The covariety of perfect numerical semigroups with fixed Frobenius number

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 ${\mathscr{C}}$ 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 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 ${\mathscr{P}}(F),$ the maximal elements of ${\mathscr{P}}(F)$ and the elements of ${\mathscr{P}}(F)$ 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 ${\mathrm{Psat}}(F)$.Comment: arXiv admin note: text overlap with arXiv:2302.09121, arXiv:2303.12470, arXiv:2305.02070, arXiv:2305.1388

The covariety of saturated numerical semigroups with fixed Frobenius number

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.Comment: arXiv admin note: text overlap with arXiv:2303.12470, arXiv:2305.0207