A characterization of arithmetical invariants by the monoid of relations

TL;DR

This paper explores the algebraic structure of methods to analyze non-unique factorizations in monoids, extending previous work to more general cases and applying it to various invariants.

Contribution

It generalizes the calculation of arithmetical invariants from finitely generated monoids to broader classes, providing new insights into their algebraic structure.

Findings

Extended the method to non-finitely generated monoids.

Applied the approach to invariants like elasticity and distances.

Provided a deeper algebraic understanding of the invariants.

Abstract

The investigation and classification of non-unique factorization phenomena have attracted some interest in recent literature. For finitely generated monoids, S.T. Chapman and P. Garcia-Sanchez, together with several co-authors, derived a method to calculate the catenary and tame degree from the monoid of relations, and they applied this method successfully in the case of numerical monoids. In this paper, we investigate the algebraic structure of this approach. Thereby, we dispense with the restriction to finitely generated monoids and give applications to other invariants of non-unique factorizations, such as the elasticity and the set of distances.