Frobenius restricted varieties in numerical semigroups

TL;DR

This paper introduces restricted Frobenius varieties in numerical semigroups, generalizing existing concepts to include more families and analyzing their tree structures.

Contribution

It defines R-varieties, extending Frobenius and pseudo-varieties, and generalizes key results to this new framework, including the study of their tree structures.

Findings

R-varieties encompass more families of numerical semigroups.

Most properties of Frobenius varieties are extended to R-varieties.

The tree structure within R-varieties is characterized.

Abstract

The common behaviour of many families of numerical semigroups led up to defining, firstly, the Frobenius varieties and, secondly, the (Frobenius) pseudo-varieties. However, some interesting families are still out of these definitions. To overcome this situation, here we introduce the concept of restricted Frobenius variety (or R-variety). We will generalize most of the results for varieties and pseudo-varieties to R-varieties. In particular, we will study the tree structure that arise within them.