# Groups of Binary Operations and Binary $G$-Spaces

Pavel S. Gevorgyan  Moscow State Pedagogical University<EMAIL_ADDRESS>
###### 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.

###### Key words and phrases:

Compact-open topology, group of homeomorphisms, group of binary operations,
binary $G$-space

###### 1991 Mathematics Subject Classification:

54H15; 57S99

##  1\. Auxiliary Results and Notation

Throughout this paper, by a space we mean a topological space. All spaces are
assumed to be Hausdorff.

We denote the category of topological spaces and continuous maps by $Top$.

By $C(X,Y)$ we denote the space of all continuous maps of $X$ to $Y$ endowed
with the compact-open topology, that is, the topology generated by the subbase
consisting of all sets of the form $W(K,U)=\\{f:X\to Y;\ f(K)\subset U\\}$,
where $K$ a compact subset of $X$ and $U$ is an open subset of $Y$.

All continuous function spaces are considered in the compact-open topology.

If $G$ is a topological group, then there is a natural group operation on
$C(X,G)$: given any continuous maps $f,g\in C(X,G)$, their product $fg\in
C(X,G)$ is defined by $(fg)(x)=f(x)g(x)$ for all $x\in X$.

######  Theorem 1 ([6]).

If $G$ is a topological group, then so is $C(X,G)$.

The group of all self-homeomorphisms of $X$ is denoted by $H(X)$. This group
is not generally a topological group. However, the following theorem is valid.

######  Theorem 2 ([1]).

If a space $X$ is locally compact and locally connected, then $H(X)$ is a
topological group.

Let $X$ be a topological space, and let $G$ be a topological group $G$ with
identity element $e$. Suppose given a continuous map $\theta:G\times X\to X$
satisfying the conditions

$(1)\quad\theta(g,\theta(h,x))=\theta(gh,x)\qquad\text{and}\qquad(2)\quad\theta(e,x)=x$

for $g,h\in G$ and $x\in X$. Then $X$ is called a $G$-space, and the
continuous map $\theta:G\times X\to X$ is called the action of the group $G$
on the $G$-space $X$. In this case, we use the notation $\theta(g,x)=gx$.

Let $X$ and $Y$ be $G$-spaces. A continuous map $f:X\to Y$ is said to be
equivariant if $f(gx)=gf(x)$ for any $g\in G$ and $x\in X$.

All $G$-spaces and their equivariant maps form a category. This category is
denoted by $G$-$Top$.

The symmetric group on a set $X$ is denoted by $S(X)$. In the case of a finite
set $X$, this group is denoted by $S_{n}(X)$ or $S_{n}$, where $n$ is the
number of elements in $X$. The order of $S_{n}(X)$ equals $n!$:
$|S_{n}(X)|=n!$

Details on these notions, as well as on all definitions, notions, and results
used in this paper without reference, can be found in [2], [3], [4] and [5].

##  2\. The Group of Continuous Binary Operations

Let $X$ be any topological space.

A continuous map $f:X^{2}\to X$ is called a continuous binary operation, or a
binary map, on $X$. We denote the set of all continuous binary operations on
$X$ by $C_{2}(X)$. On $C_{2}(X)$ we define an operation “$\ast$” by

(1) $(f*\varphi)(t,x)=f(t,\varphi(t,x))$

for all $t,x\in X$.

######  Proposition 1.

The set $C_{2}(X)$ under the operation “$\ast$” is a topological semigroup
with identity element $e(t,x)=x$, i.e., a topological monoid.

###### Proof.

Let $f$, $\varphi$, and $h$ be continuous binary operations on a topological
space $X$. Let us check the semigroup axiom:

$[f*(\varphi*h)](t,x)=f(t,(\varphi*h)(t,x))=f(t,\varphi(t,h(t,x)))\\\
=(f*\varphi)(t,h(t,x))=[(f*\varphi)*h](t,x).$

The binary operation $e:X^{2}\to X$ defined by $e(t,x)=x$ is the identity
element of the semigroup, because $f*e=e*f=f$. Indeed,

$(f*e)(t,x)=f(t,e(t,x))=f(t,x),$ $(e*f)(t,x)=e(t,f(t,x))=f(t,x).$

∎

######  Definition 1.

A continuous binary operation $f\in C_{2}(X)$ is said to be invertible if
there exists a continuous binary operation $\varphi\in C_{2}(X)$ such that

(2) $f*\varphi=\varphi*f=e.$

In this case, $f$ and $\varphi$ are said to be mutually inverse binary
operations.

