# Modular Frobenius pseudo-varieties

Aureliano M. Robles-Pérez  and José Carlos Rosales∗ Both authors are supported
by the project MTM2017-84890-P (funded by Ministerio de Economía, Industria y
Competitividad and Fondo Europeo de Desarrollo Regional FEDER) and by the
Junta de Andalucía Grant Number FQM-343.Departamento de Matemática Aplicada &
Instituto de Matemáticas (IMAG), Universidad de Granada, 18071-Granada, Spain.
E-mail<EMAIL_ADDRESS>ORCID: 0000-0003-2596-1249.Departamento de Álgebra &
Instituto de Matemáticas (IMAG), Universidad de Granada, 18071-Granada, Spain.
E-mail<EMAIL_ADDRESS>ORCID: 0000-0003-3353-4335.

###### Abstract

If $m\in\mathbb{N}\setminus\\{0,1\\}$ and $A$ is a finite subset of
$\bigcup_{k\in\mathbb{N}\setminus\\{0,1\\}}\\{1,\ldots,m-1\\}^{k}$, then we
denote by

$\displaystyle\mathscr{C}(m,A)=\Big{\\{}S\in\mathscr{S}_{m}\mid
s_{1}+\cdots+s_{k}-m\in S\mbox{ if }(s_{1},\ldots,s_{k})\in S^{k}\mbox{ and }$
$\displaystyle s_{1}\bmod m,\ldots,s_{k}\bmod m)\in A\Big{\\}}.$

In this work we prove that $\mathscr{C}(m,A)$ is a Frobenius pseudo-variety.
We also show algorithms that allows us to establish whether a numerical
semigroup belongs to $\mathscr{C}(m,A)$ and to compute all the elements of
$\mathscr{C}(m,A)$ with a fixed genus. Moreover, we introduce and study three
families of numerical semigroups, called of second-level, thin and strong, and
corresponding to $\mathscr{C}(m,A)$ when $A=\\{1,\ldots,m-1\\}^{3}$,
$A=\\{(1,1),\ldots,(m-1,m-1)\\}$, and
$A=\\{1,\ldots,m-1\\}^{2}\setminus\\{(1,1),\ldots,(m-1,m-1)\\}$, respectively.

Keywords: Modular pseudo-varieties, second-level numerical semigroups, thin
numerical semigroups, strong numerical semigroups, tree associated (with a
modular pseudo-variety).

2010 AMS Classification: 20M14.

##  1 Introduction

Let $\mathbb{N}$ be the set of non-negative integers. A _numerical semigroup_
is a subset $S$ of $\mathbb{N}$ that is closed under addition, $0\in S$ and
$\mathbb{N}\setminus S$ is finite. The _Frobenius number_ of $S$, denoted by
$\mathrm{F}(S)$, is the greatest integer that does not belong to $S$. The
cardinality of $\mathbb{N}\setminus S$, denoted by $\mathrm{g}(S)$, is the
_genus_ of $S$.

A Frobenius pseudo-variety is a non-empty family $\mathcal{P}$ of numerical
semigroups that fulfils the following conditions.

  1. 1.

$\mathcal{P}$ has a maximum element (with respect to the inclusion order).

  2. 2.

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

  3. 3.

If $S\in\mathcal{P}$ and $S\neq\max(\mathcal{P})$, then $S\cup\\{{\rm
F}(S)\\}\in\mathcal{P}$.

The _multiplicity_ of a numerical semigroup $S$, denoted by $\mathrm{m}(S)$,
is the least positive integer that belongs to $S$. If $m$ is a positive
integer, then we denote by $\mathscr{S}_{m}=\left\\{S\mid S\mbox{ is a
numerical semigroup with }\mathrm{m}(S)=m\right\\}$.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$ and let $A$ be a finite subset of
$\bigcup_{k\in\mathbb{N}\setminus\\{0,1\\}}\\{1,\ldots,m-1\\}^{k}$ (where
$X^{k}=X\times\stackrel{{{}^{(k)}}}{{\cdots}}\times
X=\left\\{(x_{1},\ldots,x_{k})\mid x_{1},\ldots,x_{k}\in X\right\\}$). We
denote by

$\displaystyle\mathscr{C}(m,A)=\Big{\\{}S\in\mathscr{S}_{m}\mid
s_{1}+\cdots+s_{k}-m\in S\mbox{ if }(s_{1},\ldots,s_{k})\in S^{k}\mbox{ and }$
$\displaystyle s_{1}\bmod m,\ldots,s_{k}\bmod m)\in A\Big{\\}}.$

Our main purpose in this work is to study the set $\mathscr{C}(m,A)$.

In Section 2 we show that $\mathscr{C}(m,A)$ is a Frobenius pseudo-variety
with maximum element given by $\Delta(m)=\left\\{0,m,\to\right\\}$ (where the
symbol $\to$ means that every integer greater than $m$ belongs to
$\Delta(m)$). Thus, we call the pseudo-varieties that arise in this way
modular Frobenius pseudo-varieties. Also, we give an algorithm that allows us
to establish whether a numerical semigroup belongs to $\mathscr{C}(m,A)$. In
addition, with the help of the results in [9], we can arrange
$\mathscr{C}(m,A)$ in a rooted tree and we find an algorithm to compute all
the elements of $\mathscr{C}(m,A)$ with a fixed genus.

If $X$ is a non-empty subset of $\mathbb{N}$, then we denote by $\langle
X\rangle$ the submonoid of $(\mathbb{N},+)$ generated by $X$, that is,

$\langle X\rangle=\big{\\{}\lambda_{1}x_{1}+\cdots+\lambda_{n}x_{n}\mid
n\in\mathbb{N}\setminus\\{0\\},\ x_{1},\ldots,x_{n}\in X,\
\lambda_{1},\ldots,\lambda_{n}\in\mathbb{N}\big{\\}}.$

It is well known (see Lemma 2.1 of [11]) that $\langle X\rangle$ is a
numerical semigroup if and only if $\gcd(X)=1$. If $M$ is a submonoid of
$(\mathbb{N},+)$ and $M=\langle X\rangle$, then we say that $X$ is a _system
of generators_ of $M$. Moreover, if $M\not=\langle Y\rangle$ for any subset
$Y\subsetneq X$, then we say that $X$ is a _minimal system of generators_ of
$M$. In Corollary 2.8 of [11] it is shown that each submonoid of
$(\mathbb{N},+)$ has a unique minimal system of generators and that such a
system is finite. We denote by $\mathrm{msg}(M)$ the minimal system of
generators of $M$. The cardinality of $\mathrm{msg}(M)$, denoted by
$\mathrm{e}(M)$, is the _embedding dimension_ of $M$.

By applying Proposition 2.10 of [11], if $S$ is a numerical semigroup, then we
know that $\mathrm{e}(S)\leq\mathrm{m}(S)$. A numerical semigroup $S$ has
_maximal embedding dimension_ if $\mathrm{e}(S)=\mathrm{m}(S)$. This family of
numerical semigroups has been extensively studied (for instance, see [2] and
[11]). Let us denote by $\mathcal{M}_{m}$ the set formed by the numerical
semigroups that have maximal embedding dimension and with multiplicity $m$. It
is easy to see that $\mathcal{M}_{m}=\mathscr{C}(m,\\{1,\ldots,m-1\\}^{2})$.

In Sections 3, 4, and 5 we study the family $\mathscr{C}(m,A)$ for

  * •

$A=\\{1,\ldots,m-1\\}^{3}$,

  * •

$A=\\{(1,1),\ldots,(m-1,m-1)\\}$,

  * •

$A=\\{1,\ldots,m-1\\}^{2}\setminus\\{(1,1),\ldots,(m-1,m-1)\\}$,

respectively. Observe that, in a certain sense, these families are
generalisation of $\mathcal{M}_{m}$.

To finish this introduction, we are going to comment several ideas (see [3, 4,
14]) that motivate the study of modular Frobenius pseudo-varieties.

First of all, observe that modular Frobenius pseudo-varieties are related to
the non-homogeneous patterns with positive coefficients that involve in their
constant parameter the multiplicity of the numerical semigroup (see [4]).

The notion of non-homogeneous pattern was introduced in [4] as a
generalisation of the notion of homogeneous pattern [3]. Thus, a _linear
pattern_ $p(x_{1},\ldots,x_{n})$ is an expression of the form
$a_{1}x_{1}+\cdots+a_{n}x_{n}+a_{0}$, with $a_{0}\in\mathbb{Z}$ (as usual,
$\mathbb{Z}$ is the set of integers numbers) and
$a_{1},\ldots,a_{n}\in\mathbb{Z}\setminus\\{0\\}$. In particular, the (linear)
pattern $p$ is _homogeneous_ if $a_{0}=0$, and _non-homogeneous_ if
$a_{0}\not=0$.

