On topological McAlister semigroups

Abstract

In this paper we investigate algebraic and topological properties of McAlister semigroups. We show that for a non-zero cardinal λ the group of automorphisms of the McAlister semigroup Mλ is isomorphic to the direct product Sym(λ)×Z2, where Sym(λ) is the group of permutations of λ. McAlister semigroups admit a unique compact Hausdorff semigroup topology. Each non-zero element of a Hausdorff semitopological McAlister semigroup is isolated. It follows that the free inverse semigroup over a singleton admits only the discrete Hausdorff shift-continuous topology. We prove that a Hausdorff locally compact semitopological semigroup M1 is either compact or discrete. However, this dichotomy does not hold for the semigroup M2. Moreover, M2 admits continuum many different Hausdorff locally compact inverse semigroup topologies.