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 .
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.
Let G be a set with a single-valued binary operation (multiplication) 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 :
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 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 . 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 .
If H is a submetagroup (or a subloop) of the metagroup G (or the loop G) and for each , then H will be called almost normal. If, in addition, and for each g and k in G, then H will be called a normal submetagroup (or a normal subloop respectively).
Henceforward, notations and 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 means the difference of them: . Henceforward, maps and functions on metagroups are supposed to be single-valued unless otherwise specified.
If G is a metagroup, then for each a and , the following identities are fulfilled:
The conditions of Equations – imply that
for each a and b in G. Using the condition of Equation and the identities of Equations and , we deduce the following:
which leads to Equation .
Let ; then, from the identities of Equations and , it follows that
which provides Equation .
Now, let ; then, the identities of Equations and imply that
which demonstrates Equation . □
Assume that G is a metagroup. Thenm for every a, , , and in G and , , and in , we have the following:
Since and for every , , in G, then
Therefore, for every , , in G and , , and in , we infer the following:
For each , a and b in G, because is the commutative group. Thus, .
From the condition in Equation , Lemma 1, and the identity of Equation , it follows that
for each , implying Equation . □
If G is a metagroup and is a subgroup in a center such that for each a, b, and c in G, then its quotient is a group.
As traditionally, the following notation is used:
for subsets A and B in G. Then from the conditions of Equations –, it follows that, for each a, b, and c in G, the following identities take place:
and . Evidently . In view of Lemmas 1 and 2 , consequently, for each a unique inverse exists. Thus the quotient of G by is a group. □
Let G be a metagroup, then and are metagroups.
At first, we consider . Let and belong to G. Then, there are unique and , since the map is single-valued (see Definition 1). Since and for each , then and are bijective and surjective maps.
We put for each and in G, where for each . This provides a single-valued map from into . Then, for each a, b, x, and y in G, the equations and are equivalent to and , respectively. That is, and are unique. On the other hand, and for each .
Then, we infer the following:
Consequently, . Evidently, and . Thus, the conditions of Equations – and are satisfied for .
Similarly, putting and for each and , the conditions of Equations – and are verified for . □
Assume that G is a metagroup and that , , and . Then
From Equations and , we deduce that
, which implies Equation . Then, from Equations and , we infer the following:
which implies Equation .
Utilizing Equations and , we get ; hence, , implying Equation .
Equations and imply that ; consequently, , and hence,
3. Smashed Products and Smashed Twisted Products of Metagroups
Let be a family of metagroups (see Definition 1), where , J is a set. Then, their direct product is a metagroup and
Let A and B be two metagroups, and let be a commutative group such that
where denotes a minimal subgroup in containing 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 , an equivalence relation Ξ is considered such that
for every v in A, b in B, and γ in .
where 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 and , then it will be written shortly instead of , where . Let also
be single-valued mappings written shortly as η, κ, and ξ correspondingly such that
for every u and v in A, b, and c in B, γ in , where e denotes the neutral element in and in A and B.
for each of and in A and of and in B.
The Cartesian product supplied with such a binary operation of Equation will be denoted by .
Then, we put
for each of and in A and of and in B.
The Cartesian product supplied with a binary operation of Equation will be denoted by .
Let the conditions of Remark 1 be fulfilled. Then, the Cartesian product supplied with a binary operation of Equation is a metagroup. Moreover, there are embeddings of A and B into such that B is an almost normal submetagroup in . If in addition , then B is a normal submetagroup.
The first part of this theorem was proven in Theorem 9 in Reference . Naturally, A is embedded into as and B is embedded into as . Let and ; then, and , since by . From , , , and Equations and , it follows that , where . Thus, B is an almost normal submetagroup in (see Definition 1). If in addition , then evidently B is a normal submetagroup (see also the condition of Equation ), since and for each g and h in G, . □
Suppose that the conditions of Remark 1 are satisfied. Then, the Cartesian product supplied with a binary operation of Equation is a metagroup. Moreover, there exist embeddings of A and B into such that B is an almost normal submetagroup in . If additionally , then B is a normal submetagroup.
The conditions of Remark 1 imply that the binary operation of Equation is single-valued.
We consider the following formulas:
and , where , , and are in A and where , , and are in B. Utilizing Equations – and , we get the following:
Consequently, for each , , . We denote
in more details by
(see Equation ).
Evidently, Equation is a consequence of Equations and .
Note that, if , then
Therefore, . Consequently, .
Then, we seek a solution of the following equation:
where , .