On the other hand, it is said that a numerical semigroup $S$ admits the
homogeneous pattern $p$ if $p(s_{1},\ldots,s_{n})\in S$ for every non-
increasing sequence $(s_{1},\ldots,s_{n})\in S^{n}$. The corresponding
definition for non-homogeneous patterns is a bit different: a numerical
semigroup $S$ admits the non-homogeneous pattern $p$ if
$p(s_{1},\ldots,s_{n})\in S$ for every non-increasing sequence
$(s_{1},\ldots,s_{n})\in(S\setminus\\{0\\})^{n}$.

Having in mind that $M(S)=S\setminus\\{0\\}$ is an ideal of the numerical
semigroup $S$ (in fact, $M(S)$ is the maximal ideal of $S$), in [14] the
concepts of the above paragraph were extended in the following way: an ideal
$I$ of a numerical semigroups $S$ admits the pattern $p$ if
$p(s_{1},\ldots,s_{n})\in S$ for every non-increasing sequence
$(s_{1},\ldots,s_{n})\in I^{n}$.

At this point we have the main difference between the above-mentioned papers
and our proposal in this work: we do not keep the non-increasing condition (on
the sequences in which we evaluate the pattern) in mind. In addition, we take
sequences in sets $A$ without structure (that is, $A$ does not have to be an
ideal of a numerical semigroup).

Now, let us denote by $\mathscr{S}_{m}(p)$ the family of numerical semigroups
with multiplicity $m$ that admit the pattern $p$. If we take the patterns
$p_{1}=2x_{1}+x_{2}-m$ and $p_{2}=x_{1}+2x_{2}-m$, then
$\mathscr{S}_{m}(p_{1},p_{2})=\mathscr{S}_{m}(p_{1})\cap\mathscr{S}_{m}(p_{2})$
(that is, the family of numerical semigroups which admit $p_{1}$ and $p_{2}$
simultaneously) is equal to $\mathscr{C}(m,A)$ for

$A=\left\\{(a_{1},a_{2},a_{3})\in\\{1,\ldots,m-1\\}^{3}\mid a_{1}\equiv
a_{2}\pmod{m}\right\\},$

being this one an easy example of the connection between modular Frobenius
pseudo-varieties and families of numerical semigroups defined by non-
homogeneous patterns (as considered in [4]).

A first question studied in [4, 14] is the next one: if
$p=a_{1}x_{1}+\cdots+a_{n}x_{n}+a_{0}$ is a non-homogeneous pattern, for which
values of $a_{0}$ we have that the family $\mathscr{S}(p)$, of numerical
semigroups that admit the pattern $p$, is non-empty? If the answer is
affirmative, then it is said that $p$ is an _admissible pattern_. In our case,
we have imposed that $a_{0}=-m$, where $m$ will be the multiplicity of all the
numerical semigroups, in order to get a relevant advantage: we want to build
in a (more or less) easy and explicit way the Apéry set of the numerical
semigroups belonging to $\mathscr{C}(m,A)$. As a consequence of this fact, we
will be able to obtain extra information about such families of numerical
semigroups.

Of course, in addition to $-m$, it is possible to consider other values that
maintain the pseudo-variety structure. Thus, for $p=x_{1}+\cdots+x_{n}+a_{0}$,
we have that, independently of the chosen set $A$, every numerical semigroup
$S$, such that $a_{0}\in S$, admits the pattern $p$.

Now, let us observe that, if $A_{1}\subseteq A_{2}$, then
$\mathscr{C}(m,A_{2})\subseteq\mathscr{C}(m,A_{1})$. This fact allow us to
have sufficient conditions on the patterns
$p=a_{1}x_{1}+\cdots+a_{n}x_{n}-km$, with $k\in\mathbb{N}\setminus\\{0,1\\}$,
from the results in [4, 14]. For example, from [4, Theorem 4.1] or [14,
Proposition 20], we can assert that $\mathscr{C}(m,A)\not=\emptyset$ if
$n-k\geq 1$.

A second question that appears in [3, 14] is about the _equivalence of
patterns_. Briefly, the pattern $p_{1}$ induces the pattern $p_{2}$ if
$\mathscr{S}(p_{2})\subseteq\mathscr{S}(p_{1})$ and, moreover, $p_{1}$ and
$p_{2}$ are equivalent patterns if they induce each other. Maybe the first
result of this type is the equivalence between the homogeneous patterns
$2x_{1}-x_{2}$ and $x_{1}+x_{2}-x_{3}$, which correspond to the family of Arf
numerical semigroups (see [5]). Since the set $A$ has an important role (not
to say the main role) in the families $\mathscr{C}(m,A)$, the large number of
possibilities in the choice of $A$ leads us to believe that this is an issue
that deserves a new work.

By the way, in Section 4 we study the family of thin numerical semigroups,
denoted by $\mathscr{T}_{m}$, and in Section 5 we consider the family of
strong numerical semigroups, denoted by $\mathscr{R}_{m}$. These families are
associated with the patterns $2x-m$ and $x+y-m$, respectively. However, the
set $A$ is different in each case and, as a consequence, we have that there is
not a inclusion relation between both of them (see Examples 4.6 and 5.6). This
fact may give an idea about the difficulty of obtaining similar results to
those seen in [3, 14].

In any case, we can show simple results about the equivalence (or, at least,
the inclusions) of the families $\mathscr{C}(m,A)$.

######  Remark 1.1.

All the patterns $p=x_{1}+\cdots+x_{n}-m$ are equivalent (independently of the
chosen set $A$) if $n\geq m$, in which case we have that
$\mathscr{C}(m,A)=\mathscr{S}_{m}$. Indeed, applying the pigeonhole principle,
if $s_{1},\ldots,s_{n}$ are elements of $S\in\mathscr{S}_{m}$ such that
$s_{i}\not\equiv 0\pmod{m}$, $1\leq i\leq n$, then there exist
$i,j\in\\{1,\ldots,n\\}$, with $i<j$, such that $s_{i}+\cdots+s_{j}=km$ for
some $k\in\mathbb{N}\setminus\\{0\\}$. Consequently, $S\in\mathscr{C}(m,A)$.

######  Remark 1.2.

Let us set $A=\\{(1,1),(3,4)\\}$, $B=\\{(1,1,2),(1,3,1),(1,3,4)\\}$, and
$m\geq 5$. Then $\mathscr{C}(m,A)\subseteq\mathscr{C}(m,B)$. In order to
verify this inclusion, we take $a_{1},a_{2},a_{3}\in S\in\mathscr{C}(m,A)$.

  * •

If $a_{1}\equiv 1\pmod{m}$, $a_{2}\equiv 1\pmod{m}$, $a_{3}\equiv 2\pmod{m}$,
then $a_{1}+a_{2}+a_{3}-m=(a_{1}+a_{2}-m)+a_{3}\in S$ .

  * •

If $a_{1}\equiv 1\pmod{m}$, $a_{2}\equiv 3\pmod{m}$, $a_{3}\equiv 1\pmod{m}$,
then $a_{1}+a_{2}+a_{3}-m=(a_{1}+a_{3}-m)+a_{2}\in S$.

  * •

If $a_{1}\equiv 1\pmod{m}$, $a_{2}\equiv 3\pmod{m}$, $a_{3}\equiv 4\pmod{m}$,
then $a_{1}+a_{2}+a_{3}-m=(a_{2}+a_{3}-m)+a_{1}\in S$.

More generally, let us suppose that, for each $b\in B$, there exists $a\in A$
such that $b$ is obtained by adding coordinates to $a$. Then
$\mathscr{C}(m,A)\subseteq\mathscr{C}(m,B)$.

As a final recommendation, it is worth mentioning that in [4, Section 2] and
in [14, Introduction] there are several motivating examples, and references,
as to why it is interesting to study (non-homogeneous) patterns of numerical
semigroups.

##  2 The pseudo-variety $\mathscr{C}\boldsymbol{(m,A)}$

In this section, $m$ is an integer greater than or equal to $2$ and $A$ is a
finite subset of
$\bigcup_{k\in\mathbb{N}\setminus\\{0,1\\}}\\{1,\ldots,m-1\\}^{k}$. Moreover,
recall that

$\displaystyle\mathscr{C}(m,A)=\Big{\\{}S\in\mathscr{S}_{m}\mid
s_{1}+\cdots+s_{k}-m\in S\mbox{ if }(s_{1},\ldots,s_{k})\in S^{k}\mbox{ and }$
$\displaystyle s_{1}\bmod m,\ldots,s_{k}\bmod m)\in A\Big{\\}},$

where $x\bmod m$ denotes the remainder after division of $x$ by $m$.

