Yuriy R. Classification of small length functional equations on quasigroup operations

A number of parastrophic equivalence invariants are founded; the full classification of quadratic functional equations containing n individual variables (n=2, 3, 4) up to parastrophic equivalence are given; it is stated that there exist two classes when n=2, four classes when n=3 and 17 classes when n=4. It is proved that every general quadratic functional equation having four individual variables, which has no self-contained subterm, is parastrophic equivalent to exactly one of the five given equations; every quadratic parastrophic irreducible functional equation having five individual variables is parastrophic equivalent to at least one given functional equations. It is stated that there exist at least two classes up to parastrophic equivalence. The existence of nonquadratic functional equations, every solution of which has a group isotope, has proved.