On groupoids with identity x(xy) = y
--------------------------------------

Lidija Goracinova-Ilieva, Smile markovski, Ana Sokolova

Abstract:
--------

The groupoid identity x(xy) = y appears in defining several classses of groupoids, such as Steiner's
loops which are closely related to Steiner's triple systems, the class of cancellative groupoids
with property (2,5), Boolean groups, and groupoids that exhibit orthogonality of quasigroups.
Its dual identity is one of the defining identities for the variety of quasigroups corresponding to strongly
2-perfect m-cycle systems. In this paper we consider several varieties of groupoids defined, among others by the
mentioned identity. Canonical constructions of free objects in each of these varieties are given and
several other structural properties are presented.