# On the quotient of affine semigroups by a positive integer

J. I. García-García, R. Tapia-Ramos, and A. Vigneron-Tenorio

###### Abstract

This work delves into the quotient of an affine semigroup by a positive
integer, exploring its intricate properties and broader implications. We
unveil an associated tree that serves as a valuable tool for further analysis.
Moreover, we successfully generalize several key irreducibility results,
extending their applicability to the more general class of
$\mathcal{C}$-semigroup quotients. To shed light on these concepts, we
introduce the novel notion of an arithmetic variety of affine semigroups,
accompanied by illuminating examples that showcase its power.

Key words: affine semigroup, arithmetic variety, $\mathcal{C}$-semigroup,
Frobenius element, irreducibility, quotient by a positive integer, rooted
tree.

2020 Mathematics Subject Classification: 20M14, 20M07, 11D07, 05C05, 13H10.

## Introduction

We consider an affine semigroup $S$ as a non-empty subset of $\mathbb{N}^{p}$
($\mathbb{N}$ denoting the set of non-negative integers) containing the zero
element, closed under the usual addition in $\mathbb{N}^{p}$, and such that
there exists a finite subset $A=\\{a_{1},\ldots,a_{q}\\}$ of $S$ with
$S=\\{\sum_{i=1}^{q}\lambda_{i}a_{i}\mid\lambda_{1},\ldots,\lambda_{q}\in\mathbb{N}\\}$.
A fundamental result states that each affine semigroup possesses a unique
minimal generating set (see [19]). When $p=1$, and the elements of $A$ are
coprime, we say that $S$ is a numerical semigroup. If $p>1$, $\mathcal{C}$ the
minimal positive integer cone containing to $S$, and $S$ satisfies that
$\mathcal{C}\setminus S$ is finite, $S$ is called $\mathcal{C}$-semigroup (see
[11]). Notably, when $\mathcal{C}=\mathbb{N}^{p}$, the semigroup $S$ is known
as generalized numerical semigroup (see [8]).

Given an affine semigroup $S\subseteq\mathbb{N}^{p}$, its quotient by a
positive integer $d$ is defined as $\frac{S}{d}=\\{x\in\mathbb{N}^{p}\mid
dx\in S\\}$. This construction plays a significant role in the study of
numerical semigroups, appearing in numerous works such as [2], [17], [20], and
[23]. However, despite its relevance in numerical contexts, a systematic
investigation of quotients of affine semigroups, in general, is surprisingly
absent in the literature. We address this gap by presenting various results
about quotients of affine semigroups by a positive integer. While some of
these results are known for numerical semigroups, others hold true only for
general affine semigroups with $p>1$ or $\mathcal{C}$-semigroups.

We present an efficient algorithm for computing a generating set of the
quotient of any affine semigroup by a positive integer. This algorithm
provides a valuable tool for further analysis of such quotients. Moreover, we
demonstrate that the families of convex body semigroups and Cohen-Macaulay
simplicial semigroups exhibit the remarkable property of being closed under
quotients. This closure property unveils a powerful tool for generating
infinite Cohen-Macaulay semigroups from a single instance, paving the way for
exploring their properties and applications.

A central focus of this work is identifying all $\mathcal{C}$-semigroups that
are quotients of a given $\mathcal{C}$-semigroup (Theorem 12). Inspired by
this set, we construct a directed graph under specific hypotheses.
Additionally, we delve into arithmetic varieties (non-empty sets closed under
intersection and quotient by any positive integer), introducing three distinct
families of affine semigroups that form three distinct arithmetic varieties.

Another key objective is to study the irreducibility of quotients of
$\mathcal{C}$-semigroups. As shown in [10], every irreducible semigroup is
either symmetric or pseudo-symmetric. We demonstrate that any
$\mathcal{C}$-semigroup is part of infinitely many symmetric
$\mathcal{C}$-semigroups and infinitely many pseudo-symmetric
$\mathcal{C}$-semigroups. Furthermore, we provide a characterization of
irreducible $\mathcal{C}$-semigroups relative to the quotient of
$\mathcal{C}$-semigroups (Theorem 24).

Our paper is structured as follows. Section 1 establishes the necessary
background for subsequent sections. In Section 2, we introduce the concept of
quotients of affine semigroups, along with their corresponding systems of
generators. Section 3 delves into the set of all $\mathcal{C}$-semigroups $T$
for which $S=\frac{T}{d}$. This section also introduces the novel concept of
arithmetic varieties of affine semigroups, employed to compute an associated
tree for these semigroups. Section 4 presents several new results on the
irreducibility of quotients of $\mathcal{C}$-semigroups, drawing inspiration
from Chapter 5, Section 2 of [21]. Finally, Section 5 revisits the exploration
of arithmetic varieties, showcasing their application through illustrative
examples.

##  1 Preliminars

Let $\mathbb{Q}$ and $\mathbb{R}_{\geq}$ denote the sets of rational and non-
negative real numbers, respectively. For any $d\in\mathbb{N}$, define
$[d]=\\{0,\ldots,d\\}$. Given $X\subseteq\mathbb{N}^{p}$, let $dX=\\{dx\mid
x\in X\\}$.

This work employs various orders on different sets. We denote by $\leq$ the
usual partial order on $\mathbb{N}^{p}$, where $x\leq y$ implies
$y-x\in\mathbb{N}^{p}$. Additionally, we consider a total order $\preceq$ on
$\mathbb{N}^{p}$ satisfying compatibility with addition and ensuring $0\preceq
c$ for any $c\in\mathbb{N}^{p}$. Throughout this work, we fix a specific total
order $\preceq$.

For a $\mathcal{C}$-semigroup $S$, any element $x\in\mathcal{C}\setminus S$ is
called a gap of $S$. We call $x\in S$ a _pseudo-Frobenius element_ if
$x+(S\setminus\\{0\\})\subset S$. The set of gaps of $S$ is denoted by
$\mathcal{H}(S)$, and its set of pseudo-Frobenius elements by $PF(S)$. The
cardinality of $\mathcal{H}(S)$ is known as the genus of $S$. Following these
definitions, the Frobenius element of $S$, denoted by $\mathrm{Fb}(S)$, is the
maximum element in $\mathcal{H}(S)$ with respect to the fixed total order on
$\mathbb{N}^{p}$. Note that $\mathrm{Fb}(S)=\max_{\preceq}PF(S)$.

The following proposition provides a useful characterization of the gaps of a
$\mathcal{C}$-semigroup in terms of its pseudo-Frobenius elements.

######  Proposition 1.

[1, Corollary 2.15] Let $S$ be a $\mathcal{C}$-semigroup and
$x\in\mathcal{C}$. Then, $x\notin S$ if and only if there exists $f\in PF(S)$
such that $f-x\in S.$

A $\mathcal{C}$-semigroup $S$ is irreducible if it cannot be expressed as an
intersection of two $\mathcal{C}$-semigroups containing $S$ properly.
Equivalently, $S$ is irreducible if and only if the set $PF(S)$ is equal to
$\\{\mathrm{Fb}(S)\\}$ or $\\{\mathrm{Fb}(S),\frac{\mathrm{Fb}(S)}{2}\\}$ (see
[16]). If $PF(S)=\\{\mathrm{Fb}(S)\\}$, we say that $S$ is symmetric, and
pseudo-symmetric when $PF(S)=\\{\mathrm{Fb}(S),\frac{\mathrm{Fb}(S)}{2}\\}$.

From the relationship between the elements belonging to $S$ and its Frobenius
element, the irreducibility of $S$ is determined as follows.

######  Proposition 2.

[1, Theorem 3.6] Let $S$ be a $\mathcal{C}$-semigroup and $x\in\mathcal{C}$.
The following conditions are equivalent:

  * (i)

$S$ is a symmetric $\mathcal{C}$-semigroup.

  * (ii)

$x\in S$ if and only if $\mathrm{Fb}(S)-x\notin S$.

######  Proposition 3.

[1, Theorem 3.7] Let $S$ be a $\mathcal{C}$-semigroup and $x\in\mathcal{C}$.
Then, $S$ is a pseudo-symmetric $\mathcal{C}$-semigroup if and only if the
following conditions hold:

  * (i)

$\mathrm{Fb}(S)\in 2\mathbb{N}^{p}$.

  * (ii)

$x\in S$ if and only if $\mathrm{Fb}(S)-x\notin S$ and
$x\neq\frac{\mathrm{Fb}(S)}{2}$.

It is well-known that the fundamental gaps of a $\mathcal{C}$-semigroup
determine it uniquely (see [10]). According to the usual literature, a
fundamental gap of a $\mathcal{C}$-semigroup $S$ is a gap $x$ of $S$ such that
$2x,3x\in S$. We show that fundamental gaps of the quotient of a
$\mathcal{C}$-semigroup by a positive integer can be obtained from the
fundamental gaps of the original $\mathcal{C}$-semigroup.

