Minimal semigroup of SH-Approximation

In the present paper, one of the important directions of modern algebra is considered: we study algebraic systems, and also the whole classes of them. Some results about approximation of a semigroup were proposed in the 1970s. The problem of finding the minimal approximation and SH-approximation of semigroups was proposed also in the 1970s. We study finding the minimal approximation and SH-approximation of semigroups with respect to different predicates of semigroup theory. For this, we prove some results for various semigroups, for instance, for commutative semigroups of three idempotents. The main theorem of this paper is the following. For the class of commutative, regular and periodic semigroups, we can choose a semigroup which is a minimal SH-approximable semigroup for this class with respect to the predicate of the possible belonging of an element to a subsemigroup. From this result, we also have found the necessary and sufficient conditions with respect to Green's relations, such as l-equivalency, D-equivalence, H-equivalence, and some other predicates.