Modular Frobenius pseudo-varieties

TL;DR

This paper proves that certain classes of numerical semigroups form Frobenius pseudo-varieties, introduces algorithms for their classification, and studies three specific families of these semigroups with particular properties.

Contribution

It establishes that the classes (m,A) are Frobenius pseudo-varieties and provides algorithms to identify and enumerate their elements, also defining three special families of semigroups.

Findings

(m,A) are Frobenius pseudo-varieties

Algorithms for membership and enumeration within (m,A)

Introduction of second-level, thin, and strong semigroup families

Abstract

If $m \in \mathbb{N}$ and $A$ is a finite subset of $\bigcup_{k \in \mathbb{N} \setminus \{0,1\}} \{1,\ldots,m-1\}^k$, then we denote by \begin{align*} \mathscr{C}(m,A) = \left\{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 }\right. \\ \left.(s_1 \bmod m, \ldots, s_k \bmod m)\in A \right\}. \end{align*} 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.