The covariety of numerical semigroups with fixed Frobenius number

TL;DR

This paper studies covarieties of numerical semigroups with fixed Frobenius number, providing algorithms to compute them, and establishing their minimal elements and structural properties.

Contribution

It introduces an algorithmic method to compute covarieties of numerical semigroups, characterizes their minimal elements, and applies these results to families with fixed Frobenius number.

Findings

Existence of smallest covariety containing a given set

Algorithm to compute all elements of a covariety

Identification of the covariety of semigroups with fixed Frobenius number

Abstract

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 of two elements of $\mathscr{C}$ is again an element of $\mathscr{C}$ and $S\backslash \{\mathrm m(S)\}\in \mathscr{C}$ for all $S\in \mathscr{C}$ such that $S\neq \min(\mathscr{C}).$ In this work we describe an algorithmic procedure to compute all the elements of $\mathscr{C}.$ We prove that there exists the smallest element of $\mathscr{C}$ containing a set of positive integers. We show that $\mathscr{A}(F)=\{S\mid S \mbox{ is a numerical semigroup with Frobenius number }F\}$ is a covariety, and we particularize the previous results in this covariety. Finally, we will see that there is the smallest covariety containing a finite set of numerical semigroups.