On semigroups defined by the identity $xxy = y$

Smile Markovski, Ana Sokolova and Lidija Goracinova Ilieva

Abstract: 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) [9,10], orthogonality in quasigroups [4] and others [12,2]. We have considered in [8] 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 ${\mathcal V}$.

Keywords: semigroup; identity; free object; variety; word problem

Classification (MSC2000): 20M05, 20M10

Full text of the article:

• Compressed PostScript file (71 kilobytes)
• PDF file (184 kilobytes)