The covariety of numerical semigroups with fixed Frobenius number

Source abstract

Abstract Denote by $${\mathrm m}(S)$$ m ( S ) the multiplicity of a numerical semigroup S . A covariety is a nonempty family $$\mathscr {C}$$ C of numerical semigroups that fulfils the following conditions: there is the minimum of $$\mathscr {C},$$ C , the intersection of two elements of $$\mathscr {C}$$ C is again an element of $$\mathscr {C}$$ C and $$S\backslash \{{\mathrm m}(S)\}\in \mathscr {C}$$ S \ { m ( S ) } ∈ C for all $$S\in \mathscr {C}$$ S ∈ C such that $$S\ne \min (\mathscr {C}).$$ S ≠ min ( C ) . In this work we describe an algorithmic procedure to compute all the elements of $$\mathscr {C}.$$ C . We prove that there exists the smallest element of $$\mathscr {C}$$ C containing a set of positive integers. We show that $$\mathscr {A}(F)=\{S\mid S \hbox { is a numerical semigroup with Frobenius number }F\}$$ A ( F ) = { S ∣ S 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.