If $S$ is a numerical semigroup and $x\in S\setminus\\{0\\}$, then the _Apéry
set of $x$ in $S$_ (see [1]) is
$\mathrm{Ap}(S,x)=\\{w(0)=0,w(1),\ldots,w(x-1)\\}$, where $w(i)$ is the least
element of $S$ that is congruent with $i$ modulus $x$. Observe that an integer
$s$ belongs to $S$ if and only if there exists $t\in\mathbb{N}$ such that
$s=w(s\bmod x)+tx$.

######  Proposition 2.1.

Let $S\in\mathscr{S}_{m}$ and
$\mathrm{Ap}(S,m)=\\{w(0)=0,w(1),\ldots,w(m-1)\\}$. Then the following
conditions are equivalent.

  1. 1.

$S\in\mathscr{C}(m,A)$.

  2. 2.

$w(i_{1})+\cdots+w(i_{k})-m\in S$ for all $(i_{1},\ldots,i_{k})\in A$.

###### Proof.

(1. $\Rightarrow$ 2.) Since $w(i_{1}),\ldots,w(i_{k})\in S$ and
$(w(i_{1})\bmod m,\ldots,w(i_{k})\bmod m)=(i_{1},\ldots,i_{k})\in A$, then
$w(i_{1})+\cdots+w(i_{k})-m\in S$.

(2\. $\Rightarrow$ 1.) Let $s_{1},\ldots,s_{k}\in S$ such that

$(s_{1}\bmod m,\ldots,s_{k}\bmod m)=(i_{1},\ldots,i_{k})\in A.$

Then there exist $t_{1},\ldots,t_{k}\in\mathbb{N}$ such that
$s_{j}=w(i_{j})+t_{j}m$, $1\leq j\leq k$, and thus,
$s_{1}+\cdots+s_{k}=(w(i_{1})+\cdots+w(i_{k})-m)+(t_{1}+\cdots+t_{k})m\in S$.
∎

By using the function AperyListOfNumericalSemigroupWRTElement(S,m) of [6], we
can compute $\mathrm{Ap}(S,m)$ from a system of generators of $S$. Thereby, we
have the following algorithm to decide if a numerical semigroup $S$ belongs or
not to $\mathscr{C}(m,A)$.

######  Algorithm 2.2.

INPUT: A finite subset $G$ of positive integers.

OUTPUT: $\langle G\rangle\in\mathscr{C}(m,A)$ or $\langle
G\rangle\notin\mathscr{C}(m,A)$.

  * (1)

If $\min$(G)$\not=m$, return $\langle G\rangle\notin\mathscr{C}(m,A)$.

  * (2)

If $\gcd(G)\not=1$, return $\langle G\rangle\notin\mathscr{C}(m,A)$.

  * (3)

Compute $\mathrm{Ap}(\langle G\rangle,m)=\\{w(0),w(1),\ldots,w(m-1)\\}$.

  * (4)

If $w(i_{1})+\cdots+w(i_{k})-m\in S$ for all $(i_{1},\ldots,i_{k})\in A$,
return $\langle G\rangle\in\mathscr{C}(m,A)$.

  * (5)

Return $\langle G\rangle\notin\mathscr{C}(m,A)$.

Let us illustrate the working of the previous algorithm through an example.

######  Example 2.3.

Let us make use of Algorithm 2.2 with $G=\\{5,7,9\\}$, $m=5$, and
$A=\\{(1,3),(2,2)\\}$.

  * •

$\min(G)=5$.

  * •

$\gcd(G)=1$.

  * •

$\mathrm{Ap}(\langle G\rangle,5)=\\{w(0)=0,w(1)=16,w(2)=7,w(3)=18,w(4)=19\\}$.

  * •

$w(1)+w(3)-5=29\in\langle G\rangle$ and $w(2)+w(2)-5=9\in\langle G\rangle$.

  * Therefore, $\langle G\rangle=\langle 5,7,9\rangle\in\mathscr{C}(5,\\{(1,3),(2,2)\\})$.

Recall that, if $m$ is an integer greater than or equal to $2$, then we denote
by $\Delta(m)=\\{0,m,\to\\}$. It is clear that $\Delta(m)\in\mathscr{S}_{m}$
and that, if $s_{1},\ldots,s_{k}\in\Delta(m)\setminus\\{0\\}$ and $k\geq 2$,
then $s_{1}+\cdots+s_{k}-m\in\Delta(m)$. Therefore, we have the next result.

######  Lemma 2.4.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$ and
$A\subseteq\bigcup_{k\in\\{2,\to\\}}\\{1,\ldots,m-1\\}^{k}$ ($A$ finite). Then
$\Delta(m)\in\mathscr{C}(m,A)$.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$. Then it is easy to show that $S\cap
T\in\mathscr{S}_{m}$ for all $S,T\in\mathscr{S}_{m}$. Moreover, $\Delta(m)$ is
the maximum of $\mathscr{S}_{m}$ and, if $S\in\mathscr{S}_{m}$ and
$S\not=\Delta(m)$, then $S\cup\\{\mathrm{F}(S)\\}\in\mathscr{S}_{m}$. From all
this, we conclude the following result.

######  Lemma 2.5.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$. Then $\mathscr{S}_{m}$ is a Frobenius
pseudo-variety with $\Delta(m)$ as maximum element.

######  Proposition 2.6.

$\mathscr{C}(m,A)$ is a Frobenius pseudo-variety.

###### Proof.

From Lemmas 2.4 and 2.5, we have that $\Delta(m)$ is the maximum of
$\mathscr{C}(m,A)$. It is also easy to see that, if $S,T\in\mathscr{C}(m,A)$,
then $S\cap T\in\mathscr{C}(m,A)$. In order to finish the proof, let us see
that, if $S\in\mathscr{C}(m,A)$ and $S\not=\Delta(m)$, then
$S\cup\\{\mathrm{F}(S)\\}\in\mathscr{C}(m,A)$. For that, we have to show that,
if $s_{1},\ldots,s_{k}\in S\cup\\{\mathrm{F}(S)\\}$ and $(s_{1}\bmod
m,\ldots,s_{k}\bmod m)\in A$, then $s_{1}+\cdots+s_{k}-m\in
S\cup\\{\mathrm{F}(S)\\}$. In effect, if
$\mathrm{F}(S)\notin\\{s_{1},\ldots,s_{k}\\}$, then the result is true because
$S\in\mathscr{C}(m,A)$. On the other hand, if
$\mathrm{F}(S)\in\\{s_{1},\ldots,s_{k}\\}$, then
$s_{1}+\cdots+s_{k}-m\geq\mathrm{F}(S)$ and, consequently,
$s_{1}+\cdots+s_{k}-m\in S\cup\\{\mathrm{F}(S)\\}$. ∎

Our next purpose in this section is to show an algorithm that allows us to
build all the elements of $\mathscr{C}(m,A)$ that have a fixed genus. To do
that, we use the concept of rooted tree.

A _graph_ $G$ is a pair $(V,E)$ where $V$ is a non-empty set (whose elements
are called _vertices_ of $G$) and $E$ is a subset of $\\{(v,w)\in V\times
V\mid v\neq w\\}$ (whose elements are called _edges_ of $G$). A _path (of
length $n$) connecting the vertices $x$ and $y$ of $G$_ is a sequence of
different edges $(v_{0},v_{1}),(v_{1},v_{2}),\ldots,(v_{n-1},v_{n})$ such that
$v_{0}=x$ and $v_{n}=y$.

We say that a graph $G$ is a _(rooted) tree_ if there exists a vertex $r$
(known as the _root_ of $G$) such that, for any other vertex $x$ of $G$, there
exists a unique path connecting $x$ and $r$. If there exists a path connecting
$x$ and $y$, then we say that $x$ is a _descendant_ of $y$. In particular, if
$(x,y)$ is an edge of the tree, then we say that $x$ is a _child_ of $y$. (See
[12].)

We define the graph $\mathrm{G}\big{(}\mathscr{C}(m,A)\big{)}$ in the
following way: $\mathscr{C}(m,A)$ is the set of vertices and
$(S,S^{\prime})\in\mathscr{C}(m,A)\times\mathscr{C}(m,A)$ is an edge if
$S\cup\\{\mathrm{F}(S)\\}=S^{\prime}$. The following result is a consequence
of Lemma 4.2 and Theorem 4.3 of [9].

