The covariety of numerical semigroups with fixed Frobenius number

M. A. Moreno-Frías; J. C. Rosales

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arXiv · 2302.09121

The covariety of numerical semigroups with fixed Frobenius number

M. A. Moreno-Frías·J. C. Rosales

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Abstract

Denote by $\mathrm m(S)$ the multiplicity of a numerical semigroup $S$. A covariety is a nonempty family $\mathscr{C}$ of numerical semigroups that fulfills the following conditions: there is the minimum of $\mathscr{C},$ the intersection of two elements of $\mathscr{C}$ is again an element of $\mathscr{C}$ and $S\backslash \{\mathrm m(S)\}\in \mathscr{C}$ for all $S\in \mathscr{C}$ such that $S\neq \min(\mathscr{C}).$ In this work we describe an algorithmic procedure to compute all the elements of $\mathscr{C}.$ We prove that there exists the smallest element of $\mathscr{C}$ containing a set of positive integers. We show that $\mathscr{A}(F)=\{S\mid S \mbox{ is a numerical semigroup with Frobenius number }F\}$ is a covariety, and we particularize the previous results in this covariety. Finally, we will see that there is the smallest covariety containing a finite set of numerical semigroups.

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M. A. Moreno-Frías, J. C. Rosales. 2023-02-17. The covariety of numerical semigroups with fixed Frobenius number. https://arxiv.org/abs/2302.09121

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