Vol. 70(84), pp. 1--8 (2001)

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On semigroups defined by the identity $xxy = y$

Smile Markovski, Ana Sokolova and Lidija Goracinova Ilieva

Faculty of Sciences and Mathematics, Institute of Informatics, p.f. 162, Skopje, Republic of Macedonia and Pedagogical Faculty, \v Stip, Republic of Macedonia

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

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Electronic fulltext finalized on: 17 Oct 2002. This page was last modified: 13 Nov 2002.

© 2002 Mathematical Institute of the Serbian Academy of Science and Arts
© 2002 ELibM and FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition