Geodesic
Classification and types of bases of all ultrafunctions on two-element set
The Bulletin of Irkutsk State University. Series Mathematics, Tome 16 (2016), pp. 58-70 Cet article a éte moissonné depuis la source Math-Net.Ru

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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
Mots-clés : maximal clone. 
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S. V. Zamaratskaya; V. I. Panteleev. Classification and types of bases of all ultrafunctions on two-element set. The Bulletin of Irkutsk State University. Series Mathematics, Tome 16 (2016), pp. 58-70. http://geodesic.mathdoc.fr/item/IIGUM_2016_16_a4/

[1] Zamaratskaya S. V., Panteleev V. I., “Maximal clones rank 2 ultrafunctions”, The bulletin of Irkutsk State University. Mathematics, 15 (2016), 26–37 (in Russia) | MR

[2] Kazimirov A. S., Panteleev V. I., Tokareva L. V., “Classification and Enumeration of Bases in Clone of All Hyperfunctions on Two-Elements Set”, The bulletin of Irkutsk State University. Mathematics, 7 (2014), 61–78 (in Russia) | MR | Zbl

[3] Panteleev V. I., “The Completeness Criterion for Certain Boolean Functions”, Vestnik of Samara State University. Natural Science Series, 2009, no. 2(68), 60–79 (in Russia)

[4] Yablonskij S. V., “On the Superpositions of Logic Functions”, Mat. Sbornik, 30:2(72) (1952), 329–348 (in Russian) | MR | Zbl

[5] M. Miyakawa, I. Stojmenović, D. Lau, I. Rosenberg, “Classification and basis enumerations in many-valued logics”, Proc. 17th International Symposium on Multi-Valued logic (Boston, 1987), 151–160

[6] M. Miyakawa, I. Stojmenović, D. Lau, I. Rosenberg, “Classification and basis enumerations of the algebras for partial functions”, Proc. 19th International Symposium on Multi-Valued logic (Rostock, 1989), 8–13

[7] L. Krnić, “Types of bases in the algebra of logic”, Glasnik matematicko-fizicki i astronomski. Ser 2, 20 (1965), 23–32 | MR

[8] D. Lau, M. Miyakawa, “Classification and enumerations of bases in $P_k(2)$”, Asian-European Journal of Mathematics, 01:02 (2008), 255–282 | DOI | MR

[9] M. Miyakawa, I. Rosenberg, I. Stojmenović, “Classification of three-valued logical functions preserving 0”, Discrete Applied Mathematics, 28 (1990), 231–249 | DOI | MR | Zbl