Topologies on Smashed Twisted Wreath Products of Metagroups

Abstract

In this article, topologies on metagroups and quasigroups are studied. Topologies on smashed twisted wreath products of metagroups are scrutinized, which are making them topological metagroups. For this purpose, transversal sets are studied. As a tool for this, semi-direct products of topological metagroups are also investigated. They have specific features in comparison with topological groups because of the nonassociativity, in general, of metagroups. A related structure of topological metagroups is investigated. Particularly, their compact subloops and submetagroups are studied. Isomorphisms of topological unital quasigroups (i.e., loops) obtained by the smashed twisted wreath products are investigated. Examples are provided.

1. Introduction

A topological group structure plays a very important role in mathematics and its applications [1,2,3]. Topologies on groupoids, semi-groups and other algebraic structures attract great attention. In the associative case for topological groups, a lot of investigations have already been made, but in regard to the nonassociative case for topological quasigroups or metagroups comparatively little is known.

On the other hand, noncommutative analysis is a very important part of mathematical analysis and it interacts with operator theory, operator algebras and algebraic analysis. In particular, analysis over octonions and generalized Cayley–Dickson algebras has developed fast in recent years (see [4,5,6,7,8,9,10,11,12,13,14,15,16,17] and references therein). It also plays a huge role in noncommutative geometry, mathematical physics, quantum field theory and quantum gravity, partial differential equations (PDEs), particle physics, operator theory, etc. [18,19,20].

It appears that a multiplicative law of their canonical bases is nonassociative and leads to a more general notion of a metagroup instead of a group [5,21,22]. They were used in [14,15,21,22,23] for investigations of partial differential operators and other unbounded operators over quaternions and octonions, also for automorphisms, derivations and cohomologies of generalized $C^{\ast }$-algebras over $\mathbf{R}$ or $\mathbf{C}$. They certainly have a lot of specific features in their derivations and (co)homology theory [21,22]. It was shown in [23] that an analog of the Stone theorem for one parameter group of unitary operators for the generalized $C^{\ast }$-algebras over quaternions and octonions becomes more complicated and multiparameter. The generalized $C^{\ast }$-algebras arise naturally while decompositions of PDEs or systems of PDEs of higher order into PDEs or their systems of order not higher than two [6,7,10,16] permits subsequently integrating them or simplifying their analysis.

In [9] different types of products of metagroups were studied, such as smashed products and smashed twisted wreath products. This also permitted construction of their abundant families different from groups. Metagroups appear naturally in noncommutative geometry and noncommutative analysis. However, topological quasigroups and topological metagroups are little studied in comparison with topological groups.

Notice also that a loop is quite a different object to a loop group considered in geometry or mathematical physics. Certainly loops are more general objects than metagroups. Note that metagroups are commonly nonassociative, having many specific features in comparison with loops and groups [9,24,25]. On the other hand, if a loop G is simple, then a subloop generated by all elements of the form $\left(\right(ab\left)c\right)/\left(a\right(bc\left)\right)$ for all a, b, c in G coincides with G [24,26]. Metagroups are intermediate between groups and quasigroups. Functions on topological metagroups were studied in [27].

In this article, topologies on metagroups and quasigroups are studied. Methods of algebraic topology are used. They have specific features in comparison with groups because of the nonassociativity, in general, of metagroups. In Section 2 topologies on smashed twisted wreath products of metagroups are scrutinized, which are making them topological unital quasigroups (i.e., loops) or metagroups (see Theorem 3 and Remark 5). For this purpose transversal sets also are studied. Their compact subloops and submetagroups are studied in Theorem 4 (see also Remark 5). Isomorphisms of topological unital quasigroups (i.e., loops) obtained by the smashed twisted wreath products are investigated in Proposition 1 and Corollary 2. Particularly, for topological metagroups, isomorphisms are provided by Proposition 1, Corollary 2 and Remark 5. A related structure of topological quasigroups and metagroups is investigated in Theorem 2 and Lemma 1. Semi-direct products of topological metagroups are studied in Theorem 1 and Corollary 1. This permits the construction of abundant families of topological nondiscrete metagroups and topological unital quasigroups for which transversal maps are continuous (see Remark 2). Examples are discussed in Remark 2. Necessary definitions and notations are recalled in the Appendix A.

All the main results of this paper were obtained for the first time. Their possible applications are outlined in Section 3.

2. Topologies on Smashed Twisted Wreath Products of Metagroups

Definition 1.

Let G be a quasigroup and let H be its subquasigroup. Let $V=V_{G,H}$ be a subset in G, such that

$G={\bigcup }_{v\in V}Hv$ and

$\left(H{v}_1\right)\cap \left(H{v}_2\right)=\varnothing$ for each $v_1\ne v_2$ in V.

Then, V is called a transversal set of H in G.

The set of all right cosets $Hg$ with $g\in G$ is denoted by $D{\backslash }_{c}A$ (this notation is used in order to distinguish it from $Di{v}_l(a,b)=ab$, 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 $V=V_{D,A}$ by Corollary 1 in [9]. According to Formulae (53) in [9] there exist single-valued surjective maps

${\psi }_A^D:D\to A$ and

${\tau }_A^D:D\to V_{D,A}$,

• such that ${\psi }_A^D\left(d\right)=a$, ${\tau }_A^D\left(d\right)=v$, where $d\in D$, $d=av$, $a\in A$, $v\in V$. It is also denoted by $d^{{\psi }_A^D}={\psi }_A^D\left(d\right)$ and $d^{{\tau }_A^D}={\tau }_A^D\left(d\right)$ or shortly $d^{\psi }=\psi \left(d\right)$ and $d^{\tau }=\tau \left(d\right)$, respectively, if D and A are specified.

