Identities with permutations leading to linearity of quasigroups

Abstract

We consider a class of identities with permutations of three variables in a quasigroup (Q,·), each of which leads to an isotopy of the quasigroup to a group (abelian group). With the use of such identities, a criterion of isotopy of a quasigroup to a group (abelian group) is formulated, and a set of identities with permutations is given which lead to a special type of linearity (alinearity) of a quasigroup over a group (abelian group). It follows from these results that in the Belousov identity, which characterises quasigroups isotopic to a group (abelian group), two out of five variables (one out of four variables) can be fixed in arbitrary way. The obtained results give a possibility to describe an infinite number of identities in a primitive quasigroup (Q,·, \, /) leading to an isotopy of a quasigroup (Q,·) to a group or to its linearity of a given type.