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Free Łukasiewicz implication algebras

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Abstract

Łukasiewicz implication algebras are the {→,1}-subreducts of MV- algebras. They are the algebraic counterpart of Super-Łukasiewicz Implicational Logics investigated in Komori (Nogoya Math J 72:127–133, 1978). In this paper we give a description of free Łukasiewicz implication algebras in the context of McNaughton functions. More precisely, we show that the |X|-free Łukasiewicz implication algebra is isomorphic to \({\bigcup_{x\in X} [x_\theta)}\) for a certain congruence θ over the |X|-free MV-algebra. As corollary we describe the free algebras in all subvarieties of Łukasiewicz implication algebras.

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References

  1. Aglianó P., Panti G.: Geometrical methods in Wajsberg hoops. J. Algebra 256, 352–374 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berman J., Blok W.J.: Free Lukasiewicz hoop residuation algebras. Stud. Log. 77, 153–180 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blok W.J., Raftery J.G.: On the quasivariety of BCK-algebras and its subquasivarieties. Algebra Universalis 33, 68–90 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burris S., Sankappanavar H.P.: A Course in Universal Algebra. Graduate texts in Mathematics, vol. 78. Springer, New York (1981)

    Google Scholar 

  5. Chang C.C.: Algebraic analysis of many-valued logics. Trans. Am. Math. Soc. 88, 467–490 (1958)

    Article  MATH  Google Scholar 

  6. Cignoli R., D’Ottaviano I., Mundici D.: Algebraic Foundations of Many-Valued Reasoning. Trends in Logics. vol. 7. Kluwer, Dordrecht (2000)

    Google Scholar 

  7. Díaz Varela, J.P., Torrens, A.: Decomposability of free Łukasiewicz implication algebras. Arch. Math. Log. 45, 1011–1020 (2006)

  8. Font J.M., Rodriguez A.J., Torrens A.: Wajsberg algebras. Stochastica 8, 5–31 (1984)

    MathSciNet  MATH  Google Scholar 

  9. Komori, Y.: The separation theorem of the \({\aleph_0}\)-valued propositional logic. Rep. Fac. Sci. Shizouka Univ. 12, 1–5 (1978)

    MathSciNet  MATH  Google Scholar 

  10. Komori Y.: Super-Łukasiewicz implicational logics. Nogoya Math. J. 72, 127–133 (1978)

    MathSciNet  MATH  Google Scholar 

  11. Mundici D.: MV-algebras are categorically equivalent to bounded commutative BCK.-algebras. Math. Jpn. 31, 889–894 (1986)

    MathSciNet  MATH  Google Scholar 

  12. Panti G.: A geometric proof of the completeness of the Łukasiewicz calculus. J. Symb. Log. 60(2), 563–578 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Panti G.: Varieties of MV-algebras. Multi Valued Log. 9(1), 141–157 (1999)

    MathSciNet  MATH  Google Scholar 

Download references

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

    José Patricio Díaz Varela

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  1. José Patricio Díaz Varela
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Correspondence to José Patricio Díaz Varela.

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The support of Universidad Nacional del Sur and CONICET is gratefully acknowledged.

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Díaz Varela, J.P. Free Łukasiewicz implication algebras. Arch. Math. Logic 47, 25–33 (2008). https://doi.org/10.1007/s00153-008-0067-5

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  • DOI: https://doi.org/10.1007/s00153-008-0067-5

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