Let ${\mathcal{C}}_A\left(D\right)=\mathcal{C}\left(D\right)\cap A$ and let ${\mathcal{C}}_1$ be a subgroup in $\mathcal{C}\left(D\right)$ such that ${\mathcal{C}}_m\left(D\right)\subset {\mathcal{C}}_1$, where ${\mathcal{C}}_m\left(D\right)$ denotes a minimal subgroup in D, such that $t_D(a,b,c)\in {\mathcal{C}}_m\left(D\right)$ for each a, b and c in D (see also Appendix A). For a topological metagroup D using the (joint) continuity of multiplication, $Di{v}_l$ and $Di{v}_r$ on D one can consider, without loss of generality, that ${\mathcal{C}}_1$ is closed in D. This and Condition $\left(A9\right)$ imply that $A{\mathcal{C}}_1$ is a submetagroup in D, since ${\mathcal{C}}_1$ is the commutative (Abelian) group and ${\mathcal{C}}_m\left(D\right)\subset {\mathcal{C}}_1$, where $PB=\{c\in D:\exists p\in P,\exists b\in B,c=pb\}$ for subsets B and P in D. Hence $D{}_c{\mathcal{C}}_1$ and $\left(A{\mathcal{C}}_1\right){}_c{\mathcal{C}}_1$ are the quotient groups by Theorem 1 in [9] (see also Definition 1 above). Therefore,

${\psi }_A^D\left(d\right)={\psi }_A^{A{\mathcal{C}}_1}\circ {\psi }_{A{\mathcal{C}}_1}^D\left(d\right)$ and

${\tau }_A^D\left(d\right)={\tau }_A^{A{\mathcal{C}}_1}({\psi }_{A{\mathcal{C}}_1}^D\left(d\right)){\tau }_{A{\mathcal{C}}_1}^D\left(d\right)$

• for each $d\in D$. Moreover, the transversal set $V_{A{\mathcal{C}}_1,A}$ can be chosen such that $V_{A{\mathcal{C}}_1,A}\subset V_{D,A}$ (see Remark 3 in [9]).

Theorem 1.

Let V and H be topological metagroups. Let a topological unital quasigroup G be the semidirect product $G=V{⨂}^{s}H$ of V with H. Then $V_{G,H}$ can be chosen as V and the mappings ${\tau }_H^G$ and ${\psi }_H^G$ are continuous.

Proof. 

Let $\mu :H\to Au{t}_c\left(H\right)$ be an injective mapping such that $V\times H\ni (v,h)\mapsto \mu \left(v\right)h\in H$ and $V\times H\ni (v,h)\mapsto {\left(\mu \left(v\right)\right)}^{-1}h\in H$ are jointly continuous and $\mu \left(e_V\right)=i{d}_H$, where $V\times H$ is supplied with the Tychonoff product topology, $Au{t}_c\left(H\right)$ is a family of all continuous automorphisms of H, $i{d}_H\left(h\right)=h$ for each h in H, where $e_V$ denotes the unit element in V.

Let $(v_1,h_1)(v_2,h_2)=(v_1{v}_2,h_1{h}_2^{v_1})$ for each $v_1$ and $v_2$ in V, $h_1$ and $h_2$ in H, where $h_2^{v_1}=\mu \left(v_1\right)h_2$. This supplies $V\times H$ with the semidirect product structure $G=V{⨂}^{s}H$, such that G is the topological unital quasigroup, where G is in the Tychonoff product topology on $V\times H$. Multiplication on G is jointly continuous, since multiplications on V and H are jointly continuous, $\mu \left(v\right)h$ is jointly continuous in $(v,h)\in V\times H$. The equation $(v_1,h_1)(x,y)=(v_2,h_2)$ is equivalent to $v_{1}x=v_2$, $h_1{y}^{v_1}=h_2$ with given $v_1$ and $v_2$ in V, $h_1$ and $h_2$ in H, where $(x,y)$ is to be calculated, since $(v_1,h_1)(x,y)=(v_{1}x,h_1{y}^{v_1})$. Therefore, $x=v_1{v}_2$, $y^{v_1}=h_1{h}_2$, hence, $y={\left(\mu \left(v_1\right)\right)}^{-1}(h_1{h}_2)$. Thus, $Di{v}_l((v_1,h_1),(v_2,h_2))=(v_1{v}_2,{\left(\mu \left(v_1\right)\right)}^{-1}(h_1{h}_2))$ is the jointly continuous mapping in $v_1$, $v_2$, $h_1$, $h_2$ variables, since $Di{v}_l(v_1,v_2)=v_1{v}_2$, $Di{v}_l(h_1,h_2)=h_1{h}_2$ and ${\left(\mu \left(v_1\right)\right)}^{-1}(h_1{h}_2)$ are the jointly continuous mappings. Symmetrically, it is proved that $Di{v}_r$ is jointly continuous. Therefore, G is the topological unital quasigroup. There are the natural embeddings $V\ni v\mapsto (v,e_H)\in G$, $H\ni h\mapsto (e_V,h)\in G$ of V and H into G, since $(v,e_H)(v_1,e_H)=(v{v}_1,e_H)$ and $(e_V,h)(e_V,h_1)=(e_V,h{h}_1)$ for each v and $v_1$ in V, h and $h_1$ in H. Then, we infer that

$(v_2,h_2)\left((v_1,h_1)(e_V,H)\right)=(v_2,h_2)(v_1,H)=(v_2{v}_1,h_{2}H)=(v_2{v}_1,H)$ and

$\left((v_2,h_2)(v_1,h_1)\right)(e_V,H)=(v_2{v}_1,h_2{h}_1^{v_2})(e_V,H)=(v_2{v}_1,H)$

• for each $v_1$ and $v_2$ in V, $h_1$ and $h_2$ in H, since $\mu \left(e_V\right)=i{d}_H$, $H^{v_2}=H$, $h_{2}H=H$. Then,

$(v_1,h_1)(e_V,H)=(v_1,h_{1}H)=(v_1,H)=(e_V,H)(v_1,h_1)$;

$(v_2,h_2)\left((e_V,H)(v_1,h_1)\right)=(v_2,h_2)(v_1,H)=(v_2{v}_1,H)$ and

$\left((v_2,h_2)(e_V,H)\right)(v_1,h_1)=(v_2,H)(v_1,h_1)=(v_2{v}_1,H{h}_1^{v_2})=(v_2{v}_1,H)$,

• hence, $(e_V,H)\left((v_2,h_2)(v_1,h_1)\right)=\left((v_2,h_2)(v_1,h_1)\right)(e_V,H)=(v_2,h_2)\left((v_1,h_1)(e_V,H)\right)=\left((v_2,h_2)(e_V,H)\right)(v_1,h_1)=\left((e_V,H)(v_2,h_2)\right)(v_1,h_1)$,
• for each $v_1$ and $v_2$ in V, $h_1$ and $h_2$ in H, since $\mu \left(e_V\right)=i{d}_H$, $H^{v_2}=H$, $h_{2}H=H=H{h}_2$.

