On congruences of finite multibasic universal algebras

Abstract

The subset of congruences of a finite multibasic universal algebra (called the set of maximal conguences) which determines the whole set of congruences is found. The optimality of some method of equation solving with the use of maximal congruences is demonstrated; congruences of quasigroups which are isotopic and cross-isotopic to groups are described. The existence of simple (having no non-trivial congruences) universal algebras defined on sets of any finite orders is proved. With the use of wreath product construction, extensions of multibasic universal algebras are described.

Homomorphic relations of algebras with a scheme of operators

“…The last proposition shows that our notion of inverse homomorphisms has some new properties, since there exist n-quasigroups that have no nontrivial homomorphisms (see [7]). It also shows that homomorphic relations cannot be reduced to homomorphisms of algebraic systems.…”

“…A homomorphic relation can be applied in the same way as homomorphisms [7], π-homomorphisms [3], and crossed homomorphisms [4], for example, for solution of equations over algebras.…”

“…In [7], the notion of a maximal congruence was introduced. Maximal congruences are sufficient to describe all congruences, and maximal congruences are often more easily studied (see, for example, [4,7]).…”

“…. At the same time, any Abelian group of prime order has no nontrivial homomorphic images (see [7]), therefore, (E; f ) is not its inverse homomorphic image. Indeed, if (ϕ 1 , .…”

Quasigroups in cryptology

“…Therefore we should have easily generated objects (cyclic group, abelian group, group), fast and complicate methods of their transformation (parastrophy, isotopy, isostrophy, crossed isotopy [109], homotopy, generalized isotopy), their glue and blowing (direct product, semi-direct product, wreath product [66], crossed product, generalized crossed product). For these aims various linear quasigroups (especially n-ary quasigrous) are quite suitable [7,85,113].…”