Groups of Invertible Binary Operations of a Topological Space

Abstract

In this paper, continuous binary operations of a topological space are studied and a criterion of their invertibility is proved. The classification problem of groups of invertible continuous binary operations of locally compact and locally connected spaces is solved. A theorem on the binary distributive representation of a topological group is also proved.

Bi-equivariant fibrations

“…This is explained, in particular, by the special role of distributive subgroups of the group H 2 (X) of all continuous invertible binary operations on X. For example, any topological group is a distributive subgroup of the group of invertible binary operations on some space [3]. This statement is a binary topological analogue of Cayley's classical theorem on the representation of any finite group by unary operations (permutations).…”

On Orbits and Bi-invariant Subsets of Binary $$G$$-Spaces

“…One of the reasons why this notion is important is the special role played by distributive subgroups of the group H 2 (X) of all invertible continuous binary operations on X. For example, any topological group is a distributive subgroup of the group of invertible binary operations of some space [4]. This statement is the binary topological counterpart of Cayley's classical theorem on the representation of any finite group by unary operations (permutations).…”