Logica Universalis

, Volume 9, Issue 1, pp 1–26 | Cite as

Existence of Finite Total Equivalence Systems for Certain Closed Classes of 3-Valued Logic Functions



The article deals with finding finite total equivalence systems for formulas based on an arbitrary closed class of functions of several variables defined on the set {0, 1, 2} and taking values in the set {0,1} with the property that the restrictions of its functions to the set {0, 1} constitutes a closed class of Boolean functions. We consider all classes whose restriction closure is either the set of all functions of two-valued logic or the set T a of functions preserving \({a, a\in\{0, 1\}}\). In each of these cases, we find a finite total equivalence system, construct a canonical type for formulas, and present a complete algorithm for determining whether any two formulas are equivalent.


Many-valued logic closed class finite total equivalence system 

Mathematics Subject Classification

03B50 06E30 


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  1. 1.
    Post, E.L.: Two-valued iterative systems of mathematical logic. Annals of Mathematical Studies, vol. 5, no. 122. Princeton University Press, Princeton (1941)Google Scholar
  2. 2.
    Lyndon R.C.: Identities in two-valued calculi. Trans. Am. Math. Soc. 71(3), 457 (1951)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Lyndon R.C.: Identities in finite algebras. Proc. Am. Math. Soc. 5(1), 8 (1954)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Vishin V.V.: Some examples of varieties of semigroups. Donetsk Ac. Sc. 150(4), 719 (1963)Google Scholar
  5. 5.
    Murskii, V.L.: Examples of varieties of semigroups. Math. Notes Ac. Sc. USSR 3(6), 423 (1968, english)Google Scholar
  6. 6.
    Lau, D.: Function Algebras on Finite Sets, vol. 688. Springer, Rostok (2006)Google Scholar
  7. 7.
    Novi Sad Algebraic Conference. http://sites.dmi.rs/events/2013/nsac2013/
  8. 8.
    Willard, R.: The finite basis problem. In: Contributions to General Algebra. Heyn, Klagenfurt 15, 199 (2004)Google Scholar
  9. 9.
    Bodnartchuk, V.G., Kaluznin, L.A., Kotov, V.N., Romov, B.A.: Galois theory for Post algebras 1–2. Kibernitika. Kiev 3, 1 (1969)Google Scholar
  10. 10.
    Geiger, D.: Closed systems of functions and predicates. Pac. J. Math. 27, 95 (1968)Google Scholar
  11. 11.
    Pöschel, R.: A general Galois theory for operations and relations and concrete characterization of related algebraic structures. Akademie der Wissenschaften der DDR Institut für Mathematik. Berlin, report 1 (1980)Google Scholar
  12. 12.
    Kayser, D.: The Place of Logic in Reasoning. Logica Universalis, SP, vol. 4. Birkhäuser Verlag, Basel, pp. 225–239 (2010)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Data Analysis and Artificial IntelligenceNational Research University Higher School of EconomicsMoscowRussia

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