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 group-like semigroup $S$ can be constructed via universal minimal dynamical system for $S$ and, moreover, $B(S)$ and $B(G(S))$ are the same.