Structural properties of universal minimal dynamical systems for discrete semigroups

Abstract

We show that for a discrete semigroup S S there exists a uniquely determined complete Boolean algebra B ( S ) B(S) - the algebra of clopen subsets of M ( S ) M(S) . M ( S ) M(S) is the phase space of the universal minimal dynamical system for S 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 ) B(S) is either atomic or atomless; that B ( S ) B(S) is weakly homogenous provided S S has a minimal left ideal; and that for countable semigroups B ( S ) 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 ) G(S) of a group-like semigroup S S can be constructed via universal minimal dynamical system for S S and, moreover, B ( S ) B(S) and B ( G ( S ) ) B(G(S)) are the same.