Arf good semigroups with fixed genus

TL;DR

This paper presents an algorithm to compute all Arf numerical semigroups with a fixed genus and extends the concept to good semigroups in higher dimensions, providing methods to enumerate them.

Contribution

It introduces a novel algorithm for Arf semigroups with fixed genus and generalizes the genus concept to higher-dimensional good semigroups, offering enumeration procedures.

Findings

Algorithm for Arf numerical semigroups with fixed genus

Generalization of genus to good semigroups in 1^r

Procedure to compute all Arf semigroups in 1^r with given genus

Abstract

In this paper we give an algorithm for the computation of all the Arf numerical semigroups with a given genus. Moreover, we generalize the concept of genus of a numerical semigroup to good semigroups of $\mathbb{N}^r$ and we give a procedure to calculate all the Arf semigroups of $\mathbb{N}^r$ with a given genus.