The covariety of saturated numerical semigroups with fixed Frobenius number

Abstract

In this work we will show that if $F$ is a positive integer, then ${\mathrm{Sat}}(F)=\{S\mid S \mbox{ is a saturated numerical semigroup with Frobenius number } F\}$ is a covariety. As a consequence, we present two algorithms: one that computes $\mathrm{Sat}(F),$ and the other which computes all the elements of $\mathrm{Sat}(F)$ with a fixed genus. If $X\subseteq S\backslash \Delta (F)$ for some $S\in \mathrm{Sat}(F),$ then we will see that there is the least element of $\mathrm{Sat}(F)$ containing a $X$. This element will denote by $\mathrm{Sat}(F)[X].$ If $S\in \mathrm{Sat}(F),$ then we define the $\mathrm{Sat}(F)$-rank of $S$ as the minimum of $\{\mbox{cardinality}(X)\mid S={\mathrm{Sat}}(F)[X]\}.$ In this paper, also we present an algorithm to compute all the element of $\mathrm{Sat}(F)$ with a given $\mathrm{Sat}(F)$-rank.

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