The Rees-Suschkewitsch Theorem for simple topological semigroups

TL;DR

This paper characterizes simple topological semigroups that are topological paragroups, extending classical results by identifying conditions involving compactness and the structure of subgroups.

Contribution

It provides new criteria under which simple topological semigroups are topological paragroups, generalizing previous results and connecting algebraic and topological properties.

Findings

Simple topological semigroups are topological paragroups under specified conditions.

Extension of Wallace's classical result to broader classes of semigroups.

Identification of compactness conditions that guarantee the paragroup structure.

Abstract

We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product of a topological group over topological spaces with a continuous sandwich function. We prove that a simple topological semigroup $S$ is a topological paragroup if one of the following conditions is satisfied: (1) $S$ is completely simple and the maximal subgroups of $S$ are topological groups, (2) $S$ contains an idempotent and the square $S\times S$ is countably compact or pseudocompact, (3) $S$ is sequentially compact or each power of $S$ is countably compact. The last item generalizes an old Wallace's result saying that each simple compact topological semigroup is a topological paragroup.