The problem of semigroups approximation with respect to various predicates has been investigated by many
scientists. Some necessary and sufficient conditions for the semigroups approximation with respect to such predicates
as “equality”, “membership of an element to a subsemigroup”, “regular conjugation relation”, “Green ratio
of L-, R-, H- and D-equivalence”, “membership of an element to a monogenic subsemigroup”, etc. were obtained.
However, there were practically no results on the conditions of approximation with respect to the predicate of
membership of an element to a subgroup of a given semigroup. The paper presents the necessary and sufficient
condition for approximation with respect to this predicate. We constructed a special semigroup acting the role of a
minimal approximation semigroup for many predicates. This semigroup has neither identity nor additive identity.
It contains an infinite number of idempotents, and the presence of each idempotent is mandatory. By this semigroup,
we have successfully solved the problem of approximation with respect to the predicate of membership of
an element to a subsemigroup. A class of semigroups is also described, for which it is the minimal approximation
semigroup. We obtained a criterion for the approximation of a semigroup with respect to the Green H-equivalence.
The problem of algebraic systems approximating with respect to a predicate consists of three components: a set
of algebraic systems (groups, semigroups, etc.); set of predicates; set of functions (homomorphisms, continuous
mappings, etc.). The change of one of these components determines a new line of research.