The covariety of numerical semigroups with fixed Frobenius number

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