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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2016. Vol. 16

Classification and Types of Bases of All Ultrafunctions on Two-Element Set.

Author(s)
S. V. Zamaratskaya, V. I. Panteleev
Abstract

This paper studies properties of ultrafunctions with respect of their inclusion in maximal clones.

The number of maximal clones for all ultrafunctions of rank 2 is equal to 11 [V. Panteleev, 2009].
All ultrafunctions are divided into 45 equivalence classes.
Based on this classification all kinds of bases are discribed. Two bases are of different kinds if there is a function in one basis with no equivalent function in the other one. We show that bases of hyperfunctions can have cardinality from 1 to 4: there is only one kind of basis with cardinality 1, 180 with cardinality 2, 686 with cardinality 3, 28 with cardinality 4.
Keywords
ultrafunction, clone, base, maximal clone
UDC
519.7

MSC

08A99,03B50

References

1. Zamaratskaya S.V., Panteleev V.I. Maximal clones rank 2 ultrafunctions (in Russia). The bulletin of Irkutsk State University. Mathematics, 2016, vol. 15, pp. 26-37.

2. Kazimirov A.S., Panteleev V.I., Tokareva L.V. Classification and Enumeration of Bases in Clone of All Hyperfunctions on Two-Elements Set (in Russia). The bulletin of Irkutsk State University. Mathematics, 2014, vol. 10, pp. 61-78.

3. Panteleev V.I. The Completeness Criterion for Certain Boolean Functions (in Russia). Vestnik of Samara State University. Natural Science Series, 2009, vol. 2, no 68, рр. 60-79.

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

5. 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.

6. Miyakawa M., Stojmenovi´c I., Lau D., Rosenberg I. Classification and basis enumerations of the algebras for partial functions. Proc. 19th International Symposium on Multi-Valued logic. Rostock, 1989, pp. 8-13.

7. Krni´c L. Types of bases in the algebra of logic/ Glasnik matematicko-fizicki i astronomski. Ser. 2, 1965, vol. 20, p. 23-32.

8. Lau D., Miyakawa M. Classification and enumerations of bases in Pk(2)/ Asian-European Journal of Mathematics, 2008, vol. 01, no 02, pp. 255-282.

9. Miyakawa M., Rosenberg I., Stojmenovi´c I. Classification of three-valued logical functions preserving 0. Discrete Applied Mathematics, 1990, vol. 28, pp. 231-249.


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