######  Theorem 2.7.

$\mathrm{G}\big{(}\mathscr{C}(m,A)\big{)}$ is a tree with root $\Delta(m)$.
Moreover, the children set of $S\in\mathscr{C}(m,A)$ is
$\big{\\{}S\setminus\\{x\\}\mid x\in\mathrm{msg}(S),\
S\setminus\\{x\\}\in\mathscr{C}(m,A),\ x>\mathrm{F}(S)\big{\\}}.$

Let $S$ be a numerical semigroup and let $x\in S$. Then it is clear that
$S\setminus\\{x\\}$ is a numerical semigroup if and only if
$x\in\mathrm{msg}(S)$. Moreover, let us observe that, if $x\in\mathrm{msg}(S)$
and $m\in S\setminus\\{0,x\\}$, then

$\mathrm{Ap}(S\setminus\\{x\\},m)=\big{(}\mathrm{Ap}(S,m)\setminus\\{x\\}\big{)}\cup\\{x+m\\}.$

In the following result we characterise the children of
$S\in\mathscr{C}(m,A)$.

######  Proposition 2.8.

Let $S\in\mathscr{C}(m,A)$, $\mathrm{Ap}(S,m)=\\{w(0),w(1),\ldots,w(m-1)\\}$
and $x\in\mathrm{msg}(S)\setminus\\{m\\}$. Then
$S\setminus\\{x\\}\in\mathscr{C}(m,A)$ if and only if
$w(i_{1})+\cdots+w(i_{k})\neq m+x$ for all $(i_{1},\ldots,i_{k})\in A$.

###### Proof.

Let us suppose that
$\mathrm{Ap}(S\setminus\\{x\\},m)=\big{(}\mathrm{Ap}(S,m)\setminus\\{x\\}\big{)}\cup\\{x+m\\}=\\{w^{\prime}(0),w^{\prime}(1),\ldots,w^{\prime}(m-1)\\}$.

(Necessity.) If $w(i_{1})+\cdots+w(i_{k})=m+x$, then
$m+x\notin\\{w(i_{1}),\ldots,w(i_{k})\\}$ and, therefore,
$w(i_{1})=w^{\prime}(i_{1}),\ldots,w(i_{k})=w^{\prime}(i_{k})$. Moreover,
$w(i_{1})+\cdots+w(i_{k})-m=x\notin S\setminus\\{x\\}$ and, consequently,
$S\setminus\\{x\\}\notin\mathscr{C}(m,A)$.

(Sufficiency.) In order to prove that $S\setminus\\{x\\}\in\mathscr{C}(m,A)$,
by Proposition 2.1, it is enough to see that, if $(i_{1},\ldots,i_{k})\in A$,
then $w^{\prime}(i_{1})+\cdots+w^{\prime}(i_{k})-m\in S\setminus\\{x\\}$.

Since $S\in\mathscr{C}(m,A)$, we easily deduce that
$w^{\prime}(i_{1})+\cdots+w^{\prime}(i_{k})-m\in S$. Now, if
$w^{\prime}(i_{1})+\cdots+w^{\prime}(i_{k})-m\notin S\setminus\\{x\\}$, then
$w^{\prime}(i_{1})+\cdots+w^{\prime}(i_{k})=m+x$. Thus,
$w(i_{1})=w^{\prime}(i_{1}),\ldots,w(i_{k})=w^{\prime}(i_{k})$ and
$w(i_{1})+\cdots+w(i_{k})=m+x$, where the last equality is in contradiction
with $S\in\mathscr{C}(m,A)$. ∎

Let us observe that a tree can be built in a recurrent way starting from its
root and connecting each vertex with its children. Let us also observe that
the elements of $\mathscr{C}(m,A)$ with genus equal to $g+1$ are precisely the
children of the elements of $\mathscr{C}(m,A)$ with genus equal to $g$.

We are ready to show the above announced algorithm.

######  Algorithm 2.9.

INPUT: A positive integer $g$.

OUTPUT: $\\{S\in\mathscr{C}(m,A)\mid\mathrm{g}(S)=g\\}$.

  * (1)

If $g<m-1$, return $\emptyset$.

  * (2)

$X=\\{\Delta(m)\\}$ and $i=m-1$.

  * (3)

If $i=g$, return $X$.

  * (4)

For each $S\in X$, compute the set

$\mathscr{B}_{S}=\big{\\{}x\in\mathrm{msg}(S)\mid x>\mathrm{F}(S),\ x\not=m,\
S\setminus\\{x\\}\in\mathscr{C}(m,A)\big{\\}}.$

  * (5)

If $\ \bigcup_{S\in X}\mathscr{B}_{S}=\emptyset$, return $\emptyset$.

  * (6)

$X=\bigcup_{S\in X}\big{\\{}S\setminus\\{x\\}\mid
x\in\mathscr{B}_{S}\big{\\}}$, $i=i+1$, and go to (3).

For computing item (4) in the above algorithm, we use Proposition 2.8 and the
following Lemma 2.10, which is a reformulation of Corollary 18 of [8], and is
useful to obtain the minimal system of generators of $S\setminus\\{x\\}$
starting from the minimal system of generators of $S$.

######  Lemma 2.10.

Let $S$ be a numerical semigroup with
$\mathrm{msg}(S)=\\{n_{1}<n_{2}<\cdots<n_{e}\\}$. If $i\in\\{2,\ldots,e\\}$
and $n_{i}>\mathrm{F}(S)$, then

$\mathrm{msg}(S\setminus\\{n_{i}\\})=\left\\{\begin{array}[]{l}\\{n_{1},\ldots,n_{e}\\}\setminus\\{n_{i}\\},\quad\mbox{if
there exists }j\in\\{2,\ldots,i-1\\}\\\ \hskip 96.73918pt\mbox{such that
}n_{i}+n_{1}-n_{j}\in S;\\\\[5.69054pt]
\big{(}\\{n_{1},\ldots,n_{e}\\}\setminus\\{n_{i}\\}\big{)}\cup\\{n_{i}+n_{1}\\},\quad\mbox{in
other case.}\end{array}\right.$

We illustrate the functioning of Algorithm 2.9 with an example.

######  Example 2.11.

Let us compute all the elements of $\mathscr{C}(5,\\{(1,1),(1,2)\\})$ with
genus equal to $6$.

  * •

$X=\\{\langle 5,6,7,8,9\rangle\\}$ and $i=4$.

  * •

$B_{\langle 5,6,7,8,9\rangle}=\\{6,9\\}$.

  * •

$X=\\{\langle 5,7,8,9,11\rangle,\langle 5,6,7,8\rangle\\}$ and $i=5$.

  * •

$B_{\langle 5,7,8,9,11\rangle}=\\{7,8,9,11\\}$ and $B_{\langle
5,6,7,8\rangle}=\emptyset$.

  * •

$X=\\{\langle 5,8,9,11,12\rangle,\langle 5,7,9,11,13\rangle,\langle
5,7,8,11\rangle,\langle 5,7,8,9\rangle\\}$ and $i=6$.

  * •

Return $\\{\langle 5,8,9,11,12\rangle,\langle 5,7,9,11,13\rangle,\langle
5,7,8,11\rangle,\langle 5,7,8,9\rangle\\}$.

Taking advantage of the above example, we finish this section building the
first three levels of the tree
$\mathrm{G}\big{(}\mathscr{C}(5,\\{(1,1),(1,2)\\})\big{)}$.

$\langle 5,6,7,8,9\rangle$69$\langle 5,7,8,9,11\rangle$$\langle
5,6,7,8\rangle$78911$\langle 5,8,9,11,12\rangle$$\langle
5,7,9,11,13\rangle$$\langle 5,7,8,11\rangle$$\langle 5,7,8,9\rangle$

Observe that the number which appear next to each edge $(S^{\prime},S)$ is the
minimal generator $x$ of $S$ such that $S^{\prime}=S\setminus\\{x\\}$.

##  3 Second-level numerical semigroups

We say that a numerical semigroup $S$ is of second-level if
$x+y+z-\mathrm{m}(S)\in S$ for all $(x,y,z)\in(S\setminus\\{0\\})^{3}$. We
denote by $\mathscr{L}_{2,m}$ the set of all the second-level numerical
semigroups with multiplicity equal to $m$.

######  Proposition 3.1.

Let $S$ be a numerical semigroup with minimal system of generators given by
$\\{m=n_{1}<n_{2}<\cdots<n_{e}\\}$. Then the following two conditions are
equivalents.

  1. 1.

$S\in\mathscr{L}_{2,m}$.

  2. 2.

If $(i,j,k)\in\\{2,\ldots,e\\}^{3}$, then $n_{i}+n_{j}+n_{k}-m\in S$.

