Equations which preserve the height of variables (Q2567993)

A linear equation on a quasigroup is the one in which each variable appears precisely once on each side. If each subterm on one side of the linear equation has a counterpart on the other side with the same variables, then the equation is called Belousov. The authors define the height of variable in a term, the height of term and the height of a linear equation. Then, they introduce a class of height preserving equation -- wider than the class of Belousov equation. One of the results states that a semigroup which satisfies a height preserving equation which is not Belousov is isotopic to an abelian group and it is a \(T\)-quasigroup. A large collection of results (generalizing the known ones), examples and further definitions is given.