One can write shortly $g=vh$ instead of $g=(v,h)$ with $v\in V$ and $h\in H$, since $(v,e_H)(e_V,h)=(v,h)=vh$. Therefore, the left transversal set $V_{G,H}$ of H in G can be chosen such that $V_{G,H}=V$ with $\tau \left(g\right)=v$ and $\psi \left(g\right)=h$, since for each $g\in G$ there exist unique $v\in V$ and $h\in H$ such that $g=vh$. That is $\tau :G\to V$ and $\psi :G\to H$ are mappings induced by projections from $V\times H$ onto V and H correspondingly. Consequently, $\tau$ and $\psi$ are continuous. □

Corollary 1.

Let the conditions of Theorem 1 be satisfied. Then the right coset space $G{}_{c}H$ exists. Moreover, if ${\mathcal{C}}_m\left(V\right)\cup {\mathcal{C}}_m\left(H\right)\subseteq \mathcal{C}\left(V\right)\cap \mathcal{C}\left(H\right)$ and $\mu \left({\mathcal{C}}_m\left(H\right)\right)\subseteq Au{t}_c\left({\mathcal{C}}_m\left(H\right)\right)$, then G is a topological metagroup.

Remark 2.

Let G be a topological metagroup and let A be a submetagroup in G and $V_{G,A}$ be a transversal set of A in G. Let ${\mathcal{T}}_{G,A}$ denote the coarsest topology on G such that the maps $m_G$, $Di{v}_l$, $Di{v}_r$, ${\psi }_A^G$, ${\tau }_A^G$ are continuous, where A, $V_{G,A}$ are considered in the topologies inherited from $(G,{\mathcal{T}}_{G,A})$, $m_G$ denotes multiplication on G. Generally, this topology may be nondiscrete. Indeed, Theorem 1 demonstrates that ${\mathcal{T}}_{G,H}={\mathcal{T}}_G$ for the semidirect product $G=V{⨂}^{s}H$. 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 $A{⨂}^{\varphi ,\eta ,\kappa ,\xi }B$ by Theorem 3 (or $A{\star }^{\varphi ,\eta ,\kappa ,\xi }B$ 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 $A=SO(n,\mathbf{R})$ of the Euclidean space ${\mathbf{R}}^n$, the special linear group $B=SL(m,\mathbf{R})$ of the Euclidean space ${\mathbf{R}}^m$, where $1<n\le m\in \mathbf{N}$, A and B are supplied with topologies induced by the operator norm topology.

(β)

Let $l_2$ be the separable Hilbert space over the complex field $\mathbf{C}$, where $\mathbf{C}$ is supplied with the standard multiplicative norm topology. We consider the unitary group $A=U\left(l_2\right)$ and the general linear group $B=GL\left(l_2\right)$ of $l_2$, where A and B are considered in the topologies inherited from the operator norm topology.

(γ)

Assume that $\mathbf{F}$ is an infinite nondiscrete spherically complete field supplied with a multiplicative norm satisfying the strong triangle inequality $|a+b|\le max\left(\right|a|,|b\left|\right)$ for each a and b in $\mathbf{F}$.

By $c_0(\alpha ,\mathbf{F})=X$ is denoted a Banach space consisting of all vectors $x=(x_j:\forall j\in \alpha ,x_j\in \mathbf{F})$ such that for each $ϵ>0$ the cardinality $card(j\in \alpha :|x_j|>ϵ)<{\aleph }_0$ and with a norm $\left|x\right|={sup}_{j\in \alpha }\left|x_j\right|$, where α is a (nonvoid) set. We consider the linear isometry group $A=IL\left(X\right)$ and the general linear group $B=GL\left(X\right)$ of X supplied with topologies inherited from the operator norm topology, where $IL\left(X\right)=\{g\in GL(X):\forall x\in X,|g\left(x\right)|=|x\left|\right\}$.

Remark 3.

In view of Corollary 1 in [9] and Remark 1

$\left(1\right)$${\psi }_A^D\left({\psi }_A^D\left(d\right)\right)={\psi }_A^D\left(d\right)$ and ${\tau }_A^D\left({\tau }_A^D\left(d\right)\right)={\tau }_A^D\left(d\right)$ and

$\left(2\right)$$d=d^{{\psi }_A^D}{d}^{{\tau }_A^D}$ for each $d\in D$.

• In particular, ${(d^{{\tau }_A^D})}^{{\psi }_A^D}=e$ and ${(d^{{\psi }_A^D})}^{{\tau }_A^D}=e$ for each $d\in D$, where D is the metagroup. Denoting $a=d^{\psi }$ and $v=d^{\tau }$, $a=a_d$, $v=v_d$, where $\psi ={\psi }_A^D$, $\tau ={\tau }_A^D$, one gets $a=d/v$.

From Theorem 1 in [9] it follows that ${\mathcal{C}}_1^{\tau }$ is isomorphic with ${\mathcal{C}}_1{}_c{\mathcal{C}}_{1,A}$, where $B^{\tau }=\{b^{\tau }:b\in B\}$ for $B\subset D$. Moreover, $\left(A{\mathcal{C}}_1\right){}_c{\mathcal{C}}_{1,A}$ and $A{}_c{\mathcal{C}}_{1,A}$ are the quotient groups, such that $A{}_c{\mathcal{C}}_{1,A}\hookrightarrow \left(A{\mathcal{C}}_1\right){}_c{\mathcal{C}}_{1,A}$, where ${\mathcal{C}}_{1,A}={\mathcal{C}}_1\cap A$.

