On halffactoriality 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 nonunit $a\in H$ 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 halffactorial 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 divisorclosed submonoid of $H$ containing $a$ is halffactorial. 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 halffactorial.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.04267
 Bibcode:
 2019arXiv191104267G
 Keywords:

 Mathematics  Commutative Algebra;
 Mathematics  Combinatorics;
 Mathematics  Group Theory;
 Mathematics  Number Theory;
 11B30;
 11R27;
 13A05;
 13F05;
 20M13
 EPrint:
 Communications in Algebra 49 (2021), No. 1, pp. 409420