Semi-covariety of numerical semigroups

The main aim of this work is to introduce and justify the study of semi-covarities. A {\it semi-covariety} is a non-empty family $\mathcal{F}$ of numerical semigroups such that it is closed under finite intersections, has a minimum, $\min (\mathcal{F}),$ and if $S\in \mathcal{F}$ being $S\mathrm{\ne }\min (\mathcal{F})$, then there is $x\in S$ such that $S\mathrm{\backslash }\{x\}\in \mathcal{F}$. 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.