If $a_1{n}_1=b\left(a_2{n}_2\right)$, where $a_1,a_2,b$ belong to A, $n_1,n_2$ are in ${\mathcal{C}}_1$, then $(a_2(b{a}_1))n_1=n_2$, and, hence, $(a_2(b{a}_1))\in A\cap {\mathcal{C}}_1={\mathcal{C}}_{1,A}$. Vice versa, if $n_1=\beta n_2$ with $\beta \in {\mathcal{C}}_{1,A}$, $n_1,n_2$ in ${\mathcal{C}}_1$, then for each $a_1,a_2$ in A there exists $b=\left(a_1\beta \right)/a_2\in A$ such that $a_1\beta =b{a}_2$ and, consequently, $a_1{n}_1=a_1\beta n_2=b\left(a_2{n}_2\right)$. Thus, the quotient groups $\left(A{\mathcal{C}}_1\right){}_{c}A$ and ${\mathcal{C}}_1{}_c{\mathcal{C}}_{1,A}$ are isomorphic. From the latter isomorphism, Remark 3 in [9], Formulae $\left(1\right)$ and $\left(2\right)$ above, it follows that $V_{D,A}$, $V_{{\mathcal{C}}_1,{\mathcal{C}}_{1,A}}$ and $V_{D,A{\mathcal{C}}_1}$ can be chosen, such that

$\left(3\right)$$V_{{\mathcal{C}}_1,{\mathcal{C}}_{1,A}}{V}_{D,A{\mathcal{C}}_1}=V_{D,A}$; $V_{{\mathcal{C}}_1,{\mathcal{C}}_{1,A}}=V_{A{\mathcal{C}}_1,A}$,

• since ${\mathcal{C}}_1$ is the invariant subgroup in D and $p_D(a,b,c)=e$ if $e\in \{a,b,c\}\subset D$. Hence,

$\left(4\right)$${\left(d^{\psi }\gamma \right)}^{\psi }=d^{\psi }{\gamma }^{\psi }$ and ${\left(d^{\psi }\gamma \right)}^{\tau }={\gamma }^{\tau }$ for each $d\in D$ and $\gamma \in {\mathcal{C}}_1$, since $d^{\psi }\gamma =d^{\psi }\left({\gamma }^{\psi }{\gamma }^{\tau }\right)=\left(d^{\psi }{\gamma }^{\psi }\right){\gamma }^{\tau }$, since ${\gamma }^{\psi }\in {\mathcal{C}}_{1,A}$ and ${\gamma }^{\tau }={\gamma }^{\psi }\gamma \in {\mathcal{C}}_1$. Remember that

$\left(5\right)$${\left(a^{\tau }\right)}^{\left[c\right]}:={\left(a^{\tau }c\right)}^{\tau }=:{\nu }_{D,A}(a,c)$

• 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 $C=D\times F$ (or $C^{\ast }=D\times F^{\ast }$) for each d, $d_1$ in D, f, $f_1$ in F (or $F^{\ast }$ respectively) a binary operation is:

$\left(6\right)$$(d_1,f_1)(d,f)=(d_{1}d,\xi ((d_1^{\psi },f_1),(d^{\psi },f))f_1{f}^{\{d_1\}})$,

• where $\xi ((d_1^{\psi },f_1),(d^{\psi },f))\left(v\right)=\xi ((d_1^{\psi },f_1\left(v\right)),(d^{\psi },f\left(v\right)))\in {\mathcal{C}}_1$ for each $v\in V_{D,A}$ (see Formula $\left(86\right)$ in [9] and Formulae $\left(3\right)$–$\left(5\right)$ above).

As a suitable reference, we formulate the following theorem. Its proof is in the Appendix A.

Theorem 2.

Let G be a $T_1$ topological quasigroup and let $\mathcal{B}={\bigcup }_{g\in G}{\mathcal{B}}_g$, where ${\mathcal{B}}_g$ is an open base at g in G. Then $\mathcal{B}$ satisfies the following properties $\left(7\right)$–$\left(14\right)$:

$\left(7\right)$$\forall g\in G,$$\forall h\in G$, ${\mathcal{B}}_{hg}=h{\mathcal{B}}_g$ and ${\mathcal{B}}_{gh}={\mathcal{B}}_{g}h$;

$\left(8\right)$$\forall g\in G$, $\forall h\in G$, ${\mathcal{B}}_{hg}=h{\mathcal{B}}_g$ and ${\mathcal{B}}_{g/h}={\mathcal{B}}_g/h$;

$\left(9\right)$$\forall g\in G$, $\forall B\in {\mathcal{B}}_g$, $g\in B$;

$\left(10\right)$$\forall g\in G$, $\forall U_g\in {\mathcal{B}}_g$, $\forall a\in G$, $\exists U_a\in {\mathcal{B}}_a$, $\exists b=ag\in G$, $\exists U_b\in {\mathcal{B}}_b$, $U_a{U}_b\subset U_g$;

$\left(11\right)$$\forall g\in G$, $\forall U_g\in {\mathcal{B}}_g$, $\forall a\in G$, $\exists U_a\in {\mathcal{B}}_a$, $\exists b=ga\in G$, $\exists U_b\in {\mathcal{B}}_b$, $U_a/U_b\subset U_g$;

$\left(12\right)$$\forall g\in G$, $\forall U_g\in {\mathcal{B}}_g$, $\forall a\in G$, $\exists U_a\in {\mathcal{B}}_a$, $\exists b=a/g\in G$, $\exists U_b\in {\mathcal{B}}_b$, $U_b{U}_a\subset U_g$;

$\left(13\right)$$\forall g\in G$, $\forall U\in {\mathcal{B}}_g$, $\forall V\in {\mathcal{B}}_g$, $\exists W\in {\mathcal{B}}_g$, $W\subset U\cap V$;

$\left(14\right)$$\forall g\in G$, ${\bigcap }_{U\in {\mathcal{B}}_g}=\left\{g\right\}$.

Conversely, let G be a quasigroup and let $\mathcal{B}$ be a family of subsets in G satisfying $\left(7\right)$–$\left(14\right)$. Then the family $\mathcal{B}$ is a base for a $T_1$ topology ${\mathcal{T}}_{\mathcal{B}}={\mathcal{T}}_{\mathcal{B}}\left(G\right)$ on G and $(G,{\mathcal{T}}_{\mathcal{B}})$ is a topological quasigroup.

