A boundary quotient diagram for right LCM semigroups

TL;DR

This paper introduces a boundary quotient diagram for right LCM semigroups with property (AR), unifying various case studies on KMS-states and exploring K-theoretical aspects inspired by integral dynamics.

Contribution

It generalizes the boundary quotient diagram for the $ax+b$-semigroup, focusing on core and core irreducible subsemigroups to unify multiple case studies.

Findings

Unified framework for KMS-states across different semigroups

Extension of boundary quotient diagrams to right LCM semigroups

Insights into K-theory related to integral dynamics

Abstract

We propose a boundary quotient diagram for right LCM semigroups with property (AR) that generalizes the boundary quotient diagram for the $ax+b$-semigroup over the natural numbers. Our approach focuses on two important subsemigroups: the core subsemigroup and the semigroup of core irreducible elements. The diagram is then employed to unify several case studies on KMS-states, and we end with a discussion on $K$-theoretical aspects of the diagram motivated by recent findings for integral dynamics.