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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, (F), and if S F being S (F), then there is x S such that S \x\ 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.
Moreno-Frías et al. (Wed,) studied this question.
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