Lemma 1.

Let B be a quasigroup, let V be a set, $F=B^V=\{f:V\to B\}$ be a family of all (single-valued) maps from V into B, $W(S,Q):=\{f\in F:f(S)\subset Q\}$, where $V\supset S\ne \varnothing$, $B\supset Q\ne \varnothing$. Then, for nonvoid subsets S, $S_1$, $S_2$, $S_i$ in V, Q, $Q_1$, $Q_i$ in B:

$\left(15\right)$$\forall S\subset V$, $\forall Q_2\subset B$, $\forall Q_1\subset Q_2$, $W(S,Q_1)\subset W(S,Q_2)$;

$\left(16\right)$$\forall b\in B$, $\forall S\subset V$, $\forall Q\subset B$, $W(S,b/Q)=b/W(S,Q)$&$W(S,Qb)=W(S,Q)b$;

$\left(17\right)$$\forall S\subset V$, $\forall Q\subset B$, $\forall Q_1\subset B$, $W(S,Q)W(S,Q_1)\subset W(S,Q{Q}_1)$&$W(S,Q)/$$W(S,Q_1)\subset W(S,Q/Q_1)$&$W(S,Q)W(S,Q_1)\subset W(S,Q{Q}_1)$;

$\left(18\right)$$\forall S_2\subset V$, $\forall S_1\subset S_2$, $\forall Q\subset B$, $W(S_2,Q)\subset W(S_1,Q)$;

$\left(19\right)$ for each set Λ, $\forall i\in \mathsf{\Lambda }$, $\forall S_i\subset V$, $\forall Q\subset B$, $W({\bigcup }_{i\in \mathsf{\Lambda }}{S}_i,Q)={\bigcap }_{i\in \mathsf{\Lambda }}W(S_i,Q)$;

$\left(20\right)$ for each set Λ, $\forall i\in \mathsf{\Lambda }$, $\forall Q_i\subset B$, $({\bigcap }_{i\in \mathsf{\Lambda }}{Q}_i\ne \varnothing )$, $\forall S\subset V$, $W(S,{\bigcap }_{i\in \mathsf{\Lambda }}{Q}_i)={\bigcap }_{i\in \mathsf{\Lambda }}W(S,Q_i)$.

Proof. 

$\left(15\right)$. From $f\left(S\right)\subset Q_1$ it follows that $f\left(S\right)\subset Q_2$.

$\left(16\right)$. $(\forall f\in W(S,b/Q),$$\forall s\in S$, $\exists q\in Q$, $\left(f\right(s)=b/q\leftrightarrow q=f(s)b))$→$(\exists f_1\in W(S,Q),$$(f=b/f_1\leftrightarrow f_1=fb)$;

$(\forall f_1\in W(S,Q)$, $\forall s\in S$, $\exists q_1\in Q$, $f_1\left(s\right)=q_1)$→$(\exists f\in W(S,b/Q),$$(f_1=fb\leftrightarrow f=b/f_1))$.

Thus $W(S,b/Q)=b/W(S,Q)$. Symmetrically it is proved $W(S,Qb)=W(S,Q)b$.

$\left(17\right)$. $(\forall f\in W(S,Q),$$\forall f_1\in W(S,Q_1),$$\forall s\in S,$$f\left(s\right)f_1\left(s\right)\in Q{Q}_1)$→$(\forall f\in W(S,Q),$$\forall f_1\in W(S,Q_1),$$f{f}_1\in W(S,Q{Q}_1))$;

$(\forall f\in W(S,Q)$, $\forall f_1\in W(S,Q_1),$$\forall s\in S$, $f\left(s\right)/f_1\left(s\right)\in Q/Q_1)$→$(\forall f\in W(S,Q),$$\forall f_1\in W(S,Q_1)$, $f/f_1\in W(S,Q/Q_1))$; since $f\left(S\right)\subset Q$ for each $f\in W(S,Q)$, $f_1\left(S\right)\subset Q_1$ for each $f_1\in W(S,Q_1)$;

$(\forall f\in W(S,Q)$, $\forall f_1\in W(S,Q_1)$, $\forall s\in S$, $f\left(s\right)f_1\left(s\right)\in Q{Q}_1)$→$(\forall f\in W(S,Q),$$\forall f_1\in W(S,Q_1),$$f{f}_1\in W(S,Q{Q}_1))$,

• since $Q{Q}_1=\bigcup (b\in B:\exists q\in Q,\exists q_1\in Q_1,b=q{q}_1)$,

$Q/Q_1=\bigcup (b\in B:\exists q\in Q,\exists q_1\in Q_1,b=q/q_1)$,

$Q{Q}_1=\bigcup (b\in B:\exists q\in Q,\exists q_1\in Q_1,b=q{q}_1)$.

$\left(18\right)$. $(\forall f\in W(S_2,Q)$, $f\left(S_2\right)\subset Q)$→$(f\left(S_1\right)\subset Q)$,

• hence $W(S_2,Q)\subset W(S_1,Q)$, since $S_1\subset S_2\subset V$.

$\left(19\right)$. $\forall f\in W({\bigcup }_{i\in \mathsf{\Lambda }}{S}_i,Q)$, $((f\left({\bigcup }_{i\in \mathsf{\Lambda }}{S}_i\right)\subset Q)$↔$(\forall i\in \mathsf{\Lambda },$$f\left(S_i\right)\subset Q\left)\right)$,

• since $f\left({\bigcup }_{i\in \mathsf{\Lambda }}{S}_i\right)\subset Q$ for each $f\in W({\bigcup }_{i\in \mathsf{\Lambda }}{S}_i,Q)$;

$\forall f\in {\bigcap }_{i\in \mathsf{\Lambda }}W(S_i,Q),$$\left(\right(\forall i\in \mathsf{\Lambda },$$f\left(S_i\right)\subset Q)$↔$(f\left({\bigcup }_{i\in \mathsf{\Lambda }}{S}_i\right)\subset Q))$,

• since $f\left(S_i\right)\subset Q$ for each $f\in W(S_i,Q)$.

