The covariety of numerical semigroups with fixed Frobenius number

Abstract 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 (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 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.