Identities with permutations and quasigroups isotopic to groups and to Abelian groups

Abstract

This paper continues the study of quasigroups isotopic to groups and to Abelian groups. For identities with permutations of a certain kind the notion of the type is introduced; identities with permutations of various types are presented, which characterize quasigroups isotopic to Abelian groups. The approach being used had allowed to establish a number of new unbalanced identities involving four variables in an equasigroup, as well as a number of identities involving three variables in an equasigroup with one nullary operation, each of which characterises a quasigroup isotopic to a group or to an Abelian group.

A-nuclei and A-centers of a quasigroup

“…on a quasigroup (Q, •), where all α i are some fixed permutations of the set Q, it is possible to reduce to the following identity [25]…”

“…In [25] using famous Four quasigroups theorem [1,16] it is proved: if a quasigroup (Q, •) satisfies identity (37) then this quasigroup is a group isotope and vice versa.…”