Boundary quotients of C^*-algebras of left cancellative monoids and their groupoid models

For a left cancellative monoid S we consider a quotient of the reduced semigroup C^*-algebra Cᵣ^*(S) known as the boundary quotient. We present two potential groupoid models for this boundary quotient, obtained as reductions of Paterson and Spielberg's groupoids associated to S, and formulate conditions on S which guarantees that either is a groupoid model. We outline how these conditions are related to the notions (strong) C^*-regularity introduced in a previous paper, and construct an example of a left cancellative monoid which is not C^*-regular, but satisfies both of the new conditions.