On two methods of constructing compactifications of topological groups

TL;DR

This paper compares two methods for constructing compactifications of topological groups, focusing on algebraic extension properties and using hyperspace representations to analyze their structures.

Contribution

It introduces and contrasts Ellis' method with a hyperspace graph approach for creating compactifications with extended algebraic operations.

Findings

Ellis' method yields all right topological semigroup compactifications.

Hyperspace graph approach allows extension of involution and multiplication.

The paper clarifies the relationship between the two construction methods.

Abstract

The classification of (proper) compactifications of topological groups with respect to the possibility of extensions of algebraic operations is presented. Ellis' method of construction compactifications of topological groups allows one to obtain all right topological semigroup compactifications on which the multiplication on the left continuously extends. Presentation of group elements as graphs of maps in the hyperspace with Vietoris topology allows one to obtain compactifications on which the involution and the multiplication on the left extend.