######  Proposition 4.

Let $S$ be a $\mathcal{C}$-semigroup and $d\in\mathbb{N}\setminus\\{0\\}$.
Then,

$FG\left(\frac{S}{d}\right)=\left\\{\frac{h}{d}\mid h\in FG(S)\text{ and
}h\equiv 0\mod d\right\\}.$

###### Proof.

Since $d\in\mathbb{N}\setminus\\{0\\}$, $x$ belongs to
$FG\left(\frac{S}{d}\right)$ if and only if $dx\notin S$, and $kdx\in S$ for
any $k\in\mathbb{N}$. Hence, $dx\in FG(S)$. ∎

The Apéry set of a $\mathcal{C}$-semigroup $S$ relative to $m\in
S\setminus\\{0\\}$ is defined as $Ap(S,m)=\\{x\in S\mid
x-m\in\mathcal{H}(S)\\}$. If $d\in\mathbb{N}\setminus\\{0\\}$ divides to $m$,
then we can describe $Ap\left(\frac{S}{d},\frac{m}{d}\right)$ in terms of
$Ap(S,m)$ as follows.

######  Proposition 5.

Let $S$ be a $\mathcal{C}$-semigroup and $m$ a non-zero element of $S$ such
that $d$ divides to $m$. Then

$Ap\left(\frac{S}{d},\frac{m}{d}\right)=\left\\{\frac{w}{d}\mid w\in
Ap(S,m)\text{ and }w\equiv 0\mod d\right\\}.$

###### Proof.

By definition $x\in Ap\left(\frac{S}{d},\frac{m}{d}\right)$ if and only if
$x=\frac{w}{d}\in\frac{S}{d}$ and $\frac{w}{d}-\frac{m}{d}\notin\frac{S}{d}$
where $w=dx,$ this means, $w\in Ap(S,m)$ and $w\equiv 0\mod d$. ∎

The above propositions 4 and 5, generalize to $\mathcal{C}$-semigroups
Proposition [21, Proposition 6.2] and [21, Proposition 6.5], respectively.

##  2 Some families of affine semigroups closed under the quotient by a
positive integer

Recall that the quotient of $S$ by $d$ is the set
$\frac{S}{d}=\\{s\in\mathbb{N}^{p}\mid dx\in S\\}$. Consequently, we say that
$\frac{S}{2}$ is one-half of $S$, and that $\frac{S}{4}$ is one-fourth of $S$.
These specific cases will be relevant later in our work.

This section introduces certain families of affine semigroups where the
quotient of any element by a positive integer remains within the same family.
Specifically, we focus on convex body semigroups and Cohen-Macaulay simplicial
semigroups. Additionally, we present an algorithm for computing the minimal
generating set of the quotient of any affine semigroup by a positive integer.

We begin by outlining the algorithm. Let $A=\\{a_{1},\ldots,a_{q}\\}$ be the
minimal generating set of an affine semigroup $S\subseteq\mathbb{N}^{p}$, and
$d\in\mathbb{N}\setminus\\{0\\}$. By definition of $\frac{S}{d}$, an element
$x=(x_{1},\ldots,x_{p})$ belongs to $\frac{S}{d}$ if and only if there exists
$\lambda=(\lambda_{1},\ldots,\lambda_{q})\in\mathbb{N}^{q}$ such that
$dx=\sum_{i=1}^{q}\lambda_{i}a_{i}$. Equivalently, $x\in\frac{S}{d}$ if and
only if $(\lambda,x)\in\mathbb{N}^{q}\times\mathbb{N}^{p}$ is a non-negative
integer solution of the system of linear equations $(M\mid
D)(\lambda,x)^{t}=0$, where $M$ is the matrix with $a_{i}$ as its $i$-th
column and $D$ is $-d$ times the identity matrix of dimension $p$.
Importantly, the set of non-negative integer solutions of any system of
homogeneous linear equations with rational coefficients forms an affine
semigroup, which can be computed as detailed in [18]. We denote by
$\mathcal{X}\subset\mathbb{N}^{q+p}$ the affine semigroup defined by the set
of non-negative integer solutions of the system, with
$L=\\{l_{1},\ldots,l_{k}\\}$ being its minimal generating set. Furthermore, we
define $\pi(y)$ as the projection onto the last $p$ coordinates of
$y=(\lambda,x)\in\mathcal{X}$.

######  Lemma 6.

Let $S$ be an affine semigroup generated by $A=\\{a_{1},\ldots,a_{q}\\}$, and
$d$ a positive integer. The quotient $\frac{S}{d}$ is generated by
$\pi(L)=\\{\pi(l_{i})\mid i=1,\ldots,k\\}$.

###### Proof.

Get $x\in\frac{S}{d}$, so $dx=\sum_{i=1}^{q}\lambda_{i}a_{i}$ for some
$\lambda_{1},\ldots,\lambda_{q}\in\mathbb{N}$. Hence,
$(\lambda,x)\in\mathcal{X}$, and there exist
$\alpha_{1},\ldots,\alpha_{k}\in\mathbb{N}$ such that
$(\lambda,x)=\sum_{i=1}^{k}\alpha_{i}l_{i}$. Thus,
$x=\pi(\sum_{i=1}^{k}\alpha_{i}l_{i})$. Analogously, if $x$ can be obtained
from $\pi(L)$ by a non-negative linear combination of its elements, then
$x\in\frac{S}{d}$. ∎

Algorithm 1 efficiently computes the minimal generating set of the quotient of
any affine semigroup $S$ by a positive integer $d$. We denoted by $I_{p}$ the
identity matrix of dimension $p$.

Input: $A=\\{a_{1},\ldots,a_{q}\\}$, the minimal generating set of $S$.

Output: A minimal generating set of $\frac{S}{d}$.

1

2$M\leftarrow(a_{1}|\cdots|a_{q})$;

3 $D\leftarrow-dI_{p}$;

4 $L\leftarrow$ the minimal generating set of the $\mathbb{N}$-solutions of
$(M|D)(\lambda,x)^{t}=0$;

5 $L\leftarrow\pi(L)$;

6 $\mathcal{L}\leftarrow$ remove non-minimal elements belonging to $L$;

return _$\mathcal{L}$_

Algorithm 1 Computing the minimal generating set of $\frac{S}{d}$.

We begin by defining convex body semigroups, a family closed under the
quotient by any positive integer. Given a convex body $F$ in
$\mathbb{R}^{p}_{\geq}$ (a compact convex subset with non-empty interior), we
define $\mathcal{C}(F)$ as the non-negative integer cone spanned by $F$. This
cone consists of all elements in $\mathbb{N}^{p}$ that can be expressed as a
non-negative rational linear combination of elements in $F$. The definition of
cone spanned by a set can be extended to every subset of
$\mathbb{R}^{p}_{\geq}$.

Furthermore, the set
$\mathcal{B}(F)=\bigcup_{i=0}^{\infty}iF\cap\mathbb{N}^{p}$ is an affine
semigroup if and only if $F\cap\tau\cap\mathbb{Q}^{p}\neq\emptyset$ for any
extremal ray of the rational cone spanned by $F$ (see [15]). In this work, we
assume every considered convex body satisfies this property. While not always
$\mathcal{C}(F)$-semigroups (see [13] and references therein), such semigroups
are called convex body (affine) semigroups.

######  Lemma 7.

Let $F\subset\mathbb{R}^{p}_{\geq}$ be a convex body, and
$d\in\mathbb{N}\setminus\\{0\\}$. Then, $\frac{\mathcal{B}(F)}{d}$ is a convex
body semigroup. Moreover, $\frac{\mathcal{B}(F)}{d}=\mathcal{B}(\frac{F}{d})$.

###### Proof.

Since $\mathcal{B}(F)$ is the convex body semigroup determined by $F$,
$x\in\frac{\mathcal{B}(F)}{d}$ if and only if $dx\in\mathcal{B}(F)$, that is,
$x\in\frac{i}{d}F$ for any $i\in\mathbb{N}$. So,
$\frac{\mathcal{B}(F)}{d}=\bigcup_{i=0}^{\infty}\frac{i}{d}F\cap\mathbb{N}^{p}=\mathcal{B}(\frac{F}{d})$.
That proves the lemma. ∎

The previous result implies that the set of the convex body semigroups is
closed under the quotient by any positive integer. In particular, the set of
the convex body semigroups generating a fixed integer cone is also closed
under the quotient. We summarise these results in two corollaries.

######  Corollary 8.

The set of all convex body semigroups is closed under the quotient by any
integer.

######  Corollary 9.

Let $\mathcal{C}$ be an integer cone and $\mathfrak{B}$ the of the set of all
of convex body semigroups such that $\mathcal{C}=\mathcal{C}(\mathcal{B})$ for
any $\mathcal{B}\in\mathfrak{B}$. Hence, $\mathfrak{B}$ is closed under the
quotient by any integer.

To introduce the Cohen-Macaulay simplicial semigroups, we need to give several
definitions. For any finitely generated commutative semigroup $S$, and a field
$\mathrm{k}$, the semigroup ring of $S$ over $\mathrm{k}$ is defined. This
ring is determined by $\bigoplus_{m\in S}\mathrm{k}\chi^{m}$ endowed with a
multiplication which is $\mathrm{k}$-linear and such that
$\chi^{m}\cdot\chi^{n}=\chi^{m+n}$, $m$ and $n\in S$ (see [4]).

