The covariety of numerical semigroups with fixed Frobenius number

Abstract

Denote by \({\mathrm m}(S)\) the multiplicity of a numerical semigroup S. A covariety is a nonempty family \(\mathscr {C}\) of numerical semigroups that fulfils 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 \(S\ne \min (\mathscr {C}).\) In this work we describe an algorithmic procedure to compute all the elements of \(\mathscr {C}.\) We prove that there exists the smallest element of \(\mathscr {C}\) containing a set of positive integers. We show that \(\mathscr {A}(F)=\{S\mid S \hbox { is a numerical semigroup with Frobenius number }F\}\) is a covariety, and we particularize the previous results in this covariety. Finally, we will see that there is the smallest covariety containing a finite set of numerical semigroups.