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On n × m-valued Łukasiewicz-Moisil algebras

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Central European Journal of Mathematics

Abstract

n×m-valued Łukasiewicz algebras with negation were introduced and investigated in [20, 22, 23]. These algebras constitute a non trivial generalization of n-valued Łukasiewicz-Moisil algebras and in what follows, we shall call them n×m-valued Łukasiewicz-Moisil algebras (or LM n×m -algebras). In this paper, the study of this new class of algebras is continued. More precisely, a topological duality for these algebras is described and a characterization of LM n×m -congruences in terms of special subsets of the associated space is shown. Besides, it is determined which of these subsets correspond to principal congruences. In addition, it is proved that the variety of LM n×m -algebras is a discriminator variety and as a consequence, certain properties of the congruences are obtained. Finally, the number of congruences of a finite LM n×m -algebra is computed.

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References

  1. Balbes R., Dwinger P., Distributive lattices, University of Missouri Press, Columbia, Mo., 1974

    MATH  Google Scholar 

  2. Boicescu V., Filipoiu A., Georgescu G., Rudeanu S., Łukasiewicz–Moisil algebras, Annals of Discrete Mathematics, 49, North-Holland Publishing Co., Amsterdam, 1991

    Google Scholar 

  3. Bulman-Fleming S., Werner H., Equational compactness in quasi-primal varieties, Algebra Universalis, 1977, 7, 33–46

    Article  MathSciNet  MATH  Google Scholar 

  4. Burris S., Sankappanavar H.P., A course in universal algebra, Graduate Texts in Mathematics, 78, Springer-Verlag, New York-Berlin, 1981

    Google Scholar 

  5. Cignoli R., Moisil algebras, Notas de Lógica Matemática, 27, Instituto de Matemática, Universidad Nacional del Sur, Bahía Blanca, 1970

    Google Scholar 

  6. Cignoli R., Proper n-valued Łukasiewicz algebras as S-algebras of Łukasiewicz n-valued propositional calculi, Studia Logica, 1982, 41, 3–16

    Article  MathSciNet  MATH  Google Scholar 

  7. Cignoli R., D’Ottaviano I., Mundici D., Algebras das logicas de Łukasiewicz, Coleção CLE, 12, Campinas UNICAMP-CLE, 1995

  8. Cignoli R., D’Ottaviano I., Mundici D., Algebraic foundations of many-valued reasoning, Trends in Logic-Studia Logica Library, 7, Kluwer Academic Publishers, Dordrecht, 2000

    Google Scholar 

  9. Cornish W., Fowler P., Coproducts of De Morgan algebras, Bull. Austral. Math. Soc., 1977, 16, 1–13

    Article  MathSciNet  MATH  Google Scholar 

  10. Figallo A.V., Pascual I., Ziliani A., Notes on monadic n-valued Łukasiewicz algebras, Math. Bohem., 2004, 129, 255–271

    MathSciNet  MATH  Google Scholar 

  11. Grigolia R., Algebraic analysis of Łukasiewicz-Tarski’s n-valued logical systems, In: Wójcicki R., Malinowski G. (Eds.), Selected papers on Łukasiewicz sentential calculi, Zaklad Narod. im. Ossolin., Wydawn. Polsk. Akad. Nauk, Wroclaw, 1977, 81–92

    Google Scholar 

  12. Iorgulescu A., Connections between MVn algebras and n-valued Łukasiewicz-Moisil algebras IV, Journal of Universal Computer Science, 2000, 6, 139–154

    MathSciNet  Google Scholar 

  13. Łukasiewicz J., On three-valued logic, Ruch Filozoficzny, 1920, 5, 160–171

    Google Scholar 

  14. Moisil Gr.C., Notes sur les logiques non-chrysippiennes, Ann. Sci. Univ. Jassy, 1941, 27, 86–98

    MathSciNet  Google Scholar 

  15. Moisil Gr.C., Essais sur les logiques non chrysippiennes, Éditions de l’Académie de la République Socialiste de Roumanie, Bucharest, 1972

    MATH  Google Scholar 

  16. Post E., Introduction to a general theory of elementary propositions, Amer. J. Math., 1921, 43, 163–185

    Article  MathSciNet  MATH  Google Scholar 

  17. Priestley H., Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc., 1970, 2, 186–190

    Article  MathSciNet  MATH  Google Scholar 

  18. Priestley H., Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc., 1972, 4, 507–530

    Article  MathSciNet  Google Scholar 

  19. Priestley H., Ordered sets and duality for distributive lattices, Ann. Discrete Math., North-Holland, Amsterdam, 1984, 23, 39–60

    Google Scholar 

  20. Sanza C., Algebras de Łukasiewicz matriciales n × m-valuadas con negación, Noticiero de la Unión Matemática Argentina, Rosario (Argentina), 2000

    Google Scholar 

  21. Sanza C., Algebras de Łukasiewicz n × m-valuadas con negación, Ph.D. thesis, Universidad Nacional del Sur. Argentina, 2004

    Google Scholar 

  22. Sanza C., Notes on n × m-valued Łukasiewicz algebras with negation, Log. J. IGPL, 2004, 12, 499–507

    Article  MathSciNet  MATH  Google Scholar 

  23. Sanza C., n × m-valued Łukasiewicz algebras with negation, Rep. Math. Logic, 2006, 40, 83–106

    MathSciNet  MATH  Google Scholar 

  24. Suchoń W., Matrix Łukasiewicz Algebras, Rep. Math. Logic, 1975, 4, 91–104

    MATH  Google Scholar 

  25. Tarski A., Logic semantics metamathematics, Clarendon Press, Oxford, 1956

    Google Scholar 

  26. Werner H., Discriminator-algebras, Algebraic representation and model theoretic properties, Akademie-Verlag, Berlin, 1978

    Google Scholar 

Download references

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  1. Universidad Nacional del Sur, Av. Alem 1253, 8000, Bahía Blanca, Argentina

    Claudia A. Sanza

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  1. Claudia A. Sanza
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Sanza, C.A. On n × m-valued Łukasiewicz-Moisil algebras. centr.eur.j.math. 6, 372–383 (2008). https://doi.org/10.2478/s11533-008-0035-7

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