Let $R$ be a Noetherian local ring, a finite $R$-module $M\neq 0$ is a Cohen-
Macaulay module if ${\rm depth}(M)=\dim(M)$. If $R$ itself is a Cohen-Macaulay
module, it is called a Cohen-Macaulay ring (see [6]). A semigroup is called
Cohen-Macaulay if its associated semigroup ring $\mathrm{k}[S]$ is Cohen-
Macaulay.

An affine semigroup $S\subset\mathbb{N}^{p}$ is simplicial if $\mathcal{C}$
has just $p$ extremal rays. We denote by $\\{\tau_{1},\ldots,\tau_{p}\\}$ the
set of all extremal rays of $S$. The following result is proved in [14],
characterising the affine simplicial semigroups.

######  Theorem 10.

[14, Theorem 2.2] Let $S\subset\mathbb{N}^{p}$ be an affine simplicial
semigroup. Then, the following conditions are equivalent:

  1. 1.

$S$ is Cohen-Macaulay.

  2. 2.

For any $a,b\in S$ with $a+n_{i}=b+n_{j}$ ($1\leq i\neq j\leq p$),
$a-n_{j}=b-n_{i}\in S$. Where $n_{i}$ is an element belonging to $\tau_{i}\cap
S$ for $i=1,\ldots,p$.

Note that Cohen-Macaulay simplicial $\mathcal{C}$-semigroup does not exist
(see [12, Corollary 11]).

The above theorem allows us to state a result that can obtain many Cohen-
Macaulay semigroups from a given one using the quotient by a positive integer.

######  Proposition 11.

Let $S$ be a Cohen-Macaulay affine simplicial semigroup. Then, the affine
simplicial semigroup $\frac{S}{d}$ is Cohen-Macaulay for any positive integer
$d$.

###### Proof.

Trivially, $S$ simplicial implies $\frac{S}{d}$ is also simplicial. Besides,
since the cone associated to $\frac{S}{d}$ and the cone associated to $S$ are
the same, $\frac{S}{d}$ is an affine semigroup (see [5, Corollary 2.10]).

Get $a,b\in\frac{S}{d}$ such that $a+n_{i}=b+n_{j}$ for any $1\leq i\neq j\leq
p$, with $n_{i}\in\tau_{i}\cap\frac{S}{d}$ and
$n_{j}\in\tau_{j}\cap\frac{S}{d}$. Hence, $d(a+n_{i})=d(b+n_{j})$. Since $S$
is Cohen-Macaulay, Theorem 10 affirms that $d(a-n_{j})=d(b-n_{i})\in S$, we
obtain that $a-n_{j}=b-n_{i}\in\frac{S}{d}$. Thus, we conclude that
$\frac{S}{d}$ is Cohen-Macaulay. ∎

##  3 Computing $\mathcal{D}_{d}(S)$ and building a tree on an arithmetic
variety

Let $S$ be a $\mathcal{C}$-semigroup and $d\in\mathbb{N}\setminus\\{0\\}$. One
of the aims of this section is to describe the set of all the
$\mathcal{C}$-semigroups $T$ such that $S=\frac{T}{d}$. From now on, this set
is denoted by $\mathcal{D}_{d}(S)$, that is, a subset of any arithmetic
variety containing $S$. In concordance with the notation given in [3], an
arithmetic variety is a non-empty family of affine semigroups $\mathcal{A}$
that fulfils the following conditions:

  1. 1.

If $G,H\in\mathcal{A}$, then $G\cap H\in\mathcal{A}$.

  2. 2.

If $G\in\mathcal{A}$ and $d\in\mathbb{N}\setminus\\{0\\}$, then
$\frac{G}{d}\in\mathcal{A}$.

Another main objective of this section is to introduce a tree of infinite and
finite arithmetic varieties.

We start to provide an explicit description of the $\mathcal{C}$-semigroups
belonging to $\mathcal{D}_{d}(S)$. Fixed $f\in\mathcal{C}$, consider
$M_{f}=\\{m\in S\setminus d\mathbb{N}^{p}\mid m\prec f\\}$, and

$\overline{M}_{f}=\Bigg{\\{}\\{\lambda_{1},\ldots,\lambda_{q}\\}\subseteq
M_{f}\mid\sum_{i=1}^{q}a_{i}\lambda_{i}\in dS\sqcup(\mathbb{N}^{p}\setminus
d\mathbb{N}^{p}),\forall a_{1},\ldots,a_{q}\in[d-1]\Bigg{\\}}.$

Note that $\overline{M}_{f}$ is finite. By convention, we consider that any
linear combination of the elements in the empty set is zero.

######  Theorem 12.

Let $S$ be a $\mathcal{C}$-semigroup and a positive integer $d$. Then, the set
$\mathcal{D}_{d}(S)$ is equal to

$\left\\{T(f,\Lambda)\mid f\in\mathcal{C}\mbox{ with }f\succeq
d\mathrm{Fb}(S),\mbox{ and
}\Lambda\in\overline{M}_{f}\cup\\{\\{\emptyset\\}\\}\right\\},$

with

$T(f,\Lambda)=\\{x\in\mathcal{C}\mid x\succ f\\}\cup\\\
\left(dS+\left\\{\sum_{i=1}^{\sharp\Lambda}a_{i}\lambda_{i}\mid
a_{i}\in[d-1]\mbox{ and }\lambda_{i}\in\Lambda,\,\forall
i\in\\{1,\ldots,\sharp\Lambda\\}\right\\}\right).$

###### Proof.

Fix $f\in\mathcal{C}$ with $f\succ d\mathrm{Fb}(S)$, and
$\Lambda=\\{\lambda_{1},\ldots,\lambda_{q}\\}\in\overline{M}_{f}$. Let us
prove that $T(f,\Lambda)$ is a $\mathcal{C}$-semigroup. Since
$0=d0+\sum_{i=1}^{q}0\lambda_{i}$, we have $0\in T(f,\Lambda)$. Let
$x,y\in\\{x\in\mathcal{C}\mid x\succ f\\}$, and
$z=2s+\sum_{i=1}^{q}a_{i}\lambda_{i}$ for some $s\in S$ and
$a_{1},\ldots,a_{q}\in[d-1]$, note that, $x+y,x+z,y+z\in\\{x\in\mathcal{C}\mid
x\succ f\\}$. Consider $x,y\in T(f,\Lambda)$ such that $x,y\prec f$,
therefore, $x=ds+\sum_{i=1}^{q}a_{i}\lambda_{i}$ and
$y=ds^{\prime}+\sum_{i=1}^{q}a^{\prime}_{i}\lambda_{i}$ with $s,s^{\prime}\in
S$ and $a_{1},a^{\prime}_{1}\ldots,a_{q},a_{q}^{\prime}\in[d-1]$. Since
$\Lambda\in\overline{M}_{f}$, we can express
$x+y=d(s+s^{\prime})+ds^{\prime\prime}+\sum_{i=1}^{q}a^{\prime\prime}_{i}\lambda_{i}$
for some $a^{\prime\prime}_{1},\ldots,a^{\prime\prime}_{q}\in[d-1]$ and
$s^{\prime\prime}\in S$. So, we deduce that $T(f,\Lambda)$ is closed under
addition. Observe that, $\\{x\in\mathcal{C}\mid x\succ f\\}\subset
T(f,\Lambda)$, thus $\mathcal{C}\setminus T(f,\Lambda)$ is finite. Hence,
$T(f,\Lambda)$ is a $\mathcal{C}$-semigroup.

Since $dS\subset T(f,\Lambda)$, we obtain $S\subset\frac{T(f,\Lambda)}{d}$.
Let $x\in\frac{T(f,\Lambda)}{d}$, by definition $dx\in T(f,\Lambda)$. If
$dx\succ f\succ d\mathrm{Fb}(S)$, then $x\succ\mathrm{Fb}(S)$, and $x\in S$.
In case that $dx=ds+\sum_{i=1}^{q}a_{i}\lambda_{i}$ for any $s\in S$ and
$a_{1},\ldots,a_{q}\in[d-1]$, and taking into account that
$\Lambda\in\overline{M}_{f}$, we deduce $\sum_{i=1}^{q}a_{i}\lambda_{i}\in
dS$. Therefore, $S=\frac{T(f,\Lambda)}{d}$.

Let $T$ be a $\mathcal{C}$-semigroup such that $S=\frac{T}{d}$, and consider
$\Lambda=M_{f}\cap T=\\{\lambda_{1},\ldots,\lambda_{q}\\}$ with
$f=\mathrm{Fb}(T)$. We now prove that $T=T(f,\Lambda)$. Note that it is
equivalent to show that $\Lambda\in\overline{M}_{f}$. Take some
$a_{1},\ldots,a_{q}\in[d-1]$. In case that
$\sum_{i=1}^{q}a_{i}\lambda_{i}\notin d\mathbb{N}^{p}$, we hold the result.
Otherwise, assume that $\sum_{i=1}^{q}a_{i}\lambda_{i}\in
d\mathbb{N}^{p}\setminus dS$. Thus,
$\frac{\sum_{i=1}^{q}a_{i}\lambda_{i}}{d}\notin S$. Since $S=\frac{T}{d}$, it
follows that $d\frac{\sum_{i=1}^{q}a_{i}\lambda_{i}}{d}\notin T$,
contradicting that $\Lambda\subset T$. ∎

Observe that $T(f,\emptyset)=\\{x\in\mathcal{C}\mid x\succ f\\}\cup dS$.
Moreover, if $f=d\mathrm{Fb}(S)$, then $T(f,\emptyset)$ is the one with the
largest genus of the $\mathcal{C}$-semigroups in $\mathcal{D}_{d}(S)$ with
Frobenius element equal to $d\mathrm{Fb}(S)$.