###### Proof.

(1. $\Rightarrow$ 2.) It is evident from the definition of second-level
numerical semigroup.

(2. $\Rightarrow$ 1.) Let $(x,y,z)\in(S\setminus\\{0\\})^{3}$. If
$0\in\\{x\bmod m,y\bmod m,z\bmod m\\}$, then it is clear that $x+y+z-m\in S$.
Now, if $0\notin\\{x\bmod m,y\bmod m,z\bmod m\\}$, then we easily deduce that
there exist $(i,j,k)\in\\{2,\ldots,e\\}^{3}$ and $(s_{1},s_{2},s_{3})\in
S^{3}$ such that $(x,y,z)=(n_{i},n_{j},n_{k})+(s_{1},s_{2},s_{3})$. Therefore,
$x+y+z-m=(n_{i}+n_{j}+n_{k}-m)+s_{1}+s_{2}+s_{3}\in S$. Consequently,
$S\in\mathscr{L}_{2,m}$. ∎

The above proposition allows us to easily decide whether a numerical semigroup
is of second-level or not.

######  Example 3.2.

Let us see that $S=\langle 5,7,16\rangle\in\mathscr{L}_{2,m}$. In effect, it
is clear that
$\\{7+7+7-5,7+7+16-5,7+16+16,16+16+16-5\\}=\\{16,25,34,43\\}\subseteq S$.
Therefore, by Proposition 3.1, we have that $S\in\mathscr{L}_{2,m}$.

Now our intention is to show that $\mathscr{L}_{2,m}$ is a modular Frobenius
pseudo-variety.

######  Proposition 3.3.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$. Then
$\mathscr{L}_{2,m}=\mathscr{C}(m,\\{1,\ldots,m-1\\}^{3})$.

###### Proof.

Let $S\in\mathscr{L}_{2,m}$ and
$\mathrm{Ap}(S,m)=\\{w(0),w(1),\ldots,w(m-1)\\}$. If
$(i,j,k)\in\\{1,\ldots,m-1\\}^{3}$, then
$(w(i),w(j),w(k))\in(S\setminus\\{0\\})^{3}$ and, therefore,
$w(i)+w(j)+w(k)-m\in S$. By applying Proposition 2.1, we have that
$S\in\mathscr{C}(m,\\{1,\ldots,m-1\\}^{3})$.

In order to see the other inclusion, let
$S\in\mathscr{C}(m,\\{1,\ldots,m-1\\}^{3})$ and
$(x,y,z)\in(S\setminus\\{0\\})^{3}$. Firstly, if $0\in\\{x\bmod m,y\bmod
m,z\bmod m\\}$, then it is clear that $x+y+z-m\in S$. Secondly, if
$0\notin\\{x\bmod m,y\bmod m,z\bmod m\\}$, then there exist
$(i,j,k)\in\\{1,\ldots,m-1\\}^{3}$ and $(p,q,r)\in\mathbb{N}^{3}$ such that
$x=w(i)+pm$, $y=w(j)+qm$, and $z=w(k)+rm$. Thus,
$x+y+z-m=(w(i)+w(j)+w(k)-m)+(p+q+r)m\in S$ and, therefore,
$S\in\mathscr{L}_{2,m}$. ∎

As an immediate consequence of Propositions 2.6 and 3.3, we have the following
result.

######  Corollary 3.4.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$. Then $\mathscr{L}_{2,m}$ is a modular
pseudo-Frobenius variety and, in addition, $\Delta(m)$ is the maximum of
$\mathscr{L}_{2,m}$.

Our next step is to build the tree associated with the pseudo-variety
$\mathscr{L}_{2,m}$. To do this, we should characterise the possible children
of each element in $\mathscr{L}_{2,m}$.

Let $S$ be a numerical semigroup with
$\mathrm{msg}(S)=\\{n_{1},\ldots,n_{e}\\}$. If $s\in S$, then we denote by

$L_{S}(s)=\max\\{a_{1}+\cdots+a_{e}\mid(a_{1},\ldots,a_{e})\in\mathbb{N}^{e}\mbox{
and }a_{1}n_{1}+\cdots+a_{e}n_{e}=s\\}.$

######  Proposition 3.5.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$, $S\in\mathscr{L}_{2,m}$, and
$x\in\mathrm{msg}(S)\setminus\\{m\\}$. Then
$S\setminus\\{x\\}\in\mathscr{L}_{2,m}$ if and only if
$L_{S\setminus\\{x\\}}(x+m)\leq 2$.

###### Proof.

(Necessity.) Let us suppose that $L_{S\setminus\\{x\\}}(x+m)\geq 3$. Then
there exists $(a,b,c)\in(S\setminus\\{0,x\\})^{3}$ such that $x+m=a+b+c$.
Thus, $a+b+c-m=x\notin S\setminus\\{x\\}$ and, therefore,
$S\setminus\\{x\\}\notin\mathscr{L}_{2,m}$.

(Sufficiency.) If $(a,b,c)\in(S\setminus\\{0,x\\})^{3}$, then $a+b+c-m\in S$,
since $S\in\mathscr{L}_{2,m}$. Now, if $a+b+c-m=x$, then
$L_{S\setminus\\{x\\}}(x+m)\geq 3$. Therefore, $a+b+c-m\not=x$ and,
consequently, $a+b+c-m\in S\setminus\\{x\\}$. Thus, we conclude that
$S\setminus\\{x\\}\in\mathscr{L}_{2,m}$. ∎

By applying Theorem 2.7, Propositions 3.3 and 3.5, and Lemma 2.10, we can
easily build the tree $\mathrm{G}(\mathscr{L}_{2,m})$.

######  Example 3.6.

The first four levels of $\mathrm{G}(\mathscr{L}_{2,4})$ appear in the
following figure.

$\langle 4,5,6,7\rangle$567$\langle 4,6,7,9\rangle$$\langle
4,5,7\rangle$$\langle 4,5,6\rangle$6797$\langle 4,7,9,10\rangle$$\langle
4,6,9,11\rangle$$\langle 4,6,7\rangle$$\langle 4,5,11\rangle$7910911$\langle
4,9,10,11\rangle$$\langle 4,7,10,13\rangle$$\langle 4,7,9\rangle$$\langle
4,6,11,13\rangle$$\langle 4,6,9\rangle$

The Frobenius problem (see [7]) consists in finding formulas that allow us to
compute the Frobenius number and the genus of a numerical semigroup in terms
of the minimal system of generators of such a numerical semigroup. This
problem was solved in [15] for numerical semigroups with embedding dimension
two. At present, the Frobenius problem is open for embedding dimension greater
than or equal to $3$. However, if we know the Apéry set $\mathrm{Ap}(S,x)$ for
some $x\in S\setminus\\{0\\}$, then we have solved the Frobenius problem for
$S$ because we have the following result from [13].

######  Lemma 3.7.

Let $S$ be a numerical semigroup and let $x\in S\setminus\\{0\\}$. Then

  1. 1.

$\mathrm{F}(S)=(\max(\mathrm{Ap}(S,x)))-x$,

  2. 2.

$\mathrm{g}(S)=\frac{1}{x}(\sum_{w\in\mathrm{Ap}(S,x)}w)-\frac{x-1}{2}$.

The knowledge of $\mathrm{Ap}(S,x)=\\{w(0),w(1),\ldots,w(x-1)\\}$ also allows
us to determine the membership of an integer to the numerical semigroup $S$.
In fact, if $n\in\mathbb{N}$, then $n\in S$ if and only if $n\geq w(n\bmod
x)$.

Now our purpose is to show that, if $S\in\mathscr{L}_{2,m}$, then is rather
easy to compute $\mathrm{Ap}(S,m)$. We need the following easy result.

######  Lemma 3.8.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$, $S\in\mathscr{L}_{2,m}$,
$\mathrm{msg}(S)=\\{n_{1}\\!=\\!m,n_{2},\ldots,n_{e}\\}$. Then
$\\{0,n_{2},\ldots,n_{e}\\}\\!\subseteq\\!\mathrm{Ap}(S,m)\\!\subseteq\\!\\{0,n_{2},\ldots,n_{e}\\}\cup\\{n_{i}+n_{j}\mid(i,j)\in\\{2,\ldots,e\\}^{2}\\}.$

As an immediate consequence of the above lemma we can formulate the following
result.

