The covariety of perfect numerical semigroups with fixed Frobenius number

Abstract

Let $S$ be a numerical semigroup. We will say that $h\in \mathbb{N}\backslash S$ is an {\it isolated gap }of $S$ if $\{h-1,h+1\}\subseteq S.$ A numerical semigroup without isolated gaps is called perfect numerical semigroup. 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 prove that the set ${\mathscr{P}}(F)=\{S\mid S \mbox{ is a perfect numerical}\ \mbox{semigroup with Frobenius number }F\}$ is a covariety. Also, we describe three algorithms which compute: the set $\mathcal{P}(F),$ the maximal elements of $\mathcal{P}(F)$ and the elements of $\mathcal{P}(F)$ with a given genus. A $\mathrm{Parf}$-semigroup (respectively, $\mathrm{Psat}$-semigroup) is a perfect numerical semigroup that in addition is an Arf numerical semigroup (respectively, saturated numerical semigroup). We will prove that the sets: ${\mathrm{Parf}}(F)=\{S\mid S \mbox{ is a ${\mathrm{Parf}}$-numerical semigroup with Frobenius number} F\}$ and ${\mathrm{Psat}}(F)=\{S\mid S \mbox{ is a ${\mathrm{Psat}}$-numerical semigroup with Frobenius number } F\}$ are covarieties. As a consequence we present some algorithms to compute $\mathrm{Parf}(F)$ and $\mathrm{Psat}(F)$.

arXiv admin note: text overlap with arXiv:2302.09121, arXiv:2303.12470, arXiv:2305.02070, arXiv:2305.13881