SH-weak duality of semigroups and minimum semi-group of SH-approximation

Background. The subject of the study are semigroups and variants of their duality and approximation. Using the operators of taking all subsemigroups of a given semigroup and all its homomorphic images, we obtain the class of semigroups (A) SH, for which we consider questions of weak duality and approximation. The aim of the paper is to describe the interconnections of SH-weak duality with its other types, as well as to find the minimal semigroup for SH-approximation of semigroups relative to the predicate of the element belonging to the subsemigroup. Materials and methods. In the paper, the general methods of the analysis and synthesis are used. Special methods of describing semigroups and methods of working with them, in particular, the method of constructing the morphism of the semigroup are also used. We construct a special semigroup playing the role of the minimal semigroup of SH-approximation with respect to several predicates. In this semigroup, there are no zero and single elements. Moreover, it contains an infinite number of idempotents. Results. A description of the interrelationships of SH-weak duality with its other types in the general case is obtained, as well as in a number of specific examples. In particular, connections between various types of weak duality with respect to the multiplicative semigroup of complex numbers equal in absolute value to 0 or 1, as well as to the multiplicative semigroup of non-negative real numbers, have been clarified. In the class of semigroups described, we have obtained the minimal from the point of view of SH-approximation with respect to the predicate of an element belonging to a semigroup: the necessary and sufficient conditions for the SH-approximation are explicitly described. Conclusion. One of the important directions in modern algebra is the study of not only the algebraic system itself, but also systems derived from it. The focus of the study of this paper is the class of semigroups (A)SH, containing any homomorphic image of any subsemigroup of the given semigroup. A study of the conditions of weak duality and approximation for this class gave a number of new theoretical results. Using the relationships established in Theorem 1, one could extend the obtained results to other classes of semigroups formed by some Birkhoff operators.