On the Transitive Binary $G$-Spaces

TL;DR

This paper studies distributive properties of binary operations on topological spaces, introduces transitive binary G-spaces, and classifies them for compact groups.

Contribution

It introduces the concept of transitive binary G-spaces and provides a classification for compact groups, expanding understanding of distributive binary actions.

Findings

Subgroups generated by distributive subsets are also distributive.

A criterion for the distributivity of binary actions is established.

Classification of transitive distributive binary G-spaces for compact G is provided.

Abstract

Distributive subsets of the group of all invertible continuous binary operations on a topological space are considered, and it is proved that the subgroups generated by them are also distributive. A criterion for the distributivity of a binary action of a topological group $G$ on a space $X$ is obtained. The concept of transitive binary $G$-space is introduced, and a classification of transitive distributive binary $G$-spaces is given in the case of a compact group $G$.