2. Topologies on Smashed Twisted Wreath Products of Metagroups
Definition 1. Let G be a quasigroup and let H be its subquasigroup. Let be a subset in G, such that
and
for each in V.
Then, V is called a transversal set of H in G.
The set of all right cosets with is denoted by (this notation is used in order to distinguish it from , see also Appendix A). Remark 1. Note that for groups, transversal sets are used to study the structure of groups. Therefore, this definition is motivated as their nonassociative analog.
For a metagroup D and a submetagroup A there exists the transversal set by Corollary 1 in [
9]
. According to Formulae (53) in [
9]
there exist single-valued surjective maps and
,
such that , , where , , , . It is also denoted by and or shortly and , respectively, if D and A are specified.
Let and let be a subgroup in such that , where denotes a minimal subgroup in D, such that for each a, b and c in D (see also Appendix A). For a topological metagroup D using the (joint) continuity of multiplication, and on D one can consider, without loss of generality, that is closed in D. This and Condition imply that is a submetagroup in D, since is the commutative (Abelian) group and , where for subsets B and P in D. Hence and are the quotient groups by Theorem 1 in [
9]
(see also Definition 1 above). Therefore, and
for each . Moreover, the transversal set can be chosen such that (see Remark 3 in [
9]
).
Theorem 1. Let V and H be topological metagroups. Let a topological unital quasigroup G be the semidirect product of V with H. Then can be chosen as V and the mappings and are continuous.
Proof. Let be an injective mapping such that and are jointly continuous and , where is supplied with the Tychonoff product topology, is a family of all continuous automorphisms of H, for each h in H, where denotes the unit element in V.
Let for each and in V, and in H, where . This supplies with the semidirect product structure , such that G is the topological unital quasigroup, where G is in the Tychonoff product topology on . Multiplication on G is jointly continuous, since multiplications on V and H are jointly continuous, is jointly continuous in . The equation is equivalent to , with given and in V, and in H, where is to be calculated, since . Therefore, , , hence, . Thus, is the jointly continuous mapping in , , , variables, since , and are the jointly continuous mappings. Symmetrically, it is proved that is jointly continuous. Therefore, G is the topological unital quasigroup. There are the natural embeddings , of V and H into G, since and for each v and in V, h and in H. Then, we infer that
and
for each and in V, and in H, since , , . Then,
;
and
,
hence, ,
for each and in V, and in H, since , , .
One can write shortly instead of with and , since . Therefore, the left transversal set of H in G can be chosen such that with and , since for each there exist unique and such that . That is and are mappings induced by projections from onto V and H correspondingly. Consequently, and are continuous. □
Corollary 1. Let the conditions of Theorem 1 be satisfied. Then the right coset space exists. Moreover, if and , then G is a topological metagroup.
Remark 2. Let G be a topological metagroup and let A be a submetagroup in G and be a transversal set of A in G. Let denote the coarsest topology on G such that the maps , , , , are continuous, where A, are considered in the topologies inherited from , denotes multiplication on G. Generally, this topology may be nondiscrete. Indeed, Theorem 1 demonstrates that for the semidirect product . While V and H can be taken as nondiscrete topological metagroups.
Examples of nonassociative metagroups are given in [
9]
. Particularly, direct products of topological metagroups are topological metagroups. On the other hand, each topological group is also a topological metagroup. Generally, metagroups may be nonassociative. Smashed products or smashed twisted products of metagroups or groups provide nonassociative metagroups. For topological metagroups or groups A and B with jointly continuous smashing factors ϕ, η, κ and ξ this provides topological metagroups by Theorem 3 (or by Theorem 4) in [
9].
In particular, as pairs of A and B the following can be taken:
- (α)
We take the special orthogonal group of the Euclidean space , the special linear group of the Euclidean space , where , A and B are supplied with topologies induced by the operator norm topology.
- (β)
Let be the separable Hilbert space over the complex field , where is supplied with the standard multiplicative norm topology. We consider the unitary group and the general linear group of , where A and B are considered in the topologies inherited from the operator norm topology.
- (γ)
Assume that is an infinite nondiscrete spherically complete field supplied with a multiplicative norm satisfying the strong triangle inequality for each a and b in .
By is denoted a Banach space consisting of all vectors such that for each the cardinality and with a norm , where α is a (nonvoid) set. We consider the linear isometry group and the general linear group of X supplied with topologies inherited from the operator norm topology, where .
Remark 3. In view of Corollary 1 in [
9]
and Remark 1 and and
for each .
In particular, and for each , where D is the metagroup. Denoting and , , , where , , one gets .
From Theorem 1 in [
9]
it follows that is isomorphic with , where for . Moreover, and are the quotient groups, such that , where . If , where belong to A, are in , then , and, hence, . Vice versa, if with , in , then for each in A there exists such that and, consequently, . Thus, the quotient groups and are isomorphic. From the latter isomorphism, Remark 3 in [
9]
, Formulae and above, it follows that , and can be chosen, such that ; ,
since is the invariant subgroup in D and if . Hence,
and for each and , since , since and . Remember that
for each a and c in D (see Formula (68) in [
9]
). Suppose that the conditions of Remark 4 in [
9]
are satisfied. Let on the Cartesian product (or ) for each d, in D, f, in F (or respectively) a binary operation is:
,
where for each (see Formula in [
9]
and Formulae – above).
As a suitable reference, we formulate the following theorem. Its proof is in the Appendix A. Theorem 2. Let G be a topological quasigroup and let , where is an open base at g in G. Then satisfies the following properties –:
, and ;
, , and ;
, , ;
, , , , , , ;
, , , , , , ;
, , , , , , ;
, , , , ;
, .
