A characterization of seminormal C-monoids

TL;DR

This paper characterizes seminormal C-monoids by their reduced class semigroup structure and provides criteria for when such monoids are half-factorial, advancing the understanding of their algebraic properties.

Contribution

It establishes a new characterization of seminormal C-monoids via their reduced class semigroup as a union of groups and offers a criterion for half-factoriality.

Findings

Seminormal C-monoids correspond to reduced class semigroups that are unions of groups.

Provides a criterion for when seminormal C-monoids are half-factorial.

Connects the structure of the class semigroup with algebraic properties of C-monoids.

Abstract

It is well-known that a C-monoid is completely integrally closed if and only if its reduced class semigroup is a group and if this holds, then the C-monoid is a Krull monoid and the reduced class semigroup coincides with the usual class group of Krull monoids. We prove that a C-monoid is seminormal if and only if its reduced class semigroup is a union of groups. Based on this characterization we establish a criterion (in terms of the class semigroup) when seminormal C-monoids are half-factorial.