Geodesic
Crossed homomorphisms of finite algebras with a scheme of binary operators
Diskretnaya Matematika, Tome 12 (2000) no. 2, pp. 66-84

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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. 
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     author = {G. A. Karpunin and I. G. Shaposhnikov},
     title = {Crossed homomorphisms of finite algebras with a scheme of binary operators},
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G. A. Karpunin; I. G. Shaposhnikov. Crossed homomorphisms of finite algebras with a scheme of binary operators. Diskretnaya Matematika, Tome 12 (2000) no. 2, pp. 66-84. http://geodesic.mathdoc.fr/item/DM_2000_12_2_a4/