## 1. Introduction

Nonassociative algebras compose a great area of algebra. In nonassociative algebra, noncommutative geometry, and quantum field theory, there frequently appear binary systems which are nonassociative generalizations of groups and related with loops, quasi-groups, Moufang loops, etc., (see References [

1,

2,

3,

4] and references therein). It was investigated and proved in the 20th century that a nontrivial geometry exists if and only if there exists a corresponding loop [

1,

5,

6].

Octonions and generalized Cayley–Dickson algebras play very important roles in mathematics and quantum field theory [

7,

8,

9,

10,

11,

12,

13]. Their structure and identities attract great attention. They are used not only in algebra and noncommutative geometry but also in noncommutative analysis, PDEs, particle physics, mathematical physics, the theory of Lie groups, algebras and their generalization, mathematical analysis, and operator theory and their applications in natural sciences including physics and quantum field theory (see References [

7,

11,

12,

14,

15,

16,

17,

18,

19] and references therein).

A multiplicative law of their canonical bases is nonassociative and leads to a more general notion of a metagroup instead of a group [

11,

20,

21]. The preposition “meta” is used to emphasize that such an algebraic object has properties milder than a group. By their axiomatics, metagroups satisfy the conditions of Equations (1)–(3) and rather mild relations (Equation (

9)). They were used in References [

20,

21] for investigations of automorphisms, derivations, and cohomologies of nonassociative algebras. In the associative case, twisted and wreath products of groups are used for investigations not only in algebra but also in algebraic geometry, geometry, coding theory, and PDEs and their applications [

22,

23,

24,

25]. Twisted structures also naturally appear in investigations in the G-N theory of wave propagation of the components of displacement, stress, temperature distribution, and change in the volume fraction field in an isotropic homogeneous thermoelastic solid with voids subjected to thermal loading due to laser pulse [

26].

In this article, nonassociative metagroups are studied. Necessary preliminary results on metagroups are described in

Section 2. Quotient groups of metagroups are investigated in Theorem 1. Identities in metagroups established in Lemmas 1, 2, and 4 are applied in

Section 3 and

Section 4.

Different types of smashed products of metagroups are investigated in Theorems 3 and 4. Besides them, direct products are also considered in Theorem 2. They provide large families of metagroups (see Remark 2).

In

Section 4, smashed twisted wreath products of metagroups and particularly also of groups are scrutinized. It appears that, generally, they provide loops (see Theorem 5). If additional conditions are imposed, they give metagroups (see Theorem 6). Their metaisomorphisms are investigated in Theorem 7. In Theorem 8 and Corollary 2, smashed splitting extensions of nontrivial central metagroups are studied.

All main results of this paper are obtained for the first time. They can be used for further studies of binary systems, nonassociative algebra cohomologies, structure of nonassociative algebras, operator theory and spectral theory over Cayley–Dickson algebras, PDEs, noncommutative analysis, noncommutative geometry, mathematical physics, and their applications in the sciences (see also the conclusions).

## 2. Nonassociative Metagroups

To avoid misunderstandings, we give necessary definitions. A reader familiar with References [

1,

20,

21] may skip Definition 1. For short, it will be written as a metagroup instead of a nonassociative metagroup.

**Definition** **1.** Let G be a set with a single-valued binary operation (multiplication) ${G}^{2}\ni (a,b)\mapsto ab\in G$ defined on G satisfying the following conditions:and a unique y∈ G exists satisfying ya = b, which are denoted by There exists a neutral (i.e., unit) element ${e}_{G}=e\in G$: If the set

G with the single-valued multiplication satisfies the conditions of Equations (1) and (2), then it is called a quasi-group. If the quasi-group

G satisfies also the condition of Equation (

3), then it is called an algebraic loop (or in short, a loop).

The set of all elements

$h\in G$ commuting and associating with

G is as follows:

We call

G a metagroup if a set

G possesses a single-valued binary operation and satisfies the conditions of Equations (1)–(3) and

for each

a,

b, and

c in

G, where

$t(a,b,c)={t}_{G}(a,b,c)\in \mathcal{C}(G)$. If

G is a quasi-group satisfying the condition of Equation (

9), then it will be called a strict quasi-group.

Then, the metagroup

G will be called a central metagroup, if it satisfies also the following condition:

for each

a and

b in

G, where

${\mathsf{t}}_{2}(a,b)\in \mathcal{C}(G)$.

If H is a submetagroup (or a subloop) of the metagroup G (or the loop G) and $gH=Hg$ for each $g\in G$, then H will be called almost normal. If, in addition, $(gH)k=g(Hk)$ and $k(gH)=(kg)H$ for each g and k in G, then H will be called a normal submetagroup (or a normal subloop respectively).

Henceforward, notations $In{v}_{l}(a)=Di{v}_{l}(a,e)$ and $In{v}_{r}(a)=Di{v}_{r}(a,e)$ will be used.

Elements of a metagroup G will be denoted by small letters; subsets of G will be denoted by capital letters. If A and B are subsets in G, then $A-B$ means the difference of them: $A-B=\{a\in A:\phantom{\rule{3.33333pt}{0ex}}a\notin B\}$. Henceforward, maps and functions on metagroups are supposed to be single-valued unless otherwise specified.

**Lemma** **1.** If G is a metagroup, then for each a and $b\in G$, the following identities are fulfilled: **Proof.** The conditions of Equations

$(1)$–

$(3)$ imply that

for each

a and

b in

G. Using the condition of Equation

$(9)$ and the identities of Equations

$(14)$ and

$(15)$, we deduce the following:

which leads to Equation

$(11)$.

Let

$c=a\backslash b$; then, from the identities of Equations

$(11)$ and

$(14)$, it follows that

which provides Equation

$(12)$.

Now, let

$d=b/a$; then, the identities of Equations

$(11)$ and

$(15)$ imply that

which demonstrates Equation

$(13)$. □

**Lemma** **2.** Assume that G is a metagroup. Thenm for every a, ${a}_{1}$, ${a}_{2}$, and ${a}_{3}$ in G and ${p}_{1}$, ${p}_{2}$, and ${p}_{3}$ in $\mathcal{C}(G)$, we have the following: **Proof.** Since

$({a}_{1}{a}_{2}){a}_{3}=t({a}_{1},{a}_{2},{a}_{3}){a}_{1}({a}_{2}{a}_{3})$ and

$t({a}_{1},{a}_{2},{a}_{3})\in \mathcal{C}(G)$ for every

${a}_{1}$,

${a}_{2}$,

${a}_{3}$ in

G, then

Therefore, for every

${a}_{1}$,

${a}_{2}$,

${a}_{3}$ in

G and

${p}_{1}$,

${p}_{2}$, and

${p}_{3}$ in

$\mathcal{C}(G)$, we infer the following:

since

For each $p\in \mathcal{C}(G)$, a and b in G, because $\mathcal{C}(G)$ is the commutative group. Thus, $t({p}_{1}{a}_{1},{p}_{2}{a}_{2},{p}_{3}{a}_{3})=t({a}_{1},{a}_{2},{a}_{3})$.

From the condition in Equation

$(9)$, Lemma 1, and the identity of Equation

$(16)$, it follows that

for each

$a\in G$, implying Equation

$(17)$. □

**Theorem** **1.** If G is a metagroup and ${\mathcal{C}}_{0}$ is a subgroup in a center $\mathcal{C}(G)$ such that $t(a,b,c)\in {\mathcal{C}}_{0}$ for each a, b, and c in G, then its quotient $G/{\mathcal{C}}_{0}$ is a group.

