Groups of Binary Operations and Binary $G$-Spaces

TL;DR

This paper explores the structure of continuous binary operations on topological spaces, their connection with homeomorphisms, and introduces the category of binary G-spaces, extending classical G-space theory.

Contribution

It establishes the relationship between binary operations and homeomorphisms and constructs the category of binary G-spaces, extending existing G-space frameworks.

Findings

Relationship between binary operations and homeomorphisms established

Category of binary G-spaces constructed and analyzed

Foundational results for binary G-spaces obtained

Abstract

The group of continuous binary operations on a topological space is studied; its relationship with the group of homeomorphisms is established. The category of binary $G$-spaces and bi-equivariant maps is constructed, which is a natural extension of the category of $G$-spaces and equivariant maps. Results related to the foundations of the theory of binary $G$-spaces are obtained.