Length-Factoriality and Pure Irreducibility

TL;DR

This paper investigates length-factoriality in commutative, cancellative monoids, exploring its properties, related concepts like PLS and bi-length-factoriality, and extending the notion beyond atomic monoids.

Contribution

It generalizes the concept of length-factoriality to broader monoid classes and examines related factorization properties in semirings.

Findings

Length-factoriality characterized in commutative, cancellative monoids.

Connections established between length-factoriality, PLS property, and bi-length-factoriality.

Extended the study of length-factoriality to semirings and non-atomic monoids.

Abstract

An atomic monoid $M$ is called length-factorial if for every non-invertible element $x \in M$, no two distinct factorizations of $x$ into irreducibles have the same length (i.e., number of irreducible factors, counting repetitions). The notion of length-factoriality was introduced by J. Coykendall and W. Smith in 2011 under the term 'other-half-factoriality': they used length-factoriality to provide a characterization of unique factorization domains. In this paper, we study length-factoriality in the more general context of commutative, cancellative monoids. In addition, we study factorization properties related to length-factoriality, namely, the PLS property (recently introduced by Chapman et al.) and bi-length-factoriality in the context of semirings.