From Theorem 12, the set $\mathcal{D}_{d}(S)$ is not finite since $f$ can be
any element belonging to $\mathcal{C}$ greater than $d\mathrm{Fb}(S)$. But,
fixed an element $f\succ d\mathrm{Fb}(S)$, the set
$\mathcal{D}_{d}(S,f)=\\{T\in\mathcal{D}_{d}(S)\mid\mathrm{Fb}(T)\preceq f\\}$
is finite. In particular, $\mathcal{D}_{d}(S,f)$ is equal to
$\\{T(f,\Lambda)\mid\Lambda\in\overline{M}_{f}\cup\\{\\{\emptyset\\}\\}\\}$.

######  Example 13.

Consider the $\mathcal{C}$-semigroup minimally generated by

$\\{(4,1),(5,2),(7,2),(9,5),(4,2),(6,2),(6,3),(7,3),(11,6)\\}.$ (1)

This semigroup is the non-negative integer cone $\mathcal{C}$ determined by
the extremal rays $\\{(4,1),(9,5)\\}$ except the points $\\{(2,1),(3,1)\\}$
(see Figure 1).

Figure 1:  $\mathcal{C}$-semigroup minimally generated by (1).

Fixed $d=3$, $f=3\mathrm{Fb}(S)=(9,3)$, and the reverse graded lexicographical
order (see [7]), we obtain that the set $M_{f}$ is equal to

$\\{(4,1),(4,2),(5,2),(6,2),(7,2),(7,3),(8,2),(8,3)\\},$

and $T(f,\emptyset)$ is the $\mathcal{C}$-semigroup showed in Figure 2.

Figure 2:  $T(f,\emptyset)$.

Moreover, the cardinality of $\mathcal{D}_{3}(S,f)$ is bounded by
$\sum_{i=0}^{8}\binom{8}{i}=256$, which is the amount of all the possible
elements in $\overline{M}_{f}$ plus one.

Since, $(4,1)+(5,2)\in 3\mathbb{N}^{2}$, no element in $\overline{M}_{f}$ can
contain $\\{(4,1),(5,2)\\}$, and, taking into account $(4,1)+(4,2),2(4,1)\in
M_{f}$, we have to remove those redundant $\mathcal{C}$-semigroups. For
example, $T(f,\\{(4,1)\\})=T(f,\\{(4,1),(8,2)\\})$. So, we obtain the
cardinality of $\mathcal{D}_{3}(S,f)$ is just equal to $151$. Figure 3 shows
some graphical examples of elements belonging to $\mathcal{D}_{3}(S,f)$.

$\begin{array}[]{cc}\includegraphics[scale={.25}]{T1_para_ejemplo_d_3.pdf}&\includegraphics[scale={.25}]{T2_para_ejemplo_d_3.pdf}\\\
\Lambda=\\{(4,1)\\}&\Lambda=\\{(4,2),(7,3)\\}\\\ \\\
\includegraphics[scale={.25}]{T3_para_ejemplo_d_3.pdf}&\includegraphics[scale={.25}]{T4_para_ejemplo_d_3.pdf}\\\
\Lambda=\\{(6,2),(7,2),(8,2)\\}&\Lambda=\\{(4,2),(5,2),(8,2),(8,3)\\}\\\
\end{array}$

Figure 3: Some elements in $\mathcal{D}_{3}(S)$.

This section ends by introducing a tree on an arithmetic variety of
$\mathcal{C}$-semigroups. Consider $\mathcal{A}$ an arithmetic variety, we can
build the tree $G_{\mathcal{A},d}$ with root $\mathcal{C}$, whose vertex set
is the arithmetic variety, and $(T,S)\in\mathcal{A}^{2}$ is a directed edge if
and only if $S=\frac{T}{d}$. It is more when $(T,S)$ is an edge, we say that
$T$ is a child of $S$.

######  Lemma 14.

For any $\mathcal{A}$ arithmetic variety of $\mathcal{C}$-semigroups,
$G_{\mathcal{A},d}$ is a tree with root $\mathcal{C}$. Furthermore, the set of
children of any $S\in\mathcal{A}$ is the set
$\mathcal{D}_{d}(S)\cap\mathcal{A}$.

###### Proof.

Let $S\in\mathcal{A}$ such that $S\neq\mathcal{C}$. Consider the sequence
$\\{S_{i}\\}_{i\in\mathbb{N}}$ defined by $S_{i}=\frac{S}{d^{i}}$. Since the
complement of $S$ in $\mathcal{C}$ is finite, there exists a positive integer
$n$ such that $S=S_{0}\subset S_{1}\subset\cdots\subset S_{n}=\mathcal{C}$.
Therefore, for every vertex $S\in\mathcal{A}$, a path exists connecting $S$
and $\mathcal{C}$. The uniqueness of the path is given by its definition. ∎

Note that the graph $G_{\mathcal{A},d}$ could be infinite. If $d=2$, and we
consider the set of all numerical semigroups as the arithmetic variety, this
result is known (see [17]).

Given $f\in\mathcal{C}$, take the set
$\mathcal{A}_{f}=\\{S\in\mathcal{A}\mid\mathrm{Fb}(S)\preceq f\\}$. Let $S$
and $T$ be two $\mathcal{C}$-semigroups and $d$ be a positive integer. Observe
that $\mathrm{Fb}(S\cap T)=\max_{\preceq}\\{\mathrm{Fb}(S),\mathrm{Fb}(T)\\}$,
and, taking into account that $S\subseteq\frac{S}{d}$, it follows
$\mathrm{Fb}(\frac{S}{d})\preceq\mathrm{Fb}(S)$. From the previous
considerations, we deduce that $\mathcal{A}_{f}$ is an arithmetic variety.
Since a finite amount of $\mathcal{C}$-semigroups with Frobenius element less
than or equal to $f$ exists, $\mathcal{A}_{f}$ is also finite. Therefore, we
can fully construct a graph $G_{\mathcal{A}_{f},d}$. In view of Lemma 14, to
build the graph $G_{\mathcal{A}_{f},d}$, it is enough to compute
$\mathcal{D}_{d}(S)\cap\mathcal{A}_{f}=\mathcal{D}_{d}(S,f)\cap\mathcal{A}=\\{T(f,\Lambda)\in\mathcal{A}\mid\Lambda\in\overline{M}_{f}\cup\\{\\{\emptyset\\}\\}\\}$,
as the following example illustrates.

######  Example 15.

As Example 13, let $\mathcal{C}$ be again the integer cone determined by the
extremal rays $\\{(4,1),(9,5)\\}$, $\preceq$ the reverse lexicographical
order, and $f=(4,2)$. Consider $\mathcal{A}$ the arithmetic variety consisting
of all $\mathcal{C}$-semigroups, hence, $\mathcal{A}_{f}$ contains twelve
elements: $S_{0}=\mathcal{C}\setminus\\{\emptyset\\}$,
$S_{1}=\mathcal{C}\setminus\\{(2,1)\\}$,
$S_{2}=\mathcal{C}\setminus\\{(3,1)\\}$,
$S_{3}=\mathcal{C}\setminus\\{(4,1)\\}$,
$S_{4}=\mathcal{C}\setminus\\{(2,1),(3,1)\\}$,
$S_{5}=\mathcal{C}\setminus\\{(2,1),(4,1)\\}$,
$S_{6}=\mathcal{C}\setminus\\{(3,1),(4,1)\\}$,
$S_{7}=\mathcal{C}\setminus\\{(2,1),(4,2)\\}$,
$S_{8}=\mathcal{C}\setminus\\{(2,1),(3,1),(4,1)\\}$,
$S_{9}=\mathcal{C}\setminus\\{(2,1),(3,1),(4,2)\\}$,
$S_{10}=\mathcal{C}\setminus\\{(2,1),(4,1),(4,2)\\}$, and
$S_{11}=\mathcal{C}\setminus\\{(2,1),(3,1),(4,1),(4,2)\\}$. Figure 4
illustrates the finite graph $G_{\mathcal{A}_{(4,2)},2}$.

$S_{0}=\mathcal{C}$$S_{1}$$S_{7}$$S_{9}$$S_{10}$$S_{11}$$S_{2}$$S_{3}$$S_{4}$$S_{5}$$S_{6}$$S_{8}$
Figure 4:  $G_{\mathcal{A}_{(4,2)},2}$.