We denote the set of all invertible elements of $C_{2}(X)$ by $H_{2}(X)$. The
set $H_{2}(X)$ is a group.

######  Example 1.

Let $X=\\{a,b\\}$ be a two-point discrete space. The symmetric group
$S_{2}(X)$ of permutations of this set is the cyclic group $\mathbb{Z}_{2}$,
and the group of all invertible binary operations on $X=\\{a,b\\}$ is the
group of order 4 with two generators $\varphi_{1}$ and $\varphi_{2}$, which
are specified by

$\varphi_{1}$: |  $b$ |  $a$ |  $a$
---|---|---|---
|  $a$ |  $b$ |  $b$
|  |  a |  b

$\varphi_{2}$: |  $b$ |  $b$ |  $a$
---|---|---|---
|  $a$ |  $a$ |  $b$
|  |  a |  b

The multiplication table for this group is as follows:

$e$ |  $\varphi_{1}$ |  $\varphi_{2}$ |  $\varphi_{3}$
---|---|---|---
$\varphi_{1}$ |  $e$ |  $\varphi_{3}$ |  $\varphi_{2}$
$\varphi_{2}$ |  $\varphi_{3}$ |  $e$ |  $\varphi_{1}$
$\varphi_{3}$ |  $\varphi_{2}$ |  $\varphi_{1}$ |  $e$

This is the Klein four-group.

In the case of a three-point set $X$, the order of $H_{2}(X)$ equals
$(3!)^{3}=216$ (see Corollary 2 below).

######  Theorem 3.

If a continuous binary operation $f:X^{2}\to X$ is invertible, then, for any
$t\in X$, the continuous map $f_{t}:X\to X$ defined by

(3) $f_{t}(x)=f(t,x)$

for $x\in X$ is a homeomorphism.

###### Proof.

Suppose that a binary operation $f:X^{2}\to X$ is invertible, i.e., there
exists a binary operation $\varphi\in C_{2}(X)$ satisfying relations (2). Take
any element $t\in X$. Let us prove that the continuous map $f_{t}:X\to X$ is a
homeomorphism.

The map $f_{t}$ is a monomorphism. Indeed, suppose that
$f_{t}(x)=f_{t}(x^{\prime})$, i.e., $f(t,x)=f(t,x^{\prime})$. Then

$x=e(t,x)=(\varphi*f)(t,x)=\varphi(t,f(t,x))=\\\
=\varphi(t,f(t,x^{\prime}))=(\varphi*f)(t,x^{\prime})=e(t,x^{\prime})=x^{\prime}.$

The continuous map $\varphi_{t}:X\to X$ defined by

$\varphi_{t}(x)=\varphi(t,x)$

is inverse to $f_{t}:X\to X$. Indeed,

$(f_{t}\circ\varphi_{t}(x)=f_{t}(\varphi_{t}(x))=f_{t}(\varphi(t,x))=f(t,\varphi(t,x))=\\\
=(f*\varphi)(t,x)=e(t,x)=x,$

i.e., $f_{t}\circ\varphi_{t}=1_{X}$. A similar argument proves that
$\varphi_{t}\circ f_{t}=1_{X}$. ∎

It follows from Theorem 3 that if an invertible binary operation $f:X^{2}\to
X$ is represented in the form of a family of homeomorphisms $\\{f_{t}\\}$,
$t\in X$, then the inverse binary operation has the form $\\{f^{-1}_{t}\\}$.

The converse of Theorem 3 is true for locally compact and locally connected
spaces.

######  Theorem 4.

Suppose that a space $X$ is locally compact and locally connected and
$f:X^{2}\to X$ is a continuous binary operation. If the map $f_{t}:X\to X$ is
a homeomorphism for each $t\in X$, then the binary operation $f$ is
invertible, and the inverse binary operation $f^{-1}$ is defined by
$f^{-1}(t,x)=f_{t}^{-1}(x)$.

