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IMI/Publicaţii/BASM/Ediţii/BASM n.2(81), 2016/

On pseudoisomorphy and distributivity of quasigroups

Authors: Fedir M. Sokhatsky

Abstract

A repeated bijection in an isotopism of quasigroups is called a \emph{companion} of the third component. The last is called a \emph{pseudoisomorphism} with the companion. Isotopy coincides with pseudoisomorphy\footnote{isotopy, pseudoisomorphy, isomorphy denote relation among groupoids and isotopism, psuedoisomorphism, isomorphism are the corresponding sequence of bijections}\; in the class of inverse property loops and with isomorphy in the class of commutative inverse property loops. This result is a generalization of the corresponding theorem for commutative Moufang loops. A notion of middle distributivity is introduced: a quasigroup is \emph{middle distributive} if all its middle translations are automorphisms. In every quasigroup two identities of distributivity (left, right and middle) imply the third. This fact and some others help us to find a short proof of a theorem which gives necessary and sufficient conditions for a quasigroup to be distributive. There is but a slight difference between this theorem and the well-known Belousov's theorem.

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