Embedding topological semigroups into the hyperspaces over topological groups

TL;DR

This paper characterizes when compact Clifford topological semigroups can be embedded into hyperspaces over topological groups, linking algebraic inverse semigroup properties with topological embedding conditions.

Contribution

It provides a necessary and sufficient condition for embedding compact Clifford topological semigroups into hyperspaces over topological groups, based on inverse semigroup and zero-dimensionality properties.

Findings

A compact Clifford topological semigroup is embeddable into exp(G) if and only if it is a topological inverse semigroup.

The idempotent semilattice must be zero-dimensional for such an embedding.

The paper establishes a precise algebraic-topological criterion for embedding into hyperspaces.

Abstract

We study algebraic and topological properties of subsemigroups of the hyperspace exp(G) of non-empty compact subsets of a topological group G endowed with the Vietoris topology and the natural semigroup operation. On this base we prove that a compact Clifford topological semigroup S is topologically isomorphic to a subsemigroup of exp(G) for a suitable topological group G if and only if S is a topological inverse semigroup with zero-dimensional idempotent semilattice.