**Proof.** As traditionally, the following notation is used:

for subsets

A and

B in

G. Then from the conditions of Equations

$(4)$–

$(8)$, it follows that, for each

a,

b, and

c in

G, the following identities take place:

$((a{\mathcal{C}}_{0})(b{\mathcal{C}}_{0}))(c{\mathcal{C}}_{0})=(a{\mathcal{C}}_{0})((b{\mathcal{C}}_{0})(c{\mathcal{C}}_{0}))$ and $a{\mathcal{C}}_{0}={\mathcal{C}}_{0}a$. Evidently $e{\mathcal{C}}_{0}={\mathcal{C}}_{0}$. In view of Lemmas 1 and 2 $(a{\mathcal{C}}_{0})\backslash e=e/(a{\mathcal{C}}_{0})$, consequently, for each $a{\mathcal{C}}_{0}\in G/{\mathcal{C}}_{0}$ a unique inverse ${(a{\mathcal{C}}_{0})}^{-1}$ exists. Thus the quotient $G/{\mathcal{C}}_{0}$ of G by ${\mathcal{C}}_{0}$ is a group. □

**Lemma** **3.** Let G be a metagroup, then $In{v}_{r}(G)$ and $In{v}_{l}(G)$ are metagroups.

**Proof.** At first, we consider $In{v}_{r}(G)$. Let ${a}_{1}$ and ${a}_{2}$ belong to G. Then, there are unique $e/{a}_{1}$ and $e/{a}_{2}$, since the map $In{v}_{r}$ is single-valued (see Definition 1). Since $In{v}_{r}\circ In{v}_{l}(a)=a$ and $In{v}_{l}\circ In{v}_{r}(a)=a$ for each $a\in G$, then $In{v}_{r}:G\to G$ and $In{v}_{l}:G\to G$ are bijective and surjective maps.

We put ${\widehat{a}}_{1}\circ {\widehat{a}}_{2}=(e/{a}_{2})(e/{a}_{1})$ for each ${a}_{1}$ and ${a}_{2}$ in G, where ${\widehat{a}}_{j}=In{v}_{r}({a}_{j})$ for each $j\in \{1,2\}$. This provides a single-valued map from $In{v}_{r}(G)\times In{v}_{r}(G)$ into $In{v}_{r}(G)$. Then, for each a, b, x, and y in G, the equations $\widehat{a}\circ \widehat{x}=\widehat{b}$ and $\widehat{y}\circ \widehat{a}=\widehat{b}$ are equivalent to $(e/x)(e/a)=e/b$ and $(e/a)(e/y)=e/b$, respectively. That is, $\widehat{x}=(e/b)/(e/a)$ and $\widehat{y}=(e/a)\backslash (e/b)$ are unique. On the other hand, $e/e=e$ and $\widehat{e}\circ \widehat{b}=e/b=\widehat{b}\circ \widehat{e}=\widehat{b}$ for each $b\in G$.

Then, we infer the following:

Consequently, ${t}_{In{v}_{r}(G)}({\widehat{a}}_{1},{\widehat{a}}_{2},{\widehat{a}}_{3})=e/{t}_{G}({\widehat{a}}_{3},{\widehat{a}}_{2},{\widehat{a}}_{1})$. Evidently, $In{v}_{r}(\mathcal{C}(G))=\mathcal{C}(G)$ and $\mathcal{C}(In{v}_{r}(G))=\mathcal{C}(G)$. Thus, the conditions of Equations $(1)$–$(3)$ and $(9)$ are satisfied for $In{v}_{r}(G)$.

Similarly, putting $In{v}_{l}({a}_{j})={\stackrel{\u02c7}{a}}_{j}$ and ${\stackrel{\u02c7}{a}}_{1}\circ {\stackrel{\u02c7}{a}}_{2}=({a}_{2}\backslash e)({a}_{1}\backslash e)$ for each ${a}_{j}\in G$ and $j\in \{1,2,3\}$, the conditions of Equations $(1)$–$(3)$ and $(9)$ are verified for $In{v}_{l}(G)$. □

**Lemma** **4.** Assume that G is a metagroup and that $a\in G$, $b\in G$, and $c\in G$. Thenand **Proof.** From Equations $(9)$ and $(15)$, we deduce that

$((e/b)(e/a))(ab)=t(e/b,e/a,ab)(e/b)((e/a)(ab))=t(e/b,e/a,ab)/t(e/a,a,b)$, which implies Equation

$(23)$. Then, from Equations

$(9)$ and

$(14)$, we infer the following:

which implies Equation

$(24)$.

Utilizing Equations $(14)$ and $(9)$, we get $b(c((bc)\backslash a))=a/t(b,c,(bc)\backslash a)$; hence, $c((bc)\backslash a)=(b\backslash a)/t(b,c,(bc)\backslash a)$, implying Equation $(26)$.

Equations

$(15)$ and

$(9)$ imply that

$((a/(bc))b)c=t(a/(bc),b,c)a$; consequently,

$(a/c)t(a/(bc),b,c)=(a/(bc))b$, and hence,

□

## 3. Smashed Products and Smashed Twisted Products of Metagroups

**Theorem** **2.** Let ${G}_{j}$ be a family of metagroups (see Definition 1), where $j\in J$, J is a set. Then, their direct product $G={\prod}_{j\in J}{G}_{j}$ is a metagroup and **Proof.** It is given in Theorem 8 in Reference [

21]. □

**Remark** **1.** Let A and B be two metagroups, and let $\mathcal{C}$ be a commutative group such thatwhere ${\mathcal{C}}_{m}(A)$ denotes a minimal subgroup in $\mathcal{C}(A)$ containing ${t}_{A}(a,b,c)$ for every a, b, and c in A. Using direct products, it is always possible to extend either A or B to get such a case. In particular, either A or B may be a group. On $A\times B$, an equivalence relation Ξ is considered such thatfor every v in A, b in B, and γ in $\mathcal{C}$.where $\mathcal{A}(B)$ denotes a family of all bijective surjective single-valued mappings of B onto B subjected to the conditions of Equations (31)–(34) given below. If $a\in A$ and $b\in B$, then it will be written shortly ${b}^{a}$ instead of $\varphi (a)b$, where $\varphi (a):B\to B$. Let alsoandbe single-valued mappings written shortly as η, κ, and ξ correspondingly such thatand $\kappa (u,\gamma ,b)=\kappa (u,b,\gamma )=e$;andfor every u and v in A, b, and c in B, γ in $\mathcal{C}$, where e denotes the neutral element in $\mathcal{C}$ and in A and B. We putfor each of ${a}_{1}$ and ${a}_{2}$ in A and of ${b}_{1}$ and ${b}_{2}$ in B. The Cartesian product $A\times B$ supplied with such a binary operation of Equation $(36)$ will be denoted by $A{\otimes}^{\varphi ,\eta ,\kappa ,\xi}B$.

Then, we putfor each of ${a}_{1}$ and ${a}_{2}$ in A and of ${b}_{1}$ and ${b}_{2}$ in B. The Cartesian product $A\times B$ supplied with a binary operation of Equation $(37)$ will be denoted by $A{\star}^{\varphi ,\eta ,\kappa ,\xi}B$.

**Theorem** **3.** Let the conditions of Remark 1 be fulfilled. Then, the Cartesian product $A\times B$ supplied with a binary operation of Equation $(36)$ is a metagroup. Moreover, there are embeddings of A and B into $A{\otimes}^{\varphi ,\eta ,\kappa ,\xi}B={C}_{1}$ such that B is an almost normal submetagroup in ${C}_{1}$. If in addition ${\mathcal{C}}_{m}({C}_{1})\subseteq {\mathcal{C}}_{m}(B)\subseteq \mathcal{C}$, then B is a normal submetagroup.

**Proof.** The first part of this theorem was proven in Theorem 9 in Reference [

21]. Naturally,

A is embedded into

${C}_{1}$ as

$\{(a,e):\phantom{\rule{3.33333pt}{0ex}}a\in A\}$ and

B is embedded into

${C}_{1}$ as

$\{(e,b):\phantom{\rule{3.33333pt}{0ex}}b\in B\}$. Let

$a\in A$ and

${b}_{0}\in B$; then,

$(a,{b}_{0})B=\{(a,\xi ((a,{b}_{0}),(e,b)){b}_{0}{b}^{a}):\phantom{\rule{3.33333pt}{0ex}}b\in B\}$ and

$B(a,{b}_{0})=\{(a,\xi ((e,b),(a,{b}_{0}))b{b}_{0}):\phantom{\rule{3.33333pt}{0ex}}b\in B\}$, since

${b}_{0}^{e}={b}_{0}$ by

$(31)$. From

${B}^{a}=B$,

${b}_{0}B=B$,

$B{b}_{0}=B$,

$\mathcal{C}\subset \mathcal{C}(B)$ and Equations

$(30)$ and

$(35)$, it follows that

$(a,{b}_{0})B=B(a,{b}_{0})$, where

${B}^{a}=\{{b}^{a}:\phantom{\rule{3.33333pt}{0ex}}b\in B\}$. Thus,

B is an almost normal submetagroup in

${C}_{1}$ (see Definition 1). If in addition

${\mathcal{C}}_{m}({C}_{1})\subseteq {\mathcal{C}}_{m}(B)\subseteq \mathcal{C}$, then evidently

B is a normal submetagroup (see also the condition of Equation

$(29)$), since

${t}_{{C}_{1}}(g,b,h)\in {\mathcal{C}}_{m}({C}_{1})$ and

${t}_{{C}_{1}}(h,g,b)\in {\mathcal{C}}_{m}({C}_{1})$ for each

g and

h in

G,

$b\in H$. □

**Theorem** **4.** Suppose that the conditions of Remark 1 are satisfied. Then, the Cartesian product $A\times B$ supplied with a binary operation of Equation $(37)$ is a metagroup. Moreover, there exist embeddings of A and B into $A{\star}^{\varphi ,\eta ,\kappa ,\xi}B={C}_{2}$ such that B is an almost normal submetagroup in ${C}_{2}$. If additionally ${\mathcal{C}}_{m}({C}_{2})\subseteq {\mathcal{C}}_{m}(B)\subseteq \mathcal{C}$, then B is a normal submetagroup.