From Equations and , it follows that
Consequently, . Therefore, Equations and imply that
Thus, and prescribed by Equations and provide a unique solution of Equation .
Analogously for the following equation
where , , we deduce that
Consequently, , and hence, . From Equations and , it follows that ; consequently,
Thus, a unique solution of Equation is given by Equations and .
Then, we have and and get the following:
and , where . This means that the properties of Equations – and are fulfilled for .
Evidently, there are embeddings of A and B into as and , respectively. Suppose that and , then
Therefore, by the conditions of Equations (30) and (35), since and . Thus, B is an almost normal submetagroup in (see Definition 1). If additionally , then apparently B is a normal submetagroup (see also the condition of Equation ), since and for each g and h in G, . □
We call the metagroup provided by Theorem 3 (or 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.
From Theorems 2–4, it follows that, taking nontrivial η, κ, and ξ and starting even from groups with nontrivial or , it is possible to construct new metagroups with nontrivial and ranges of 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 for an infinite family of elements a in or in . 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 instead of or instead of on the right sides of Equations and , correspondingly, etc.
4. Smashed Twisted Wreath Products of Metagroups
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 , where , that is,
The cases and are trivial. Let and , and let be a center of D. From the conditions of Equations (4)–(8), it follows that implies .
Assume that and are such that . It is equivalent to (, , ). From Equation , it follows that because . Thus,
Suppose now that , and . This is equivalent to (, , ). By the identity of Equation , the latter is equivalent to . On the other hand,
by Equations , , and . Together with this gives the equivalence:
Let be a family of subsets K in D such that for each in K. Let be directed by inclusion. Then, , since and . Therefore, from Equations and and the Kuratowski-Zorn lemma (see Reference ), the assertion of this lemma follows, since a maximal element V in gives Equations and . □
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.
Let D be a metagroup, A be a submetagroup in D, and V be a right transversal of A in D. Then,
We denote b in the decomposition of Equation by and , where τ and ψ are the shortened notations of and , respectively. That is, there are single-valued maps
These maps are idempotent and for each .
According to Equation , ; hence, . From Equation , it follows that ; consequently, by Lemma 2,
Notice that the metagroup need not be power-associative. Then, and can be calculated with the help of the identity of Equation . Suppose that a and y belong to D, , , , and . Then, . According to Equation there exists a unique decomposition , where , ; hence, . On the other hand, by Equation . We denote a subgroup in by or shortly , when D is specified. From Lemma 2 and Equation , it follows that
Let be a minimal subgroup in generated by a set . From Equation , it follows that and is a submetagroup in D. By virtue of Theorem 1, and are groups such that . For each , there exists a unique decomposition
by Equation . Take in particular ; then, , where , . Therefore, and there exists a subset in V such that , since is a subgroup in (see Equation ). Equation implies that and for each . Using this, we subsequently deduce that
for each and . Hence,
where and . From a uniqueness of this representation, it follows that
Using Equation we infer that
On the other hand, if , then and Equations and imply particularly that
since for each a and d in D and . Then, from , , it follows that and ; consequently, by Lemma 2 and Equation ,
for each a and y in D. Particularly,
From Equations and , it follows that the metagroup D acts on V transitively by right shift operators , where for each a and y in D. Therefore, we put
Then from Equations , , , and and Lemma 2, we deduce that, for each a, c, and d in D
In particular, for each . Next, we put . It is convenient to choose . Hence, for each . Thus, the submetagroup A is the stabilizer of e and Equation implies that
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 – be satisfied for A and B. By Theorem 2, there exists a metagroup
It contains a submetagroup
where is a support of and denotes the cardinality of a set Ω.
Let for each and . We put
where , , for each , and . Then, for each , we put
(see also Equations and ). Hence,
since and .
Let the conditions of Remark 4 be satisfied. Then, for each of f and in F and of d and in D, ,
where , , .
Equations and imply the identity of Equation .
Let , d and belong to D, and ; then, from Equations and , it follows that
From Equations , , , , , and and Lemma 2, we deduce that
where , where
Then Equations (73), (66), (25), (13), and (62) and Lemma 2 imply that
where by , and Lemma 2;
by Equations and , where
by Equations , , , and , where . Hence,
Thus, the identities of Equations – imply that
By Lemmas 1 and 2 and Equation , representations of simplify:
Therefore, for each d and in D and , since , , and belong to A. Then, from Equations , , , , we infer that
, . Equations and imply that for each and , ; consequently, , and hence, for each , d and in D, by Equation , since . Thus Equation follows from Equations and . □
Suppose that the conditions of Remark 4 are satisfied and on the Cartesian product (or ) a binary operation is given by the following formula:
where for every d and in D, f and in F (or respectively), and .
