The closure operator with the equality predicate branching on the set of hyperfunctions on two-element set

In this work we consider the closure operator with the equality predicate branching ($E$-operator) on the set of hyperfunctions on two-element set.
With respect to this operator closed classes of hyperfunctions are generated. We show that there are four submaximal classes and prove the criterion of functional completeness.
The relation of equivalence is considered on the set of hyperfunctions obtained by their membership in $E$-submaximal classes. All hyperfunctions are divided into 14 equivalence classes.
In closed sets the minimal closed subsets named basis are derived. We show that basis of hyperfunctions can have cardinality from 1 to 3 and there is no basis with cardinality more than 3. There is only one kind of basis with cardinality 1. The function from that basis does not belong to any of four $E$-submaximal classes. We obtain 23 kind of basis with cardinality 2 and 11 with cardinality 3.