Crossed homomorphisms of finite algebras with a scheme of binary operators

A notion of a right (left) crossed homomorphism of finite algebras with a scheme of binary operators is introduced. This notion generalizes the notion of a right (left) crossed isotopy of quasigroups introduced by V. D. Belousov. A theorem on crossed homomorphisms (an analogue of the classical theorem on homomorphisms) is proved. The description of crossed homomorphisms of an algebra with a scheme of operators is reduced to the description of its crossed congruences. Crossed congruences of quasigroups that are isotopic to groups and cross-isotopic to groups are studied. The possibility of applying crossed congruences to constructing algorithms for solving equations over algebras is shown.