alexa On the structure of left and right F-, SM-, and E-quasigroups | OMICS International
ISSN: 1736-4337

Journal of Generalized Lie Theory and Applications
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Review Article

On the structure of left and right F-, SM-, and E-quasigroups

Victor SHCHERBACOV*

Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, str. Academiei 5, MD-2028 Chisinau, Moldova
E-mail: [email protected]

Received Date: November 08, 2008; Accepted Date: January 25, 2009

Abstract

It is proved that any left F-quasigroup is isomorphic to the direct product of a left F-quasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right F-quasigroups, left and right SM- and E-quasigroups. Information on simple quasigroups from these quasigroup classes is given; for example, nite simple F-quasigroup is a simple group or a simple medial quasigroup. It is proved that any left F-quasigroup is isotopic to the direct product of a group and a left S-loop. Some properties of loop isotopes of F-quasigroups (including M-loops) are pointed out. A left special loop is an isotope of a left F-quasigroup if and only if this loop is isotopic to the direct product of a group and a left S-loop (this is an answer to Belousov \1a" problem). Any left E-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(1) problem). As corollary it is obtained that any left FESM-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(2) problem). New proofs of some known results on the structure of commutative Moufang loops are presented.

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