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

Abstract

Orbits and bi-invariant subsets of binary G-spaces are studied. The problem of the distributivity of a binary action of a group G on a space X, which was posed in 2016 by one of the authors, is solved. 2020 Mathematics Subject Classification. 54H15; 57S99. Key words and phrases. binary operation, topological group, groups of homeomorphisms, representations of a topological groups.

On Orbit Spaces of Distributive Binary $$G$$-Spaces

“…Note that if X and Y are G-spaces, then any equivariant map f : X → Y is biequivariant with respect to the action (5). Therefore, the category G − T op 2 is a natural extension of the category G − T op of all G-spaces and equivariant maps.…”

Bi-equivariant fibrations

“…However, in distributive binary G-spaces the sets G(x, x) are bi-invariant, and hence [x] = G(x, x) [2, Theorem 9]. Moreover, the orbits of a distributive binary G-space either are disjoint or coincide [7,Proposition 6]; therefore, the space X is partitioned into disjoint classes. We denote the corresponding quotient set by X|G.…”

“…All notions, definitions, and results used in the paper without references, as well as all those mentioned above, can be found in [2]- [7].…”

“…The bi-equivariance of the homotopy F follows from that of the maps λ, f , and F and relations (5), (7), and (8):…”

“…All binary G-spaces and bi-equivariant maps form a category, which is a natural extension of the category of G-spaces and equivariant maps. Set-theoretic questions related to bi-equivariant topology were also considered in [2]- [7].…”