Abstract
In this article, nonassociative metagroups are studied. Different types of smashed products and smashed twisted wreath products are scrutinized. Extensions of central metagroups are studied.
Highlights
Nonassociative algebras compose a great area of algebra
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
Twisted products are used for creating complicated codes [22]
Summary
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). 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. It appears that, generally, they provide loops (see Theorem 5). 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 the conclusions)
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