On a functional equation with a group isotopy property

Abstract

The set of all solutions of functional equation F1(F2(z; x);F3(y; z)) = F4(F5(x; y); x) on quasigroup operations of an arbitrary fixed set are found. The result implies W. Dudek’s theorem [1], which presents the operation in a quasigroup satisfying the identity xy · x = zx · yz.