##  4 Some results about irreducibility over the quotients of
$\mathcal{C}$-semigroups

This section establishes two key results regarding $\mathcal{C}$-semigroups.
Firstly, we demonstrate that each $\mathcal{C}$-semigroup constitutes
precisely one-half of infinitely many distinct symmetric
$\mathcal{C}$-semigroups. Secondly, we prove that every
$\mathcal{C}$-semigroup forms one-fourth of a unique pseudo-symmetric
$\mathcal{C}$-semigroup.

Denote by $\mathcal{O}$ the set $\mathbb{N}\setminus 2\mathbb{N}$ of odd non-
negative integers.

Inspired by [21, Theorem 6.7], and by the study of the irreducibility of
$\mathcal{C}$-semigroups given in [10], we have the following result.

######  Theorem 16.

Let $S$ be a $\mathcal{C}$-semigroup and $f\in\mathcal{C}\cap\mathcal{O}^{p}$
such that $f-f_{i}-f_{j}\in S$ for all $f_{i},f_{j}\in
PF(S)=\\{f_{1},\ldots,f_{t}\\}.$ Then,

$\displaystyle T$ $\displaystyle=$ $\displaystyle 2S\cup\\{x\in\mathcal{C}\mid
f-x\notin\mathcal{C}\\}$ $\displaystyle\cup$
$\displaystyle\bigcup_{i=1}^{t}\Big{(}(f-2f_{i})+2S\Big{)}$
$\displaystyle\cup$
$\displaystyle\left\\{x\in\mathcal{C}\setminus(2\mathbb{N}^{p}\cup\mathcal{O}^{p})\mid
x\succ\tfrac{f}{2},\,f-x\in\mathcal{C}\right\\}$

is a symmetric $\mathcal{C}$-semigroup, with Frobenius element $f$, and such
that $S=\frac{T}{2}.$

###### Proof.

We start by proving that $T$ is a $\mathcal{C}$-semigroup. To show that $T$ is
closed under addition, we distinguish the following cases:

  * •

The sum of two elements of $2S$ belongs to $2S$.

  * •

If $x,y\in\mathcal{C}$ such that $f-x,f-y\notin\mathcal{C}$, then
$f-(x+y)\notin\mathcal{C}$, otherwise $f-x=y+c\in\mathcal{C}$ for some
$c\in\mathcal{C}$, which is impossible.

  * •

For any $f_{i},f_{j}\in PF(S)$, we have that
$(f-2f_{i}+2S)+(f-2f_{j}+2S)=2(f-f_{i}-f_{j})+2S\subset 2S$.

  * •

If
$x,y\in\left\\{x\in\mathcal{C}\setminus(2\mathbb{N}^{p}\cup\mathcal{O}^{p})\mid
x\succ\tfrac{f}{2},\,f-x\in\mathcal{C}\right\\}$, then $f-(x+y)\preceq 0$, and
this implies that $f-(x+y)\notin\mathcal{C}$.

  * •

Let $x\in 2S$ and $y\in\\{x\in\mathcal{C}\mid f-x\notin\mathcal{C}\\}$. Hence,
$x+y\in\\{x\in\mathcal{C}\mid f-x\notin\mathcal{C}\\}$, otherwise, there
exists $c\in\mathcal{C}$ such that $f-y=x+c\in\mathcal{C}$, which it is not
possible.

  * •

For any $s\in 2S$ then $2s+((f-2f_{i})+2S)\subset(f-2f_{i})+2S$ for every
$i\in[t]\setminus\\{0\\}$.

  * •

Let $x\in 2S$ and
$y\in\left\\{x\in\mathcal{C}\setminus(2\mathbb{N}^{p}\cup\mathcal{O}^{p})\mid
x\succ\tfrac{f}{2},\,f-x\in\mathcal{C}\right\\}$. So, if
$f-(x+y)\notin\mathcal{C}$, then $x+y\in\\{x\in\mathcal{C}\mid
f-x\notin\mathcal{C}\\}$. In the other case, if $f-(x+y)\in\mathcal{C}$, since
$x+y\in\mathcal{C}\setminus(2\mathbb{N}^{p}\cup\mathcal{O}^{p})$, and
$x+y\succ y\succ f/2$, we have
$x+y\in\left\\{x\in\mathcal{C}\setminus(2\mathbb{N}^{p}\cup\mathcal{O}^{p})\mid
x\succ\tfrac{f}{2},\,f-x\in\mathcal{C}\right\\}$.

  * •

Consider $x\in\\{x\in\mathcal{C}\mid f-x\notin\mathcal{C}\\}$ and
$y\in\bigcup_{i=1}^{t}\left((f-2f_{i})+2S\right)\cup\left\\{x\in\mathcal{C}\setminus(2\mathbb{N}^{p}\cup\mathcal{O}^{p})\mid
x\succ\tfrac{f}{2},\,f-x\in\mathcal{C}\right\\}$. Observe that
$x+y\in\\{x\in\mathcal{C}\mid f-x\notin\mathcal{C}\\}$. Otherwise,
$f-x=y+t\in\mathcal{C}$ for some $t\in\mathcal{C}$, which is not possible.

  * •

If $y\in\bigcup_{i=1}^{t}\left((f-2f_{i})+2S\right)$ and
$x\in\left\\{x\in\mathcal{C}\setminus(2\mathbb{N}^{p}\cup\mathcal{O}^{p})\mid
x\succ\tfrac{f}{2},\,f-x\in\mathcal{C}\right\\}$ then,
$x+y\in\\{x\in\mathcal{C}\mid
f-x\notin\mathcal{C}\\}\cup\left\\{x\in\mathcal{C}\setminus(2\mathbb{N}^{p}\cup\mathcal{O}^{p})\mid
x\succ\tfrac{f}{2},\,f-x\in\mathcal{C}\right\\}$.

Thus, we conclude that $T$ is a semigroup of $\mathbb{N}^{p}$. Besides, since
$\mathcal{H}(T)\subset\\{x\in\mathcal{C}\mid f-x\in\mathcal{C}\\}$, this set
is finite, and $Fb(T)=f$.

Now, let us prove that $S=\frac{T}{2}$. Trivially, $S\subset\frac{T}{2}$. For
the other inclusion, we first show that if $x\in\mathcal{H}(S)\cup
2\mathcal{H}(S)$, then $f-x\in\mathcal{C}$. Let $x\in\mathcal{H}(S)$. From
Proposition 1 there exists $f_{i}\in PF(S)$ such that $f_{i}-x\in S$, so
$f-x=f-f_{i}+f_{i}-x$. If $f-f_{i}\notin S$, and by hypothesis
$f-f_{i}-f_{j}\in S$ with $f_{j}\in PF(S)$, it would lead
$f-f_{i}=f_{j}+s\notin S$, for some $s\in S$, contradicting the definition of
$f_{j}$. Hence, $f-x=f-f_{i}+f_{i}-x\in S\subseteq\mathcal{C}$. Analogously,
if $x\in 2\mathcal{H}(S)$, then $x=2y$ with $y\in\mathcal{H}(S),$ again by
Proposition 1 we obtain that $f-x=f-2f_{i}+2(f_{i}-y)\in S$. Whence,
$f-x\in\mathcal{C}$. Let $x\in\frac{T}{2}$, so $2x\in T$. Since $2x\in
2\mathbb{N}^{p}$, $2x$ have to belong to $2S\cup\\{x\in\mathcal{C}\mid
f-x\notin\mathcal{C}\\}$. If $2x\in 2S$, we have done. Assume that
$2x\in\mathcal{C}\setminus 2S$ and $f-2x\notin\mathcal{C}$, hence
$2x\notin\mathcal{H}(S)\cup 2\mathcal{H}(S)$. Thus, $x\in S$. Therefore,
$S=\frac{T}{2}.$

To prove that $T$ is symmetric, take $x\in\mathcal{C}$. By applying
Proposition 2, we must show that $x\in\mathcal{H}(T)$ if and only if $f-x\in
T.$ We distinguish three cases depending on the parity of $x$:

  * •

If $x\in 2\mathbb{N}^{p}$ then as $x\notin T,$ we have that $\frac{x}{2}\notin
S.$ In view of Proposition 1 there exists $f_{i}\in PF(S)$ such that
$f_{i}-\frac{x}{2}\in S$. Thus, $2f_{i}-x\in 2S$, hence
$f-x=f-2f_{i}+2f_{i}-x\in T.$

  * •

If $x\in\mathcal{O}^{p}$, then $f-x\in 2\mathbb{N}^{p}.$ Thus, if $f-x\notin
T$, by using the preceding case, we obtain that $f-(f-x)=x\in T$. A
contradiction.

  * •

If $x\in\mathcal{C}\setminus\\{2\mathbb{N}^{p}\cup\mathcal{O}^{p}\\}$, then
$x\prec\frac{f}{2}$, otherwise, since $x\notin T$, we get that
$f-x\in\mathcal{C}$ and $f-x\notin\mathcal{C},$ which is impossible. Observe
that $f-x\notin 2\mathbb{N}^{p}\cup\mathcal{O}^{p}$ and $f-x\succ\frac{f}{2}$.
Hence, as $x\notin T$, we know that $f-x\in\mathcal{C}$. Therefore, $f-x\in
T$.