###### Proof.

Consider the binary operation $\varphi$ given by $\varphi(t,x)=f_{t}^{-1}(x)$.
Let us prove that this binary operation is continuous and inverse to
$f:X^{2}\to X$: $\varphi=f^{-1}$.

First, we prove the continuity of the map $\varphi:X^{2}\to X$. Let
$(t_{0},x_{0})\in X^{2}$ be any point, and let
$\varphi(t_{0},x_{0})=f_{t_{0}}^{-1}(x_{0})=y_{0}$. Consider any open
neighborhood $W\subset X$ of $y_{0}$ such that the closure $\overline{W}$ is
compact. There exists a compact connected neighborhood $K$ of $x_{0}$ for
which

(4) $f_{t_{0}}^{-1}(K)\subset W.$

We denote the interior of $K$ by $K^{\circ}$. We have

(5) $f_{t_{0}}(y_{0})=x_{0}\in K^{\circ}.$

Inclusion (4) implies

(6) $f_{t_{0}}(W^{C}\cap\overline{W})\subset K^{C},$

where $W^{C}$ and $K^{C}$ are the complements of $W$ and $K$, respectively.

Since $f:X^{2}\to X$ is a continuous binary operation, $\\{y_{0}\\}$ and
$W^{C}\cap\overline{W}$ are compact supsets of $X$, and $K^{\circ}$ and
$K^{C}$ are open subsets, it follows from (5) and (6) that the point $t_{0}$
has an open neighborhood $U$ such that, for any $t\in U$, we have

(7) $f_{t}(y_{0})\in K^{\circ}$

and

(8) $f_{t}(W^{C}\cap\overline{W})\subset K^{C}.$

Inclusion (8) implies that

$K\subset f_{t}(W\cup\overline{W}^{C})$

for any $t\in U$. Therefore,

$f_{t}^{-1}(K)\subset W\cup\overline{W}^{C}.$

Since $f_{t}^{-1}(K)$ is connected and $W$ and $\overline{W}^{C}$ are disjoint
open sets, it follows from the last inclusion that $f_{t}^{-1}(K)$ is
contained in one of the sets $W$ and $\overline{W}^{C}$. However, by virtue of
(7), we obviously have $f_{t}^{-1}(K)\subset W$. Hence

(9) $f_{t}^{-1}(K^{\circ})\subset W$

for all $t\in U$.

Thus, given any open neighborhood $W$ of $y_{0}=f_{t_{0}}^{-1}(x_{0})$, we
have found open neighborhoods $U$ of $t_{0}$ and $K^{\circ}$ of $x_{0}$ for
which (9) holds. This proves the continuity of the binary operation
$\varphi(t,x)=f_{t}^{-1}(x)$.

It is easy to show that the continuous binary operation
$\varphi(t,x)=f_{t}^{-1}(x)$ is inverse to $f:X^{2}\to X$. Indeed,

$(f*\varphi)(t,x)=f(t,\varphi(t,x))=f(t,f_{t}^{-1}(x))=f_{t}(f_{t}^{-1}(x))=x,$

i.e. $f*\varphi=e$. The relation $\varphi*f=e$ is proved in a similar way. ∎

Theorems 3 and 4 imply the following criterion for the invertibility of
continuous binary operations on locally compact and locally connected spaces.

######  Theorem 5.

A continuous binary operation $f:X^{2}\to X$ on a locally compact locally
connected space $X$ is invertible if and only if the continuous map
$f_{t}:X\to X$ defined by (3) is a homeomorphism for any $t\in X$.

######  Proposition 2.

The group $H(X)$ of all self-homeomorphisms of a topological space $X$ is
isomorphic (algebraically and topologically) to a subgroup of the group
$H_{2}(X)$ of invertible binary operations.

###### Proof.

To each $f\in H(X)$ we assign the continuous map $\tilde{f}:X^{2}\to X$
defined by $\tilde{f}(t,x)=f(x)$, $t,x\in X$. Obviously,
$\widetilde{f^{-1}}=\tilde{f}^{-1}$. Thus, $\tilde{f}$ is a continuous
invertible binary operation, i.e., $\tilde{f}\in H_{2}(X)$. The correspondence
$f\to\tilde{f}$ is the required isomorphism between the group $H(X)$ and a
subgroup of $H_{2}(X)$. ∎

######  Theorem 6.

For any locally compact locally connected space $X$, the group $H_{2}(X)$ is
isomorphic (algebraically and topologically) to $C(X,H(X))$.

