The boundary quotient for algebraic dynamical systems

TL;DR

This paper introduces accurate foundation sets for right LCM semigroups, providing explicit boundary quotient presentations and conditions for pure infiniteness and simplicity in algebraic dynamical systems.

Contribution

It develops the accurate refinement property for right LCM semigroups and applies it to algebraic dynamical systems to characterize their boundary quotients.

Findings

Explicit boundary quotient presentations for certain semigroups

Sharp conditions for pure infiniteness and simplicity

Analysis of finiteness properties of foundation sets

Abstract

We introduce the notion of accurate foundation sets and the accurate refinement property for right LCM semigroups. For right LCM semigroups with this property, we derive a more explicit presentation of the boundary quotient. In the context of algebraic dynamical systems, we also analyse finiteness properties of foundation sets which lead us to a very concrete presentation. Based on Starling's recent work, we provide sharp conditions on certain algebraic dynamical systems for pure infiniteness and simplicity of their boundary quotient.