Semi-covariety of numerical semigroups

The main aim of this work is to introduce and justify the study of semi-covarities. A { semi-covariety} is a non-empty family F of numerical semigroups such that it is closed under finite intersections, has a minimum, min(F), and if S∈ F being S≠ min(F), then there is x∈ S such that S \∈ F. As examples, we will study the semi-covariety formed by all the numerical semigroups containing a fixed numerical semigroup, and the semi-covariety composed by all the numerical semigroups of coated odd elements and fixed Frobenius number.