$ on $\mathbb{N}^{p}$ and decompose the set
$I_{\mathcal{C}}(\mathbf{f})$ as
$I_{\mathcal{C}}(\mathbf{f})=I_{1}(\mathbf{f})\sqcup I_{2}(\mathbf{f})$ with
$I_{1}(\mathbf{f})=\\{\mathbf{x}\in
I_{\mathcal{C}}(\mathbf{f})\mid\mathbf{0}\neq\mathbf{x}\preceq\mathbf{f}/2\\}$
and $I_{2}(\mathbf{f})=\\{\mathbf{x}\in
I_{\mathcal{C}}(\mathbf{f})\mid\mathbf{x}\succ\mathbf{f}/2\\}$ (when
$\mathbf{f}/2\notin\mathbb{N}^{p}$, consider $\preceq$ as the monomial order
extended to $\mathbb{Q}_{\geq}^{p}$). We define the $\mathcal{C}$-semigroup
$S(\mathbf{f})$ as
$\big{(}\mathcal{C}\setminus\\{\mathbf{f}\\}\big{)}\setminus
I_{1}(\mathbf{f})$. 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 $\mathbf{f}=(4,2)$.
Then, $\mathbf{f}/2=(2,1)$ and

$I_{\mathcal{C}}(\mathbf{f})=\\{(1,0),(1,1),(1,2),(2,0),(2,1),(2,2),(3,0),(3,1),(3,2),(4,2)\\}.$

Let $\prec_{1}$ and $\prec_{2}$ be the orders defined by the matrices
$\left(\begin{array}[]{cc}1&1\\\ 1&0\end{array}\right)$ and
$\left(\begin{array}[]{cc}1&1\\\ 0&1\end{array}\right)$, respectively. In the
first case,
$I_{1}(\mathbf{f})_{\prec_{1}}=\\{(1,0),(1,1),(1,2),(2,0),(2,1)\\}$ and

$S(\mathbf{f})_{\prec_{1}}=\langle(3,0),(4,0),(5,0),(3,1),(4,1),\\\
(5,1),(2,2),(3,2),(2,3),(3,3),(4,3),(2,4),(3,4),(3,5),(3,6)\rangle.$

In the other one,
$I_{1}(\mathbf{f})_{\prec_{2}}=\\{(1,0),(1,1),(2,0),(2,1),(3,0)\\}$ and

$S(\mathbf{f})_{\prec_{2}}=\langle(4,0),(5,0),(6,0),(7,0),(3,1),\\\
(4,1),(5,1),(6,1),(1,2),(2,2),(3,2),(2,3),(3,3)\rangle.$

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

######  Lemma 12.

The $\mathcal{C}$-semigroup $S(\mathbf{f})$ is irreducible. Moreover,
$\mathbf{f}$ is the Frobenius vector of $S(\mathbf{f})$ for any monomial
order, and $S(\mathbf{f})$ is the unique irreducible $\mathcal{C}$-semigroup
satisfying all its gaps belong to $I_{1}(\mathbf{f})\cup\\{\mathbf{f}\\}$.

###### Proof.

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

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

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

The following proposition gives us an irreducible $\mathcal{C}$-semigroups
from an existing one, such that both have the same Frobenius vector.

######  Proposition 13.

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

  1. 1.

$2\mathbf{x}-\mathbf{f}\notin S$.

  2. 2.

$3\mathbf{x}\neq 2\mathbf{f}$.

  3. 3.

$4\mathbf{x}\neq 3\mathbf{f}$.

Then, $S^{\prime}=(S\setminus\\{\mathbf{x}\\})\cup\\{\mathbf{f}-\mathbf{x}\\}$
is a $\mathcal{C}$-semigroup irreducible with Frobenius vector $\mathbf{f}$.

###### Proof.

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

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

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

Since $\sharp I_{S}(\mathbf{f})=\sharp I_{S^{\prime}}(\mathbf{f})$ and
$\sharp\mathcal{H}(S)=\sharp\mathcal{H}(S^{\prime})$, $S^{\prime}$ is
irreducible by the propositions 3 and 4. ∎

From this point forward, we will use the notation $m(S)$ to represent the
minimum element (with respect to the partial order $\preceq$) in the minimal
generating set of $S$. This element is often referred to as the multiplicity
of $S$.

