Skip to main content
SpringerLink
Log in

A glimpse of deductive systems in algebra

  • Research Article
  • Published:
Central European Journal of Mathematics

Abstract

The concept of a deductive system has been intensively studied in algebraic logic, per se and in connection with various types of filters. In this paper we introduce an axiomatization which shows how several resembling theorems that had been separately proved for various algebras of logic can be given unique proofs within this axiomatic framework. We thus recapture theorems already known in the literature, as well as new ones. As a by-product we introduce the class of pre-BCK algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abbott J.C., Implicational algebras, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 1967, 11(59), 3–23

    MathSciNet  Google Scholar 

  2. Abbott J.C., Semi-boolean algebra, Mat. Vesnik., 1967, 4(19), 177–198

    MathSciNet  Google Scholar 

  3. Birkhoff G., Lattice Theory, 3rd ed., American Mathematical Society, Providence, 1967

    MATH  Google Scholar 

  4. Boicescu V., Filipoiu A., Georgescu G., Rudeanu S., Łukasiewicz-Moisil Algebras, North-Holland, Amsterdam, 1991

    MATH  Google Scholar 

  5. Buşneag D., Contribuţi la studiul algebrelor Hilbert, Ph.D. thesis, Univ. Bucharest, 1985

  6. Buşneag D., On the maximal deductive systems of a bounded Hilbert algebra, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 1987, 31(79)(1), 9–21

    MathSciNet  Google Scholar 

  7. Buşneag D., Hertz algebras of fractions and maximal Hertz algebra of quotients, Math. Japon., 1994, 39(3), 461–469

    MATH  MathSciNet  Google Scholar 

  8. Buşneag D., Categories of Algebraic Logic, Editura Academiei Române, Bucharest, 2006

    Google Scholar 

  9. Buşneag D., Piciu D., On the lattice of deductive systems of a BL-algebra, Cent. Eur. J. Math., 2003, 1(2), 221–237

    Article  MATH  MathSciNet  Google Scholar 

  10. Celani S.A., Cabrer L.M., Montangie D., Representation and duality for Hilbert algebras, Cent. Eur. J. Math., 2009, 7(3), 463–478

    Article  MATH  MathSciNet  Google Scholar 

  11. Chajda I., The lattice of deductive systems on Hilbert algebras, Southeast Asian Bull. Math., 2002, 26(1), 21–26

    Article  MATH  MathSciNet  Google Scholar 

  12. Chajda I., Halaš R., Algebraic properties of pre-logics, Math. Slovaca, 2002, 52(2), 157–175

    MATH  MathSciNet  Google Scholar 

  13. Chajda I., Halaš R., Abbott groupoids, J. Mult.-Valued Logic Soft Comput., 2004, 10(4), 385–394

    MATH  MathSciNet  Google Scholar 

  14. Chajda I., Halaš R., Deductive systems and Galois connections, In: Galois Connections and Applications, Kluwer, Dordrecht, 2004, 399–411

    Google Scholar 

  15. Chajda I., Halaš R., Distributive implication groupoids, Cent. Eur. J. Math., 2007, 5(3), 484–492

    Article  MATH  MathSciNet  Google Scholar 

  16. Chajda I., Halaš R., Kuhr J., Semilattice Structures, Heldermann, Lemgo, 2007

    MATH  Google Scholar 

  17. Cignoli R., Algebras de Moisil de order n, Ph.D. thesis, Universidad Nacional del Sur, Bahía Blanca, 1969

    Google Scholar 

  18. Diego A., Sobre álgebras de Hilbert, Notas de Lógica Matematica, 12, Instituto de Matemática, Univ. Nacional del Sur, Bahía Blanca, 1965

  19. Diego A., Sur les algèbres de Hilbert, Collection de Logique Math., Sér. A, 21, Gauthier-Villars, Paris, 1966

    Google Scholar 

  20. Figallo A., Ziliani A., Remarks on Hertz algebras and implicative semilattices, Bull. Sect. Logic Univ. Łódź, 2005, 34(1), 37–42

