On topological McAlister semigroups

Abstract

In this paper we consider McAlister semigroups over arbitrary cardinals and investigate their algebraic and topological properties. We show that the group of automorphisms of a McAlister semigroup $\mathcal{M}_{\lambda}$ is isomorphic to the direct product $Sym(\lambda){\times}\mathbb{Z}_2$, where $Sym(\lambda)$ is the group of permutations of the cardinal $\lambda$. This fact correlates with the result of Mashevitzky, Schein and Zhitomirski which states that the group of automorphisms of the free inverse semigroup over a cardinal $\lambda$ is isomorphic to the wreath product of $Sym(\lambda)$ and $\mathbb{Z}_2$. Each McAlister semigroup admits a compact semigroup topology. Consequently, the Green's relations $\mathscr D$ and $\mathscr J$ coincide in McAlister semigroups. The latter fact complements results of Lawson. We showed that each non-zero element of a Hausdorff semitopological McAlister semigroup is isolated. This fact is an analogue of the result of Mesyan, Mitchell, Morayne and P\'{e}resse, who proved that each non-zero element of Hausdorff topological polycyclic monoid is isolated. Also, it follows that the free inverse semigroup over a singleton admits only the discrete Hausdorff shift-continuous topology. We proved that a Hausdorff locally compact semitopological semigroup $\mathcal{M}_1$ is either compact or discrete. This fact is similar to the result of Gutik, who showed that a Hausdorff locally compact semitopological polycyclic monoid $\mathcal{P}_1$ is either compact or discrete. However, this dichotomy does not hold for the semigroup $\mathcal{M}_2$. Moreover, $\mathcal{M}_2$ admits continuum many different Hausdorff locally compact inverse semigroup topologies.