We denote by $\mathfrak{I}(\mathbf{f})$ the set of the irreducible
$\mathcal{C}$-semigroups with Frobenius vector $\mathbf{f}$. Given
$S\in\mathfrak{I}(\mathbf{f})$, consider $S_{0}=S,$ and
$S_{n}=(S_{n-1}\setminus\\{m(S_{n-1})\\})\cup\\{\mathbf{f}-m(S_{n-1})\\}$ when
$m(S_{n-1})\in I_{1}(\mathbf{f})$, or $S_{n}=S_{n-1}$ in other case, for
$n>1$. Note that $S_{n}=S_{n-1}$ if $S_{n}=S(\mathbf{f})$. Since
$I_{1}(\mathbf{f})$ 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
$V=\mathfrak{I}(\mathbf{f})$, and edge set $E=\big{\\{}(A,B)\in V\times V\mid
m(A)\prec\mathbf{f}/2\text{ and
}B=(A\setminus\\{m(A)\\})\cup\\{\mathbf{f}-m(A)\\}\big{\\}}$.

######  Theorem 14.

Let $\preceq$ be a monomial order on $\mathbb{N}^{p}$,
$\mathcal{C}\subset\mathbb{N}^{p}$ be an integer cone, and
$\mathbf{f}\in\mathcal{C}$ 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 $\mathfrak{I}(\mathbf{f})$. If $m(S)\notin
I_{1}(\mathbf{f})$, then $S=S(\mathbf{f})$. Assume that $m(S)\in
I_{1}(\mathbf{f})$. In that case, $m(S)\prec\mathbf{f}/2$, that is,
$2m(S)-\mathbf{f}\notin S$, $3m(S)\neq 2\mathbf{f}$, and $4m(S)\neq
3\mathbf{f}$. By Proposition 13,
$S_{1}=(S\setminus\\{m(S)\\})\cup\\{\mathbf{f}-m(S)\\}$ is irreducible. That
means $(S,S_{1})\in E$. Following this construction, $G$ is a tree whose root
is $S(\mathbf{f)}$. ∎

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).

Input: A monomial order $\preceq$ on $\mathbb{N}^{p}$, an integer cone
$\mathcal{C}$ and $\textbf{f}\in\mathcal{C}$.

Output: A tree of irreducible $\mathcal{C}$-semigroups with Frobenius vector
f.

begin

$X\leftarrow\\{S(\textbf{f})\\}$;

$Y\leftarrow\emptyset$;

while  _$X\neq\emptyset$_ do

$S\leftarrow\mbox{First}(X)$;

$A\leftarrow\\{x\in S\mid x\in
I_{2}(\textbf{f})\cap\Lambda_{S},2x-\textbf{f}\notin S,3x\neq\textbf{f},4x\neq
3\textbf{f},\textbf{f}-x\prec m(S)\\}$;

if  _$A=\emptyset$_ then

$Y\leftarrow Y\cup\\{S\\}$;

else

for  _$x\in A$_ do

$H\leftarrow(\mathcal{H}(S)\setminus\\{\textbf{f}-x\\})\cup\\{x\\}$;

$S^{\prime}\leftarrow\mathcal{C}$-semigroup with $\mathcal{H}(S^{\prime})=H$;

$X\leftarrow X\cup\\{S^{\prime}\\}$;

$X\leftarrow X\setminus\\{S\\}$;

return _$Y$_

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

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

Applying Algorithm 1, we have that
$I_{2_{\prec_{1}}}(\textbf{f})=\\{(2,2),(3,0),(3,1),(3,2),(4,2)\\}$. 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 Figure 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 $\prec_{2}$, Figure 2
appears.

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

Figure 1: Tree of irreducible $\mathcal{C}$-semigroups with $\prec_{1}$.

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

Figure 2: Tree of irreducible $\mathcal{C}$-semigroups with $\prec_{2}$.

##  4 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 [15, 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 $\\{\mathbf{a}\in\mathcal{C}\mid n\mathbf{a}\in X\text{ for
some }n\in\mathbb{N}\\}$.

######  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 $\mathbf{x}-\mathbf{s}\in X$ for every
$(\mathbf{x},\mathbf{s})\in(X,\mathcal{C}\setminus X)$ with
$\mathbf{s}\leq_{\mathcal{C}}\mathbf{x}$.

###### Proof.

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

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

From 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).

Input: $\mathcal{C}\subset\mathbb{N}^{p}$ an integer cone, and $X$ a finite
subset of $\mathcal{C}\setminus\\{0\\}$.

