Approximated class

Background. In this paper, we select a predicate P and consider the approximation of a semigroups class with respect to this predicate P when using mappings only from a set F. We determine the necessary and sufficient conditions for this approximation. In addition, we consider the minimization problem, i.e. the search for a minimal structure of B such that the class of semigroups A still approximates with respect to the predicate P when using only homomorphisms to the semigroup B. Materials and methods. We consider different pairs consisting of some classes of semigroups K and predicates P. For each such pair, we find an approximation class, i.e. a class of semigroups K$_1$, such that each semigroup of K is approximated by maps from K to K$_1$ with respect to the predicate P. We use constructive methods to construct homomorphisms and semigroups. Results. Some theorems related to the problem of constructing an approximation class are obtained. The problem in question is much more complicated than the approximation problem. 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 positively solved. Even more difficult is the problem of determining the necessary conditions that class K$_1$ is an approximation class for a given class K. Conclusion. One of the important directions in modern algebra is the study of not only the algebraic system itself, but also the systems derived from it. Using the relationships established in the paper, it is possible to extend the obtained results to other classes of algebraic structures, other sets of homomorphisms and other predicates.