Betti elements and full atomic support in rings and monoids
Scott T. Chapman, Pedro García-Sánchez, Christopher O'Neill, Vadim Ponomarenko
TL;DR
The paper extends Betti-element methods from affine monoids to broader rings and monoids, and develops a structural theory for Betti elements under full atomic support. It introduces the multiplicity shadow to classify single-Betti monoids up to isomorphism and shows that, in the full atomic support case, the catenary degree, omega-primality, and tame degree all coincide and can be computed from the Betti set. A key contribution is the Carlitz-like characterization of block monoids: having a single Betti element with full atomic support is equivalent to being length factorial, with the finite-group instances precisely when $G\cong \mathbb{Z}_3$ or $\mathbb{Z}_2\oplus \mathbb{Z}_2$ and $D(G)=3$. The results yield computable criteria for central factorization invariants and connect the theory to concrete classes such as Puiseux monoids, block monoids, and algebraic number rings of small class number, highlighting both breadth and limitations via varied examples.
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.