The covariety of saturated numerical semigroups with fixed Frobenius number

TL;DR

This paper studies the structure of saturated numerical semigroups with a fixed Frobenius number, showing they form a covariety, and introduces algorithms to compute these semigroups and their elements based on genus and rank.

Contribution

It proves that the set of saturated numerical semigroups with a fixed Frobenius number forms a covariety and provides algorithms for their enumeration and rank-based classification.

Findings

${ m Sat}(F)$ is a covariety for fixed Frobenius number $F$.

Algorithms are developed to compute all elements of ${ m Sat}(F)$.

Methods to determine elements with a given ${ m Sat}(F)$-rank.

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.