The covariety of numerical semigroups with fixed Frobenius number

Abstract

Denote by $\mathrm{m}(S)$ the multiplicity of a numerical semigroup $S$. A covariety is a nonempty family $\mathcal{C}$ of numerical semigroups that fulfills the following conditions: there is the minimum of $\mathcal{C},$ the intersection of two elements of $\mathcal{C}$ is again an element of $\mathcal{C}$ and $S\backslash \{\mathrm{m}(S)\}\in \mathcal{C}$ for all $S\in \mathcal{C}$ such that $S\ne \min (\mathcal{C}).$ In this work we describe an algorithmic procedure to compute all the elements of $\mathcal{C}.$ We prove that there exists the smallest element of $\mathcal{C}$ 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.

arXiv comment13 pages