Let C, , D, F, and be the same as in Definition 4. Then, C and are loops and there are natural embeddings , , , and such that F (or ) is an almost normal subloop in C (or respectively).
The operation of Equation is single-valued. Let and , where d and are in D and where f and are in F (or ).
The equation is equivalent to and
where , (or respectively), , for each . Therefore, , by Equation and Theorem 2. On the other hand, by Equation and by Equation , where , , , and by Equation . Thus, using Equation , we get that
belongs to C (or respectively), giving Equation .
Then, we seek a solution (or respectively) of the equation . It is equivalent to two equations: and
for each , where , (or respectively), and . Therefore, and . Thus,
belongs to C (or respectively), giving Equation .
Moreover, and for each , (or respectively) by Equations and . Therefore, the condition of Equation is also satisfied. Thus, C and are loops.
Evidently and (or respectively) provide embeddings of D and F (or D and respectively) into C (or respectively).
It remains to verify that F (or respectively) is an almost normal subloop in C (or respectively). Assume that , . Then,
Using the embedding and Equation , we infer that , since by Equation , Lemma 5, and Equation . It can be verified similarly that is the almost normal subloop in . □
The product Equation in the loop C (or ) of Theorem 5 is called a smashed twisted wreath product of D and F (or a restricted smashed twisted wreath product of D and respectively) with smashing factors ϕ, η, κ, and ξ, and it will be denoted by (or respectively). The loop C (or ) is also called a smashed splitting extension of F (or of respectively) by D.
Let the conditions of Remark 4 be satisfied and , where is as in Equation . Then, C and supplied with the binary operation of Equation are metagroups.
In view of Theorem 5, C and are loops. To each element b in B, there corresponds an element in F which can be denoted by b also. From the conditions of Equations –, we deduce that
Hence, Equations and imply that . On the other hand, with and with ,…, in (see Equation ); hence, the condition implies that Equation simplifies to
for each , , and d and in D, since by Equation . Next, we consider the following products:
Then, Equations , , and – imply that
From Equations , , , and , we infer that
where . Therefore, from Equations and , we infer that
for every f, , in F, d, , in D, and . Then from Equation , (see Theorem 2) and Equation , it follows that the loops C and satisfy the condition of Equation , since . Thus, C and are metagroups. □
Generally, if and , 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 of Theorem 5 can be non-metagroups. If Equation drops the conditions and for each and , then the proofs of Theorems 3–5 demonstrate that and are strict quasi-groups and that C and are quasi-groups.
Let and be two loops with centers and . Let also
for each a and b in , where . Then, μ will be called a metamorphism of into . If in addition μ is surjective and bijective, then it will be called a metaisomorphism and it will be said that is metaisomorphic to .
Suppose that A, B, and D are metagroups and that , , and are right transversals of A in D, ,
Then, is metaisomorphic to and to .
By virtue of Theorem 5, and are loops, where , . From Equations and , it follows that
for each , , and , where , , , and correspond to , . Then, Equations and imply that
for each , , and , . Therefore, from the identities of Equations , , and and Lemma 2, we infer that
for each of d and in D, , and in , and .
For each of and , we put
From Lemma 5, it follows that and for each in , where . Then, Equations , , , and and Lemma 1 imply that
for each , , where , (see also Equation ). From the identity of Equation and the conditions of Equations and , we infer that
for each of d and in D, f and in , and , where , . Hence,
for each (see also Equations and ). Thus, is metaisomorphic to and to . □
Suppose that D is a nontrivial metagroup. Then, there exists a smashed splitting extension of a nontrivial central metagroup H by D such that , where for each a and b in .
Let be an arbitrary fixed element in . Assume that A is a submetagroup in D such that A is generated by and a subgroup contained in a center of D, , where is a minimal subgroup in a center of D such that for each of a, b, and c in D. Therefore,
for each , k, and n in , where the following notation is used: , and , and for each and . Hence, in particular, A is a central metagroup. Then, is a cyclic element in the quotient group (see Theorem 1). Then, we choose a central metagroup B generated by an element and a commutative group such that , and and the quotient group is of finite order . Then, let satisfy the condition of Equation and be such that
To satisfy the condition of Equation , a natural number l can be chosen as a divisor of if the order of in is positive; otherwise, l can be taken as any fixed odd number if is infinite.
