2021/03/04 by Serhii Bardyla, Bardyla, Serhii
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2103.03301
openalex publication_date 2021/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Mλ is isomorphic to the direct product Sym(λ)×ℤ2, where Sym(λ) is the group of permutations of the cardinal λ. 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 λ is isomorphic to the wreath product of Sym(λ) and ℤ2. 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 Péresse, 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 M1 is either compact or discrete. This fact is similar to the result of Gutik, who showed that a Hausdorff locally compact semitopological polycyclic monoid P1 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.