On half-factoriality of transfer Krull monoids

TL;DR

This paper investigates conditions under which transfer Krull monoids over abelian groups exhibit half-factoriality, providing new criteria based on element powers and subgroup structures.

Contribution

It establishes novel sufficient conditions for half-factoriality in transfer Krull monoids related to element powers and subgroup properties.

Findings

If a power of an element has a unique factorization length, the submonoid generated by that element is half-factorial.

For finite G_0, a specific product of elements with their orders yields a half-factorial monoid.

The paper links the structure of the group G and the set G_0 to the factorization properties of the monoid.

Abstract

Let $H$ be a transfer Krull monoid over a subset $G_0$ of an abelian group $G$ with finite exponent. Then every non-unit $a\in H$ can be written as a finite product of atoms, say $a=u_1 \cdot \ldots \cdot u_k$. The set $\mathsf L(a)$ of all possible factorization lengths $k$ is called the set of lengths of $a$, and $H$ is said to be half-factorial if $|\mathsf L(a)|=1$ for all $a\in H$. We show that, if $a \in H$ and $|\mathsf L(a^{\lfloor (3\exp(G) - 3)/2 \rfloor})| = 1$, then the smallest divisor-closed submonoid of $H$ containing $a$ is half-factorial. In addition, we prove that, if $G_0$ is finite and $|\mathsf L(\prod_{g\in G_0}g^{2\mathsf{ord}(g)})|=1$, then $H$ is half-factorial.