$\left(20\right)$. $\forall s\in S$, $\left(\right(\forall i\in \mathsf{\Lambda },$$f\left(s\right)\in Q_i)$↔$(f\left(s\right)\in {\bigcap }_{i\in \mathsf{\Lambda }}{Q}_i))$,

• hence $W(S,{\bigcap }_{i\in \mathsf{\Lambda }}{Q}_i)={\bigcap }_{i\in \mathsf{\Lambda }}W(S,Q_i)$. □

Remark 4.

Let the conditions of Theorem 5 in [9] be satisfied. We consider the topology ${\mathcal{T}}_{D,A{\mathcal{C}}_1,A}$ on the topological metagroup D, such that multiplication $m_D$, maps $Di{v}_l$, $Di{v}_r$ are jointly continuous, maps ${\psi }_{A{\mathcal{C}}_1}^D$, ${\psi }_A^D$, ${\tau }_{A{\mathcal{C}}_1}^D$, ${\tau }_A^D$ are continuous, where A, ${\mathcal{C}}_1$, $A{\mathcal{C}}_1$ and the transversal sets $V=V_{D,A}$, $V_{D,A{\mathcal{C}}_1}$ are in the topology inherited from $(D,{\mathcal{T}}_{D,A{\mathcal{C}}_1,A})$. Assume that $(D,{\mathcal{T}}_{D,A{\mathcal{C}}_1,A})$ and $(B,{\mathcal{T}}_B)$ are $T_1$ as topological spaces, where B is a topological metagroup with a topology ${\mathcal{T}}_B$. From Theorem 1 and Remark 2 it follows that there is an abundant family of such topological metagroups D with submetagroups $A{\mathcal{C}}_1$ and A.

Let $C(V,B)$ denote a family of all continuous maps $f:V\to B$. As usual, U is a canonical closed subset (i.e., a closed domain) in V if and only if $U=c{l}_V\left(In{t}_{V}U\right)$, where $In{t}_{V}U$ denotes the interior of U in V, while $c{l}_{V}S$ denotes the closure of S in V, where $S\subset V$.

Let $\mathcal{F}$ be a family of nonvoid canonical closed subsets U in $(V,{\mathcal{T}}_{D,A{\mathcal{C}}_1,A}\cap V)$, such that

$\forall U_1\in \mathcal{F}$, $\forall U_2\in \mathcal{F}$, $U_1\cup U_2\in \mathcal{F}$;

$\forall v\in V$, $\forall v\in P\in ({\mathcal{T}}_{D,A{\mathcal{C}}_1,A}\cap V)$, $\exists U_1\in \mathcal{F}$, $v\in Int\left(U_1\right)$&$U_1\subset P$.

We put $C_t=\bigcup ((d,f)\in C:d\in D,f\in C(V,B))$, where $C=D{\Delta }_A^{\varphi ,\eta ,\kappa ,\xi }F$ is the smashed twisted wreath product of D and F with smashing factors ϕ, η, κ, ξ, where $F=B^V$. The smashed twisted wreath product of D and F is also shortly denoted by $D{\Delta }^{\varphi ,\eta ,\kappa ,\xi }F$ if A is specified (see Definition 5 in [9]). Let $\mathcal{W}$ be a family of all subsets $W(S,Q)$ in $C(V,B)$ such that $S\in \mathcal{F}$ and $Q\ne \varnothing$ is open in $(B,{\mathcal{T}}_B)$.

Theorem 3.

Let the conditions of Remark 4 be satisfied and let the maps $\varphi ,\eta ,\kappa ,\xi$ be jointly continuous (see Remark 1 in [9]). Then ${\mathcal{T}}_{D,A{\mathcal{C}}_1,A}\times \mathcal{W}$ is a base of a topology ${\mathcal{T}}_{C_t}$ on $C_t$ relative to which $C_t$ is a topological $T_1\cap T_3$ loop.

Proof. 

Evidently, each constant map $f_b:V\to B$ belongs to $C(V,B)$, where b in B is arbitrarily fixed, $f_b\left(v\right)=b$ for each $v\in V$. This induces the natural embedding of B into $C(V,B)$, and, consequently, $C(V,B)$ is the nonvoid metagroup with pointwise multiplication and $Di{v}_l$ and $Di{v}_r$. If $U_1$ and $U_2$ belong to $\mathcal{F}$, then $U_1\cup U_2\in \mathcal{F}$ (see §I.8.8 in [28], 1.1.C [29]).

By the conditions of Remark 4 $\{In{t}_V\left(U\right):U\in \mathcal{F}\}$ is a base of the topology on $(V,{\mathcal{T}}_{D,A{\mathcal{C}}_1,A}\cap V)$. From D and B being the $T_1$ topological metagroups and, hence, topological quasigroups, it follows that they are $T_3$ as the topological spaces. Therefore, for each open subset $S_0$ in D (or V) and each $d_0\in S_0$ there exists a canonical closed subset (i.e., closed domain) S in D (or V, respectively), such that $d_0\in In{t}_{D}S$ (or $d_0\in In{t}_{V}S$, respectively) and $S\subset S_0$ by Proposition 1.5.5 [29]. Hence $\mathcal{W}$ is a base of a topology on $C(V,B)$. By virtue of Theorem 5 in [9] and Remark 4 the maps ${\tau }_A^D$ and ${\psi }_A^D$ are continuous on D and $(D,{\mathcal{T}}_{D,A{\mathcal{C}}_1,A})$ is the topological metagroup.

