Toplogical semigroups embedded into topological groups

TL;DR

This paper establishes conditions for embedding certain classes of topological semigroups into compact topological groups and explores properties like cellularity, advancing understanding of their structure and topological features.

Contribution

It provides new criteria for embedding topological semigroups into groups and characterizes when such semigroups are topological groups, especially under compactness and regularity conditions.

Findings

Feebly compact regular first countable cancellative semigroups with open shifts are topological groups.

Connected locally compact Hausdorff cancellative monoids with open shifts are topological groups.

Conditions are identified that ensure a topological semigroup has countable cellularity.

Abstract

In this paper we give conditions under which a topological semigroup can be embedded algebraically and topologically into a compact topological group. We prove that every feebly compact regular first countable cancellative commutative topological semigroup with open shifts is a topological group, as well as every connected locally compact Hausdorff cancellative commutative topological monoid with open shifts. Finally, we use these results to give sufficient conditions on a topological semigroup that guarantee it to have countable cellularity.