On the problem of finding minimum semigroup of approximation

Background. The research subject is semigroups and some predicates, given in semigroups, in particular, the equality predicate and the predicate of element's entering a sub-semigroup. The study deals with the problem of finding minimum semigroup of approximation for some classes of П semigroups and Q predicates. The authors considered characters as approximation homomorphisms. Therefore, apparently, among the semigroups for the given problem it is necessary to consider only commutative semigroups. Under a complex character of semigroup one should understand a homomorphism of the given semigroup into a multiplicative semigroup, consisting of complex numbers, equaling 1 in absolute value, and a zero. The aim of the work is to describe the known semigroups of approximation, methods for proving the given fact, to calrify the issues of existence and uniqueness of minimum semigroups of approximation for some classes and predicates. Materials and methods. The authors used general methods of analysis and synthesis, as well as approximation methods, in particular, the method of homomorphism building, the method of decomposition of a commutative regular semigroup into a semilattice of maximum sub-groups, the method of sub-group homomorphism extension to a homomorphism of the whole semigroup. Results. The minimum semigroup of approximation relative to a certain given predicate is determined for a random class of semigroups. The work reviews the known results, specifies minimum semigroups of approximation for some classes of П semigroups and Q predicates, in particular, for the calss of commutative regular periodic semigroups relative to the predicate of element's entering a sub-semigroup. Approximation allows to substitute one objects by other, either more compact or better studied ones. In this case, one can judge about the true value of the predicate, set on a certain class of semigroups, by its value on the corresponding elements of minimum semigroups of approximation. Conclusions. The article gives an example of a group, for which it is impossible to find a minimum semigroup of approximaton. Moreover, the issues of existence and uniqueness of a minimum subgroup of approximation for some classes of G semigroups and Q predicates are solved negatively. However, in special cases it is possible to find a minimum semigroup of approximation - the article describes and proves the examples of such minimum semigroups of approximation.