In view of Lemma 6 in [9] the map $w_j:D\times D\times V_{D,A}\to {\mathcal{C}}_1$ is (jointly) continuous as the composition of jointly continuous maps for each $j\in \{1,2,3\}$. According to Remark 4 in [9] $f^{\left\{d\right\}}\left(v\right)=f^{s(d,v)}(v^{[de]})=\varphi (s(d,v))f(v^{[de]})$ for each $d\in D$, $v\in V$, $f\in F$. The map $A\times B\mapsto \varphi \left(a\right)b\in B$ is jointly continuous by the conditions of this theorem. Lemma 1 imply that $\mathcal{W}$ is the base of a topology on $C(V,B)$. We take the topology ${\mathcal{T}}_{C_t}$ generated by the base ${\mathcal{B}}_{D,A{\mathcal{C}}_1,A}\times \mathcal{W}$, where ${\mathcal{B}}_{D,A{\mathcal{C}}_1,A}$ denotes the base of the topology ${\mathcal{T}}_{D,A{\mathcal{C}}_1,A}$.

Hence, Lemma 1 and Formula $\left(6\right)$ imply that multiplication $m_{C_t}:C_t\times C_t\to C_t$ is (jointly) continuous. From Formulae in the proof of Theorem 5 in [9] it follows that $Di{v}_l$ and $Di{v}_r$ are jointly continuous from $C_t\times C_t$ into $C_t$.

The base $\mathcal{W}$ generates the $T_1$ topology ${\mathcal{T}}_{\mathcal{W}}$ on $C(V,B)$ by Lemma 1 and Remark 4. Hence ${\mathcal{T}}_{C_t}$ is the $T_1$ topology on $C_t$, since $(D,{\mathcal{T}}_{D,A{\mathcal{C}}_1,A})$ is $T_1$ by the conditions of this theorem. This implies that $(C_t,{\mathcal{T}}_{C_t})$ is the $T_3$ topological loop. □

Theorem 4.

Let the conditions of Theorem 3 be satisfied, let $(D,{\mathcal{T}}_{D,A{\mathcal{C}}_1,A})$ be locally compact (or compact) and $\mathcal{F}$ be a family of all canonical closed compact subsets in $(V,{\mathcal{T}}_{D,A{\mathcal{C}}_1,A}\cap V)$, let $F_0$ be closed in $(C(V,B),{\mathcal{T}}_{\mathcal{W}})$, let also $c{l}_{C(V,B)}{F}_0\left(v\right)$ be compact for each $v\in V$, where $F_0\left(v\right)=\{f\left(v\right):f\in F_0\}$, let for each compact subset Z in V the restriction $F_0{|}_Z$ be evenly continuous, let $D{\Delta }_A^{\varphi ,\eta ,\kappa ,\xi }{F}_0=:C_0$ be a subloop in $C_t$. Then $C_0$ is a locally compact (or compact respectively) loop.

Proof. 

Since $V={\psi }^{-1}\left(e\right)$ and D is $T_1\cap T_3$ 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 $\mathcal{F}$ it follows that $\mathcal{W}$ induces a compact–open topology ${\mathcal{T}}_{\mathcal{W}}$ on $C(V,B)$. For each compact subset Z in V and open $U_D$ in D the conditions of this theorem imply that $\bigcup \{F_0^{\left\{d\right\}}:d\in U_D\}{|}_Z$ is evenly continuous, since $F_0^{\left\{d\right\}}\subset F_0$ for each $d\in D$. By virtue of Theorem 3.4.21 [29] $F_0$ is compact. Since $(D,{\mathcal{T}}_{D,A{\mathcal{C}}_1,A})$ is locally compact, it is sufficient to take any open $U_D$ in D with the compact closure $c{l}_D\left(U_D\right)$ in the ${\mathcal{T}}_{D,A{\mathcal{C}}_1,A}$ topology. Moreover, $c{l}_{C(V,B)}(\bigcup \{F_0^{\left\{d\right\}}\left(v\right):d\in U_D\})$ is closed in $F_0$ and, hence, compact for each $v\in V$. From B being $T_1$ it follows that B is $T_3$.

From the compactness of $F_0$ and Theorem 3 it follows that $C_0$ 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 $i:D\to D$ and $j:B\to B$ be automorphisms of the metagroups D and B such that ${i|}_{{\mathcal{C}}_1}{=j|}_{{\mathcal{C}}_1}$. Then there exists a loop $C_{i,j}$ and an isomorphism

${\theta }_{i,j}:C\to C_{i,j}$ of C onto $C_{i,j}$ such that ${\theta }_{i,j}{|}_D=i$ and ${\theta }_{i,j}{|}_B=j$.

Proof. 

By the conditions of this proposition $i\left(ab\right)=i\left(a\right)i\left(b\right)$, $i(a/b)=i\left(a\right)/i\left(b\right)$ and $i(ab)=i\left(a\right)i\left(b\right)$ for each a and b in D, similarly for $i^{-1}$. Therefore, $i:A\to i\left(A\right)$ is an isomorphism of metagroups and $i:{\mathcal{C}}_1\to i\left({\mathcal{C}}_1\right)$ is an isomorphism of groups such that $i\left({\mathcal{C}}_1\right)=j\left({\mathcal{C}}_1\right)$. In view of Corollary 1 in [9] $V_{D,i\left(A\right)}=V_{i\left(D\right),i\left(A\right)}=i\left(V_{D,A}\right)$, $V_{i\left({\mathcal{C}}_1\right),i\left({\mathcal{C}}_{1,A}\right)}=i\left(V_{{\mathcal{C}}_1,{\mathcal{C}}_{1,A}}\right)$, $V_{D,i\left(A{\mathcal{C}}_1\right)}=i\left(V_{D,A{\mathcal{C}}_1}\right)$, $i\left({\mathcal{C}}_{1,A}\right)=i\left({\mathcal{C}}_1\right)\cap i\left(A\right)$.

