Related papers: Symmetric (not Complete Intersection) Semigroups G…
In this work we will show that if $F$ is a positive integer, then the set ${\mathrm{Arf}}(F)=\{S\mid S \mbox{ is an Arf numerical semigroup with Frobenius number } F\}$ verifies the following conditions: 1) $\Delta(F)=\{0,F+1,\rightarrow\}$…
We introduce a module-theoretic approach and a linear-programming method to compute the matricial dimension of numerical semigroups. We use these to compute the matricial dimension of every numerical semigroup with Frobenius number at most…
Given a numerical semigroup $S$ and a positive integer $p$, the quotient $\frac{S}{p}=\{x\in \mathbb{N} \mid px\in S\}$ also forms a numerical semigroup. In this paper, we first characterize the Ap\'ery set for a class of quotients of…
The pseudo-Frobenius numbers of a numerical semigroup are those gaps of the numerical semigroup that are maximal for the partial order induced by the semigroup. We present a procedure to detect if a given set of integers is the set of…
A numerical semigroup is an additive subsemigroup of the non-negative integers. In this paper, we consider parametrized families of numerical semigroups of the form $P_n = \langle f_1(n), \ldots, f_k(n) \rangle$ for polynomial functions…
In this paper we study the regular semigroups weakly generated by a single element x, that is, with no proper regular subsemigroup containing x. We show there exists a regular semigroup $F_1$ weakly generated by x such that all other…
Denote by $\mathrm m(S)$ the multiplicity of a numerical semigroup $S$. A covariety is a nonempty family $\mathscr{C}$ of numerical semigroups that fulfills the following conditions: there is the minimum of $\mathscr{C},$ the intersection…
In this paper we study numerical semigroups containing a given positive integer and closed with respect to the action of an affine map. For such semigroups we find a minimal set of generators, their embedding dimension, their genus and…
Fixed two positive integers m and e, some algorithms for computing the minimal Frobenius number and minimal genus of the set of numerical semigroups with multiplicity m and embedding dimension e are provided. Besides, the semigroups where…
We study almost symmetric numerical semigroups and semigroup rings. We describe a characteristic property of the minimal free resolution of the semigroup ring of an almost symmetric numerical semigroup. For almost symmetric semigroups…
Let $S,T$ be two numerical semigroups. We study when $S$ is one half of $T$, with $T$ almost symmetric. If we assume that the type of $T$, $t(T)$, is odd, then for any $S$ there exist infinitely many such $T$ and we prove that $1 \leq t(T)…
In this paper we present a procedure to build the set of irreducible numerical semigroups with a fixed Frobenius number. The construction gives us a rooted tree structure for this set. Furthermore, by using the notion of Kunz-coordinates…
This paper presents a new methodology to count the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius…
Let n_g denote the number of numerical semigroups of genus g. Bras-Amoros conjectured that n_g possesses certain Fibonacci-like properties. Almost all previous attempts at proving this conjecture were based on analyzing the semigroup tree.…
We study statistical properties of random numerical semigroups of a given genus. We analyze the graph of a typical numerical semigroup, understood as a function from $\mathbb{N}$ to $\mathbb{N}$. If $S$ is a numerical semigroup of genus…
For a numerical semigroup, we encode the set of primitive elements that are larger than its Frobenius number and show how to produce in a fast way the corresponding sets for its children in the semigroup tree. This allows us to present an…
Let $\mathcal{C}\subseteq \mathbb{N}^p$ be an integer cone. A $\mathcal{C}$-semigroup $S\subseteq \mathcal{C}$ is an affine semigroup such that the set $\mathcal{C}\setminus S$ is finite. Such $\mathcal{C}$-semigroups are central to our…
We examine properties of random numerical semigroups under a probabilistic model inspired by the Erdos-Renyi model for random graphs. We provide a threshold function for cofiniteness, and bound the expected embedding dimension, genus, and…
A generalised numerical semigroup (GNS) is a submonoid $S$ of $\mathbb{N}^d$ for which the complement $\mathbb{N}^d\setminus S$ is finite. The points in the complement $\mathbb{N}^d\setminus S$ are called gaps. A gap $F$ is considered…
This paper is a continuation of the paper "Numerical Semigroups: Ap\'ery Sets and Hilbert Series". We consider the general numerical AA-semigroup, i.e., semigroups consisting of all non-negative integer linear combinations of relatively…