2. Basic Definitions and Facts
First, we recall the definitions of basic notions and some known facts which will be used in the following.
A quasigroup is a groupoid in which for every two elements every of the equations and has a unique solution in X. Instead of we write . If a quasigroup X contains an identity element, then X is called a loop. Evidently, every associative loop is a group. A quasigroup is called an IP-quasigroup (or a quasigroup with the invertibility property), if there exist mappings and such that, for any it holds and . An IP-quasigroup with an identity element is called an IP-loop. In any IP-loop every element x in X has an inverse and it holds for every It is easy to see that IP-loop is a groupoid with an identity element and with the following property: for each there exists an element such that and for every .
The theory of quasigroups and loops was established by Bruck [
7,
8], see also [
9,
10,
11]. Moufang loops [
15,
16] are a very important case of IP-loops. Another interesting example of an IP-loop is the octonion (Cayley) algebra
[
17]. Every octonion
x can be written in the form:
where
are the unit octonions,
is the scalar element (the real number 1) and
are real coefficients. By linearity and distributivity, multiplication of octonions is completely determined once given a multiplication table for the unit octonions (see, e.g., [
17]). The conjugate of an octonion:
is given by:
Conjugation is an involution of
O and satisfies
This octonionic multiplication is neither commutative (
if
are distinct and non-zero) nor associative. On other hand, the nonzero elements of
O form an IP-loop. The norm of the octonion
x is defined as:
where
is the conjugate of
x. So this norm agrees with standard Euclidean norm on
The existence of a norm on
O implies the existence of inverses for all nonzero elements of
O. The inverse of
x,
, is given by the equality
It satisfies
The couple
, where the operation
is defined by the equality:
is also an IP-loop.
A metric d on a groupoid is said to be left-invariant, if for every . A right-invariant metric is defined analogously and a metric is said to be bi-invariant if it is both left and right invariant. The term “invariant” hence means that the distance is unchanged when you translate by a fixed element a. If X is abelian, then both left and right invariance implies bi-invariance and we simply say that d is invariant.
Example 1. Let be a set of all octonions with a unit norm. Let d be the metric induced by the norm on . Let x, y, a be octonions with a unit norm:Since:the metric is left-invariant. Analogously we obtain that is right-invariant. Example 2. Let where is a set of octonions with a unit norm, Put:
where is the metric from Example 1. It is easy to verify that is also a metric. Since is left-invariant, we obtain:This means that the metric is left-invariant. Analogously we obtain that the metric is right-invariant. A topological IP-loop is an IP-loop
with Hausdorff topology such that the transformation
X X X defined by (
x,
y)
x−1y is continuous. A topological IP-loop is said to be connected, totally disconnected, compact, locally compact, etc., if the corresponding property holds for its underlying topological space. Topological IP-loops are studied, e.g., in [
18,
19,
20,
21].
In the following section, we will deal with topological IP-loops with a left-invariant uniformity. Therefore, we recall the definition of a uniform topology and remind the facts which will be further used.
A uniformity of a set X is non-empty system W of subsets of the Cartesian product X X such that the following conditions are satisfied:
(1) Every element of W contains the diagonal
(2) If then
(3) If then there exists such that where there exists such that and are in
(4) If then
(5) If and then
The above described couple
is called a uniform space and its elements entourages. Every uniform space
X becomes a topological space by defining a subset
U of
X to be open if and only if for every
there exists an entourage
V such that
is a subset of
U. A basis of a uniformity
W is any system
of entourages of
W such that every entourage of
W contains a set belonging to
. Every uniform space has a basis of entourages consisting of symmetric entourages. It is known (see [
22,
23]) that any system
of subsets of the Cartesian product
is a basis of some uniformity of
X if and only if the following conditions are satisfied:
(6) Every element of contains the diagonal
(7) If , then there exists such that
(8) If , then there exists such that
(9) If , then there exists such that
We will use throughout this paper the following notations. If and are any two subsets of a groupoid then is the set of all elements of the form where and If it is customary to write and in place of and respectively. The set (or ) is called a left translation (or right translation) of E by the element
A uniformity of a groupoid is called left-invariant, if it has a left-invariant basis , i.e., a basis such that, for every it holds (a, a) = , where (a, a) (x, y) = (ax, ay). A right-invariant uniformity is defined analogously. An invariant uniformity is a left-invariant uniformity which also satisfies a right-invariant condition.