Conversely, let $x\in T$, we prove that $f-x\notin T$. We distinguish four
cases depending on $x$:

  * •

Let $x=2s$ for some $s\in S$. Since $f-2s\in\mathcal{O}^{p}$ it follows that
$f-x\notin 2S$. Besides, $f-(f-x)=x\in\mathcal{C}$. If
$f-2s=f-2f_{i}+2s^{\prime}$ for some $f_{i}\in PF(S)$ and $s^{\prime}\in S$,
then $2f_{i}=2s^{\prime\prime}$, with $s^{\prime\prime}=s+s^{\prime}\in S$ and
thus, $f_{i}\in S$ which is not possible. Therefore, $f-x\notin T$.

  * •

If $x\in\\{x\in\mathcal{C}\mid f-x\notin\mathcal{C}\\}$, then $f-x\notin T$.

  * •

Let $x=f-2f_{i}+2s$ for some $f_{i}\in PF(S)$ and $s\in S$, and assume that
$f-x\in\mathcal{C}$. Notice that, $f-x=2f_{i}-2s=2(f_{i}-s)\notin 2S$. If
$f-x=f-2f_{j}+2s^{\prime}$ for some $s^{\prime}\in S$ and $f_{j}\in PF(S)$,
then $x\in 2\mathbb{N}^{p}$, but $x=f-2f_{i}+2s\notin 2\mathbb{N}^{p}$.
Considering that $f-x\in 2\mathbb{N}^{p}$, we conclude that $f-x\notin T$.

  * •

If
$x\in\left\\{x\in\mathcal{C}\setminus(2\mathbb{N}^{p}\cup\mathcal{O}^{p})\mid
x\succ\tfrac{f}{2},\,f-x\in\mathcal{C}\right\\}$, then
$f-x\in\mathcal{C}\setminus(2\mathbb{N}^{p}\cup\mathcal{O}^{p})$ and
$f-x\prec\frac{f}{2}$. Notice that if $f-x=f-2f_{i}+2s$ for some $f_{i}\in
PF(S)$ and $s\in S$, we obtain that $x=2f_{i}-2s$, which contradicts the fact
that $x\notin 2\mathbb{N}^{p}$.

∎

######  Remark 17.

Due to Theorem 16, we can choose infinitely many $T$ for every semigroup $S$.
This fact implies that there exist infinitely many symmetric
$\mathcal{C}$-semigroups $T$ such that $S=\frac{T}{2}$, which is also true to
numerical semigroups ([21, Corollary 6.8]).

The following example provides a symmetric $\mathcal{C}$-semigroup obtained
from Theorem 16 for the $\mathcal{C}$-semigroup minimally generated by
$\eqref{semigrupo_ejemplo1}$.

######  Example 18.

Let $S$ be the $\mathcal{C}$-semigroup given in Example 13, and consider
$f=(13,5)\in\mathcal{C}\cap\mathcal{O}^{2}$. Trivially, the set
$PF(S)=\\{f_{1}=(2,1),f_{2}=(3,1)\\}$, and $f-2f_{1},f-2f_{2},f-f_{1}-f_{2}\in
S$. Figure 5 shows the symmetric $\mathcal{C}$-semigroup $T$ obtained from
Theorem 16 for $S$ and $f$. This $\mathcal{C}$-semigroup is minimally
generated by

$\\{(6,3),(7,3),(8,2),(8,3),(8,4),(9,3),(9,4),(9,5),(10,3),(10,4),\\\
(10,5),(11,3),(11,4),(11,5),(11,6),(12,3),(12,4),(12,5),(13,4),\\\
(13,7),(14,4),(15,4)\\}.$

Figure 5: A symmetric semigroup $T$ with $S=\frac{T}{2}$ obtained from Theorem
16.

Let us see there is not a parallelism between the symmetric and pseudo-
symmetric cases. Previously, we can extend to $\mathcal{C}$-semigroups Lemma
6.9 appearing in [21].

######  Lemma 19.

Let $S$ be $\mathcal{C}$-semigroup which Frobenius element belongs to
$2\mathbb{N}^{p}$. Then

$\mathrm{Fb}\left(\frac{S}{2}\right)=\frac{\mathrm{Fb}(S)}{2}.$

###### Proof.

It follows by Proposition 4, and by considering that the Frobenius element of
a $\mathcal{C}$-semigroup is the maximum of the fundamental gaps. ∎

Note that if $S$ is a $\mathcal{C}$-semigroup and T is a pseudo-symmetric
$\mathcal{C}$-semigroup such that $S=\frac{T}{2}$, then from the Lemma 19 we
deduce that $\mathrm{Fb}(T)=2\mathrm{Fb}(S)$ and there exist finitely many
$\mathcal{C}$-semigroups with Frobenius element $2\mathrm{Fb}(S)$. Hence, we
cannot obtain a result similar to Remark 17 for the pseudo-symmetric case.

######  Proposition 20.

Let $T$ be a pseudo-symmetric $\mathcal{C}$-semigroup. Then, $\frac{T}{2}$ is
an irreducible $\mathcal{C}$-semigroup.

###### Proof.

Let $S=\frac{T}{2}$ and suppose that $PF(S)=\\{f_{1},\ldots,f_{t}\\}$. Assume
that $\mathrm{Fb}(S)=f_{t}.$ Thus, by applying Lemma 19, we get
$\mathrm{Fb}(T)=2f_{t}$, and $PS(T)=\\{f_{t},2f_{t}\\}$. Given any
$i\in\\{1,\ldots,t\\}$, since $f_{i}\in\mathcal{H}(S)$, then
$2f_{i}\in\mathcal{H}(T)$. From Proposition 1, and considering that $T$ is a
pseudo-symmetric $\mathcal{C}$-semigroup, we have that either
$2f_{t}-2f_{i}\in T$ or $f_{t}-2f_{i}\in T$. If $2f_{t}-2f_{i}\in T$, then
$f_{t}-f_{i}\in S$, and this is possible if and only if $f_{i}=f_{t}$. If
$f_{t}-2f_{i}\in T$, then $f_{t}=2f_{i}+x$ with $x\in T\subset S$. When $x\neq
0$, we obtain that $f_{t}\in S$, which is impossible. Hence, $x=0$ and
$f_{i}=f_{t}/2$. This proves that
$PF(S)\subseteq\\{\mathrm{Fb}(S),\frac{\mathrm{Fb}(S)}{2}\\}$. Therefore, $S$
is either a symmetric or a pseudo-symmetric $\mathcal{C}$-semigroup. In both
cases, $S$ is an irreducible $\mathcal{C}$-semigroup. ∎

In particular, it is not true that any $\mathcal{C}$-semigroup is one-half of
a pseudo-symmetric $\mathcal{C}$-semigroup.

We can sharpen the preceding result a bit further to distinguish which cases
one-half of a pseudo-symmetric $\mathcal{C}$-semigroup is symmetric or pseudo-
symmetric.

######  Corollary 21.

Let $T$ be a pseudo-symmetric $\mathcal{C}$-semigroup and $S=\frac{T}{2}$.
Then:

  1. 1.

$S$ is symmetric if and only if $\mathrm{Fb}(S)\not\equiv 0\mod 4$.

  2. 2.

$S$ is pseudo-symmetric if and only if $\mathrm{Fb}(S)\equiv 0\mod 4$.

###### Proof.

We prove the first item. Suppose that $S=\frac{T}{2}$ a pseudo-symmetric
$\mathcal{C}$-semigroup, then
$PF(S)=\\{\mathrm{Fb}(S),\frac{\mathrm{Fb}(S)}{2}\\}$. By Lemma 19 we know
that $\mathrm{Fb}(S)=\frac{\mathrm{Fb}(T)}{2}$, hence, $\mathrm{Fb}(S)\equiv
0\mod 4$. Conversely, from Proposition 20 and Lemma 19, we deduce that
$\\{\frac{\mathrm{Fb}(T)}{2},\frac{\mathrm{Fb}(T)}{4}\\}\subseteq PF(S).$ In
order to prove the other inclusion, take $f\in PF(S)$, then $f\notin S$ and
thus $2f\notin T.$ From Proposition 1, we get that $\mathrm{Fb}(T)-2f\in T$ or
$\frac{\mathrm{Fb}(T)}{2}-2f\in T$. If $\mathrm{Fb}(T)-2f\in T$, then
$2(\mathrm{Fb}(S)-f)\in T$, hence $\mathrm{Fb}(S)-f\in S$, and this is
possible if and only if $f=\frac{\mathrm{Fb}(S)}{2}=\frac{\mathrm{Fb}(T)}{4}$.
Analogously to the proof of Proposition 20, if $\frac{\mathrm{Fb}(T)}{2}-2f\in
T$, then $f=\frac{\mathrm{Fb}(S)}{2}=\mathrm{Fb}(T)$. Therefore
$PF(S)=\\{\mathrm{Fb}(S),\frac{\mathrm{Fb}(S)}{2}\\}$.

For the second item, we know that $S$ is irreducible (Proposition 20), and $S$
is symmetric if and only if $S$ is not pseudo-symmetric. By the previous item,
this is possible if and only if $\mathrm{Fb}(S)\equiv 0\mod 4$. ∎

To finish this section, we characterise irreducible $\mathcal{C}$-semigroups
relative to one-half of $\mathcal{C}$-semigroups. Previously, we proved that
any $\mathcal{C}$-semigroup can be expressed as infinitely many one-fourth of
pseudo-symmetric $\mathcal{C}$-semigroups.