######  Proposition 3.9.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$, $S\in\mathscr{L}_{2,m}$,
$\mathrm{msg}(S)=\\{n_{1}\\!=\\!m,n_{2},\ldots,n_{e}\\}$. Then
$\mathrm{Ap}(S,m)=\\{w(0),w(1),\ldots,w(m-1)\\}$ where $w(i)$ is the least
element of
$\\{0,n_{2},\ldots,n_{e}\\}\cup\\{n_{i}+n_{j}\mid(i,j)\in\\{2,\ldots,e\\}^{2}\\}$
that is congruent to $i$ modulo $m$.

######  Corollary 3.10.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$ and $S\in\mathscr{L}_{2,m}$. Then
$m=\mathrm{m}(S)\leq\frac{\mathrm{e}(S)\left(\mathrm{e}(S)+1\right)}{2}$.

Following the notation introduced in [10], we say that
$x\in\mathbb{Z}\setminus S$ is a pseudo-Frobenius number of $S$ if $x+s\in S$
for all $s\in S\setminus\\{0\\}$. We denote by $\mathrm{PF}(S)$ the set of all
the pseudo-Frobenius numbers of $S$. The cardinality of $\mathrm{PF}(S)$ is an
important invariant of $S$ (see [2]) that is the so-called type of $S$ and it
is denoted by $\mathrm{t}(S)$.

Let $S$ be a numerical semigroup. Then we define the following binary relation
over $\mathbb{Z}$: $a\leq_{S}b$ if $b-a\in S$. In [11] it is shown that
$\leq_{S}$ is a partial order (that is, reflexive, transitive, and
antisymmetric). Moreover, Proposition 2.20 of [11] is the next result.

######  Proposition 3.11.

Let $S$ be a numerical semigroup and $x\in S\setminus\\{0\\}$. Then

$\mathrm{PF}(S)=\\{w-x\mid
w\in\mathrm{Maximals}_{\leq_{S}}(\mathrm{Ap}(S,x))\\}$

Let us observe that, if $w,w^{\prime}\in\mathrm{Ap}(S,x)$, then
$w^{\prime}-w\in S$ if and only if $w^{\prime}-w\in\mathrm{Ap}(S,x)$.
Therefore, $\mathrm{Maximals}_{\leq_{S}}(\mathrm{Ap}(S,x))$ is the set

$\\{w\in\mathrm{Ap}(S,x)\mid
w^{\prime}-w\notin\mathrm{Ap}(S,x)\setminus\\{0\\}\mbox{ for all
}w^{\prime}\in\mathrm{Ap}(S,x)\\}.$

We finish this section with an example that illustrates the above results.

######  Example 3.12.

Having in mind that $S=\langle 5,6,13\rangle$ is a second-level numerical
semigroup, it is easy to compute $\mathrm{Ap}(S,5)$. In fact, by applying
Lemma 3.8, we have that
$\mathrm{Ap}(S,5)\subseteq\\{0,6,13\\}\cup\\{12,19,26\\}$ and, by Proposition
3.9, we conclude that $\mathrm{Ap}(S,5)=\\{0,6,12,13,19\\}$. On the other
hand, by Lemma 3.7, we know that $\mathrm{F}(S)=19-5=14$ and
$\mathrm{g}(S)=\frac{1}{5}(6+12+13+19)-\frac{5-1}{2}=8$. Finally, since
$\mathrm{Maximals}_{\leq_{S}}(\mathrm{Ap}(S,5))=\\{12,19\\}$, Proposition 3.11
allows us to claim that $\mathrm{PF}(S)=\\{7,12\\}$ and $\mathrm{t}(S)=2$.

######  Remark 3.13.

The definition of second-level numerical semigroup can be easily generalised
to greater levels. Thus, we say that a numerical semigroup is of $n$th-level
if $x_{1}+\cdots+x_{n+1}-\mathrm{m}(S)\in S$ for all
$(x_{1},\ldots,x_{n+1})\in(S\setminus\\{0\\})^{n+1}$ and denote by
$\mathscr{L}_{n,m}$ the set of all the $n$th-level numerical semigroups with
multiplicity equal to $m$.

It is clear that for $n$th-level we obtain similar results to those of second-
level. In particular,
$\mathscr{L}_{n,m}=\mathscr{C}(m,\\{1,\ldots,m-1\\}^{n+1})$ and Proposition
3.5 remains true taking $L_{S\setminus\\{x\\}}(x+m)\leq n$.

On the other hand, having in mind that $\mathscr{L}_{1,m}$ is the family of
numerical semigroups with maximal embedding dimension and that (by Remark 1.1)
$\mathscr{L}_{m-1,m}=\mathscr{S}_{m}$, we can observe that

$\mathscr{L}_{1,m}\subseteq\mathscr{L}_{2,m}\subseteq\cdots\subseteq\mathscr{L}_{m-1,m}=\mathscr{S}_{m},$

where all the inclusions are strict, as we can deduce from the following
example.

######  Example 3.14.

Let us set $m=5$ and

$S_{1}=\langle 5,6\rangle=\\{0,5,6,10,11,12,15,16,17,18,20,\to\\}.$

Then,

  * •

$S_{1}\in\mathscr{L}_{4,5}\setminus\mathscr{L}_{3,5}=\mathscr{S}_{5}\setminus\mathscr{L}_{3,5}$,
because $4\times 6-5\not\in S_{1}$ and $s_{1}+\cdots+s_{5}-5\geq 5\times
6-5\geq F(S_{1})+1$;

  * •

$S_{2}=S_{1}\cup\\{19\\}=\langle
5,6,19\rangle\in\mathscr{L}_{3,5}\setminus\mathscr{L}_{2,5}$, because $3\times
6-5\not\in S_{2}$ and $s_{1}+\cdots+s_{4}-5\geq 4\times 6-5\geq F(S_{2})+1$;

  * •

$S_{3}=S_{2}\cup\\{13,14\\}=\langle
5,6,13,14\rangle\in\mathscr{L}_{2,5}\setminus\mathscr{L}_{1,5}$, because
$2\times 6-5\not\in S_{2}$ and $s_{1}+s_{2}+s_{3}-5\geq 3\times 6-5\geq
F(S_{3})+1$.

Generalising this example to other values of $m$ is trivial.

######  Remark 3.15.

Let us observe that the chain obtained in Remark 3.13 is reminiscent, in some
sense, of the chain associated with subtraction patterns (see [3, Section 6]).

##  4 Thin numerical semigroups

We say that a numerical semigroup $S$ is thin if $2x-\mathrm{m}(S)\in S$ for
all $x\in S\setminus\\{0\\}$. We denote by $\mathscr{T}_{m}$ the set of all
the thin numerical semigroups with multiplicity equal to $m$.

######  Proposition 4.1.

Let $S$ be a numerical semigroup with minimal system of generators given by
$\\{m=n_{1}<n_{2}<\cdots<n_{e}\\}$. Then the following two conditions are
equivalents.

  1. 1.

$S\in\mathscr{T}_{m}$.

  2. 2.

$2n_{i}-m\in S$ for all $i\in\\{2,\ldots,e\\}$.

###### Proof.

(1. $\Rightarrow$ 2.) It follows from the definition of thin numerical
semigroup.

(2. $\Rightarrow$ 1.) Let $x\in S\setminus\\{0\\}$. If $x\equiv 0\pmod{m}$,
then it is clear that $2x-m\in S$. Now, if $x\not\equiv 0\pmod{m}$, then we
have that there exist $i\in\\{2,\ldots,e\\}$ and $s\in S$ such that
$x=n_{i}+s$. Therefore, $2x-m=(2n_{i}-m)+2s\in S$ and, consequently,
$S\in\mathscr{T}_{m}$. ∎

The above proposition allows us to easily decide whether a numerical semigroup
is thin or not.

######  Example 4.2.

Let us see that $S=\langle 4,6,7\rangle\in\mathscr{T}_{4}$. In effect, it is
clear that $\\{2\cdot 6-4,2\cdot 7-4\\}=\\{8,10\\}\subseteq S$. Therefore, by
Proposition 4.1, we have that $S\in\mathscr{T}_{m}$.

Now we want to show that $\mathscr{T}_{m}$ is a modular Frobenius pseudo-
variety.

######  Proposition 4.3.

$\mathscr{T}_{m}=\mathscr{C}(m,\\{(1,1),(2,2)\ldots,(m-1,m-1)\\})$ for all
$m\in\mathbb{N}\setminus\\{0,1\\}$.

###### Proof.

If $S\in\mathscr{T}_{m}$ and $\mathrm{Ap}(S,m)=\\{w(0),w(1),\ldots,w(m-1)\\}$,
then it is clear that $\\{w(1)+w(1)-m,\ldots,w(m-1)+w(m-1)-m\\}\subseteq S$.
Therefore, by applying Proposition 2.1, we have that
$S\in\mathscr{C}(m,\\{(1,1),(2,2)\ldots,(m-1,m-1)\\})$.

