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Hyperidentities in algebras and varieties. (English. Russian original) Zbl 0921.08005
Russ. Math. Surv. 53, No. 1, 57-108 (1998); translation from Usp. Mat. Nauk 53, No. 1, 61-114 (1998).
The article consists of 4 sections. Let us list some results. In Section 1 semigroups, groups and multiplicative groups of fields are represented (and characterized) as semigroups and groups of binary functions with hyperidentities, respectively. In Section 2 it is proved that pairs of groups \((G,H)\), where \(H\trianglelefteq G\), can be represented as pairs of group automorphisms \((\varphi,\widetilde\psi)\) and \((\varphi,\widetilde\varepsilon)\), \(\varepsilon\) being the identity mapping, of one and the same algebra. This section also contains a characterization of subdirectly indecomposable algebras of the hypervarieties under consideration. This is used for the characterization of hyperidentities of certain varieties of classical algebras. Section 4 is devoted to non-trivial hyperidentities of associativity and distributivity in invertible algebras and algebras related to invertible algebras. In particular, the author characterizes invertible algebras with the non-trivial hyperidentity of left (right) distributivity in the presence of the trivial hyperidentity of right (left) distributivity.

MSC:
08B99 Varieties
20M07 Varieties and pseudovarieties of semigroups
08B26 Subdirect products and subdirect irreducibility
08A35 Automorphisms and endomorphisms of algebraic structures
20M20 Semigroups of transformations, relations, partitions, etc.
20E10 Quasivarieties and varieties of groups
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