Output: True if $\mathcal{C}\setminus X$ is a $\mathcal{C}$-semigroup, and
False in other case.

begin

if  _$X\neq D(X)$_ then

return False

while  _$X\neq\emptyset$_ do

$\mathbf{x}\leftarrow\text{First}(X)$;

$A\leftarrow\\{\mathbf{s}\in\mathcal{C}\setminus
X\mid\mathbf{s}\leq_{\mathcal{C}}\mathbf{x}\\}$;

$s\leftarrow\text{First}(A)$;

while  _$A\neq\emptyset$_ do

if  _$\mathbf{x}-\mathbf{s}\notin X$_ then

return False

$s\leftarrow\text{First}(A\setminus\\{\mathbf{s}\\})$;

$X\leftarrow X\setminus\\{\mathbf{x}\\}$;

return True.

Algorithm 2 Checking if $\mathcal{C}\setminus X$ is a $\mathcal{C}$-semigroup.

Since, for each $\mathbf{x}\in X$, the set
$\\{\mathbf{s}\in\mathcal{C}\setminus
X\mid\mathbf{s}\leq_{\mathcal{C}}\mathbf{x}\\}$ can be very very large, the
condition $\mathbf{x}-\mathbf{s}\notin X$ 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 [12,
Lemma 3].

######  Lemma 17.

Fix a total order $\preceq$ on $\mathbb{N}^{p}$, and let
$X=\\{\mathbf{x}_{1}\preceq\mathbf{x}_{2}\preceq\cdots\preceq\mathbf{x}_{t}\\}$
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,
$S_{i}=S_{i-1}\setminus\\{\mathbf{x}_{i}\\}$ is a $\mathcal{C}$-semigroup, and
$\mathbf{x}_{i}$ is a minimal generator of $S_{i-1}$, for every $i\in[t]$.

###### Proof.

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

Input: A total order $\preceq$ on $\mathbb{N}^{p}$, $\Lambda_{\mathcal{C}}$
the minimal generating set of the integer cone
$\mathcal{C}\subset\mathbb{N}^{p}$, and
$X=\\{\mathbf{x}_{1}\preceq\cdots\preceq\mathbf{x}_{t}\\}\subset\mathcal{C}\setminus\\{0\\}$.

Output: If $\mathcal{C}\setminus X$ is a $\mathcal{C}$-semigroup, its minimal
generating set, and the empty set in another case.

begin

if  _$X\subset\Lambda_{\mathcal{C}}$_ then

return the minimal generating set of $\mathcal{C}\setminus X$

if  _$X\neq D(X)$_ then

return $\\{\\}$

$\Lambda\leftarrow\Lambda_{\mathcal{C}}$;

for  _$1\leq i\leq t$_ do

if  _$\mathbf{x_{i}}\notin\Lambda$_ then

return $\\{\\}$

$\Lambda\leftarrow\text{the minimal generating set of
}\langle\Lambda\rangle\setminus\\{\mathbf{x}_{i}\\}$;

$X\leftarrow X\setminus\\{\mathbf{x}_{i}\\}$;

if  _$X\subset\Lambda$_ then

return the minimal generating set of $\langle\Lambda\rangle\setminus X$

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,

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

Fix $S\subset\mathbb{N}^{p}$ a $\mathcal{C}$-semigroup minimally generated by
$\Lambda=\\{\mathbf{s}_{1},\ldots,\mathbf{s}_{q},\mathbf{s}_{q+1},\ldots,\mathbf{s}_{t}\\}$,
and consider
$\Lambda_{\mathcal{C}}=\\{\mathbf{a}_{1},\ldots,\mathbf{a}_{q},\mathbf{a}_{q+1},\ldots,\mathbf{a}_{m}\\}$
the minimal generating set of $\mathcal{C}$, with
$\mathbf{s}_{i},\mathbf{a}_{i}\in\tau_{i}$ 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 $\mathbf{x}$ of $\mathrm{SG}(S)$ are those elements in
$\mathcal{H}(S)$ such that $S\cup\\{\mathbf{x}\\}$ is again a
$\mathcal{C}$-semigroup. These gaps play an important role in decomposing a
$\mathcal{C}$-semigroups into irreducible $\mathcal{C}$-semigroups ([9]).