**Proof.** The conditions of Remark 1 imply that the binary operation of Equation $(37)$ is single-valued.

We consider the following formulas:

${I}_{1}=(({a}_{1},{b}_{1})\star ({a}_{2},{b}_{2}))\star ({a}_{3},{b}_{3})$ and

${I}_{2}=({a}_{1},{b}_{1})\star (({a}_{2},{b}_{2})\star ({a}_{3},{b}_{3}))$, where

${a}_{1}$,

${a}_{2}$, and

${a}_{3}$ are in

A and where

${b}_{1}$,

${b}_{2}$, and

${b}_{3}$ are in

B. Utilizing Equations

$(31)$–

$(35)$ and

$(37)$, we get the following:

and

Consequently,

$t(({a}_{1},{b}_{1}),({a}_{2},{b}_{2}),({a}_{3},{b}_{3}))\in \mathcal{C}$ for each

${a}_{j}\in A$,

${b}_{j}\in B$,

$j\in \{1,2,3\}$. We denote

in more details by

(see Equation

$(39)$).

Evidently, Equation $(3)$ is a consequence of Equations $(35)$ and $(37)$.

Note that, if

$\gamma \in \mathcal{C}$, then

Therefore, $\gamma \in \mathcal{C}(A{\star}^{\varphi ,\eta ,\kappa ,\xi}B)$. Consequently, $\mathcal{C}\subseteq \mathcal{C}(A{\star}^{\varphi ,\eta ,\kappa ,\xi}B)$.

Then, we seek a solution of the following equation:

where

$a\in A$,

$b\in B$.

From Equations

$(2)$ and

$(37)$, it follows that

Consequently,

$\xi ((e/a,{b}_{1}),(a,b)){b}^{(e/a)}{b}_{1}=e$. Therefore, Equations

$(1)$ and

$(35)$ imply that

Thus, ${a}_{1}\in A$ and ${b}_{1}\in B$ prescribed by Equations $(41)$ and $(42)$ provide a unique solution of Equation $(40)$.

Analogously for the following equation

where

$a\in A$,

$b\in B$, we deduce that

Consequently,

$\xi ((a,b),(a\backslash e,{b}_{2})){b}_{2}^{a}b=e$, and hence,

${b}_{2}^{a}=e/[\xi ((a,b),(a\backslash e,{b}_{2}))b]$. From Equations

$(31)$ and

$(32)$, it follows that

${({b}_{2}^{a})}^{e/a}=\eta (e/a,a,{b}_{2}){b}_{2}$; consequently,

Thus, a unique solution of Equation $(43)$ is given by Equations $(44)$ and $(45)$.

Then, we have

$({a}_{1},{b}_{1})=(e,e)/(a,b)$ and

$({a}_{2},{b}_{2})=(a,b)\backslash (e,e)$ and get the following:

and

${e}_{G}=(e,e)$, where

$G=A{\star}^{\varphi ,\eta ,\kappa ,\xi}B$. This means that the properties of Equations

$(1)$–

$(3)$ and

$(9)$ are fulfilled for

$A{\star}^{\varphi ,\eta ,\kappa ,\xi}B$.

Evidently, there are embeddings of A and B into ${C}_{2}$ as $(A,e)$ and $(e,B)$, respectively. Suppose that $a\in A$ and ${b}_{0}\in B$, then

$(a,{b}_{0})\star B=\{(a,\xi ((a,{b}_{0}),(e,b)){b}^{a}{b}_{0}):\phantom{\rule{3.33333pt}{0ex}}b\in B\}$ and

$B\star (a,{b}_{0})=\{(a,\xi ((e,b),(a,{b}_{0})){b}_{0}b):\phantom{\rule{3.33333pt}{0ex}}b\in B\}$.

Therefore, $(a,{b}_{0})\star B=B\star (a,{b}_{0})$ by the conditions of Equations (30) and (35), since ${B}^{a}=B$ and $\mathcal{C}\subset \mathcal{C}(B)$. Thus, B is an almost normal submetagroup in ${C}_{2}$ (see Definition 1). If additionally ${\mathcal{C}}_{m}({C}_{2})\subseteq {\mathcal{C}}_{m}(B)\subseteq \mathcal{C}$, then apparently B is a normal submetagroup (see also the condition of Equation $(29)$), since ${t}_{{C}_{2}}(g,b,h)\in {\mathcal{C}}_{m}({C}_{2})$ and ${t}_{{C}_{2}}(h,g,b)\in {\mathcal{C}}_{m}({C}_{2})$ for each g and h in G, $b\in B$. □

**Definition** **2.** We call the metagroup $A{\otimes}^{\varphi ,\eta ,\kappa ,\xi}B$ provided by Theorem 3 (or $A{\star}^{\varphi ,\eta ,\kappa ,\xi}B$ by Theorem 4) a smashed product (or a smashed twisted product correspondingly) of metagroups A and B with smashing factors ϕ, η, κ, and ξ.

**Remark** **2.** From Theorems 2–4, it follows that, taking nontrivial η, κ, and ξ and starting even from groups with nontrivial $\mathcal{C}({G}_{j})$ or $\mathcal{C}(A)$, it is possible to construct new metagroups with nontrivial $\mathcal{C}(G)$ and ranges ${t}_{G}(G,G,G)$ of ${t}_{G}$ that may be infinite.

With suitable smashing factors ϕ, η, κ, and ξ and with nontrivial metagroups or groups A and B, it is easy to get examples of metagroups in which $e/a\ne a\backslash e$ for an infinite family of elements a in $A{\otimes}^{\varphi ,\eta ,\kappa ,\xi}B$ or in $A{\star}^{\varphi ,\eta ,\kappa ,\xi}B$. Evidently, smashed products and smashed twisted products (see Definition 2) are nonassociative generalizations of semidirect products. Combining Theorems 3 and 4 with Lemmas 3 and 4 provides other types of smashed products by taking ${\widehat{b}}_{1}\circ {\widehat{b}}_{2}^{{a}_{1}}$ instead of ${b}_{1}{b}_{2}^{{a}_{1}}$ or ${\stackrel{\u02c7}{b}}_{2}^{{a}_{1}}\circ {\stackrel{\u02c7}{b}}_{1}$ instead of ${b}_{2}^{{a}_{1}}{b}_{1}$ on the right sides of Equations $(36)$ and $(37)$, correspondingly, etc.

## 4. Smashed Twisted Wreath Products of Metagroups

**Lemma** **5.** Let D be a metagroup and A be a submetagroup in D. Then, there exists a subset V in D such that D is a disjoint union of $vA$, where $v\in V$, that is,and **Proof.** The cases $A=\{e\}$ and $A=D$ are trivial. Let $A\ne \{e\}$ and $A\ne D$, and let $\mathcal{C}(D)$ be a center of D. From the conditions of Equations (4)–(8), it follows that $z\in \mathcal{C}(D)\cap A$ implies $z\in \mathcal{C}(A)$.

