Betti elements and full atomic support in rings and monoids

TL;DR

This paper investigates the structure of Betti elements in rings and monoids, extending existing factorization theory, and characterizes monoids with full atomic support and their algebraic properties.

Contribution

It generalizes the study of Betti elements to broader rings and monoids, providing new characterizations and computational insights for monoids with specific Betti element structures.

Findings

Computed Betti elements for algebraic number rings of class number two

Established that certain invariants coincide for monoids with a single Betti element

Characterized block monoids with full atomic support as having a unique Betti element

Abstract

Several papers in the recent literature have studied factorization properties of affine monoids using the monoid's Betti elements. In this paper, we extend this study to more general rings and monoids. We open by demonstrating the issues with computing the complete set of Betti elements of a general commutative cancellative monoid, and as an example compute this set for an algebraic number ring of class number two. We specialize our study to the case where the monoid has a single Betti element, before examining monoids with full atomic support (that is, when each Betti element is divisible by every atom). For such a monoid, we show that the catenary degree, tame degree, and omega value agree and can be computed using the monoid's set of Betti elements. We close by considering Betti elements in block monoids, giving a "Carlitz-like" characterization of block monoids with full atomic support and proving that these are precisely the block monoids having a unique Betti element.