######  Lemma 22.

Let $S$ be an irreducible $\mathcal{C}$-semigroup, $\preceq$ a monomial order
and the set

$A=\\{x\in(\mathbb{N}^{p}\setminus
2\mathbb{N}^{p})\cap\mathcal{C}\mid\mathrm{Fb}(S)\prec x\prec
2\mathrm{Fb}(S)\\}.$

Then,

$T=2S\cup A\cup\\{x\in\mathcal{C}\mid x\succ 2\mathrm{Fb}(S)\\}$

is a pseudo symmetric $\mathcal{C}$-semigroup, with Frobenius element
$2\mathrm{Fb}(S)$, and such that $S=\frac{T}{2}$.

###### Proof.

Observe that 2S is closed under addition, and for any $a,a^{\prime}\in A$, we
have that $a+a^{\prime}\in\\{x\in\mathcal{C}\mid x\succ 2\mathrm{Fb}(S)\\}$.
If $x\in 2S$ and $a\in A$, then $x+a\in A\cup\\{x\in\mathcal{C}\mid x\succ
2\mathrm{Fb}(S)\\}$. Given $x\in\\{x\in\mathcal{C}\mid x\succ
2\mathrm{Fb}(S)\\}$ and $t\in T$ then, $x+t\in\\{x\in\mathcal{C}\mid x\succ
2\mathrm{Fb}(S)\\}$. Therefore, $T$ is a $\mathcal{C}$-semigroup. Besides, by
construction of $T$, $\mathrm{Fb}(T)=2\mathrm{Fb}(S)$.

Next we show that $S=\frac{T}{2}$. Let $x\in\frac{T}{2}$, then $2x\in T$.
Since the elements of $A$ are not in $2\mathbb{N}^{p}$, $2x\in
2S\cup\\{x\in\mathcal{C}\mid x\succ 2\mathrm{Fb}(S)\\}$. If $2x\in 2S$ then
$x\in S$. Otherwise, $2x\succ 2\mathrm{Fb}(S)$ and thus $x\in S$. Conversely,
if $x\in S$, then $2x\in 2S\subset T$ so, $x\in\frac{T}{2}$.

Now, we prove that $T$ is pseudo-symmetric. Let $x\in\mathcal{C}$, applying
Proposition 3, it is enough to prove that $x\in T$ if and only if
$\mathrm{Fb}(T)-x\notin T$ and $x\neq\frac{\mathrm{Fb}(T)}{2}$. Trivially,
$x\in T$ implies $x\neq\frac{\mathrm{Fb}(T)}{2}$. Suppose that $x\in T$, we
distinguish cases depending on $x$:

  * •

If $x\in 2S$, then $x=2s$ for some $s\in S$. In the case that
$\mathrm{Fb}(T)-x=2(\mathrm{Fb}(S)-s)$ belongs to $T$, we have
$\mathrm{Fb}(S)-s\in S$, which is not possible since $S$ is irreducible.

  * •

Let $x\in A$ and assume that $\mathrm{Fb}(T)-x\in T$. Since
$\mathrm{Fb}(T)-x\notin 2\mathbb{N}^{p}$, it follows that
$\mathrm{Fb}(T)-x\succ\mathrm{Fb}(T)$ or
$\mathrm{Fb}(T)-x\succ\mathrm{Fb}(S)$. If
$\mathrm{Fb}(T)-x\succ\mathrm{Fb}(T)$ we deduce that $-x\succ 0$, which is not
possible. If $\mathrm{Fb}(T)-x\succ\mathrm{Fb}(S)$, then
$\mathrm{Fb}(S)-x\succ 0$, this implies that $\mathrm{Fb}(S)\succ x$, a
contradiction, by hypothesis $x\in A$. We conclude that
$\mathrm{Fb}(T)-x\notin T$.

  * •

If $x\in\\{x\in\mathcal{C}\mid x\succ 2\mathrm{Fb}(S)\\}$, then
$\mathrm{Fb}(T)-x\prec 0$ and thus $\mathrm{Fb}(T)-x\notin T$.

Conversely, let $x\in\mathcal{C}\setminus\\{\mathrm{Fb}(S)\\}$ and
$\mathrm{Fb}(T)-x\notin T$. Let us show that $x\in T$. If $x\in
2\mathbb{N}^{p}$, then $x=2z$, for some $z\in\mathcal{C}$. Therefore,
$\mathrm{Fb}(T)-x=2(\mathrm{Fb}(S)-z)\notin 2S$. Since $S$ is irreducible, we
deduce that $z\in S$. Whence, $x=2z\in 2S\subset T$. If
$x\in\mathbb{N}^{p}\setminus 2\mathbb{N}^{p}$ we assume
$x\prec\mathrm{Fb}(T)$, otherwise $x\in T$ and the lemma holds. Since
$\mathrm{Fb}(T)-x\notin T$ it follows that $\mathrm{Fb}(T)-x\notin A$ and,
taking into account that $\mathrm{Fb}(T)-x\notin 2\mathbb{N}^{p}$, we deduce
that $\mathrm{Fb}(T)-x\prec\mathrm{Fb}(S)$. Thus $x\succ\mathrm{Fb}(S)$ and
$x\in A\subset T$. ∎

######  Theorem 23.

Every $\mathcal{C}$-semigroup is one-fourth of infinitely many pseudo-
symmetric $\mathcal{C}$-semigroups.

###### Proof.

Let $S$ be a $\mathcal{C}$-semigroup. Applying Remark 17 we obtain that there
exists infinitely many symmetric $\mathcal{C}$-semigroups $T$ such that
$S=\frac{T}{2}$. For each $T$, Lemma 22 guarantees the existence of a pseudo-
symmetric $\mathcal{C}$-semigroup $T^{\prime}$ such that
$S=\frac{T^{\prime}}{4}$. ∎

To sum it up, as a consequence of Corollary 21 and Lemma 22, we obtain the
announced result.

######  Theorem 24.

A $\mathcal{C}$-semigroup is irreducible if and only if it is one-half of a
pseudo-symmetric $\mathcal{C}$-semigroup.

##  5 Arithmetic varieties of affine semigroups

Building upon our investigation of affine semigroup quotients, we explore
arithmetic varieties defined by such quotients, which provide valuable
insights into the structure of semigroups. This section introduces two
intriguing families: one arising from systems of modular Diophantine
inequalities, where integer solutions modulo a positive integer reveal key
information about the underlying semigroup; and the other generalizing the
crucial Arf property from numerical semigroups to a broader class of affine
semigroups, offering new avenues for analysis (see [21] for an introduction to
the Arf property).

Recall that an arithmetic variety is a non-empty family of affine semigroups
$\mathcal{A}$ that satisfy:

  * •

If $S,T\in\mathcal{A}$, then $S\cap T\in\mathcal{A}$.

  * •

If $S\in\mathcal{A}$ and $d\in\mathbb{N}\setminus\\{0\\}$, then
$\frac{S}{d}\in\mathcal{A}$.

It could be deduced that the intersection of arithmetic varieties is again an
arithmetic variety. In [3, Proposition 4], it is proved that given a family of
numerical semigroups, the intersection of all the arithmetic varieties
containing this family is an arithmetic variety. Furthermore, it is the
smallest that satisfies the previous condition. This also holds for affine
semigroups.

A modular Diophantine inequality is an expression of the form $ax\mod b\leq
cx$ with $a$, $b$, $c$ integers such that $b\neq 0$. This kind of inequality
is introduced in [22], and the set of its non-negative solutions is a
numerical semigroup, usually named proportionally modular numerical semigroup.
If a system of modular Diophantine inequations is considered, the set of its
non-negative solutions is also a numerical semigroup.

In [9], the concept of modular Diophantine inequality is generalised to more
than one variable: a modular Diophantine inequality with $p$ variables is an
expression $f(x)\mod b\leq g(x)$ for some $f,g:\mathbb{Q}^{p}\to\mathbb{Q}$
two non-null linear functions, and a non-zero natural number $b$. In that
case, its non-negative solutions make up an affine semigroup whose associated
integer cone is $\mathbb{N}^{p}$. The same holds if you take any system of
modular Diophantine inequations with $p$ variables. A semigroup $S$ is a
proportionally modular affine semigroup if and only if there exist
$a_{1},\ldots,a_{p}\in\mathbb{N}$, and $g_{1},\ldots,g_{p}\in\mathbb{Z}$, and
a natural number $b$ such that
$S=\\{x\in\mathbb{N}^{p}\mid\sum_{i=1}^{p}a_{i}x_{i}\mod
b\leq\sum_{i=1}^{p}g_{i}x_{i}\\}$. These semigroups are not necessarily
$\mathbb{N}^{p}$-semigroups. For example, if one fix $a_{1}=0$ and $g_{1}\geq
0$, then $S$ includes the infinite set $\\{(x,0)\in\mathbb{N}^{p}\mid
x\in\mathbb{N}\\}$. We use the same designation (proportionally modular affine
semigroup) when $S$ is defined for more than one modular Diophantine
inequality. In that case, such semigroup can be expressed as
$S=\\{x\in\mathbb{N}^{p}\mid Ax\mod b\leq Gx\\}$, where $A$ and $G$ are two
integers $(k\times p)$-matrices and $b$ is a positive integer matrix with $k$
entries. We can assume that all the entries in the $i$-th row of $A$ are non-
negative integers lesser than $b_{i}$.

