Weak associativity and quasigroup units

Abstract

We investigate a family of identities similar to weak associativity: x(y/y) · z = x · (y/y)z which might imply the existence of the unit in a quasigroup. A partial solution to Krapež, Shcherbacov Problem concerning such identities and consequently to similar well known Belousov's Problem is obtained. Another problem by Krapež and Shcherbacov is solved affirmatively, showing that there are many single identities determining unipotent loops among quasigroups.