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IMCS/Publications/QRS/Issues/QRS v.11, n.1 (11), 2004/

On groupoids with identity x(xy) = y

Authors: L. Goracinova-Ilieva, S. Markovski, A. Sokolova

Abstract

The groupoid identity x(xy) = y appears in de_ning several classes 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 which exhibit orthogonality of quasigroups. Its dual identity is one of the de_ning identities for the variety of quasigroups corresponding to strongly 2-perfect m-cycle systems. In this paper we consider the following varieties of groupoids: V = Var(x(xy) = y), Vc = Var(x(xy) = y; xy = yx), Vu = Var(x(xy) = y; (xy)y = xy), Vi = Var(x(xy) = y; (xy)y = yx). Suitable canonical constructions of free objects in each of these varieties are given and several other structural properties are presented. Some problems of enumeration of groupoids are also resolved. It is shown that each Vi-groupoid defines a Steiner quintuple system and vice versa, implying existence of Steiner quintuple systems of enough large finite cardinality.

S. Markovski,
A. Sokolova
Faculty of Sciences and Mathematics,
Institute of Informatics,
p.f.162 Skopje,
Republic of Macedonia.
E-mail:
__________________
L. Goracinova-Ilieva
Pedagogical Faculty,
Stip,
Republic of Macedonia.
E-mail:



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