On half-factoriality of transfer Krull monoids

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 $a=u_1\cdot \dots \cdot u_k$. 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.