Similarly to numerical semigroups, given two $\mathcal{C}$-semigroups $S$ and
$T$ with $S\subsetneq T$, any
$\mathbf{x}\in\max_{\leq_{\mathcal{C}}}(T\setminus S)$ belongs to
$\mathrm{SG}(S)$, that is to say $S\cup\\{\mathbf{x}\\}$ is a
$\mathcal{C}$-semigroup. Note that if
$\mathbf{x}\in\max_{\leq_{\mathcal{C}}}(T\setminus S)$, then $2\mathbf{x}\in
S$. 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 $\mathcal{C}$-semigroup}\mid
G\subseteq\mathcal{H}(T)\\}$ if and only if $\mathrm{SG}(S)\subseteq G$.

###### Proof.

We know that $\mathbf{x}\in\mathrm{SG}(S)$ if and only if
$S\cup\\{\mathbf{x}\\}$ is a $\mathcal{C}$-semigroup. So, if
$S\in\max_{\subseteq}\\{T\text{ is a $\mathcal{C}$-semigroup}\mid
G\subseteq\mathcal{H}(T)\\}$, then $\mathbf{x}\in G$. In other case,
$S\subsetneq S\cup\\{\mathbf{x}\\}$ and $S$ is not maximal.

Assume that $S$ is not maximal but $\mathrm{SG}(S)\subseteq G$, so there
exists $T$ a $\mathcal{C}$-semigroup such that $S\subsetneq T$ and
$G\subseteq\mathcal{H}(T)$. Let
$\mathbf{x}\in\max_{\leq_{\mathcal{C}}}(T\setminus S)$, thus
$\mathbf{x}\in\mathrm{SG}(S)\cap T$, but it is not possible
($\mathrm{SG}(S)\subseteq G\subseteq T$). ∎

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 $\mathcal{C}$-semigroup}\mid
X\subseteq\mathcal{H}(T)\\}$. These subsets were introduced in [16] 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
$\Lambda_{\mathcal{C}}=\\{\mathbf{a}_{1},\ldots,\mathbf{a}_{q},\mathbf{a}_{q+1},\ldots,\mathbf{a}_{m}\\}$
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 $S^{\prime}$ be the non-empty set

$\\{0\\}\cup\bigcup_{i=1}^{q}\Big{\\{}h_{i}\mathbf{a}_{i}+\mathcal{C}\mid
h_{i}=\min_{n\in\mathbb{N}}\\{(n\mathbf{a}_{i}+\mathcal{C})\cap
X=\emptyset\\}\Big{\\}}.$

Note that $S^{\prime}$ is a $\mathcal{C}$-semigroup. Let $\mathbf{a}$ and
$\mathbf{b}$ be two elements in $S^{\prime}$, so
$\mathbf{a}=h_{i}\mathbf{a}_{i}+\sum_{k=1}^{m}\alpha_{k}\mathbf{a}_{k}$, and
$\mathbf{b}=h_{j}\mathbf{a}_{j}+\sum_{k=1}^{m}\beta_{k}\mathbf{a}_{k}$ 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,
$\mathbf{a}+\mathbf{b}=h_{i}\mathbf{a}_{i}+(h_{j}\mathbf{a}_{j}+\sum_{k=1}^{m}(\alpha_{k}+\beta_{k})\mathbf{a}_{k})\in
h\mathbf{a}_{i}+\mathcal{C}$. Furthermore, $\mathcal{C}\setminus S^{\prime}$
is finite. Get any $\mathbf{a}\in\Lambda_{\mathcal{C}}$, then
$\mathbf{a}=\sum_{i=1}^{q}\alpha_{i}\mathbf{a}_{i}$ for some
$\alpha_{1},\ldots,\alpha_{q}\in\mathbb{Q}_{\geq}$, and hence
$k\mathbf{a}=\sum_{i=1}^{q}\beta_{i}\mathbf{a}_{i}$ for some
$\beta_{1},\ldots,\beta_{q},k\in\mathbb{N}$. We can assume that $\beta_{i}\geq
h_{i}$. In that case, $\mathcal{C}\setminus S^{\prime}$ is a subset of the
finite set $\\{\sum_{i=1}^{q}\gamma_{i}\mathbf{a}_{i}\mid
0\leq\gamma_{i}\leq\beta_{i}\\}$. We obtain that $S^{\prime}$ is a finitely
generated $\mathcal{C}$-semigroup; let $\Lambda_{S^{\prime}}$ its minimal
generating set. The set $X$ is a subset of $\mathcal{H}(S^{\prime})$ by
construction.