Assume that

$b\in D$ and

$z\in \mathcal{C}(D)$ are such that

$zbA\cap bA\ne \varnothing $. It is equivalent to (

$\exists {s}_{1}\in A$,

$\exists {s}_{2}\in A$,

$zb{s}_{1}=b{s}_{2}$). From Equation

$(14)$, it follows that

$(zb{s}_{1}=b{s}_{2})\iff (z{s}_{1}={s}_{2})\iff (z={s}_{2}/{s}_{1}\in A)$ because

$z\in \mathcal{C}(D)$. Thus,

Suppose now that

${b}_{1}\in D$,

${b}_{2}\in D$ and

${b}_{1}A\cap {b}_{2}A\ne \varnothing $. This is equivalent to (

$\exists {s}_{1}\in A$,

$\exists {s}_{2}\in A$,

${b}_{1}{s}_{1}={b}_{2}{s}_{2}$). By the identity of Equation

$(15)$, the latter is equivalent to

${b}_{1}=({b}_{2}{s}_{2})/{s}_{1}$. On the other hand,

by Equations

$(9)$,

$(13)$, and

$(15)$. Together with

$(50)$ this gives the equivalence:

Let

${\rm Y}$ be a family of subsets

K in

D such that

${k}_{1}A\cap {k}_{2}A=\varnothing $ for each

${k}_{1}\ne {k}_{2}$ in

K. Let

${\rm Y}$ be directed by inclusion. Then,

${\rm Y}\ne \varnothing $, since

$A\subset D$ and

$A\ne D$. Therefore, from Equations

$(50)$ and

$(51)$ and the Kuratowski-Zorn lemma (see Reference [

27]), the assertion of this lemma follows, since a maximal element

V in

${\rm Y}$ gives Equations

$(48)$ and

$(49)$. □

**Definition** **3.** A set V from Lemma 5 is called a right transversal (or complete set of right coset representatives) of A in D.

The following corollary is an immediate consequence of Lemma 5.

**Corollary** **1.** Let D be a metagroup, A be a submetagroup in D, and V be a right transversal of A in D. Then, **Remark** **3.** We denote b in the decomposition of Equation $(52)$ by $b=\tau (a)={a}^{\tau}$ and $s=\psi (a)={a}^{\psi}$, where τ and ψ are the shortened notations of ${\tau}_{A,D,V}$ and ${\psi}_{A,D,V}$, respectively. That is, there are single-valued maps These maps are idempotent $\tau (\tau (a))=\tau (a)$ and $\psi (\psi (a))=\psi (a)$ for each $a\in D$. According to Equation $(2)$, $s=a/b$; hence, ${a}^{\psi}=a/{a}^{\tau}$. From Equation $(13)$, it follows that $a/b=a(e/b)t(e/b,b,b\backslash e)/t(a/b,b,b\backslash e)$; consequently, by Lemma 2, Notice that the metagroup need not be power-associative. Then, $e/s$ and $s\backslash e$ can be calculated with the help of the identity of Equation $(11)$. Suppose that a and y belong to D, $s={a}^{\psi}$, $b={a}^{\tau}$, ${s}_{2}={y}^{\psi}$, and ${b}_{2}={y}^{\tau}$. Then, $({a}^{\tau}y)=b({s}_{2}{b}_{2})$. According to Equation $(52)$ there exists a unique decomposition $b({s}_{2}{b}_{2})={s}_{3}{b}_{3}$, where ${s}_{3}\in A$, ${b}_{3}\in V$; hence, ${({a}^{\tau}y)}^{\tau}={b}_{3}$. On the other hand, by Equation $(9)$ $ay=s(b({s}_{2}{b}_{2}))t(s,b,y)=(s{s}_{3}){b}_{3}t(s,b,y)/t(s,{s}_{3},{b}_{3})$. We denote a subgroup $\mathcal{C}(D)\cap A$ in $\mathcal{C}(D)$ by ${\mathcal{C}}_{A}(D)$ or shortly ${\mathcal{C}}_{A}$, when D is specified. From Lemma 2 and Equation $(51)$, it follows thatwhere $\mathcal{C}{(D)}^{\tau}=\{{a}^{\tau}:\phantom{\rule{3.33333pt}{0ex}}a\in \mathcal{C}(D)\}$. Let ${\mathcal{C}}_{m}(A)$ be a minimal subgroup in $\mathcal{C}(A)$ generated by a set $\{{t}_{A}(a,b,c):\phantom{\rule{3.33333pt}{0ex}}a\in A,\phantom{\rule{3.33333pt}{0ex}}b\in A,\phantom{\rule{3.33333pt}{0ex}}c\in A\}$. From Equation $(9)$, it follows that ${\mathcal{C}}_{m}(A)\subset {\mathcal{C}}_{A}(D)$ and $A\mathcal{C}(D)$ is a submetagroup in D. By virtue of Theorem 1, $(A\mathcal{C}(D))/{\mathcal{C}}_{A}(D)$ and $A/{\mathcal{C}}_{A}(D)$ are groups such that $A/{\mathcal{C}}_{A}(D)\hookrightarrow (A\mathcal{C}(D))/{\mathcal{C}}_{A}(D)$. For each $d\in D$, there exists a unique decompositionby Equation $(53)$. Take in particular $\gamma \in \mathcal{C}(D)$; then, $\gamma ={\gamma}^{\psi}{\gamma}^{\tau}$, where ${\gamma}^{\psi}\in {\mathcal{C}}_{A}(D)$, ${\gamma}^{\tau}\in V$. Therefore, $\mathcal{C}(D)/{\mathcal{C}}_{A}(D)\subset V$ and there exists a subset ${V}_{0}$ in V such that $(\mathcal{C}(D)/{\mathcal{C}}_{A}(D)){V}_{0}=V$, since $\mathcal{C}(D)/{\mathcal{C}}_{A}(D)$ is a subgroup in $(A\mathcal{C}(D))/{\mathcal{C}}_{A}(D)$ (see Equation $(56)$). Equation $(57)$ implies that ${({d}^{\tau})}^{\psi}=e$ and ${({d}^{\psi})}^{\tau}=e$ for each $d\in D$. Using this, we subsequently deduce thatfor each $d\in D$ and $\gamma \in \mathcal{C}(D)$. Hence,where ${d}^{\psi}{\gamma}^{\psi}\in A$ and ${d}^{\tau}{\gamma}^{\tau}\in V$. From a uniqueness of this representation, it follows thatand Using Equation $(63)$ we infer that On the other hand, if $\gamma \in \mathcal{C}(D)$, then ${\gamma}^{\psi}=\gamma /{\gamma}^{\tau}$ and Equations $(64)$ and $(60)$ imply particularly thatsince ${t}_{D}(a,d,\gamma )=e$ for each a and d in D and $\gamma \in \mathcal{C}(D)$. Then, from $s={a}^{\psi}$, ${a}^{\tau}y={s}_{3}{b}_{3}$, it follows that ${a}^{\psi}{({a}^{\tau}y)}^{\psi}=s{s}_{3}$ and ${(ay)}^{\psi}={[(s{s}_{3}){b}_{3}{t}_{D}(s,b,y)/{t}_{D}(s,{s}_{3},{b}_{3})]}^{\psi}$; consequently, by Lemma 2 and Equation $(62)$,for each a and y in D. Particularly, From Equations $(64)$ and $(65)$, it follows that the metagroup D acts on V transitively by right shift operators ${R}_{y}$, where ${R}_{y}a=ay$ for each a and y in D. Therefore, we put Then from Equations $(64)$, $(65)$, $(68)$, and $(9)$ and Lemma 2, we deduce that, for each a, c, and d in D In particular, ${({a}^{\tau})}^{[e]}={a}^{\tau}$ for each $a\in D$. Next, we put ${e}^{\tau}={b}_{*}$. It is convenient to choose ${b}_{*}=e$. Hence, ${b}_{*}^{[s]}={({e}^{\tau})}^{[s]}={({e}^{\tau}s)}^{\tau}={s}^{\tau}=e={e}^{\tau}$ for each $s\in A$. Thus, the submetagroup A is the stabilizer of e and Equation $(68)$ implies that **Remark** **4.** Let B and D be metagroups, A be a submetagroup in D, and V be a right transversal of A in D. Let also the conditions of Equations $(28)$–$(35)$ be satisfied for A and B. By Theorem 2, there exists a metagroup It contains a submetagroupwhere $\sigma (f)=\{v\in V:\phantom{\rule{3.33333pt}{0ex}}f(v)\ne e\}$ is a support of $f\in F$ and $card(\Omega )$ denotes the cardinality of a set Ω. Let ${T}_{h}f={f}^{h}$ for each $f\in F$ and $h:V\to A$. We putwhere $J:V\times F\to B$, $\phantom{\rule{3.33333pt}{0ex}}J(f,v)=fJv$, $\phantom{\rule{3.33333pt}{0ex}}{S}_{d}Jv=J{v}^{[d\backslash e]}$ for each $d\in D$, $f\in F$ and $v\in V$. Then, for each $f\in F$, $d\in D$ we putwhere(see also Equations $(52)$ and $(68)$). Hence,since ${v}^{e\backslash e}=v$ and $s(e,v)=e$. **Lemma** **6.** Let the conditions of Remark 4 be satisfied. Then, for each of f and ${f}_{1}$ in F and of d and ${d}_{1}$ in D, $v\in V$,andwhere ${w}_{j}={w}_{j}(d,{d}_{1},v)\in \mathcal{C}(D)$, $j\in \{1,2,3\}$, ${w}_{1}^{\tau}={w}_{1}$. **Proof.** Equations $(72)$ and $(33)$ imply the identity of Equation $(75)$.