###### Proof.

Consider the map $p:C(X,H(X))\to H_{2}(X)$ defined by

(10) $p(f)(t,x)=f(t)(x)$

for $f\in C(X,H(X))$ and $t,x\in X$. The map $f(t):X\to X$ is a homeomorphism
for each $t\in X$. Therefore, by virtue of Theorem 5, the binary operation
$p(f):X\times X\to X$ is invertible, that is, indeed belongs to the group
$H_{2}(X)$.

Let us prove that $p$ is a monomorphism. Take $f,g\in C(X,H(X))$, $f\neq g$.
There exists a point $t_{0}\in X$ such that $f(t_{0})\neq g(t_{0})$. Since
$f(t_{0}),g(t_{0})\in H(X)$, it follows that $f(t_{0})(x_{0})\neq
g(t_{0})(x_{0})$ for some $x_{0}\in X$. Thus, $p(f)(t_{0},x_{0})\neq
p(g)(t_{0},x_{0})$, and hence $p(f)\neq p(g)$.

The map $p$ is also an epimorphism. Indeed, let $\varphi\in H_{2}(X)$ be any
continuous binary operation. By virtue of Theorem 5, the map $\varphi_{t}:X\to
X$ defined by $\varphi_{t}(x)=\varphi(t,x)$, $t,x\in X$, is a homeomorphism.
It is easy to see that the element $f\in C(X,H(X))$ determined by the
condition $f(t)=\varphi_{t}$ is the preimage of the binary operation
$\varphi$. Indeed, we have $p(f)(t,x)=f(t)(x)=\varphi_{t}(x)=\varphi(t,x)$.

Thus, the map $p^{-1}:H_{2}(X)\to C(X,H(X))$ defined by

$p^{-1}(\varphi)(t)(x)=\varphi(t,x)$

for $\varphi\in H_{2}(X)$ and $t,x\in X$ is inverse to $p:C(X,H(X))\to
H_{2}(X)$.

The map $p$ is a homomorphism, that is, $p(f\circ g)=p(f)*p(g)$. Indeed, for
any $t,x\in X$, we have

$p(f\circ g)(t,x)=(f\circ g)(t)(x)=(f(t)\circ g(t))(x)=f(t)(g(t)(x))\\\
=f(t)(p(g)(t,x))=p(f)(t,p(g)(t,x))=(p(f)*p(g))(t,x).$

Let us prove the continuity of $p$. Take any element $W(K\times K^{\prime},U)$
of the subbase of the compact-open topology on $H_{2}(X)$, where $U\subset X$
open and $K,K^{\prime}\subset X$ are compact subsets of $X$. Let us show that
the preimage of $W(K\times K^{\prime},U)$ is the set $W(K,W(K^{\prime},U))$,
which is an element of the subbase of the compact-open topology on
$C(X,H(X))$. Indeed, for any $\varphi\in W(K\times K^{\prime},U)$ and
$f=p^{-1}(\varphi)\in C(X,H(X))$, we have

$\varphi\in W(K\times K^{\prime},U)\iff\varphi(t,x)\in U\iff p(f)(t,x)\in U\\\
\iff f(t)(x)\in U\iff f\in W(K,W(K^{\prime},U)),$

where $t\in K$ and $x\in K^{\prime}$ are arbitrary elements.

The continuity of the inverse map $p^{-1}:H_{2}(X)\to C(X,H(X))$ is proved in
precisely the same way. ∎

######  Corollary 1.

If $X$ is a locally compact locally connected space, then $H_{2}(X)$ is a
topological group.

###### Proof.

By Theorem 2, $H(X)$ is a topological group. Thus, $C(X,H(X))$ is a
topological group as well (by Theorem 1). According to Theorem 6, $H_{2}(X)$
is a topological group. ∎

######  Corollary 2.

If $|X|=n<\infty$, then $|H_{2}(X)|=(n!)^{n}$.

