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.

1. Kazimirov A.S., Panteleyev V.I., Tokareva L.V. Classification and Enumeration of Bases in Clone of All Hyperfunctions on Two-Elements Set (in Russian). IIGU Ser. Matematika, 2014, vol. 7, pp. 61–78.

2. Lo Czu Kai. Maximal closed classes on the Set of Partial Many-valued Logic Functions (in Russian). Kiberneticheskiy Sbornik, Moscow, Mir, 1988, vol. 25, pp. 131–141.

3. Lo Czu Kai. Completeness theory on Partial Many-valued Logic Functions (in Russian). Kiberneticheskiy Sbornik, Moscow, Mir, 1988, vol. 25, pp. 142–157.

4. Marchenkov S.S. Closure Operators with Predicate Branching (in Russian). Vestnik Moskov. Univ. Ser. I Mat. Mekh, 2003, no. 6, pp. 37–39.

5. Marchenkov S.S. The Closure Operator With the Equality Predicate Branching on the Set of Partial Boolean Functions (in Russian). Discrete Mathematics, 2008, vol. 20, no. 3, pp. 80–88.

6. Marchenkov S.S. The E-closure Operator on the Set of Partial Many-valued Logic Functions (in Russian). Matematicheskie voprosy kibernetiki, Moscow, Fizmatlit, 2013, vol. 19, pp. 227–238.

7. Tarasov V.V. Completeness Criterion for Partial Logic Functions (in Russian). Problemy Kibernetiki, Moscow, Nauka, 1975, vol. 30, pp. 319–325.

8. Yablonskij S.V. On the Superpositions of Logic Functions (in Russian). Mat. Sbornik, 1952, vol. 30, no. 2(72), pp. 329–348.

9. Miyakawa M., Stojmenovi´c I., Lau D., Rosenberg I. Classification and Basis Enumerations in Many-Valued Logics. Proc. 17th International Symposium on Multi-Valued Logic, Boston, 1987, pp. 151–160.

10. Krni´c L. Types of Bases in the Algebra of Logic. Glasnik matematicko-fizicki i astronomski. Ser 2, 1965, vol. 20, pp. 23–32.

11. Lau D., Miyakawa M. Classification and Enumerations of Bases in Pk(2). Asian-European Journal of Mathematics, 2008, vol. 1, no. 2, pp. 255–282.

12. Lau D. Function algebras on finite sets. A basic course on many-valued logic and clone theory. Berlin: Springer-Verlag, 2006, 668 p.

13. Machida H. Hyperclones on a Two-Element Set. Multiple-Valued Logic. An International Journal, 2002, no. 8(4), pp. 495–501.

14. Machida H., Pantovic J. On Maximal Hyperclones on {0, 1} — a new approach. Proceedings of 38th IEEE International Symposium on Multiple- Valued Logic (ISMVL 2008), 2008, pp. 32–37.

15. Romov B. A. Hyperclones on a Finite Set. Multiple-Valued Logic. An International Journal, 1998, vol.3(2), pp. 285–300.