The covariety of numerical semigroups with fixed Frobenius number

Abstract

Denote by m(S) themultiplicity of a numerical semigroup S.A covariety is a nonempty family C of numerical semigroups that fulfils the following conditions: there is the minimum of C , the intersection of two elements of C is again an element of C and S\{m(S)} ∈ C for all S ∈ C such that S = min(C ). In this work we describe an algorithmic procedure to compute all the elements of C . We prove that there exists the smallest element of C containing a set of positive integers. We show that 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.