Conversely, let G be a quasigroup and let be a family of subsets in G satisfying –. Then the family is a base for a topology on G and is a topological quasigroup.
Lemma 1. Let B be a quasigroup, let V be a set, be a family of all (single-valued) maps from V into B, , where , . Then, for nonvoid subsets S, , , in V, Q, , in B:
, , , ;
, , , &;
, , , &&;
, , , ;
for each set Λ, , , , ;
for each set Λ, , , , , .
Proof. . From it follows that .
. , , →;
, , , →.
Thus . Symmetrically it is proved .
. →;
, , →, ; since for each , for each ;
, , , →,
,
.
. , →,
. , ↔,
↔,
. , ↔,
Remark 4. Let the conditions of Theorem 5 in [
9]
be satisfied. We consider the topology on the topological metagroup D, such that multiplication , maps , are jointly continuous, maps , , , are continuous, where A, , and the transversal sets , are in the topology inherited from . Assume that and are as topological spaces, where B is a topological metagroup with a topology . From Theorem 1 and Remark 2 it follows that there is an abundant family of such topological metagroups D with submetagroups and A. Let denote a family of all continuous maps . As usual, U is a canonical closed subset (i.e., a closed domain) in V if and only if , where denotes the interior of U in V, while denotes the closure of S in V, where .
Let be a family of nonvoid canonical closed subsets U in , such that
, , ;
, , , &.
We put , where is the smashed twisted wreath product of D and F with smashing factors ϕ, η, κ, ξ, where . The smashed twisted wreath product of D and F is also shortly denoted by if A is specified (see Definition 5 in [
9]
). Let be a family of all subsets in such that and is open in . Theorem 3. Let the conditions of Remark 4 be satisfied and let the maps be jointly continuous (see Remark 1 in [
9]
). Then is a base of a topology on relative to which is a topological loop. Proof. Evidently, each constant map
belongs to
, where
b in
B is arbitrarily fixed,
for each
. This induces the natural embedding of
B into
, and, consequently,
is the nonvoid metagroup with pointwise multiplication and
and
. If
and
belong to
, then
(see §I.8.8 in [
28], 1.1.C [
29]).
By the conditions of Remark 4
is a base of the topology on
. From
D and
B being the
topological metagroups and, hence, topological quasigroups, it follows that they are
as the topological spaces. Therefore, for each open subset
in
D (or
V) and each
there exists a canonical closed subset (i.e., closed domain)
S in
D (or
V, respectively), such that
(or
, respectively) and
by Proposition 1.5.5 [
29]. Hence
is a base of a topology on
. By virtue of Theorem 5 in [
9] and Remark 4 the maps
and
are continuous on
D and
is the topological metagroup.
In view of Lemma 6 in [
9] the map
is (jointly) continuous as the composition of jointly continuous maps for each
. According to Remark 4 in [
9]
for each
,
,
. The map
is jointly continuous by the conditions of this theorem. Lemma 1 imply that
is the base of a topology on
. We take the topology
generated by the base
, where
denotes the base of the topology
.
Hence, Lemma 1 and Formula
imply that multiplication
is (jointly) continuous. From Formulae in the proof of Theorem 5 in [
9] it follows that
and
are jointly continuous from
into
.
The base generates the topology on by Lemma 1 and Remark 4. Hence is the topology on , since is by the conditions of this theorem. This implies that is the topological loop. □
Theorem 4. Let the conditions of Theorem 3 be satisfied, let be locally compact (or compact) and be a family of all canonical closed compact subsets in , let be closed in , let also be compact for each , where , let for each compact subset Z in V the restriction be evenly continuous, let be a subloop in . Then is a locally compact (or compact respectively) loop.
Proof. Since
and
D is
and locally compact, then
V is closed in
D and, hence, locally compact by Theorem 3.3.8 [
29], and, consequently,
V is a
k-space. From Lemma 1, Remark 4 and the conditions on
it follows that
induces a compact–open topology
on
. For each compact subset
Z in
V and open
in
D the conditions of this theorem imply that
is evenly continuous, since
for each
. By virtue of Theorem 3.4.21 [
29]
is compact. Since
is locally compact, it is sufficient to take any open
in
D with the compact closure
in the
topology. Moreover,
is closed in
and, hence, compact for each
. From
B being
it follows that
B is
.
From the compactness of and Theorem 3 it follows that is either the locally compact loop, if D is locally compact, or the compact loop if D is compact. □
Proposition 1. Let the conditions of Theorem 5 in [
9]
be satisfied and let and be automorphisms of the metagroups D and B such that . Then there exists a loop and an isomorphism of C onto such that and .
Proof. By the conditions of this proposition
,
and
for each
a and
b in
D, similarly for
. Therefore,
is an isomorphism of metagroups and
is an isomorphism of groups such that
. In view of Corollary 1 in [
9]
,
,
,
.
We put
with
, , ,
, ,
such that
,
,
for each , in , , in B; , such that
, for each . Let
for each and , where , . Therefore, and imply that
and
for each and in C. Since C and are loops, then and imply that and for each g and in C. Thus is the isomorphism of these loops. □
Corollary 2. Assume that the conditions of Proposition 1 are satisfied and i, , j, are continuous relative to and topologies on D and B, respectively. Then, and are continuous relative to and topologies on and , respectively.
Proof. This follows from Proposition 1, Formula and Theorem 3. □
Remark 5. If the conditions of Theorem 6 instead of that of Theorem 5 in [
9]
are satisfied, then in Theorems 3 and 4, Corollary 2 , , are topological metagroups; in Proposition 1 C and are metagroups.