A uniform topology is a generalization of a metric topology because if
is a metric space, then the system
=
where
is a basis of some uniformity of
X. If a metric
d of a quasigroup
is left-invariant, then a uniformity induced by the metric
d is left-invariant, too. Indeed, for all
and any
, we have:
Analogously, if a metric d of a quasigroup is right-invariant, then a uniformity induced by the metric d is also right-invariant.
3. Results
The following example shows that a topological quasigroup is not metrizable by invariant metric, in general.
Example 3. Let be the set of all real numbers and be a binary operation on defined, for every by the prescription Evidently, the couple is a topological quasigroup with standard topology. We shall show that this topological quasigroup is not metrizable topological space by an invariant metric. Suppose that there is a left-invariant metric d in and the metric topology induced by d coincides with standard topology. So the metric and standard metric on are equivalent. We will derive a contradiction. Let Since d is left-invariant, for every element a in we have:for any Put The sequences converge in the standard metric and then also in the metric to element We derive a contradiction:This means that the topological quasigroup with standard topology is not metrizable topological space by left-invariant metric, and also by right-invariant metric because the operation is commutative. Proposition 1. Let be an IP-loop and be a sequence of subsets of such that:
(1)
(2) for
(3) for
(4) for and every
Then there exists a non-negative real-valued function with the following properties:
(5) d(x, z) ≤ d(x, y) + d(y, z), for every elements x, y, z
(6) d(x, y) = d(ax, ay), for every elements a, x, y
(7) {(x, y); d(x, y) < } for
If, moreover, every set is symmetric, then in there exists a pseudometric d with the above properties.
Proof. Let us define a non-negative real-valued function
as follows:
Then the searched function
d has the following form:
where the infimum is taken over all elements
We shall prove that the function
has the required properties. For every
x,
y,
z we have:
This means that the first property holds.
Now, we shall prove the second property. Since is an IP-loop, the condition is equivalent to the condition for and every Therefore if and only if Thus, if and only if
we get f(ax, ay) = f(x, y), for every a, x, y
Let
be any positive real number. Then there are elements
such that:
so
d(
x,
y) ≥
d(
ax,
ay).
Analogously, there are elements
such that:
The couple
is an IP-loop, and therefore there exist the elements
such that
for
Hence:
so
Therefore, we can conclude that
for every
a,
x,
y It remains to prove the third property. The inclusions are the consequence of the previous two conditions. If then and < This means that {(x, y); d(x, y) < }.
Let us prove the inclusion {(
x,
y);
d(
x,
y) <
}
for
If
<
then there are the elements
where
and
such that:
By induction on k we prove that
For we have < and therefore If we have < what implies the inequalities Hence we get Since we have
Assume that it holds for some i.e., < such that and Now, if < then and < There exists such that and So we can conclude that
Since, by means of the second condition, we have for it holds for every and so for every If is a sequence of symmetric subsets of then for every and therefore for every The constructed function d is a pseudometric. The proof is completed. ☐
Theorem 1. Let be an IP-loop. A uniform space is pseudometrizable by a left-invariant pseudometric if and only if the uniformity of a set X has a left-invariant countable base. A uniform space is metrizable by a left-invariant metric if and only if the corresponding topological space is Hausdorff and the uniformity W of X has a left-invariant countable base.
Proof. If a uniformity W of a set X has a left-invariant countable base, then we can construct a system of symmetric subsets of a set satisfying the condition of the preceding proposition. Therefore, the uniform space is pseudometrizable by a left-invariant pseudometric. The second part of assertion of the theorem is obvious. ☐
If
is any topological IP-loop, one can consider the topological IP-loop
dual to
The topological IP-loop
has, by definition, the same elements and same topology as
the product
in
is defined , for every
by
Since, for every
it holds:
and:
we see that
is in fact an IP-loop.
Let
be a right-invariant uniform IP-loop. Consider the topological IP-loop
dual to
If
is a right-invariant base of the uniformity
of
then
is a left-invariant base of a uniformity
of the groupoid
Indeed, for every
where
is any entourage in
, and for every
we have:
Thus the topological IP-loop has a topology induced by a left-invariant uniformity. If is a left-invariant metric in then, for every it holds and hence the metric is right-invariant in
From the preceding considerations we obtain the following assertion.
Theorem 2. Let be an IP-loop. A uniform space is metrizable by a right-invariant metric if and only if the corresponding topological space is Hausdorff and the uniformity of X has a right-invariant countable base.