We put

$\left(21\right)$$C_{i,j}=D{\Delta }_A^{{\varphi }_{i,j},{\eta }_{i,j},{\kappa }_{i,j},{\xi }_{i,j}}{F}_{i,j}$ with

$F_{i,j}=\{j\left(f\left(i\left(v\right)\right)\right):v\in V_{D,A},f\in F\}$, ${\varphi }_{i,j}=j\circ \varphi \circ i^{-1}$, ${\varphi }_{i,j}:i\left(A\right)\to \mathcal{A}\left(B\right)$,

${\eta }_{i,j}:i\left(A\right)\times i\left(A\right)\times B\to i\left(\mathcal{C}\right)$, ${\kappa }_{i,j}:i\left(A\right)\times B\times B\to i\left(\mathcal{C}\right)$,

${\xi }_{i,j}:(i\left(A\right)\times B)\times (i\left(A\right)\times B)\to i\left(\mathcal{C}\right)$ such that

${\eta }_{i,j}(v^{\prime },u^{\prime },b^{\prime })=i\left(\eta (i^{-1}\left(v^{\prime }\right),i^{-1}\left(u^{\prime }\right),j^{-1}\left(b^{\prime }\right))\right)$,

${\kappa }_{i,j}(u^{\prime },c^{\prime },b^{\prime })=i\left(\kappa (i^{-1}\left(u^{\prime }\right),j^{-1}\left(c^{\prime }\right),j^{-1}\left(b^{\prime }\right))\right)$,

${\xi }_{i,j}((u^{\prime },c^{\prime }),(v^{\prime },b^{\prime }))=i\left(\xi ((i^{-1}\left(u^{\prime }\right),j^{-1}\left(c^{\prime }\right)),(i^{-1}\left(v^{\prime }\right),j^{-1}\left(b^{\prime }\right)))\right)$ for each $u^{\prime }$, $v^{\prime }$ in $i\left(A\right)$, $b^{\prime }$, $c^{\prime }$ in B; ${\psi }_i:D\to i\left(A\right)$, ${\tau }_i:D\to V_{D,i\left(A\right)}$ such that

${\psi }_i\left(d^{\prime }\right)=i\left(\psi \left(i^{-1}\left(d^{\prime }\right)\right)\right)$, ${\tau }_i\left(d^{\prime }\right)=i\left(\tau \left(i^{-1}\left(d^{\prime }\right)\right)\right)$ for each $d^{\prime }\in D$. Let

$\left(22\right)$${\theta }_{i,j}\left((d,f)\left(v\right)\right)=(i\left(d\right),j\left(f\right))\left(i\left(v\right)\right)$ for each $(d,f)\in C$ and $v\in V_{D,A}$, where $d\in D$, $f\in F$. Therefore, $\left(6\right)$ and $\left(22\right)$ imply that

$\left({\theta }_{i,j}(d_1,f_1\left(v\right))\right)\left({\theta }_{i,j}(d,f\left(v\right))\right)=$

$(i(d_{1}d),{\xi }_{i,j}(({(i(d_1))}^{{\psi }_i},j(f_1)),({(i\left(d\right))}^{{\psi }_i},j\left(f\right)))j(f_1){(j\left(f\right))}^{\{i(d_1)\}})(i\left(v\right))$ and

${\theta }_{i,j}\left((d_1,f_1\left(v\right))\right)(d,f\left(v\right)))=$

$(i(d_{1}d),i(\xi ((d_1^{\psi },f_1),(d^{\psi },f)))j(f_1)j(f^{\{i(d_1)\}}))(i\left(v\right))$

• for each $v\in V_{D,A}$, consequently,

$\left(23\right)$${\theta }_{i,j}\left((d_1,f_1)(d,f)\right)=\left({\theta }_{i,j}(d_1,f_1)\right)\left({\theta }_{i,j}(d,f)\right)$

• for each $(d,f)=g$ and $(d_1,f_1)=g_1$ in C. Since C and $C_{i,j}$ are loops, then $\left(21\right)$ and $\left(23\right)$ imply that ${\theta }_{i,j}(g/g_1)={\theta }_{i,j}\left(g\right)/{\theta }_{i,j}\left(g_1\right)$ and ${\theta }_{i,j}(g{g}_1)={\theta }_{i,j}\left(g\right){\theta }_{i,j}\left(g_1\right)$ for each g and $g_1$ in C. Thus ${\theta }_{i,j}:C\to C_{i,j}$ is the isomorphism of these loops. □

Corollary 2.

Assume that the conditions of Proposition 1 are satisfied and i, $i^{-1}$, j, $j^{-1}$ are continuous relative to ${\mathcal{T}}_{D,A{\mathcal{C}}_1,A}$ and ${\mathcal{T}}_B$ topologies on D and B, respectively. Then, ${\theta }_{i,j}$ and ${\theta }_{i,j}^{-1}$ are continuous relative to ${\mathcal{T}}_{C_t}$ and ${\mathcal{T}}_{C_{i,j,t}}$ topologies on $C_t$ and $C_{i,j,t}$, respectively.

Proof. 

This follows from Proposition 1, Formula $\left(22\right)$ 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 $C_t$, $C_0$, $C_{i,j,t}$ are topological metagroups; in Proposition 1 C and $C_{i,j}$ are metagroups.

3. Conclusions

It is worth mentioning that, in the associative case for topological groups, a lot of investigations have already been conducted. Apart from this, in the nonassociative case, topological metagroups compose a new area for investigation. The results obtained in this paper can be used for subsequent investigations of topological metagroups. Moreover, they can be applied to subsequent studies of quasigroups, loops, topological algebras, generalized $C^{\ast }$-algebras, noncommutative geometry associated with them [23,24,25,28,29,30,31,32,33,34,35]. For this purpose, smashed products and smashed twisted wreath products of topological metagroups can be used. Furthermore, topological metagroups and quasigroups appear naturally as transformations of noncommutative manifolds [25]. Therefore, smashed products and smashed twisted products of topological metagroups would be helpful for studies of twisted and smashed structures of noncommutative manifolds.

Then they can be used for further studies of sheaves on metagroups, deformations of manifolds, mathematical physics, and their applications in other sciences [1,7,36,37].

Then, it will be interesting to investigate relations of metagroup topologies with topological models of rough sets, because the latter are defined with the help of binary operations, and left and right adhesion sets [38,39,40]. Besides mathematics and computer sciences, they have applications in medicine, economics, etc.

The method of Theorem 4 can be potentially applicable to studies of a cluster consensus. Notice that the cluster consensus analysis has applications in algorithm theory and neurocomputing [41]. On the other hand, relations between a structure of a graph and that of a group and its subgroups are actively investigated [42]. Therefore, some other future directions for studies may be noncommuting graphs of a metagroup relative to its submetagroups.