Elementary transformations of systems of equations over quasigroups and generalized identities

The paper is devoted to the study of equations with the left-hand side having the form of a composition of operations which belong to given sets ${{\mathbf{S}}_1},\ldots,{{\mathbf{S}}_n},\ldots\,$ of quasigroup operations. Elementary transformations are described which allow reducing systems of this kind to the form where all equations except one do not depend essentially on the variable ${x_n}\,$. A class of systems is said to be Gaussian if every system obtained via such transformations also belongs to this class. It is evident that for Gaussian classes of systems of equations there is an efficient solving algorithm. This motivates the problem of finding conditions under which the class is Gaussian. In this work it is shown that for a class of systems to be Gaussian the operations in the sets ${{\mathbf{S}}_i}\,$ should satisfy the generalized distributivity law. Sets of operations obeying this condition are to be investigated in the future.