The ratio-covariety of numerical semigroups having maximal embedding dimension with fixed multiplicity and Frobenius number

Year 2024, Early Access, 1 - 17

Maria Angeles Moreno Frias , José Carlos Rosales

https://doi.org/10.24330/ieja.1575996

Abstract

In this paper we will show that
MED$(F,m)=\{S\mid S \mbox{ is a numerical semigroup with maximal embedding dimension, Frobenius number} ~F~ \mbox{and multiplicity}~ m\}$ is a ratio-covariety. As a consequence, we present two algorithms: one that computes MED$(F,m)$ and another one that calculates the elements of MED$(F,m)$ with a given genus.
If $X\subseteq S\backslash (\langle m \rangle \cup \{F+1,\rightarrow\})$ for some $S\in $ MED$(F,m)$, then there exists the smallest element of MED$(F,m)$ containing $X$. This element will be denoted by MED$(F,m)[X]$ and we will say that $X$ one of its MED$(F,m)$-system of generators. We will prove that every element $S$ of MED$(F,m)$ has a unique minimal MED$(F,m)$-system of generators and it will be denoted by MED$(F,m)$msg$(S).$ The cardinality of MED$(F,m)$msg$(S)$, will be called MED$(F,m)$-rank of $S.$ We will also see in this work, how all the elements of MED$(F,m)$ with a fixed MED$(F,m)$-rank are.

Keywords

Numerical semigroup, ratio-covariety, Frobenius number, genus, multiplicity, algorithm

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