  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.
