  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.