######  Proposition 25.

The set of the proportionally modular affine semigroups in $p$ variables is an
arithmetic variety.

###### Proof.

Let $S$ and $S^{\prime}$ be two proportionally modular affine semigroups in
$p$ variables. Trivially, $S\cap S^{\prime}$ is the proportionally modular
affine semigroup defined by the union of the modular Diophantine inequalities
of $S$ and $S^{\prime}$.

If $S$ is determined by $\\{x\in\mathbb{N}^{p}\mid Ax\mod b\leq Gx\\}$, and
$d$ is a positive integer, then $\frac{S}{d}$ is also a proportionally modular
affine semigroup defined by $\\{x\in\mathbb{N}^{p}\mid dAx\mod b\leq dGx\\}$.
So, the proposition holds. ∎

We say that an affine semigroup $S$ has the Arf property or is an Arf (affine)
semigroup if, for any $x,y,z\in S$ with $x\geq y\geq z$, then $x+y-z\in S$. We
say that an affine semigroup $S$ is saturated if $s,s_{1},\ldots,s_{r}\in S$
are such that $s_{i}\leq s$ for all $i\in[r]\setminus\\{0\\}$ and
$z_{1},\ldots,z_{r}\in\mathbb{Z}$ are such that
$z_{1}s_{1}+\cdots+z_{r}s_{r}\in\mathbb{N}^{p}$ then,
$s+z_{1}s_{1}+\cdots+z_{r}s_{r}\in S$. Note that, as occurs for numerical
semigroups, a saturated affine semigroup has the Arf property [21, Lemma
3.31].

######  Proposition 26.

The set of Arf affine semigroups is an arithmetic variety.

###### Proof.

Observe that $S=\mathbb{N}^{p}$ is an Arf semigroup. On the one hand, for
every two semigroups $S$ and $T$ with the Arf property, $S\cap T$ is an Arf
semigroup. On the other hand, note for any positive integer $d$ and any Arf
semigroup $S$, since $S$ is an Arf semigroup, it follows that $d(x+y-z)\in S$,
for any $x,y,z\in\frac{S}{d}$ such that $x\geq y\geq z$. Whence, $\frac{S}{d}$
has the Arf property. ∎

######  Proposition 27.

The set of saturated affine semigroups is an arithmetic variety.

###### Proof.

Observe that $S=\mathbb{N}^{p}$ is a saturated semigroup. As occurs for Arf
semigroups, the intersection of saturated semigroups is again a saturated
semigroup. Consider a positive integer $d$ and a saturated semigroup $S$.
Given $x,x_{1},\ldots,x_{r}\in\frac{S}{d}$ such that $x_{i}\leq x$, for all,
$i\in[r]\setminus\\{0\\}$ and $z_{1},\ldots,z_{r}\in\mathbb{Z}$ such that
$z_{1}x_{1}+\cdots+z_{r}x_{r}\in\mathbb{N}^{p}$,
$d(s+z_{1}s_{1}+\cdots+z_{r}s_{r})\in S$ since $S$ is saturated. Whence,
$\frac{S}{d}$ is a saturated semigroup. ∎

### Funding

The first and third-named authors were supported partially by Junta de
Andalucía research group FQM 343.

The last author is partially supported by grant PID2022-138906NB-C21 funded by
MCIN/AEI/10.13039/501100011033 and by ERDF ”A way of making Europe”.

Consejería de Universidad, Investigación e Innovación de la Junta de Andalucía
project ProyExcel_00868 also partially supported all the authors.

Proyectos de investigación del Plan Propio – UCA 2022-2023 (PR2022-011 and
PR2022-004) partially supported the first, and third-named 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>
R. Tapia-Ramos. Departamento de Matemáticas, Universidad de Cádiz, E-11406
Jerez de la Frontera (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>
## References

  * [1] Bhardwaj, O. P.; Goel, K.; Sengupta, I.,  _Affine semigroups of maximal projective dimension_.  Collectanea Mathematica, 74 (2023), 703–727.
  * [2] Bogart, T.; O’Neill, C.; Woods, K.,  _When is a numerical semigroup a quotient?_ Bull. Aust. Math. Soc.109(2024), no.1, 67–76.
  * [3] Branco, M. B.; Ojeda, I.; Rosales, J. C.,  _Arithmetic varieties of numerical semigroups_.  arXiv:2311.13500. https://doi.org/10.48550/arXiv.2311.13500
  * [4] Briales, E.; Campillo, A.; Marijuán, C.; Pisón, P.,  _Minimal systems of generators for ideals of semigroups._ J. Pure Appl. Algebra 124 (1998), no. 1-3, 7-–30.
  * [5] Bruns, W.; Gubeladze, J.,  Polytopes, rings, and K-theory,  Springer Monographs in Mathematics. Springer, Dordrecht, 2009.
  * [6] Bruns, W.; Herzog, J.,  Cohen-Macaulay rings.  Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, 1993.
  * [7] Cox, D. A.; Little, J.; O’Shea, D.,  Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra.  Undergraduate Texts in Mathematics. Springer, Cham, 2015.
  * [8] Failla G.; Peterson C.; Utano, R.,  _Algorithms and basic asymptotics for generalized numerical semigroups in $\mathbb{N}^{p}$_,  Semigroup Forum (2016) 92, 460–473.
  * [9] García-García, J. I.; Moreno-Frías, M. A.; Vigneron-Tenorio, A.,  _Proportionally modular affine semigroups_.  Journal of Algebra and its Applications, Vol. 17, No. 1 (2018) 1850017 (17 pages).
  * [10] García-García, J. I.; Marín-Aragón, D.; Sánchez-Loureiro, A.; Vigneron-Tenorio, A.,  _Some Properties of Affine $\mathcal{C}$-semigroups_.  Results Math, 79 (2024), no. 52.
  * [11] García-García J. I.; Marín-Aragón D.; Vigneron-Tenorio A.,  _An extension of Wilf’s conjecture to affine semigroups_ ,  Semigroup Forum (2018), vol. 96, Issue 2, 396–408.
  * [12] García-García, J. I.; Marín-Aragón, D.; Vigneron-Tenorio, A.,  _A characterization of some families of Cohen–Macaulay, Gorenstein and/or Buchsbaum rings_.  Discrete Applied Mathematics, Vol. 263 (2019), 166–176.
  * [13] García-García, J. I.; Sánchez-R.-Navarro, A.; Vigneron-Tenorio, A.,  _Multiple Convex Body Semigroups and Buchsbaum Rings_.  Journal of Commutative Algebra, Vol. 12 (2020), no. 3, 309-318.
  * [14] Goto, S.; Suzuki, N.; Watanabe, K.,  _On affine semigroup rings._ Japan. J. Math. (N.S.) 2 (1976), no. 1, 1–12.
  * [15] Herzog, J.; Hibi, T.; Viêt Trung, N.,  _Symbolic powers of monomial ideals and vertex cover algebras_.  Adv. Math., 210 (2007), no. 1, 304–322.
  * [16] García-García, J.I.; Ojeda, I.; Rosales, J.C.; Vigneron-Tenorio, A.,  _On Pseudo-Frobenius elements of submonoids of $\mathbb{N}^{q}$_.  Collectanea Mathematica 71 (2020), 189–204.
  * [17] Ojeda, I.; Rosales, J. C.,  _The arithmetic extensions of a numerical semigroup_.  Communications in Algebra, 48:9 (2020), 3707–3715.
  * [18] Pisón-Casares, P.; Vigneron-Tenorio, A.,  _$\mathbb{N}$ -solution to linear systems over $\mathbb{Z}$_.  Linear Algebra and its Applications, 384 (2004), 135–154.
  * [19] Rosales, J. C.; and García-Sánchez, P. A.,  Finitely Generated Commutative Monoids.  Nova Science Publishers, Inc., Commack, NY, 1999.
  * [20] Rosales, J. C.; García-Sánchez, P. A.  _Every numerical semigroup is one half of infinitely many symmetric numerical semigroups_.  Comm. Algebra 36 (2008), no. 8, 2910–2916.
  * [21] Rosales, J. C.; García-Sánchez, P. A.,  Numerical Semigroups.  Dev. Math., 20 Springer, New York, 2009.
  * [22] Rosales, J. C.; García-Sánchez P. A.; Urbano-Blanco, J. M.,  _Modular Diophantine inequalities and numerical semigroups_.  Pacific J. Math., 218 (2005), no. 2, 379–398.
  * [23] Swanson, I.,  Every numerical semigroup is one over $d$ of infinitely many symmetric numerical semigroups.  Commutative algebra and its applications, 383–386. Walter de Gruyter GmbH & Co. KG, Berlin, 2009