Ratio-Covarieties of Numerical Semigroups

Abstract

In this work, we will introduce the concept of ratio-covariety, as a family R of numerical semigroups that has a minimum, denoted by min(R), is closed under intersection, and if S ∈ R and S ̸= min(R), then S\{r(S)} ∈ R, where r(S) denotes the ratio of S. The notion of ratiocovariety will allow us to: (1) describe an algorithmic procedure to compute R; (2) prove the existence of the smallest element of R that contains a set of positive integers; and (3) talk about the smallest ratio-covariety that contains a finite set of numerical semigroups. In addition, in this paper we will apply the previous results to the study of the ratio-covariety R(F,m) = {S | S is a numerical semigroup with Frobenius number F and multiplicity m}.