Abstract S. Markovski, A. Sokolova, L.
Goracinova Ilieva

On semigroups defined by identity $xxy = y$

The groupoid identity x(xy)=y appears in  definitions of several
classes of groupoids, such as Steiner loops (which are closely related to
Steiner triple systems),  orthogonality in quasigroups
and others. We have considered before
several varieties of groupoids that include this identity
among their defining identities, and here we consider the variety
${\mathcal V}$ of semigroups defined by the same identity. The main results
are: the decomposition of a ${\mathcal V}$ semigroup as a direct product
of a Boolean group and a left unit semigroup; decomposition of
the variety ${\mathcal V}$ as a direct product of the variety of
Boolean groups and the variety of left unit semigroups;
constructions of free objects in ${\mathcal V}$ and the solution of the
word problem in ${\mathcal V}$.