Let

$v\in V$,

d and

${d}_{1}$ belong to

D, and

$f\in F$; then, from Equations

$(72)$ and

$(73)$, it follows that

and

From Equations

$(24)$,

$(69)$,

$(58)$,

$(61)$,

$(11)$, and

$(13)$ and Lemma 2, we deduce that

and

where

${w}_{1}(d,{d}_{1},v)={\gamma}^{\tau}$, where

Then Equations (73), (66), (25), (13), and (62) and Lemma 2 imply that

where

${\gamma}_{1}={t}_{D}(v/(d{d}_{1}),d,{d}_{1}){t}_{D}(e/d,d,d\backslash e)/{t}_{D}(v/(d{d}_{1}),d,d\backslash e)$ by

$(13)$,

$(25)$ and Lemma 2;

where

Note that

by Equations

$(81)$ and

$(23)$, where

Then,

by Equations

$(11)$,

$(13)$,

$(63)$, and

$(68)$, where

${\gamma}_{4}\in {\mathcal{C}}_{m}(D)$. Hence,

since

where

${\gamma}_{5}={t}_{D}({v}^{[{d}_{1}\backslash e]}/d,d,d\backslash e)/{t}_{D}(e/d,d,d\backslash e)$.

Thus, the identities of Equations

$(80)$–

$(83)$ imply that

where

By Lemmas 1 and 2 and Equation

$(73)$, representations of

${\gamma}_{j}$ simplify:

Therefore,

${w}_{2}(d,{d}_{1},v)\in \mathcal{C}(D)\cap A$ for each

d and

${d}_{1}$ in

D and

$v\in V$, since

$s(d,{d}_{1},v)$,

$s(d,{v}^{[{d}_{1}\backslash e]})$, and

$s({d}_{1},v)$ belong to

A. Then, from Equations

$(77)$,

$(84)$,

$(31)$,

$(32)$, we infer that

where

${w}_{3}(d,{d}_{1},v)=e/[\eta (s(d,{v}^{[{d}_{1}\backslash e]}){w}_{2},s({d}_{1},v),b)\eta (s(d,{v}^{[{d}_{1}\backslash e]}),{w}_{2},{b}^{s({d}_{1},v)})]$,

${w}_{2}={w}_{2}(d,{d}_{1},v)$, $\phantom{\rule{3.33333pt}{0ex}}b=f({({v}^{[{d}_{1}\backslash e]})}^{[d\backslash e]}{w}_{1}(d,{d}_{1},v))$. Equations $(68)$ and $(61)$ imply that ${(v\gamma )}^{[a]}={v}^{[a]}{\gamma}^{\tau}$ for each $v\in V$ and $\gamma \in \mathcal{C}(D)$, $a\in D$; consequently, ${({(v{w}_{1})}^{[{d}_{1}\backslash e]})}^{[d\backslash e]}={({v}^{[{d}_{1}\backslash e]})}^{[d\backslash e]}{w}_{1}$, and hence, $v{w}_{1}\in V$ for each $v\in V$, d and ${d}_{1}$ in D, ${w}_{1}={w}_{1}(d,{d}_{1},v)$ by Equation $(79)$, since ${w}_{1}={\gamma}^{\tau}$. Thus Equation $(76)$ follows from Equations $(78)$ and $(85)$. □

**Definition** **4.** Suppose that the conditions of Remark 4 are satisfied and on the Cartesian product $C=D\times F$ (or ${C}^{*}=D\times {F}^{*}$) a binary operation is given by the following formula:where $\xi (({d}_{1}^{\psi},{f}_{1}),({d}^{\psi},f))(v)=\xi (({d}_{1}^{\psi},{f}_{1}(v)),({d}^{\psi},f(v)))$ for every d and ${d}_{1}$ in D, f and ${f}_{1}$ in F (or ${F}^{*}$ respectively), and $v\in V$. **Theorem** **5.** Let C, ${C}^{*}$, D, F, and ${F}^{*}$ be the same as in Definition 4. Then, C and ${C}^{*}$ are loops and there are natural embeddings $D\hookrightarrow C$, $F\hookrightarrow C$, $D\hookrightarrow {C}^{*}$, and ${F}^{*}\hookrightarrow {C}^{*}$ such that F (or ${F}^{*}$) is an almost normal subloop in C (or ${C}^{*}$ respectively).

**Proof.** The operation of Equation $(86)$ is single-valued. Let $a=(d,f)$ and $b=({d}_{0},{f}_{0})$, where d and ${d}_{0}$ are in D and where f and ${f}_{0}$ are in F (or ${F}^{*}$).

The equation

$ay=b$ is equivalent to

$d{d}_{2}={d}_{0}$ and

where

${d}_{2}\in D$,

${f}_{2}\in F$ (or

${f}_{2}\in {F}^{*}$ respectively),

$y=({d}_{2},{f}_{2})$,

$\phantom{\rule{3.33333pt}{0ex}}\xi (({d}^{\psi},f),({d}_{2}^{\psi},{f}_{2}))(v)=\xi (({d}^{\psi},f(v)),({d}_{2}^{\psi},{f}_{2}(v)))$ for each

$v\in V$. Therefore,

${d}_{2}=d\backslash {d}_{0}$,

$\phantom{\rule{3.33333pt}{0ex}}{f}_{2}^{\{d\}}=[\xi (({d}^{\psi},f),({(d\backslash {d}_{0})}^{\psi},{f}_{2}))f]\backslash {f}_{0}$ by Equation

$(1)$ and Theorem 2. On the other hand,

${f}_{2}^{\{e\}}={f}_{2}$ by Equation

$(74)$ and

${f}_{2}(v)=\{{[{({f}_{2}^{\{d\}})}^{\{{d}_{3}\}}]}^{{w}_{2}}(v{w}_{1})\}{w}_{3}$ by Equation

$(76)$, where

${w}_{j}={w}_{j}(d,{d}_{3},v)$,

$\phantom{\rule{3.33333pt}{0ex}}j\in \{1,2,3\}$,

$\phantom{\rule{3.33333pt}{0ex}}{d}_{3}=d\backslash e$, and

$d{d}_{3}=e$ by Equation

$(14)$. Thus, using Equation

$(35)$, we get that

belongs to

C (or

${C}^{*}$ respectively), giving Equation

$(1)$.

Then, we seek a solution

$x\in C$ (or

$x\in {C}^{*}$ respectively) of the equation

$xa=b$. It is equivalent to two equations:

${d}_{1}d={d}_{0}$ and

for each

$v\in V$, where

${d}_{1}\in D$,

${f}_{1}\in F$ (or

${f}_{1}\in {F}^{*}$ respectively), and

$x=({d}_{1},{f}_{1})$. Therefore,

${d}_{1}={d}_{0}/d$ and

${f}_{1}(v)={f}_{0}(v)/[\xi (({({d}_{0}/d)}^{\psi},{f}_{1}(v)),(d,f(v))){f}^{\{{d}_{0}/d\}}(v)]$. Thus,

belongs to

C (or

${C}^{*}$ respectively), giving Equation

$(2)$.

Moreover, $(e,e)(d,f)=(d,f)$ and $(d,f)(e,e)=(d,f)$ for each $d\in D$, $f\in F$ (or $f\in {F}^{*}$ respectively) by Equations $(35)$ and $(86)$. Therefore, the condition of Equation $(3)$ is also satisfied. Thus, C and ${C}^{*}$ are loops.

Evidently $D\ni d\mapsto (d,e)$ and $F\ni f\mapsto (e,f)\in C$ (or ${F}^{*}\ni f\mapsto (e,f)\in {C}^{*}$ respectively) provide embeddings of D and F (or D and ${F}^{*}$ respectively) into C (or ${C}^{*}$ respectively).

It remains to verify that

F (or

${F}^{*}$ respectively) is an almost normal subloop in

C (or

${C}^{*}$ respectively). Assume that

${d}_{1}\in D$,

${f}_{1}\in F$. Then,

and

Using the embedding ${\mathcal{C}}^{V}\hookrightarrow F$ and Equation $(35)$, we infer that $({d}_{1},{f}_{1})F=F({d}_{1},{f}_{1})$, since ${F}^{\{{d}_{1}\}}=F$ by Equation $(68)$, Lemma 5, and Equation $(30)$. It can be verified similarly that ${F}^{*}$ is the almost normal subloop in ${C}^{*}$. □

**Definition** **5.** The product Equation $(86)$ in the loop C (or ${C}^{*}$) of Theorem 5 is called a smashed twisted wreath product of D and F (or a restricted smashed twisted wreath product of D and ${F}^{*}$ respectively) with smashing factors ϕ, η, κ, and ξ, and it will be denoted by $C=D{\Delta}^{\varphi ,\eta ,\kappa ,\xi}F$ (or ${C}^{*}=D{\Delta}^{\varphi ,\eta ,\kappa ,\xi}{F}^{*}$ respectively). The loop C (or ${C}^{*}$) is also called a smashed splitting extension of F (or of ${F}^{*}$ respectively) by D.

**Theorem** **6.** Let the conditions of Remark 4 be satisfied and ${\mathcal{C}}_{m}(D)\subseteq \mathcal{C}$, where $\mathcal{C}$ is as in Equation $(28)$. Then, C and ${C}^{*}$ supplied with the binary operation of Equation $(86)$ are metagroups.

**Proof.** In view of Theorem 5,

C and

${C}^{*}$ are loops. To each element

b in

B, there corresponds an element

$\{b(v):\forall v\in V,\phantom{\rule{3.33333pt}{0ex}}b(v)=b\}$ in

F which can be denoted by

b also. From the conditions of Equations

$(29)$–

$(35)$, we deduce that

Hence, Equations

$(87)$ and

$(86)$ imply that

$(\mathcal{C}(A),\mathcal{C}(F))\subseteq \mathcal{C}(C)$. On the other hand,

${w}_{1}={\gamma}^{\tau}$ with

$\gamma \in {\mathcal{C}}_{m}(D)$ and

${w}_{2}={\gamma}_{3}/({\gamma}_{1}^{\psi}{\gamma}_{2}^{\psi}{\gamma}_{5}^{\psi})$ with

${\gamma}_{1}$,…,

${\gamma}_{5}$ in

${\mathcal{C}}_{m}(D)$ (see Equation

$(84)$); hence, the condition

${\mathcal{C}}_{m}(D)\subset \mathcal{C}$ implies that Equation

$(76)$ simplifies to

for each

$f\in F$,

$v\in V$, and

d and

${d}_{1}$ in

D, since

$\mathcal{C}\subseteq \mathcal{C}(A)$ by Equation

$(28)$. Next, we consider the following products:

and

Then, Equations

$(86)$,

$(90)$, and

$(33)$–

$(35)$ imply that

From Equations

$(88)$,

$(89)$,

$(76)$, and

$(35)$, we infer that

where

${w}_{3}={w}_{3}({d}_{1},{d}_{2},v)$. Therefore, from Equations

$(91)$ and

$(92)$, we infer that

where

for every

f,

${f}_{1}$,

${f}_{2}$ in

F,

d,

${d}_{1}$,

${d}_{2}$ in

D, and

$v\in V$. Then from Equation

$(93)$,

$\mathcal{C}(F)={(\mathcal{C}(B))}^{V}$ (see Theorem 2) and Equation

$(28)$, it follows that the loops

C and

${C}^{*}$ satisfy the condition of Equation

$(9)$, since

$(\mathcal{C},{\mathcal{C}}^{V})\subseteq \mathcal{C}(C)$. Thus,

C and

${C}^{*}$ are metagroups. □

**Remark** **5.** Generally, if $A\ne \{e\}$ and $A\ne D$, B, ϕ, η, κ, and ξ are nontrivial, where A, B, and D are metagroups or particularly may be groups, then the loops C and ${C}^{*}$ of Theorem 5 can be non-metagroups. If Equation $(35)$ drops the conditions $\xi ((e,e),(v,b))=e$ and $\xi ((v,b),(e,e))=e$ for each $v\in V$ and $b\in B$, then the proofs of Theorems 3–5 demonstrate that ${C}_{1}$ and ${C}_{2}$ are strict quasi-groups and that C and ${C}^{*}$ are quasi-groups.

**Definition** **6.** Let ${P}_{1}$ and ${P}_{2}$ be two loops with centers $\mathcal{C}({P}_{1})$ and $\mathcal{C}({P}_{2})$. Let alsofor each a and b in ${P}_{1}$, where $\nu (a,b)\in \mathcal{C}({P}_{2})$. Then, μ will be called a metamorphism of ${P}_{1}$ into ${P}_{2}$. If in addition μ is surjective and bijective, then it will be called a metaisomorphism and it will be said that ${P}_{1}$ is metaisomorphic to ${P}_{2}$. **Theorem** **7.** Suppose that A, B, and D are metagroups and that $A\subset D$, ${V}_{1}$, and ${V}_{2}$ are right transversals of A in D, ${F}_{j}={B}^{{V}_{j}}$, Then, ${P}_{1}$ is metaisomorphic to ${P}_{2}$ and ${P}_{1}^{*}$ to ${P}_{2}^{*}$.

**Proof.** By virtue of Theorem 5,

${P}_{j}$ and

${P}_{j}^{*}$ are loops, where

$j\in \{1,2\}$,

$\phantom{\rule{3.33333pt}{0ex}}{\mathcal{C}}^{{V}_{j}}\subset \mathcal{C}({P}_{j})$. From Equations

$(62)$ and

$(73)$, it follows that

for each

$d\in D$,

$v\in {V}_{j}$, and

$\delta \in \mathcal{C}(D)$, where

${s}_{j}$,

${v}^{{[a]}_{j}}$,

${d}^{{\tau}_{j}}$, and

${d}^{{\psi}_{j}}$ correspond to

${V}_{j}$,

$j\in \{1,2\}$. Then, Equations

$(68)$ and

$(63)$ imply that

for each

$d\in D$,

$v\in {V}_{j}$, and

$\delta \in \mathcal{C}(D)$,

$j\in \{1,2\}$. Therefore, from the identities of Equations

$(98)$,

$(99)$, and

$(84)$ and Lemma 2, we infer that

for each of

d and

${d}_{1}$ in

D,

$\delta $,

${\delta}_{1}$ and

${\delta}_{2}$ in

$\mathcal{C}(D)$, and

$v\in V$.

For each of

$f\in {F}_{1}$ and

$v\in {V}_{2}$, we put

From Lemma 5, it follows that

${V}_{2}^{{\tau}_{1}}={V}_{1}$ and

${v}_{1}^{{\tau}_{1}}\ne {v}_{2}^{{\tau}_{1}}$ for each

${v}_{1}\ne {v}_{2}$ in

${V}_{2}$, where

${V}_{2}^{{\tau}_{1}}=\{{v}^{{\tau}_{1}}:v\in {V}_{2}\}$. Then, Equations

$(87)$,

$(62)$,

$(100)$, and

$(101)$ and Lemma 1 imply that

for each

$f\in {F}_{1}$,

$d\in D$, where

${f}^{\mu \{d\}}={(\mu f)}^{\{d\}}$,

${f}^{\{d\}\mu}=\mu ({f}^{\{d\}})$ (see also Equation

$(72)$). From the identity of Equation

$(102)$ and the conditions of Equations

$(33)$ and

$(34)$, we infer that

for each of

d and

${d}_{1}$ in

D,

f and

${f}_{1}$ in

${F}_{1}$, and

$v\in {V}_{2}$, where

$\mu (d,f)=(d,\mu f)$,

$(d,f)(v)=(d,f(v))$. Hence,

for each

$v\in {V}_{2}$ (see also Equations

$(28)$ and

$(30)$). Thus,

${P}_{1}$ is metaisomorphic to

${P}_{2}$ and

${P}_{1}^{*}$ to

${P}_{2}^{*}$. □

**Theorem** **8.** Suppose that D is a nontrivial metagroup. Then, there exists a smashed splitting extension ${C}^{*}$ of a nontrivial central metagroup H by D such that $[H,{C}^{*}]\mathcal{C}(H)=H$, where $[a,b]=(e/a)((e/b)(ab))$ for each a and b in ${C}^{*}$.

