The multiplicative semigroup of a Dedekind domain

TL;DR

This paper explores the structure of the multiplicative semigroup of Dedekind domains, revealing that quotients of such rings form a finite Boolean algebra of stratified extensions of groups, extending previous semigroup theory.

Contribution

It extends the theory of stratified semigroups to the multiplicative structure of Dedekind domains, showing they form a Boolean algebra of stratified extensions of groups.

Findings

Quotients of Dedekind domains have a Boolean algebra structure.

The multiplicative semigroup of Dedekind domains is a semilattice of stratified extensions.

Extension of previous semigroup concepts to algebraic rings.

Abstract

In 1995 Grillet defined the concept of a stratified semigroup and a stratified semigroup with zero. The present authors extended that idea to include semigroups with a more general base and proved, amongst other things, that finite semigroups in which the H-classes contain idempotents, are semilattices of stratified extensions of completely simple semigroups, and every strict stratified extension of a Clifford semigroup is a semilattice of stratified extensions of groups. We continue this work here by considering the multiplicative semigroup of Dedekind domains and show in particular that quotients of such rings have a multiplicative structure that is a (finite) Boolean algebra of stratified extensions of groups.