Some issues of semigroups approximation

Background. The subjects of the study are semigroups and some predicates defined on them, in particular the equality predicate, the predicate of the occurrence of an element in a subsemigroup and a more complicated special predicate defined on subsets of the set of a free monoid. Materials and methods. To solve this and similar problems, we describe a special semigroup that plays the role of a minimal semigroup for the whole class of predicates under consideration. Moreover, the semigroup considered here does not often contain either one or zero, but in this case it contains an infinite number of idempotents, and the presence of each of them is mandatory. Results. In the described class of semigroups, we obtained the minimal one from the point of view of approximation with respect to the whole class of predicates. Examples of semigroups from various fields of mathematics are given. Conclusion. The problem of approximation of semigroups consists of three components. The first is the set of algebraic structures used - such as groups, semigroups, etc. The second component is the set of predicates considered above these structures. And the third component is various variants of describing the homomorphism over the objects under consideration; some examples from different fields of mathematics are given in this article. Changing any one of these three components, we always get a new direction for further research.