###### Proof.

For a finite set $X$, we have $H(X)=S_{n}(X)$, where $S_{n}(X)$ is the
symmetric group of permutations of $X$. Since $|S_{n}(X)|=n!$, the corollary
follows directly from Theorem 6. ∎

##  3\. Binary Actions of Groups

######  Definition 2.

Let $G$ be a topological group, and let $X$ be a space. A continuous map
$\alpha:G\times X^{2}\to X$ is called a binary action of $G$ on $X$ if the
following conditions hold:

(11) $\alpha(gh,t,x)=\alpha(g,t,\alpha(h,t,x)),$ (12) $\alpha(e,t,x)=x,$

where $e$ is the identity element of $G$, $g,h\in G$, and $t,x\in X$.

We refer to a space $X$ together with a fixed binary action of a group $G$,
that is, to a triple $(G,X,\alpha)$, as a binary $G$-space.

Note that in the notation

(13) $g(t,x)=\alpha(g,t,x)$

relations (11) and (12) take the form

$gh(t,x)=g(t,h(t,x)),\quad e(t,x)=x.$

######  Example 2.

The continuous map $\alpha:G\times G^{2}\to G$ defined by

(14) $\alpha(g,h_{1},h_{2})=h_{1}gh_{1}^{-1}h_{2},\quad\text{or}\quad
g(h_{1},h_{2})=h_{1}gh_{1}^{-1}h_{2},$

is a binary action of the topological group $G$ on itself.

Indeed, conditions (11) and (12) in Definition 2 are satisfied: we have

$\alpha(gh,h_{1},h_{2})=h_{1}ghh_{1}^{-1}h_{2}=h_{1}gh_{1}^{-1}h_{1}hh_{1}^{-1}h_{2}=\\\
=\alpha(g,h_{1},h_{1}hh_{1}^{-1}h_{2})=\alpha(g,h_{1},\alpha(h,h_{1},h_{2}))$

and

$\alpha(e,h_{1},h_{2})=h_{1}eh_{1}^{-1}h_{2}=h_{2}$

for all $g,h,h_{1},h_{2}\in G$.

######  Example 3.

Let $GL(n,\mathbf{R})$ be the general linear group of degree $n$. Consider the
continuous map
$\alpha:GL(n,\mathbf{R})\times\mathbf{R}^{n}\times\mathbf{R}^{n}\to\mathbf{R}^{n}$
defined by

(15) $\alpha(A,\mathbf{x},\mathbf{y})=(E-A)\mathbf{x}+A\mathbf{y},$

or, equivalently,

(16) $A(\mathbf{x},\mathbf{y})=(E-A)\mathbf{x}+A\mathbf{y},$

where $A\in GL(n,\mathbf{R})$, $E$ is the identity matrix, and
$\mathbf{x},\mathbf{y}\in\mathbf{R}^{n}$.

Note that, for any $A,B\in GL(n,\mathbf{R})$, we have

$A(\mathbf{x},B(\mathbf{x},\mathbf{y}))=(E-A)\mathbf{x}+A(B(\mathbf{x},\mathbf{y}))=(E-A)\mathbf{x}+A((E-B)\mathbf{x}+B\mathbf{y})\\\
=(E-A)\mathbf{x}+A(E-B)\mathbf{x}+AB\mathbf{y}=(E-A+A-AB)\mathbf{x}+AB\mathbf{y}\\\
=(E-AB)\mathbf{x}+AB\mathbf{y}=AB(\mathbf{x},\mathbf{y})$

and

$E(\mathbf{x},\mathbf{y})=(E-E)\mathbf{x}+E\mathbf{y}=\mathbf{y}.$

Therefore, relation (16) defines a binary action of the general linear group
$GL(n,\mathbf{R})$ on the $n$-dimensional vector space $\mathbf{R}^{n}$.

Let $\alpha$ be a binary action of a topological group $G$ on a space $X$. For
each $g\in G$, we define a continuous map $\alpha_{g}:X^{2}\to X$ as

(17) $\alpha_{g}(t,x)=\alpha(g,t,x).$

Considitions (11) and (12) imply that $\alpha_{gh}=\alpha_{g}*\alpha_{h}$ and
$\alpha_{e}$ is the identity element in the monoid of binary operations on
$X$. Thus, the following proposition is valid.

