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Keywords:
perfect numerical semigroup; saturated numerical semigroup; Arf numerical semigroup; covariety; Frobenius number; genus; algorithm
perfect numerical semigroup; saturated numerical semigroup; Arf numerical semigroup; covariety; Frobenius number; genus; algorithm
Summary:
Let $S$ be a numerical semigroup. We say that $h\in \mathbb {N} \backslash S$ is an isolated gap of $S$ if $\{h-1,h+1\}\subseteq S.$ A numerical semigroup without isolated gaps is called a perfect numerical semigroup. Denote by ${\rm m} (S)$ the multiplicity of a numerical semigroup $S$. A covariety is a nonempty family $\scr {C}$ of numerical semigroups that fulfills the following conditions: there exists the minimum of $\scr {C},$ the intersection of two elements of $\scr {C}$ is again an element of $\scr {C}$, and $S\backslash \{{\rm m}(S)\}\in \scr {C}$ for all $S\in \scr {C}$ such that $S\neq \min (\scr {C}).$ We prove that the set $\scr {P}(F)=\{S\colon S$ is a perfect numerical semigroup with Frobenius number $F\}$ is a covariety. Also, we describe three algorithms which compute: the set $\scr {P}(F),$ the maximal elements of $\scr {P}(F)$, and the elements of $\scr {P}(F)$ with a given genus. A ${\rm Parf}$-semigroup (or ${\rm Psat}$-semigroup) is a perfect numerical semigroup that in addition is an Arf numerical semigroup (or saturated numerical semigroup), respectively. We prove that the sets ${\rm Parf}(F)=\{S\colon S$ is a ${\rm Parf}$-numerical semigroup with Frobenius number $F\}$ and ${\rm Psat}(F)=\{S\colon S$ is a ${\rm Psat}$-numerical semigroup with Frobenius number $F\}$ are covarieties. As a consequence we present some algorithms to compute ${\rm Parf}(F)$ and ${\rm Psat}(F).$
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