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.

Reviewer: B.Vernikov (Ekaterinburg)

##### 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 |