On pseudoisomorphy and distributivity of quasigroups

A repeated bijection in an isotopism of quasigroups is called a companion of the third component. The last is called a pseudoisomorphism with the companion. Isotopy coincides with pseudoisomorphy 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 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.