On topological McAlister semigroups

TL;DR

This paper explores the algebraic and topological properties of McAlister semigroups over arbitrary cardinals, revealing their automorphism groups, topological characteristics, and classification of their semitopological structures.

Contribution

It provides a detailed analysis of the automorphism groups, topological properties, and classification of semitopological structures of McAlister semigroups, extending known results to arbitrary cardinals.

Findings

Automorphism group is isomorphic to Sym(λ)×Z₂.

In McAlister semigroups, Green's relations D and J coincide.

Non-zero elements in Hausdorff semitopological McAlister semigroups are isolated.

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.