**Proof.** Let

${d}_{0}$ be an arbitrary fixed element in

$D-\mathcal{C}(D)$. Assume that

A is a submetagroup in

D such that

A is generated by

${d}_{0}$ and a subgroup

${\mathcal{C}}_{0}$ contained in a center

$\mathcal{C}(D)$ of

D,

$\phantom{\rule{3.33333pt}{0ex}}{\mathcal{C}}_{m}(D)\subseteq {\mathcal{C}}_{0}\subseteq \mathcal{C}(D)$, where

${\mathcal{C}}_{m}(D)$ is a minimal subgroup in a center

$\mathcal{C}(D)$ of

D such that

${t}_{D}(a,b,c)\in {\mathcal{C}}_{m}(D)$ for each of

a,

b, and

c in

D. Therefore,

for each

$a\in A$,

k, and

n in

$\mathcal{C}=\{0,-1,1,-2,2,\dots \}$, where the following notation is used:

${a}^{2}=aa$,

${a}^{n+1}={a}^{n}a$ and

${a}^{-n}=e/{a}^{n}$, and

${a}^{0}=e$ for each

$n\in \mathbf{N}$ and

$p(k,n,a)\in {\mathcal{C}}_{m}(A)$. Hence, in particular,

A is a central metagroup. Then,

${d}_{0}{\mathcal{C}}_{m}(A)$ is a cyclic element in the quotient group

$A/{\mathcal{C}}_{m}(A)$ (see Theorem 1). Then, we choose a central metagroup

B generated by an element

${b}_{0}$ and a commutative group

${\mathcal{C}}_{1}$ such that

${b}_{0}\notin {\mathcal{C}}_{1}$,

$\phantom{\rule{3.33333pt}{0ex}}{\mathcal{C}}_{m}(D)\hookrightarrow {\mathcal{C}}_{1}$ and

$\mathcal{C}(A)\hookrightarrow {\mathcal{C}}_{1}$ and the quotient group

$B/{\mathcal{C}}_{m}(B)$ is of finite order

$l>1$. Then, let

$\varphi :A\to \mathcal{A}(B)$ satisfy the condition of Equation

$(30)$ and be such that

To satisfy the condition of Equation $(106)$, a natural number l can be chosen as a divisor of ${2}^{|{d}_{0}{\mathcal{C}}_{m}(A)|}-1$ if the order $|{d}_{0}{\mathcal{C}}_{m}(A)|$ of ${d}_{0}{\mathcal{C}}_{m}(A)$ in $A/{\mathcal{C}}_{m}(A)$ is positive; otherwise, l can be taken as any fixed odd number $l>1$ if $A/{\mathcal{C}}_{m}(A)$ is infinite.

Then, we take a right transversal V of A in D so that A is represented in V by e. Let $\Xi $, $\eta $, $\kappa $, and $\xi $ be chosen to satisfy the conditions of Equations $(29)$–$(35)$, where ${\mathcal{C}}_{m}(B)\hookrightarrow \mathcal{C}$, ${\mathcal{C}}_{m}(A)\hookrightarrow \mathcal{C}$, ${\mathcal{C}}_{0}\hookrightarrow \mathcal{C}$, and ${\mathcal{C}}_{1}\hookrightarrow \mathcal{C}$. With these data, according to Theorem 6, ${C}^{*}$ is a metagroup, since ${\mathcal{C}}_{m}(D)\hookrightarrow {\mathcal{C}}_{1}$ and ${\mathcal{C}}_{m}(D)\hookrightarrow {\mathcal{C}}_{0}$. That is, ${C}^{*}$ is a smashed splitting extension of the central metagroup ${F}^{*}$ by D.

Apparently, there exists ${f}_{0}\in {F}^{*}$ such that ${f}_{0}(e)={b}_{0}$, ${f}_{0}(v)=e$ for each $v\in V-\{e\}$. Therefore, ${f}_{0}^{\{v\}}(v)={b}_{0}$ for each $v\in V$, since $s(v,v)=e$, ${v}^{[v\backslash e]}={[v(v\backslash e)]}^{\tau}=e$.

Let ${v}_{1}\ne {v}_{2}$ belong to V. Then, ${({v}_{2}({v}_{1}\backslash e))}^{\tau}={v}_{3}\in V$. Assume that ${v}_{3}=e$. The latter is equivalent to ${v}_{2}({v}_{1}\backslash e)=a\in A$. From Equation $(13)$, it follows that ${v}_{2}=a/({v}_{1}\backslash e)=\gamma a{v}_{1}$, where $\gamma ={t}_{D}({v}_{1},{v}_{1}\backslash e,{v}_{1})/{t}_{D}(a{v}_{1},{v}_{1}\backslash e,{v}_{1})$ by Equation $(11)$ and Lemma 2, since $e/({v}_{1}\backslash e)={v}_{1}$. Hence, ${v}_{2}={v}_{2}^{\tau}={(\gamma a{v}_{1})}^{\tau}={\gamma}^{\tau}{v}_{1}$ by Equation $(63)$, and consequently, ${({v}_{2}({v}_{1}\backslash e))}^{\tau}={\gamma}^{\tau}=e$, contradicting the supposition ${v}_{1}\ne {v}_{2}$. Thus, ${v}_{3}\ne e$, and consequently, ${f}_{0}^{\{{v}_{1}\}}({v}_{2})={e}^{s({v}_{1},{v}_{2})}=e$ by Equation $(31)$. This implies that $\{{f}_{0}^{\{v\}}:v\in V\}\mathcal{C}({F}^{*})$ generates ${F}^{*}$.

Evidently,

${[v({d}_{0}\backslash e)]}^{\tau}\ne e$ for each

$v\in V-\{e\}$, since

${d}_{0}\backslash e\in A$ and the following conditions

$s\in D$,

$sq\in A$, and

$q\in A$ imply that

$s\in A$ because

A is the submetagroup in

D. Note that

$e/d=(d\backslash e)/{t}_{A}(e/d,d,d\backslash e)$ for each

$d\in A$ by Equation

$(11)$; consequently,

$s(d,e)=d{t}_{A}(e/d,d,d\backslash e)$. On the other hand,

${t}_{A}(a,b,c)\in \mathcal{C}$ for each of

a,

b, and

c in

A and

by Equation

$(87)$; hence,

${f}_{0}^{\{{d}_{0}\}}(e)=\varphi ({d}_{0}){b}_{0}={b}_{0}^{2}$, and consequently,

since

Therefore, we deduce using Equation

$(107)$ that

where

Thus, $w=w(v)\in \mathcal{C}$ for each $v\in V$ and ${f}_{0}\in [{F}^{*},{C}^{*}]$, since ${\mathcal{C}}^{V}\cap {F}^{*}\subset \mathcal{C}({F}^{*})$. Hence, ${F}^{*}\subseteq [{F}^{*},{C}^{*}]\mathcal{C}({F}^{*})$, since ${F}^{*}\hookrightarrow {C}^{*}$ and $\mathcal{C}({C}^{*})\cap {F}^{*}\subseteq \mathcal{C}({F}^{*})$. On the other hand, ${\mathcal{C}}_{m}(A)\hookrightarrow \mathcal{C}$, ${\mathcal{C}}_{m}(B)\hookrightarrow \mathcal{C}$, ${\mathcal{C}}_{m}(D)\hookrightarrow {\mathcal{C}}_{j}$, and ${\mathcal{C}}_{j}\hookrightarrow \mathcal{C}$ for each $j\in \{0,1\}$. Therefore, Equations $(107)$, $(108)$, and $(88)$ imply that $c{F}^{*}={F}^{*}c$ and $c[{F}^{*},{C}^{*}]\mathcal{C}({F}^{*})=[{F}^{*},{C}^{*}]\mathcal{C}({F}^{*})c$ for each $c\in {C}^{*}$. Hence, $[{F}^{*},{C}^{*}]\mathcal{C}({F}^{*})\subseteq {F}^{*}$. Taking $H={F}^{*}$, we get the assertion of this theorem. □

**Corollary** **2.** Let the conditions of Theorem 8 be satisfied and D be generated by ${\mathcal{C}}_{m}(D)$ and at least two elements ${d}_{1}$, ${d}_{2}$,… such that ${d}_{1}\ne e$ and $[{d}_{2}\backslash e,{d}_{1}\backslash e]=e$. Then, the smashed splitting extension ${C}^{*}$ can be generated by $\mathcal{C}({F}^{*})$ and elements ${c}_{1}$, ${c}_{2}$,… such that ${d}_{j}\backslash e\in {F}^{*}{c}_{j}$ for each j.

