Arithmetic of seminormal weakly Krull monoids and domains

TL;DR

This paper investigates the arithmetic properties of seminormal weakly Krull monoids and domains, providing precise results on factorizations and characterizations of half-factoriality, extending known results from Krull monoids.

Contribution

It offers new arithmetical results for seminormal weakly Krull monoids with finite class group, including unions of sets of lengths as intervals and criteria for half-factoriality.

Findings

Unions of sets of lengths are intervals.

Characterization of half-factoriality.

Extension of Krull monoid results to seminormal weakly Krull monoids.

Abstract

We study the arithmetic of seminormal $v$-noetherian weakly Krull monoids with nontrivial conductor which have finite class group and prime divisors in all classes. These monoids include seminormal orders in holomorphy rings in global fields. The crucial property of seminormality allows us to give precise arithmetical results analogous to the well-known results for Krull monoids having finite class group and prime divisors in each class. This allows us to show, for example, that unions of sets of lengths are intervals and to provide a characterization of half-factoriality.