About the class of approximation by generalized characters

Given a certain class  K1 of algebraic structures. We study a problem of finding a class  K2 of algebraic structures such that the class K1  is approximable into K2 with respect to various predicates by generalized characters from  K1 to K2. The problem of minimization of approximation is also considered. Some theorems related to the problem of constructing an approximation class are obtained. The problem in question is much more complicated and actual than the approximation problem we have been studying before (see [2]-[6]). The results of the description of the approximation class play an important role in studying the solvability problem of the predicate P in the class of semigroups K. In particular, if the approximation class consists of finite semigroups, then this problem is solved positively. Even more difficult is the problem of determining the necessary conditions that class  is an approximation class for a given class K.