Algebra universalis

, Volume 68, Issue 3–4, pp 295–320 | Cite as

The cardinality of the set of all clones containing a given minimal clone on three elements

  • Dmitriy N. Zhuk


All minimal clones on three elements were found by B. Csákány. In this paper, for each minimal clone the cardinality of the set of all clones containing this clone is found.

2010 Mathematics Subject Classification

Primary: 03B50 Secondary: 08A40 

Key words and phrases

cardinality clones minimal clones three-valued logic 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baker K.A., Pixley A.F.: Polynomial interpolation and the Chinese remainder theorem for algebraic systems. Math. Z. 143, 165–174 (1975)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bodnarchuk, V.G., Kaluzhnin, L.A., Kotov, V.N., Romov, B.A.: Galois theory for Post algebras. I, II. Kibernetika (Kiev) 3, 1–10 (1969); 5, 1–9 (1969) (Russian; English translation in Cybernetics)Google Scholar
  3. 3.
    Csákány B.: All minimals clones on the three-element set. Acta Cybernet. 6, 227–238 (1983)MathSciNetMATHGoogle Scholar
  4. 4.
    Janov, Ju., Muchnik, A.: Existence of k-valued closed classes without a finite basis. Dokl. Akad. Nauk SSSR 127, 44–46 (1959) (Russian)Google Scholar
  5. 5.
    Lau, D.: Function Algebras on Finite Sets. Springer (2006)Google Scholar
  6. 6.
    Mal’tsev, A.I.: Iterative algebras and Post’s varieties. Algebra i Logika (Sem.) 5, 5–24 (1966) (Russian)Google Scholar
  7. 7.
    Marchenkov, S.S.: Closed classes in P k containing homogeneous functions. Akad. Nauk SSSR Inst. Prikl. Mat. Preprint 35 (1984) (Russian)Google Scholar
  8. 8.
    Pantović J., Voivodić D.: The cardinality of the set of clones containing unary minimal clones on the three element set. Mult.-Valued Log. 5, 367–371 (2000)MathSciNetMATHGoogle Scholar
  9. 9.
    Pantović J., Voivodić D.: The cardinality of the set of clones containing minimal clones generated by semiprojections on a three element set. Novi Sad J. Math. 30, 155–158 (2000)Google Scholar
  10. 10.
    Pantović J., Voivodić D.: Note on the cardinality of some sets of clones. Acta Cybernet. 14, 491–495 (2000)MathSciNetMATHGoogle Scholar
  11. 11.
    Post, E.: Determination of all closed systems of truth tables. Bull. Amer. Math. Soc. 26, 427 (1920)Google Scholar
  12. 12.
    Post, E.: Two-Valued Iterative Systems of Mathematical Logic. Princeton Univ. Press, Princeton (1941)Google Scholar
  13. 13.
    Romov, B.A.: The lattice of subalgebras of direct products of Post algebras of finite degree. In: Mathematical Models of Complex Systems, pp. 156–168. Akad. Nauk Ukrain. SSR Inst. Kibernet., Kiev (1973) (Russian)Google Scholar
  14. 14.
    Rosenberg, I.G.: Minimal clones I: the five types. In: Lectures in Universal Algebra. Colloq. Math. Soc. Janos Bolyai, vol. 43, pp. 405–427. North-Holland, Amsterdam (1986)Google Scholar
  15. 15.
    Rosenberg I.G.: Clones on a prime cardinality universe containing an affine essential operation. Tatra Mt. Math. Publ. 5, 201–215 (1995)MathSciNetMATHGoogle Scholar
  16. 16.
    Taymanov, V.A.: On cartesian powers of P 2. Dokl. Akad. Nauk SSSR 270, 1327–1330 (1983) (Russian)Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of Mathematics and MechanicsMoscow State UniversityMoscowRussia

Personalised recommendations