For every $\mathbf{a}\in\mathcal{C}\setminus D(X)$ and we can define
$S_{\mathbf{a}}$ as the semigroup generated by
$\\{\mathbf{a}\\}\cup\Lambda_{S^{\prime}}$. Since
$X\subset\mathcal{H}(S_{\mathbf{a}})$, and $X$ determines $\mathcal{H}(S)$, we
have that $S_{\mathbf{a}}\subset S$. 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 $\mathcal{C}\setminus D(X)$. ∎

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 $\mathrm{FG}(S)\subseteq
X$.

###### Proof.

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

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

Analogously to the case of numerical semigroups, it happens that
$\mathrm{FG}(S)$ 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,
$\mathrm{SG}(S)=\max_{\leq_{S}}\mathrm{FG}(S)$.

###### Proof.

Trivially, for any $\mathbf{x}\in\mathrm{SG}(S)$, $2\mathbf{x},3\mathbf{x}\in
S$, and then $\mathrm{SG}(S)\subseteq\mathrm{FG}(S)$. Assume that for a
$\mathbf{x}\in\mathrm{SG}(S)$, there exists some $\mathbf{y}\in\mathrm{FG}(S)$
with $\mathbf{x}\leq_{S}\mathbf{y}$. So, $\mathbf{x}+\mathbf{s}=\mathbf{y}$
for some $\mathbf{s}\in S$. Since $\mathbf{x}$ is a pseudo-Frobenius element
of $S$, $\mathbf{y}\in S$. It is not possible, then
$\mathbf{x}\in\max_{\leq_{S}}\mathrm{FG}(S)$. ∎

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

######  Corollary 23.

$S$ is a $\mathcal{C}$-irreducible semigroup if and only if the cardinality of
$\max_{\leq_{S}}\mathrm{FG}(S)$ is equal to one.

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

$\\{(x,s)\in(D(X),\mathcal{C}\setminus D(X))\mid
s\leq_{\mathcal{C}}D(X)\\}=\\\
\\{((0,0),(1,1)),((0,0),(3,0)),((0,0),(3,1)),((0,0),(3,2)),\\\
((0,0),(5,1)),((0,0),(5,2)),((2,0),(3,0)),((2,0),(3,1)),((2,0),(5,1)),\\\
((2,0),(5,2)),((2,1),(3,2)),((2,1),(5,1)),((2,1),(5,2)),\\\
((2,2),(3,2)),((2,2),(5,2)),((4,0),(5,1))\\}$

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,
$\mathrm{FG}(S)=X$. Moreover, we can compute the set of pseudo-Frobenius
elements of $S$, and we get $\mathrm{PF}(S)=\\{(5,1),(5,2)\\}$, so
$\mathrm{SG}(S)=\\{(5,1),(5,2)\\}$. On the other hand,
$\mathrm{FG}(S)=\\{(1,1),(3,0),(3,1),(3,2),(5,1),(5,2)\\}$ and
$\max_{\leq_{S}}\mathrm{FG}(S)=\\{(5,1),(5,2)\\}$, as we knew by Lemma 22.

##  5 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 $\mathbf{s}$ fufilling that $\mathbf{s}\preceq F$.
Performing this process as many times as possible, we obtain all the
$\mathcal{C}$-semigroups with Frobenius vector $F$. Note that this process is
finite due to the finitiness of the set
$\\{\mathbf{s}\in\mathcal{C}\mid\mathbf{s}\preceq F\\}$. 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.

Input: A total order $\preceq$ on $\mathbb{N}^{p}$, $\Lambda_{\mathcal{C}}$
the minimal generating set of the integer cone
$\mathcal{C}\subset\mathbb{N}^{p}$, and $F\in\mathcal{C}\setminus\\{0\\}$.

