Classification of quadratic functional equations for five object variables on quasigroups

Abstract

We consider functional equations over quasigroup operations. The functional equation w = u is called general if the functional variables contained in its record are pair wise different, balanced if each object variable has exactly one appearance on the left-hand side and on the right-hand side of the equation, quadratic if each object variable has exactly two appearances in the equation. We prove that every quadratic functional equation for five object variables is commutative equivalent to just one of the functional equations (1) – (23).