######  Proposition 3.

The map $g\to\alpha_{g}$ is a continuous homomorphism from $G$ to the group
$H_{2}(X)$ of all invertible continuous binary operations.

Thus, the elements of the group $G$ can be treated as continuous binary
operations on the space $X$ which act by the rule
$g(t,x)=\alpha_{g}(t,x)=\alpha(g,t,x)$.

Take any $t\in X$. Consider the continuous map $\alpha_{t}:G\times X\to X$
defined by

(18) $\alpha_{t}(g,x)=\alpha(g,t,x).$

######  Proposition 4.

The map $\alpha_{t}$ is an action of the group $G$ on the space $X$.

###### Proof.

The map $\alpha_{t}$ satisfies the conditions in the definition of an action
of a group on a topological space. Indeed, we have

(1)
$\alpha_{t}(gh,x)=\alpha(gh,t,x)=\alpha(g,t,\alpha(h,t,x))=\alpha(g,t,\alpha_{t}(h,x))=\alpha_{t}(g,\alpha_{t}(h,x))$
and

(2) $\alpha_{t}(e,x)=\alpha(e,t,x)=x$

for all $g,h\in G$ and $x\in X$. ∎

Thus, a binary action $\alpha$ of a group $G$ on a space $X$ induces the
family $\\{\alpha_{t}\\}$, $t\in X$, of “ordinary” actions of $G$ on $X$.

In the case of the binary action (14) (see Example 2), all induced actions of
$G$ on itself are equivalent. To be more precise, the homeomorphism $f:G\to G$
given by $f(g)=h^{-1}gh$, where $h$ is any fixed element of $G$, is an
equivalence with respect to the actions $\alpha_{e}$ and $\alpha_{h}$. Indeed,

$f(\alpha_{e}(g,\tilde{g}))=f(e^{-1}ge\tilde{g})=f(g\tilde{g})=h^{-1}g\tilde{g}=\\\
=h^{-1}ghh^{-1}\tilde{g}h=h^{-1}ghf(\tilde{g})=\alpha_{h}(g,f(\tilde{g})).$

The binary action of the general linear group $GL(n,\mathbf{R})$ on
$\mathbf{R}^{n}$ defined by (15) (see Example 3) induces the family
$\\{\alpha_{\textbf{a}};\textbf{a}\in\mathbf{R}^{n}\\}$ of ordinary actions of
the group $GL(n,\mathbf{R})$ on $\mathbf{R}^{n}$. It is easy to show that all
these actions are equivalent. To be more precise, the map
$f(\textbf{x})=\textbf{x}-\textbf{a}$ is an equivariant self-homeomorphism of
the space $\mathbf{R}^{n}$ with respect to the actions $\alpha_{\textbf{a}}$
and $\alpha_{\textbf{0}}$. Moreover, the action $\alpha_{\textbf{0}}$ is
multiplication of a given matrix by a given element of $\mathbf{R}^{n}$:
$\alpha_{\textbf{0}}(A,\textbf{y})=\alpha(A,\textbf{0},\textbf{y})=\textbf{0}+A\textbf{y}=A\textbf{y}$.

Given $A\subset X$ and $g\in G$, we set $gA=\\{g(a,a^{\prime});a,a^{\prime}\in
A\\}$. Similarly, given $K\subset G$, we set $KA=\\{g(a,a^{\prime});g\in
K,a,a^{\prime}\in A\\}$.

######  Theorem 7.

Let $A$ be a subset of a binary $G$-space $X$, and let $K$ be a subset of a
compact topological group $G$. Then the following assertions hold:

(i) if $A$ is open, then so is $KA$;

(ii) if $A$ is compact and $K$ is closed, then $KA$ is compact.

###### Proof.

(i) Note that $KA=\bigcup\limits_{g\in K}\bigcup\limits_{a\in A}g_{a}(A)$,
where $g_{a}:X\to X$ is the map defined by $g_{a}(x)=g(a,x)$, $x\in X$. By
virtue of Theorem 3, $g_{a}:X\to X$ is a homeomorphism; therefore, the set
$g_{a}(A)$ is open. Hence $KA$ is open as well.