    MATH  MathSciNet  Google Scholar 

  21. Georgescu G., Algebra logicii — logica algebrică (I), Revista de Logică, 2009, 2, available at: http://egovbus.net/rdl/articole/No1Art34.pdf

  22. Grätzer G., Universal Algebra, 2nd ed., Springer, New York-Heidelberg, 1979

    MATH  Google Scholar 

  23. Halaš R., Ideals and D-systems in orthoimplication algebras, J. Mult.-Valued Logic Soft Comput., 2005, 11(3–4), 309–316

    MATH  MathSciNet  Google Scholar 

  24. Iorgulescu A., Algebras of Logic as BCK Algebras, Editura ASE, Bucharest, 2008

    MATH  Google Scholar 

  25. Iséki K., Tanaka S., Ideal theory of BCK-algebras, Math. Japon., 1976, 21(4), 351–366

    MATH  MathSciNet  Google Scholar 

  26. Jun Y.B., Deductive systems of Hilbert algebras, Math. Japon., 1996, 43(1), 51–54

    MATH  MathSciNet  Google Scholar 

  27. Katriňák T., Bemerkung über pseudokomplementaren halbgeordneten Mengen, Acta Fac. Rerum Natur. Univ. Comenian. Math., 1968, 19, 181–185

    MATH  MathSciNet  Google Scholar 

  28. Liu L.Z., Li K.T., Boolean filters and positive implicative filters of residuated lattices, Inform. Sci., 2007, 177(24), 5725–5738

    Article  MATH  MathSciNet  Google Scholar 

  29. Monteiro A., L’arithmétique des filtres et les espaces topologiques, In: De Segundo Symposium de Matematicas-Villavicencio (Mendoza, Buenos Aires), 21–25 July 1954, Centro di Cooperacion UNESCO para America Latina, Montevideo, 129–172; Notas de Lógica Matemática, 1974, 30, 157

  30. Monteiro A., Sur la définition des algebres de Łukasiewicz trivalentes, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine, 1963, 7(55), 3–12

    MathSciNet  Google Scholar 

  31. Nemitz W.C., On the lattice of filters of an implicative semi-lattice, J. Math. Mech., 1968/69, 18, 683–688

    MathSciNet  Google Scholar 

  32. Pałasiński M., On ideal and congruence lattices of BCK algebras, Math. Japon., 1981, 26(5), 543–544

    MATH  MathSciNet  Google Scholar 

  33. Piciu D., Algebras of Fuzzy Logic, Ed. Universitaria Craiova, 2007

  34. Rasiowa H., An Algebraic Approach to Non-Classical Logics, North-Holland, Amsterdam, 1974

    MATH  Google Scholar 

  35. Roman S., Lattices and Ordered Sets, Springer, New York, 2008

    MATH  Google Scholar 

  36. Rudeanu S., On relatively pseudocomplemented posets and Hilbert algebras, An. Śtiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat., 1985, 31(suppl.), 74–77

    MathSciNet  Google Scholar 

  37. Rudeanu S., Localizations and fractions in algebra of logic, J. Mult.-Valued Logic Soft Comput., 2010, 16(3–5), 467–504

    Google Scholar 

  38. Turunen E., Mathematics Behind Fuzzy Logic, Physica-Verlag, Heidelberg, 1999

    MATH  Google Scholar 

  39. Turunen E., BL-algebras of basic fuzzy logic, Mathware Soft Comput., 1999, 6(1), 49–61

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Informatics, University of Craiova, Craiova, Romania

    Dumitru Buşneag

  2. Faculty of Mathematics and Informatics, University of Bucharest, Bucharest, Romania

    Sergiu Rudeanu

Authors
  1. Dumitru Buşneag
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergiu Rudeanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dumitru Buşneag.

About this article

Cite this article

Buşneag, D., Rudeanu, S. A glimpse of deductive systems in algebra. centr.eur.j.math. 8, 688–705 (2010). https://doi.org/10.2478/s11533-010-0041-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11533-010-0041-4

MSC

Keywords

Search

Navigation