On topological McAlister semigroups

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 M _λ is isomorphic to the direct product Sym(λ)×Z ₂, 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 M ₁ is either compact or discrete. However, this dichotomy does not hold for the semigroup M ₂. Moreover, M ₂ admits continuum many different Hausdorff locally compact inverse semigroup topologies.