Semi-covariety of numerical semigroups

TL;DR

This paper introduces the concept of semi-covarieties of numerical semigroups, exploring their properties and specific examples, and establishing a new framework for studying families of numerical semigroups with particular closure properties.

Contribution

It defines semi-covarieties of numerical semigroups and investigates their structure and examples, providing a new perspective in the study of numerical semigroups.

Findings

Semi-covarieties are closed under finite intersections.

Existence of a minimum element in semi-covarieties.

Examples include families containing a fixed semigroup and those with specific Frobenius numbers.

Abstract

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 $S\neq \min(\mathcal{F})$, then there is $x\in S$ such that $S\backslash \{x\}\in \mathcal{F}$. 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.