Arithmetic varieties of numerical semigroups

TL;DR

This paper introduces the concept of arithmetic varieties for numerical semigroups, explores their properties, and develops algorithms for related computations, with implementations in GAP.

Contribution

It defines arithmetic varieties for numerical semigroups, analyzes their structure, and provides algorithms with GAP implementations for practical computation.

Findings

The root tree associated with an arithmetic variety is not locally finite.

Fixing the Frobenius number yields a finite tree with computable algorithms.

Algorithms are implemented in GAP for practical use.

Abstract

In this paper we present the notion of arithmetic variety for numerical semigroups. We study various aspects related to these varieties such as the smallest arithmetic that contains a set of numerical semigroups and we exhibit the root three associated with an arithmetic variety. This tree is not locally finite; however, if the Frobenius number is fixed, the tree has finitely many nodes and algorithms can be developed. All algorithms provided in this article include their (non-debugged) implementation in GAP.