**Proof.** We take

${d}_{0}={d}_{1}$ in the proof of Theorem 8; thus,

${c}_{1}=({d}_{1}\backslash e,e)$,

${c}_{2}=({d}_{2}\backslash e,{f}_{0})$, and

${c}_{j}=({d}_{j}\backslash e,e)$ for each

$j\ge 3$. Therefore Equations

$(66)$,

$(108)$, and

$(35)$ imply that

since

$[{d}_{2}\backslash e,{d}_{1}\backslash e]=e$ and

$e/({d}_{2}\backslash e)={d}_{2}$. Thus, the submetagroup of

${C}^{*}$ which is generated by

${\mathcal{C}}_{m}(D)$ and

$\{{c}_{j}:j\}$ contains the metagroup

D and

$(e,p{f}_{0})$. Therefore, the following set

$\{{f}^{\{d\}}:d\in D\}\mathcal{C}({F}^{*})$ generates the central metagroup

${F}^{*}$, since

$V\subset D$ and

$\{{f}^{\{v\}}:v\in V\}\mathcal{C}({F}^{*})$ generate

${F}^{*}$. Notice that

${\mathcal{C}}_{m}(D)\hookrightarrow \mathcal{C}({F}^{*})$. Hence,

$\{{c}_{j}:j\}\mathcal{C}({F}^{*})$ generates

${C}^{*}$. □

**Example** **1.** Assume that A is a unital algebra over a commutative associative unital ring F supplied with a scalar involution $a\mapsto \overline{a}$ so that its norm N and trace T maps have values in F and fulfil conditions:for each a and b in A. We remind that, if a scalar $f\in F$ satisfies the condition $\forall a\in A\phantom{\rule{3.33333pt}{0ex}}fa=0\Rightarrow a=0$, then such element f is called cancelable. For such a cancelable scalar f, the Cayley–Dickson doubling procedure induces a new algebra $C(A,f)$ over F such thatandfor each a and b in A. Such an element l is called a doubling generator. From Equations $(113)$–$(115)$, it follows that $\forall a\in A,\forall b\in A$$\phantom{\rule{3.33333pt}{0ex}}T(a)=T(a+bl)$ and $N(a+bl)=N(a)+fN(b)$. Apparently, the algebra A is embedded into $C(A,f)$ as $A\ni a\mapsto (a,0)$, where $(a,b)=a+bl$. It is put by induction ${A}_{n}({f}_{(n)})=C({A}_{n-1},{f}_{n})$, where ${A}_{0}=A$, ${f}_{1}=f$, $n=1,2,\dots $, and $\phantom{\rule{3.33333pt}{0ex}}{f}_{(n)}=({f}_{1},\dots ,{f}_{n})$. Then, ${A}_{n}({f}_{(n)})$ is a generalized Cayley–Dickson algebra, when F is not a field, or a Cayley–Dickson algebra, when F is a field. There is an algebra ${A}_{\infty}(f):={\bigcup}_{n=1}^{\infty}{A}_{n}({f}_{(n)})$, where $f=({f}_{n}:n\in \mathbf{N})$. In the case of $char(F)\ne 2$, let $Im(z)=z-T(z)/2$ be the imaginary part of a Cayley–Dickson number z and, hence, $N(a):={N}_{2}(a,\overline{a})/2$, where ${N}_{2}(a,b):=T(a\overline{b})$.

If the doubling procedure starts from $A=F1=:{A}_{0}$, then ${A}_{1}=C(A,{f}_{1})$ is a *-extension of F. If ${A}_{1}$ has a basis $\{1,u\}$ over F with the multiplication table ${u}^{2}=u+w$, where $w\in F$ and $4w+1\ne 0$, with the involution $\overline{1}=1$, $\overline{u}=1-u$, then ${A}_{2}$ is the generalized quaternion algebra and ${A}_{3}$ is the generalized octonion (Cayley–Dickson) algebra.

Particularly, for $F=\mathbf{R}$ and ${f}_{n}=1$ for each n by ${\mathcal{A}}_{r}$ the real Cayley-Dickson algebra with generators ${i}_{0},\dots ,{i}_{{2}^{r}-1}$ will be denoted such that ${i}_{0}=1$, $\phantom{\rule{3.33333pt}{0ex}}{i}_{j}^{2}=-1$ for each $j\ge 1$, and $\phantom{\rule{3.33333pt}{0ex}}{i}_{j}{i}_{k}=-{i}_{k}{i}_{j}$ for each $j\ne k\ge 1$. Note that the Cayley–Dickson algebra ${\mathcal{A}}_{r}$ for each $r\ge 3$ is nonassociative, for example, $({i}_{1}{i}_{2}){i}_{4}=-{i}_{1}({i}_{2}{i}_{4})$, etc. Moreover, for each $r\ge 4$, the Cayley–Dickson algebra ${\mathcal{A}}_{r}$ is nonalternative (see References [7,11,12]). Frequently, $\overline{a}$ is also denoted by ${a}^{*}$ or $\tilde{a}$. Then, ${G}_{r}=\{{i}_{j},\phantom{\rule{3.33333pt}{0ex}}-{i}_{j}:\phantom{\rule{3.33333pt}{0ex}}j=0,1,\dots ,{2}^{r}-1\}$ is a finite metagroup for each $3\le r<\infty $. Equation $(117)$ is an example of the smashed product.

Then, one can take a Cayley–Dickson algebra ${A}_{n}$ over a commutative associative unital ring $\mathcal{R}$ of characteristic different from two such that ${A}_{0}=\mathcal{R}$, $n\ge 2$. There are basic generators ${i}_{0},{i}_{1},\dots ,{i}_{{2}^{n}-1}$, where ${i}_{0}=1$. Choose Ψ as a multiplicative subgroup contained in the ring $\mathcal{R}$ such that ${f}_{j}\in \Psi $ for each $j=0,\dots ,n$. Put ${G}_{n}=\{{i}_{0},{i}_{1},\dots ,{i}_{{2}^{n}-1}\}\times \Psi $. Then, ${G}_{n}$ is a central metagroup because, in this case, Ψ is commutative.

**Example** **2.** More generally, suppose that H is a group such that $\Psi \subset H$, with relations $h{i}_{k}={i}_{k}h$ and $(hg){i}_{k}=h(g{i}_{k})$ for each $k=0,1,\dots ,{2}^{n}-1$ and each h and g in H. Then, ${G}_{n}=\{{i}_{0},{i}_{1},\dots ,{i}_{{2}^{n}-1}\}\times H$ is also a metagroup. If the group H is noncommutative, then the latter metagroup can be noncentral (see the condition of Equation $(10)$ in Definition 1). Utilizing the notation of Example 1, we get that the Cayley–Dickson algebra ${\mathcal{A}}_{\infty}$ over the real field $\mathbf{R}$ with ${f}_{n}=1$ for each n provides a pattern of a metagroup ${G}_{\infty}=\{{i}_{j},\phantom{\rule{3.33333pt}{0ex}}-{i}_{j}:\phantom{\rule{3.33333pt}{0ex}}0\le j\in \mathbf{Z}\}$, where $\mathbf{Z}$ denotes the ring of integers.

**Example** **3.** Certainly, in general, metagroups need not be central. On the other hand, if a metagroup is associative, then it is a group [1]. Apparently, each group is a metagroup also. For a group G, its associativity evidently means that ${t}_{G}(a,b,c)=e$ [1]. From the given metagroups, new metagroups can be constructed using their direct, semidirect products, smashed products, and smashed twisted wreath products. Therefore, there are abundant families of noncentral metagroups and also of central metagroups different from groups.

Equations $(39)$, $(46)$, $(47)$, $(85)$, $(86)$, and $(94)$–$(96)$ provide examples of metagroups with complicated nonassociative noncommutative structures. The presented above theorems also permit to construct different examples of nonassociative quasi-groups and loops.