To see the other inclusion, let
$S\in\mathscr{C}(m,\\{(1,1),(2,2)\ldots,(m-1,m-1)\\})$ and $x\in
S\setminus\\{0\\}$. On the one hand, if $x\equiv 0\pmod{m}$, then it is clear
that $2x-m\in S$. On the other hand, if $x\not\equiv 0\pmod{m}$, then there
exist $i\in\\{1,\ldots,m-1\\}$ and $t\in\mathbb{N}$ such that $x=w(i)+tm$.
Therefore, $2x-m=(w(i)+w(i)-m)+2tm\in S$ and in consequence
$S\in\mathscr{T}_{m}$. ∎

From Propositions 2.6 and 4.3, we get the following result.

######  Corollary 4.4.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$. Then $\mathscr{T}_{m}$ is a modular
pseudo-Frobenius variety and, in addition, $\Delta(m)$ is the maximum of
$S\in\mathscr{T}_{m}$.

In order to build the tree associated with the pseudo-variety
$S\in\mathscr{T}_{m}$, we are going to characterise the possible children of
each $S\in\mathscr{T}_{m}$.

######  Proposition 4.5.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$, $S\in\mathscr{T}_{m}$, and
$x\in\mathrm{msg}(S)\setminus\\{m\\}$. Then
$S\setminus\\{x\\}\in\mathscr{T}_{m}$ if and only if $\frac{x+m}{2}\notin S$.

###### Proof.

(Necessity.) If $\frac{x+m}{2}\in S$, then $\frac{x+m}{2}\in
S\setminus\\{0,x\\}$ and $2\frac{x+m}{2}-m=x\notin S\setminus\\{x\\}$.
Therefore, $S\setminus\\{x\\}\notin\mathscr{T}_{m}$.

(Sufficiency.) If $a\in S\setminus\\{0,x\\}$, then $2a-m\in S$ because
$S\in\mathscr{T}_{m}$. Now, if $2a-m=x$, then $\frac{x+m}{2}=a\in S$.
Therefore, $2a-m\not=x$ and, consequently, $2a-m\in S\setminus\\{x\\}$.
Thereby, $S\setminus\\{x\\}\in\mathscr{T}_{m}$. ∎

By applying Theorem 2.7, Propositions 4.3 and 4.5, and Lemma 2.10, we can
build the tree $\mathrm{G}(\mathscr{T}_{m})$. Let us see an example.

######  Example 4.6.

In the next figure we have the first four levels of
$\mathrm{G}(\mathscr{T}_{4})$.

$\langle 4,5,6,7\rangle$57$\langle 4,6,7,9\rangle$$\langle
4,5,6\rangle$679$\langle 4,7,9,10\rangle$$\langle 4,6,9,11\rangle$$\langle
4,6,7\rangle$79911$\langle 4,9,10,11\rangle$$\langle 4,7,10,13\rangle$$\langle
4,6,11,13\rangle$$\langle 4,6,9\rangle$

We finish this section describing the Apéry set for $S\in\mathscr{T}_{m}$.

######  Proposition 4.7.

Let $m\in\mathbb{N}\setminus\\{0\\}$, $S\in\mathscr{T}_{m}$,
$\mathrm{msg}(S)=\\{m=n_{1},n_{2},\ldots,n_{e}\\}$. Then
$\mathrm{Ap}(S,m)=\\{w(0),w(1),\ldots,w(m-1)\\}$, where $w(i)$ is the least
element of the set
$\\{a_{2}n_{2}+\ldots+a_{e}n_{e}\mid(a_{2},\ldots,a_{e})\in\\{0,1\\}^{e-1}\\}$
that is congruent to $i$ modulo $m$.

######  Corollary 4.8.

Let $m\in\mathbb{N}\setminus\\{0\\}$ and $S\in\mathscr{T}_{m}$. Then
$m=\mathrm{m}(S)\leq 2^{\mathrm{e}(S)-1}$.

######  Remark 4.9.

Let us observe that we can generalise the concept of thin numerical semigroups
in the following way: setting $n\in\mathbb{N}\setminus\\{0,1\\}$, we say that
a numerical semigroup $S$ is $n$-thin if $nx-\mathrm{m}(S)\in S$ for all $x\in
S\setminus\\{0\\}$, and denote by $\mathscr{T}_{n,m}$ the set of all $k$-thin
numerical semigroups with multiplicity equal to $m$. It is clear that
$\mathscr{T}_{m}=\mathscr{T}_{2,m}$.

Again, $\mathscr{T}_{n,m}$ is a modular pseudo-variety, Proposition 4.5 is
valid for the condition $\frac{x+m}{n}\not\in S$, and we can build the chain

$\mathscr{T}_{2,m}\subsetneq\mathscr{T}_{3,m}\subsetneq\cdots\subsetneq\mathscr{T}_{m,m}=\mathscr{S}_{m}.$

Note that Example 3.14 also gives us the construction that ensures the strict
inclusions in this case.

##  5 Strong numerical semigroups

We say that a numerical semigroup $S$ is strong if $x+y-\mathrm{m}(S)\in S$
for all $(x,y)\in(S\setminus\\{0\\})^{2}$ such that $x\not\equiv
y\pmod{\mathrm{m}(S)}$. We denote by $\mathscr{R}_{m}$ the set of all the
strong numerical semigroups with multiplicity equal to $m$.

######  Proposition 5.1.

Let $S$ be a numerical semigroup with minimal system of generators given by
$\\{m=n_{1}<n_{2}<\cdots<n_{e}\\}$. Then the following two conditions are
equivalents.

  1. 1.

$S\in\mathscr{R}_{m}$.

  2. 2.

$\\{n_{i}+n_{j}-m,3n_{i}-m\\}\subseteq S$ for all $i\in\\{2,\ldots,e\\}$ and
for all $(i,j)\in\\{2,\ldots,e\\}^{2}$ such that $i\not=j$.

###### Proof.

(1. $\Rightarrow$ 2.) It is enough to observe that $n_{i}\not\equiv
n_{j}\pmod{m}$ and that $2n_{i}\not\equiv n_{i}\pmod{m}$.

(2. $\Rightarrow$ 1.) Let $x,y\in S\setminus\\{0\\}$ such that $x\not\equiv
y\pmod{\mathrm{m}}$. If $x\equiv 0\pmod{m}$ or $y\equiv 0\pmod{m}$, then it is
clear that $x+y-m\in S$. Now, if $x\not\equiv 0\pmod{m}$ and $y\not\equiv
0\pmod{m}$, then there exists $(i,j)\in\\{1,\ldots,m-1\\}^{2}$, with
$i\not=j$, and there exists $(p,q)\in\mathbb{N}^{2}$ such that $x=w(i)+pm$ and
$y=w(j)+qm$. Moreover, if there exists $(a,b)\in\\{2,\ldots,e\\}^{2}$ such
that $a\not=b$ and $w(i)-n_{a},w(j)-n_{b}\in S$, then it is easy to see that
$x+y-m\in S$. In other case, there exists
$(r,t)\in(\mathbb{N}\setminus\\{0\\})^{2}$ such that $w(i)=r\cdot n_{a}$ and
$w(j)=t\cdot n_{a}$ for some $a\in\\{2,\ldots,e\\}$. Then, since $i\not=j$, we
deduce that $r+t\geq 3$ and, consequently, $x+y-m\in S$. In conclusion,
$S\in\mathscr{R}_{m}$. ∎

The above proposition allows us to easily decide whether a numerical semigroup
is strong or not.

######  Example 5.2.

If $S=\langle 4,5,7\rangle$, then $\\{5+7-4,3\cdot 5-4,3\cdot
7-4\\}=\\{8,11,17\\}\subseteq S$. Therefore, by Proposition 5.1, we have that
$S\in\mathscr{R}_{m}$.

Now we want to show that $\mathscr{R}_{m}$ is a modular Frobenius pseudo-
variety. Let us denote by
$A=\\{1,\ldots,m-1\\}^{2}\setminus\\{(1,1),(2,2),\ldots,(m-1,m-1)\\}$.

######  Proposition 5.3.

$\mathscr{R}_{m}=\mathscr{C}(m,A)$ for all $m\in\mathbb{N}\setminus\\{0,1\\}$.

###### Proof.

Let $S\in\mathscr{R}_{m}$, $\mathrm{Ap}(S,m)=\\{w(0),w(1),\ldots,w(m-1)\\}$,
and $(i,j)\in A$. Then $\\{(w(i),w(j)\\}\in(S\setminus\\{0\\})^{2}$ and
$w(i)\not\equiv w(j)\pmod{m}$. Therefore, $w(i)-w(j)-m\in S$ and, by applying
Proposition 2.1, we have that $S\in\mathscr{C}(m,A)$.