(ii) If $K$ is closed, then it is also compact, because $G$ is a compact
group. Thus, the set $K\times A^{2}$ is compact, and so is its continuous
image $KA=\alpha(K\times A^{2})$. ∎

######  Corollary 3.

Suppose that $G$ is a compact group, $X$ is a binary $G$-space, and $A\subset
X$ is a subset of $X$. If $A$ is open (compact), then so is $GA$.

##  4\. The Category of Binary $G$-Spaces

Let $(G,X,\alpha)$ and $(G,Y,\beta)$ be two binary $G$-spaces.

######  Definition 3.

We say that a continuous map $f:X\to Y$ is bi-equivariant if

$f(\alpha(g,t,x))=\beta(g,f(t),f(x)),$

or, equivalently,

$f(g(t,x))=g(f(t),f(x)),$

for all $g\in G$ and $t,x\in X$.

We refer to a bi-equivariant map $f:X\to Y$ which is simultaneously a
homeomorphism as a bi-equivalence of binary $G$-spaces. Note that the inverse
map $f^{-1}:Y\to X$ is bi-equivariant as well. Indeed, we have

$f^{-1}(g(y,y^{\prime}))=f^{-1}(g(f(x),f(x^{\prime}))=f^{-1}(f(g(x,x^{\prime})))=\\\
=g(x,x^{\prime})=g(f^{-1}(y),f^{-1}(y^{\prime})),$

where $x,x^{\prime}\in X$ and $y,y^{\prime}\in Y$.

The following assertion is valid.

######  Theorem 8.

The binary $G$-spaces and bi-equivariant maps form a category.

We denote the category of binary $G$-spaces and bi-equivariant maps by
$G$-$Top^{2}$.

######  Proposition 5.

Any bi-equivariant map $f:X\to Y$ is equivariant with respect to the induced
actions $\alpha_{t}$ and $\beta_{f(t)}$ for any $t\in X$.

###### Proof.

Indeed,

$f(\alpha_{t}(g,x))=f(\alpha(g,t,x))=\beta(g,f(t),f(x))=\beta_{f(t)}(g,f(x)).$

∎

Now, let $X$ be any $G$-space. On $X$ we define a binary $G$-action by

(19) $g(x,x^{\prime})=g\cdot x^{\prime}$

for all $g\in G$ and $x,x^{\prime}\in X$.

Note that if $X$ and $Y$ are $G$-spaces, then any equivariant map $f:X\to Y$
is bi-equivariant with respect to the action (19). Indeed,

$f(g(x,x^{\prime})=f(g\cdot x^{\prime})=g\cdot
f(x^{\prime})=g(f(x),f(x^{\prime})).$

Thus, the category $G$-$Top$ is a subcategory of the category $G$-$Top^{2}$.
We have the following chain of natural extensions of categories:

$Top\subset G\text{-}Top\subset G\text{-}Top^{2}.$

Let $(G,X,\alpha)$ be a binary $G$-space. Consider the $G$-space $(G,X\times
X,\tilde{\alpha})$ on which the group $G$ acts as

$\tilde{\alpha}(g,x,x^{\prime})=(x,\alpha(g,x,x^{\prime})).$

Using the notations $\tilde{\alpha}(g,x,x^{\prime})=g\cdot(x,x^{\prime})$ and
$\alpha(g,x,x^{\prime})=g(x,x^{\prime})$, we rewrite this formula as

$g\cdot(x,x^{\prime})=(x,g(x,x^{\prime})).$

Note that if $X$ and $Y$ are binary $G$-spaces, then any bi-equivariant map
$f:X\to Y$ generates the equivariant map $\tilde{f}:X\times X\to Y\times Y$
defined by

$\tilde{f}(x,x^{\prime})=(f(x),f(x^{\prime})),$

where $x,x^{\prime}\in X$. Indeed,

$\tilde{f}(g\cdot(x,x^{\prime}))=\tilde{f}(x,g(x,x^{\prime}))=(f(x),f(g(x,x^{\prime})))\\\
=(f(x),g(f(x),f(x^{\prime})))=g\cdot(f(x),f(x^{\prime}))=g\cdot\tilde{f}(x,x^{\prime}).$

This correspondence is a covariant functor from the category $G$-$TOP^{2}$ to
the category $G$-$TOP$.

