On topological McAlister semigroups

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