A characterization of arithmetical invariants by the monoid of relations II: The monotone catenary degree and applications to semigroup rings

TL;DR

This paper extends the algebraic framework for analyzing non-unique factorizations in monoids, focusing on the monotone catenary degree and applying these concepts to compute invariants of semigroup rings.

Contribution

It introduces a new approach to characterize the monotone catenary degree and applies it to explicit calculations in semigroup rings, broadening previous methods.

Findings

Extended the algebraic structure to the monotone catenary degree.

Provided explicit computations of arithmetical invariants for semigroup rings.

Generalized previous results to non-finitely generated monoids.

Abstract

The investigation and classification of non-unique factorization phenomena has attracted some interest in recent literature. For finitely generated monoids, S.T. Chapman and P.A. Garc\'ia-S\'anchez, together with several co-authors, derived a method to calculate the catenary and tame degree from the monoid of relations. Then, in [1], the algebraic structure of this approach was investigated and the restriction to finitely generated monoids was removed. We now extend these ideas further to the monotone catenary degree and then apply all these results to the explicit computation of arithmetical invariants of semigroup rings. [1] A. Philipp. A characterization of arithmetical invariants by the monoid of relations. Semigroup Forum, 81:424-434, 2010.