##  5\. Invariant Sets

######  Definition 4.

We say that a subset $A\subset X$ is invariant with respect to the binary
action of a group $G$ if $GA=A$.

It is easy to see that the intersection $A\cap B$ of invariant sets
$A,B\subset X$ is invariant. However, the union $A\cup B$ of two invariant
sets is not generally invariant. Indeed, any one-point set $\\{x\\}$,
$x\in\mathbf{R}^{n}$, is invariant with respect to the binary action (15) of
the general linear group $GL(n,\mathbf{R})$ on the $n$-dimensional vector
space $\mathbf{R}^{n}$ (see Example 3), since $A(x,x)=(E-A)x+Ax=x$ for all
$A\in GL(n,\mathbf{R})$. However, the union $\\{x\\}\cup\\{y\\}$ of two one-
point sets is not invariant, because
$GL(n,\mathbf{R})\\{x,y\\}=\mathbf{R}^{n}$.

######  Question 1.

What are binary $G$-spaces in which the union $A\cup B$ of any invariant
subsets $A,B\subset X$ is invariant?

######  Definition 5.

The orbit of an element $x\in X$ of a binary $G$-space $X$ is defined as the
minimal invariant set $[x]\subset X$ containing $x$.

Obviously, $x\in G(x,x)\subset[x]$ for all $x\in X$. Therefore, if $G(x,x)$ is
invariant, then $G(x,x)=[x]$. The following problem naturally arises.

######  Question 2.

In what binary $G$-spaces is the set $G(x,x)$ invariant?

The solution of this problem in a special case is given by Theorem 9 below.

######  Definition 6.

A binary $G$-space $X$ is said to be distributive if

(20)
$g(h(x,x^{\prime}),h(x,x^{\prime\prime}))=h(x,g(x^{\prime},x^{\prime\prime}))$

for all $x,x^{\prime},x^{\prime\prime}\in X$ and $g,h\in G$.

######  Example 4.

A group $G$ with binary action (14) (see Example 2) is a distributive binary
$G$-space. Indeed, for any $g,h,k,k_{1},k_{2}\in G$, we have

$g(h(k,k_{1}),h(k,k_{2}))=g(khk^{-1}k_{1},khk^{-1}k_{2})=\\\
=khk^{-1}k_{1}gk_{1}^{-1}kh^{-1}k^{-1}khk^{-1}k_{2}=khk^{-1}k_{1}gk_{1}^{-1}k_{2}=\\\
=h(k,k_{1}gk_{1}^{-1}k_{2})=h(k,g(k_{1},k_{2})).$

######  Theorem 9.

If $X$ is a distributive binary $G$-space, then the set $G(x,x)$ is invariant
for any $x\in X$.

###### Proof.

We must show that $g(h(x,x),k(x,x))\in G(x,x)$ for any $h(x,x)$, $k(x,x)\in
G(x,x)$ and any $g\in G$. By virtue of (20), we have

$g(h(x,x),k(x,x))=g(h(x,x),h(x,h^{-1}k(x,x)))=\\\
=h(x,g(x,h^{-1}k(x,x)))=h(x,gh^{-1}k(x,x))=\\\ =hgh^{-1}k(x,x)\in G(x,x),$

because $hgh^{-1}k\in G$. ∎

######  Question 3.

Is the converse of Theorem 9 true?

######  Question 4.

Given a distributive $G$-space $X$ and $x^{\prime}\notin G(x,x)$, is the set
$G(x,x^{\prime})$ invariant?

## References

  * [1] R. Arens.  Topologies for homeomorphism groups.  Amer. J. Math., 68:593–610, 1946.
  * [2] G. E. Bredon.  Introduction to Compact Transformation Groups.  New York, 1972.
  * [3] R. Engelking.  General topology.  Heldermann Verlag, Berlin, 1989.
  * [4] P. S. Gevorgyan.  Equivariant movability of topological groups.  Topology and its Applications, 159(7):1761–1766, 2012.
  * [5] P. S. Gevorgyan.  On binary g-spaces.  Mathematical Notes, 96(4):600–602, 2014.
  * [6] R. A. McCoy and I. Ntantu.  Topological properties of spaces of continuous functions.  Berlin: Springer-Verlag, 1988.