Then, we take a right transversal V of A in D so that A is represented in V by e. Let , , , and be chosen to satisfy the conditions of Equations –, where , , , and . With these data, according to Theorem 6, is a metagroup, since and . That is, is a smashed splitting extension of the central metagroup by D.
Apparently, there exists such that , for each . Therefore, for each , since , .
Let belong to V. Then, . Assume that . The latter is equivalent to . From Equation , it follows that , where by Equation and Lemma 2, since . Hence, by Equation , and consequently, , contradicting the supposition . Thus, , and consequently, by Equation . This implies that generates .
Evidently, for each , since and the following conditions , , and imply that because A is the submetagroup in D. Note that for each by Equation ; consequently, . On the other hand, for each of a, b, and c in A and
by Equation ; hence, , and consequently,
Therefore, we deduce using Equation that
Thus, for each and , since . Hence, , since and . On the other hand, , , , and for each . Therefore, Equations , , and imply that and for each . Hence, . Taking , we get the assertion of this theorem. □
Let the conditions of Theorem 8 be satisfied and D be generated by and at least two elements , ,… such that and . Then, the smashed splitting extension can be generated by and elements , ,… such that for each j.
We take in the proof of Theorem 8; thus, , , and for each . Therefore Equations , , and imply that
since and . Thus, the submetagroup of which is generated by and contains the metagroup D and . Therefore, the following set generates the central metagroup , since and generate . Notice that . Hence, generates . □
Assume that A is a unital algebra over a commutative associative unital ring F supplied with a scalar involution 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 satisfies the condition , then such element f is called cancelable. For such a cancelable scalar f, the Cayley–Dickson doubling procedure induces a new algebra over F such that
for each a and b in A. Such an element l is called a doubling generator. From Equations –, it follows that and . Apparently, the algebra A is embedded into as , where . It is put by induction , where , , , and . Then, 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 , where . In the case of , let be the imaginary part of a Cayley–Dickson number z and, hence, , where .
If the doubling procedure starts from , then is a *-extension of F. If has a basis over F with the multiplication table , where and , with the involution , , then is the generalized quaternion algebra and is the generalized octonion (Cayley–Dickson) algebra.
Particularly, for and for each n by the real Cayley-Dickson algebra with generators will be denoted such that , for each , and for each . Note that the Cayley–Dickson algebra for each is nonassociative, for example, , etc. Moreover, for each , the Cayley–Dickson algebra is nonalternative (see References [7,11,12]). Frequently, is also denoted by or .
Then, is a finite metagroup for each . Equation is an example of the smashed product.
Then, one can take a Cayley–Dickson algebra over a commutative associative unital ring of characteristic different from two such that , . There are basic generators , where . Choose Ψ as a multiplicative subgroup contained in the ring such that for each . Put . Then, is a central metagroup because, in this case, Ψ is commutative.
More generally, suppose that H is a group such that , with relations and for each and each h and g in H. Then, is also a metagroup. If the group H is noncommutative, then the latter metagroup can be noncentral (see the condition of Equation in Definition 1). Utilizing the notation of Example 1, we get that the Cayley–Dickson algebra over the real field with for each n provides a pattern of a metagroup , where denotes the ring of integers.
Certainly, in general, metagroups need not be central. On the other hand, if a metagroup is associative, then it is a group . Apparently, each group is a metagroup also. For a group G, its associativity evidently means that .
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 , , , , , and – 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.
The results of this article can be used for further studies of metagroups, quasi-groups, loops, and noncommutative manifolds related with them. Besides applications of metagroups, loops, and quasi-groups outlined in the introduction, it is interesting to mention possible applications in mathematical coding theory and classification of information flows and their technological implementations [28,29,30] because, frequently, codes are based on binary systems. Moreover, twisted products are used for creating complicated codes . In view of this, to study creating more complicated codes with the help of smashed twisted products of metagroups, Equations and – provide additional options in the nonassociative case in comparison with the associative case.
Wreath products of groups are used for studies of varieties , so it will be interesting to investigate noncommutative varieties using metagroups. Then, twisted products are utilized for investigations of Lie groups and semi-Riemann manifolds [23,25]. Therefore, we will study their nonassociative metagroup analogs that can be used in noncommutative geometry and quantum field theory [16,31,32,33,34,35] because Lie groups and manifolds are actively used in these areas.
This research received no external funding.
Conflicts of Interest
The author declares no conflict of interest.
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