Quadratic level quasigroup equations with four variables I

Abstract

We consider a class of functional equations with one operational symbol which is assumed to be a quasigroup. Equations are quadratic, level and have four variables each. Therefore, they are of the form x1x2 · x3x4 = x5x6 · x7x8 with xi ∈ {x, y, u, v} (I ≤ i ≤ 8) with each of the variables occurring exactly twice in the equation. There are 105 such equations. They separate into 19 equivalence classes defining 19 quasigroup varieties. The paper (partially) generalizes the results of some retent papers of Förg-Rob and Krapeẑ, and Polonijo.