Output: The set of $\mathcal{C}$-semigroups with Frobenius vector equal to
$F$.

begin

$\Lambda_{\preceq
F}\leftarrow\\{\mathbf{s}\in\Lambda_{\mathcal{C}}\mid\mathbf{s}\preceq F\\}$;

$\mathcal{T}\leftarrow\\{(\Lambda_{\mathcal{C}},\Lambda_{\preceq F})\\}$;

$\mathcal{G}\leftarrow\emptyset$;

while  _$\mathcal{T}\neq\emptyset$_ do

$(\Lambda,\Lambda^{\prime})\leftarrow\textrm{First}(\mathcal{T})$;

$D\leftarrow\Lambda^{\prime}$;

while  _$D\neq\emptyset$_ do

$\mathbf{s}\leftarrow\textrm{First}(D)$;

$\Lambda^{\prime\prime}\leftarrow\\{\text{minimal generating set of
}\langle\Lambda\rangle\setminus\\{\mathbf{s}\\}\\}$;

if  _$\Lambda^{\prime\prime}\notin\pi_{1}(\mathcal{T})$_ then

if  _$\mathbf{s}=F$_ then

$\mathcal{G}\leftarrow\mathcal{G}\cup\Lambda^{\prime\prime}$;

$\Lambda^{\prime\prime}_{\preceq
F}\leftarrow\\{\mathbf{s}\in\Lambda^{\prime\prime}\mid\mathbf{s}\preceq F\\}$;

if  _$\Lambda^{\prime\prime}_{F}\neq\emptyset$_ then

$\mathcal{T}\leftarrow\mathcal{T}\cup\\{(\Lambda^{\prime\prime},\Lambda^{\prime\prime}_{\preceq
F})\\}$;

$D\leftarrow D\setminus\\{\mathbf{s}\\}$;

$\mathcal{T}\leftarrow\mathcal{T}\setminus\\{(\Lambda,\Lambda^{\prime})\\}$;

return $\mathcal{G}$

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 4, we get that the set of all
$\mathcal{C}$-semigroups with Frobenius vector $(2,1)$ is
$\\{\\{(2,0),(3,0),(1,1),(1,2)\\},\\{(1,0),(3,1),(1,2),$
$(2,3)\\},\\{(3,0),(4,0),$
$(5,0),(1,1),(3,1),(1,2),(3,2)\\},\\{(2,0),(3,0),(3,1),$ $(4,1),$
$(1,2),(2,2),$ $(2,3),$ $(3,3)\\},$ $\\{(3,0),(4,0),(5,0),$
$(3,1),(4,1),(5,1),(1,2),(2,2),$ $(3,2),$ $(2,3),(3,3)\\},$
$\\{(3,0),(4,0),(5,0),(3,1),(4,1),(5,1),(2,2),(3,2),(4,2),$ $(2,3),(3,3),$
$(4,3),$ $(2,4),$ $(3,4),(3,5),(3,6)\\}\\}$. These semigroups are shown in
Table 1.

|  |
---|---|---
|  |
Table 1: All $\mathcal{C}$-semigroups with
$\mathcal{C}=\langle(1,0),(1,1),(1,2)\rangle$, and Frobenius vector equal to
$(2,1)$; $\circ\equiv\text{gap}$, $\blacksquare\equiv\text{minimal
generator}$,
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}\equiv\text{element
in }S$.

### Funding

The first, second, and last authors were partially supported by Junta de
Andalucía research group FQM-343, and by Consejería de Universidad,
Investigación e Innovación de la Junta de Andalucía project ProyExcel_00868.
Proyecto de investigación del Plan Propio – UCA 2022-2023 (PR2022-004)
partially supported the second and last authors. Proyecto de investigación del
Plan Propio – UCA 2022-2023 (PR2022-011) also partially supported all the
authors. This publication and research have been partially granted by INDESS
(Research University Institute for Sustainable Social Development),
Universidad de Cádiz, Spain.

### Author information

J. I. García-García. Departamento de Matemáticas/INDESS (Instituto
Universitario para el Desarrollo Social Sostenible), Universidad de Cádiz,
E-11510 Puerto Real (Cádiz, Spain). E-mail<EMAIL_ADDRESS>
D. Marín-Aragón. Departamento de Matemáticas, Universidad de Cádiz, E-11510
Puerto Real (Cádiz, Spain). E-mail<EMAIL_ADDRESS>
A. Sánchez-Loureiro. Departamento de Matemáticas, Universidad de Cádiz,
E-11510 Puerto Real (Cádiz, Spain). E-mail<EMAIL_ADDRESS>
A. Vigneron-Tenorio. Departamento de Matemáticas/INDESS (Instituto
Universitario para el Desarrollo Social Sostenible), Universidad de Cádiz,
E-11406 Jerez de la Frontera (Cádiz, Spain). E-mail<EMAIL_ADDRESS>
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