To see the other inclusion, let $S\in\mathscr{C}(m,A)$ and
$(x,y)\in(S\setminus\\{0\\})^{2}$. On the one hand, if $0\in\\{x\bmod m,y\bmod
m\\}$, then it is clear that $x+y-m\in S$. On the other hand, if
$0\notin\\{x\bmod m,y\bmod m\\}$, then there exist $(i,j)\in A$ and
$(p,q)\in\mathbb{N}^{2}$ such that $x=w(i)+pm$ and $y=w(j)+qm$. Therefore,
$x+y-m=(w(i)+w(i)-m)+(p+q)m\in S$ and, consequently, $S\in\mathscr{R}_{m}$. ∎

From Propositions 2.6 and 5.3, we get the following result.

######  Corollary 5.4.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$. Then $\mathscr{R}_{m}$ is a modular
pseudo-Frobenius variety and, in addition, $\Delta(m)$ is the maximum of
$S\in\mathscr{R}_{m}$.

We are now interested in the description of the tree associated with the
pseudo-variety $S\in\mathscr{R}_{m}$. In order to do that, we are going to
characterise the children of an arbitrary $S\in\mathscr{R}_{m}$.

######  Proposition 5.5.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$, $S\in\mathscr{R}_{m}$, and
$x\in\mathrm{msg}(S)\setminus\\{m\\}$. Then
$S\setminus\\{x\\}\in\mathscr{R}_{m}$ if and only if $x+m\notin\\{a+b\mid
a,b\in\mathrm{msg}(S)\setminus\\{m,x\\},\;a\not=b\\}\cup\\{3a\mid
a\in\mathrm{msg}(S)\setminus\\{m,x\\}\\}$.

###### Proof.

(Necessity.) If $a,b\in\mathrm{msg}(S)\setminus\\{m,x\\}$ and $a\not=b$, then
$(a,b)\in(S\setminus\\{0,x\\})^{2}$ and $a\not\equiv b\pmod{m}$. Since
$S\setminus\\{x\\}\in\mathscr{R}_{m}$, we have that $a+b-m\in
S\setminus\\{x\\}$ and, therefore, $a+b-x\not=x$. Thus, $x+m\notin\\{a+b\mid
a,b\in\mathrm{msg}(S)\setminus\\{m,x\\},\;a\not=b\\}$.

On the other hand, if $a\in\mathrm{msg}(S)\setminus\\{m,x\\}$, then
$(a,2a)\in(S\setminus\\{0,x\\})^{2}$ and $a\not\equiv 2a\pmod{m}$. Once again,
since $S\setminus\\{x\\}\in\mathscr{R}_{m}$, we have that $3a-m\in
S\setminus\\{x\\}$ and, therefore, $a+b-x\not=x$. Thus, $x+m\notin\\{3a\mid
a\in\mathrm{msg}(S)\setminus\\{m,x\\}\\}$.

(Sufficiency.) Let $a,b\in S\setminus\\{0,x\\}$ such that $a\not=b$. Since
$S\in\mathscr{R}_{m}$, we have that $a+b-m\in S$ and $3a-m\in S$. Now, by
Lemma 2.10, we know that
$\mathrm{msg}(S)\setminus\\{x\\}\subseteq\mathrm{msg}(S\setminus\\{x\\})\subseteq(\mathrm{msg}(S)\setminus\\{x\\})\cup\\{x+m\\}$.
Thus, from this fact and the hypothesis, it easily follows that $a+b-m\not=x$
and $3a-m\not=x$, that is, $a+b-m,3a-m\in S\setminus\\{x\\}$. By applying
Proposition 5.1, we conclude that $S\setminus\\{x\\}\in\mathscr{R}_{m}$. ∎

From Theorem 2.7, Propositions 5.3 and 5.5, and Lemma 2.10, we can build the
tree $\mathrm{G}(\mathscr{R}_{m})$. Let us see an example.

######  Example 5.6.

In the next figure we have the first four levels of
$\mathrm{G}(\mathscr{R}_{4})$.

$\langle 4,5,6,7\rangle$56$\langle 4,6,7,9\rangle$$\langle
4,5,7\rangle$677$\langle 4,7,9,10\rangle$$\langle 4,6,9,11\rangle$$\langle
4,5,11\rangle$79109$\langle 4,9,10,11\rangle$$\langle
4,7,10,13\rangle$$\langle 4,7,9\rangle$$\langle 4,6,11,13\rangle$

We finish this section describing the Apéry set for $S\in\mathscr{R}_{m}$.

######  Proposition 5.7.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$, $S\in\mathscr{R}_{m}$,
$\mathrm{msg}(S)=\\{m=n_{1},n_{2},\ldots,n_{e}\\}$. Then
$\mathrm{Ap}(S)=\\{w(0),w(1),\ldots,w(n)\\}$, where $w(i)$ is the least
element of the set $\\{0,n_{2},\ldots,n_{e}\\}\cup\\{2n_{2},\ldots,2n_{e}\\}$
that is congruent to $i$ modulo $m$.

######  Corollary 5.8.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$ and $S\in\mathscr{R}_{m}$. Then
$m=\mathrm{m}(S)\leq 2\mathrm{e}(S)-1$.

It is interesting to observe that, under the hypotheses of Proposition 5.7,
$n_{i}\in\mathrm{Maximals}_{\leq_{S}}(\mathrm{Ap}(S,m))$ if and only if
$2n_{i}\notin\mathrm{Ap}(S,m)$. In addition, $2n_{i}\in\mathrm{Ap}(S,m)$ if
and only if $2n_{i}\in\mathrm{Maximals}_{\leq_{S}}(\mathrm{Ap}(S,m))$.
Therefore, from Propositions 3.11 and 5.7, we get the next result.

######  Corollary 5.9.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$ and $S\in\mathscr{R}_{m}$. Then
$\mathrm{t}(S)=\mathrm{e}(S)-1$.

It is well known that, if $S$ is a numerical semigroup, then
$\mathrm{e}(S)\leq\mathrm{m}(S)$ and $\mathrm{t}(S)\leq\mathrm{m}(S)-1$ (see
Proposition 2.10 and Corollary 2.23 in [11]). Combining these facts with
Corollaries 5.8 and 5.9, we have the next result.

######  Corollary 5.10.

Let $m\in\mathbb{N}\setminus\\{0,1\\}$ and $S\in\mathscr{R}_{m}$. Then
$\frac{\mathrm{m}(S)-1}{2}\leq\mathrm{t}(S)\leq\mathrm{m}(S)-1$ (or,
equivalently, $\frac{\mathrm{m}(S)+1}{2}\leq\mathrm{e}(S)\leq\mathrm{m}(S)$).

######  Remark 5.11.

Contrary to what happens in Sections 3 and 4, the concept of strong numerical
semigroup does not have a natural generalisation. In fact, we have, at least,
two possibilities.

  1. 1.

$x_{1}+\cdots x_{n}-\mathrm{m}(S)\in S$ for all
$(x_{1},\ldots,x_{n})\in(S\setminus\\{0\\})^{n}$ such that $x_{i}\not\equiv
x_{j}\pmod{\mathrm{m}(S)}$, for all $i\not=j$. In this case,
$(x_{1},\ldots,x_{n})\in A_{n}$, where

$A_{n}=\left\\{(\alpha_{1},\ldots,\alpha_{n})\in\\{1,\ldots,m-1\\}^{n}\mid\alpha_{i}\not=\alpha_{j}\mbox{
for all }i\not=j\right\\}.$

  2. 2.

$x_{1}+\cdots x_{n}-\mathrm{m}(S)\in S$ for all $(x_{1},\ldots,x_{n})\in
B_{n}$, where

$B_{n}=\\{1,\ldots,m-1\\}^{n}\setminus\\{(1,\ldots,1),(2,\ldots,2),\ldots,(m-1,\ldots,m-1)\\}.$

It is clear that, if $n\geq 3$, then $A_{n}\subsetneq B_{n}$ and, in
consequence, $\mathscr{C}(m,B_{n})\subseteq\mathscr{C}(m,A_{n})$. In addition,
it is guessed that, if $n<m$, then the second inclusion will be strict.

## Acknowledgement

This version of the article has been accepted for publication, after peer
review but is not the Version of Record and does not reflect post-acceptance
improvements, or any corrections. The Version of Record is available online
at: http://dx.doi.org/10.1007/s13348-021-00339-0.

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