**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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