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

Combinatorial properties and characterization of glued semigroups

This work focuses on the combinatorial properties of glued semigroups and provides its combinatorial characterization. Some classical results for affine glued semigroups are generalized and some methods to obtain glued semigroups are developed.Comment: 13 pages, 6 figures, This paper was initially presented at the "Centennial Congress RSME2011" http://campus.usal.es/~rsme2011