Structural Properties of Universal Minimal Dynamical Systems for Discrete Semigroups

We show that for a discrete semigroup S there exists a uniquely determined complete Boolean algebra B(S) the algebra of clopen subsets of M(S). M(S) is the phase space of the universal minimal dynamical system for S and it is an extremally disconnected compact Hausdorff space. We deal with this connection of semigroups and complete Boolean algebras focusing on structural properties of these algebras. We show that B(S) is either atomic or atomless; that B(S) is weakly homogenous provided S has a minimal left ideal; and that for countable semigroups B(S) is semi-Cohen. We also present a class of what we call group-like semigroups that includes commutative semigroups, inverse semigroups, and right groups. The group reflection G(S) of a grouplike semigroup S can be constructed via universal minimal dynamical system for S and, moreover, B(S) and B(G(S)) are the same.