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Pseudo-BCK Algebras: An Extension of BCK Algebras

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Combinatorics, Computability and Logic

Abstract

We extend BCK algebras to pseudo-BCK algebras, as MV algebras and BL algebras were extended to pseudo-MV algebras and pseudo-BL algebras, respectively. We make the connection with pseudo-MV algebras and with pseudo-BL algebras.

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References

  1. C.C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc., 88 (1958), 467–490.

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Cignoli, I.M.L. D’Ottaviano, D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer, Volume 7, 2000.

    Google Scholar 

  3. R. Cignoli, A. Torrens, An algebraic analysis of product logic, Centre de Recerca Matematica, Barcelona, Preprint No. 363, 1997.

    Google Scholar 

  4. A. Di Nola, G. Georgescu, A. lorgulescu, Pseudo-BL algebras: Part I, Mult. Val. Logic,to appear.

    Google Scholar 

  5. A. Di Nola, G. Georgescu, A. lorgulescu, Pseudo-BL algebras: Part II, Mult. Val. Logic,to appear.

    Google Scholar 

  6. A. Dvurečnskij, Commutativity of atomic pseudo MV-algebras, manuscript.

    Google Scholar 

  7. A. Dvurel;enskij, Pseudo MV-algebras are intervals in l-groups, submitted.

    Google Scholar 

  8. P. Flondor, G. Georgescu, A. lorgulescu, Pseudo-t-norms and pseudo-BL algebras, Soft Computing,to appear.

    Google Scholar 

  9. G. Georgescu, A. lorgulescu, Pseudo-MV algebras: a noncommutative extension of MV algebras, The Proceedings of the Fourth International Symposium on Economic Informatics, Bucharest, Romania, May (1999), 961–968.

    Google Scholar 

  10. G. Georgescu, A. Iorgulescu, Pseudo-MV algebras, Mult. Val. Logic (A special issue dedicated to the memory of Cr.C. Moisil), 6 1–2 (2001), 95–135.

    MathSciNet  MATH  Google Scholar 

  11. G. Georgescu, A. Iorgulescu, Pseudo-BL algebras: a noncommutative extension of BL algebras, Abstracts of The Fifth International Conference FSTA 2000, Slovakia, February (2000), 90–92.

    Google Scholar 

  12. P. Hájek, Metamathematics of fuzzy logic, Inst. of Comp. Science, Academy of Science of Czech Rep., Technical report 682 (1996).

    Google Scholar 

  13. P. Hájek, Metamathematics of Fuzzy Logic, Kluwer Acad. Publ., Dordrecht, 1998.

    Google Scholar 

  14. P. Hájek, Basic fuzzy logic and BL-algebras, Soft computing, 2 (1998), 124–128.

    Google Scholar 

  15. P. Hájek, L. Godo, F. Esteva, A complete many-valued logic with product-conjunction, Arch. Math. Logic, 35 (1996), 191–208.

    MathSciNet  MATH  Google Scholar 

  16. Y. Imai, K. Iséki, On axiom systems of propositional calculi XIV, Proc. Japan Academy, 42 (1966), 19–22.

    Article  MATH  Google Scholar 

  17. K. Iséki, S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japonica, 23 1 (1978), 1–26.

    MATH  Google Scholar 

  18. Gr.C. Moisil, Essais sur les Logiques Non-chryssippiennes, Bucarest, 1972.

    Google Scholar 

  19. D. Mundici, MV-algebras are categorically equivalent to bounded commutative BCK-algebras, Math. Japonica, 31 6 (1986), 889–894.

    MathSciNet  MATH  Google Scholar 

  20. D. Mundici, Interpretation of AF C’-algebras in Lukasiewicz sentential calculus, J. Funct. Anal., 65 (1986), 15–63.

    Article  MathSciNet  MATH  Google Scholar 

  21. A.N. Prior, Formal Logic, Oxford, 2nd ed. 1962.

    Google Scholar 

  22. E. Turunen, S. Sessa, Local BL-algebras, Mult. Val. Logic (A special issue dedicated to the memory of Gr.C. Moisil), 6 1–2 (2001), 229–250.

    Google Scholar 

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© 2001 Springer-Verlag London Limited

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Georgescu, G., Iorgulescu, A. (2001). Pseudo-BCK Algebras: An Extension of BCK Algebras. In: Calude, C.S., Dinneen, M.J., Sburlan, S. (eds) Combinatorics, Computability and Logic. Discrete Mathematics and Theoretical Computer Science. Springer, London. https://doi.org/10.1007/978-1-4471-0717-0_9

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  • DOI: https://doi.org/10.1007/978-1-4471-0717-0_9

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-526-7

  • Online ISBN: 978-1-4471-0717-0

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