Betti elements and non-unique factorizations

Abstract: Let $M$ be a commutative cancellative reduced atomic monoid with set of atoms (or irreducibles) $\mathcal{A}(M)$. Given a nonunit $x$ in $M$, let $Z(x)$ represent the set of factorizations of $x$ into atoms. Define a graph $\nabla_x$ whose vertex set is $Z(x)$ where two vertices are joined by an edge if these factorizations share an atom. Call $x$ a \textit{Betti element} of $M$ if the graph $\nabla_x$ is disconnected. Betti elements have proven to be a powerful tool in the study of nonunique factorizations of elements in monoids. In particular, over the past several years many papers have used Betti elements to study factorizaton properties in \textit{affine monoids} (i.e., finitely generated additive submonoids of $\mathbb{N}_0^k$ for some positive integer $k$). Several strong results have been obtained when $M$ is a numerical monoid (i.e., $k=1$ above). In this talk, we will review the basic properties of Betti elements and some of the results regarding affine monoids mentioned above. We will then extend this study to more general rings and monoids which are commutative and cancellative. We focus on two cases: (I) when the monoid $M$ has a single Betti element, (II) when each atom of $M$ divides every Betti element. We call those monoids satisfying condition (II) as having \textit{full atomic support}. We show using elementary arguments that a monoid of type (I) is actually of full atomic support. We close by showing for a monoid of full atomic support that the catenary degree, the tame degree, and the omega primality constant (three well studied invariants in the nonunique factorization literature) can be easily computed from the monoid's set of Betti elements.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)