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On The Properties Of A Generalized Class Of T-Norms In Interval-Valued Fuzzy Logics

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Author Info

  • BART VAN GASSE

    ()
    (Fuzziness and Uncertainty Modelling Research Unit, Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), B-9000 Ghent, Belgium)

  • CHRIS CORNELIS

    ()
    (Fuzziness and Uncertainty Modelling Research Unit, Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), B-9000 Ghent, Belgium)

  • GLAD DESCHRIJVER

    ()
    (Fuzziness and Uncertainty Modelling Research Unit, Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), B-9000 Ghent, Belgium)

  • ETIENNE E. KERRE

    ()
    (Fuzziness and Uncertainty Modelling Research Unit, Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), B-9000 Ghent, Belgium)

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    Abstract

    Since it does not generate any MTL-algebra (prelinear residuated lattice), the lattice $\mathcal{L}^I$ of closed subintervals of [0, 1] falls outside the mainstream of research on formal fuzzy logics. However, due to the intimate connection between logical connectives on $\mathcal{L}^I$ and those on [0, 1], many relevant logical properties can still be maintained, sometimes in a slightly weaker form. In this paper, we focus on a broad class of parametrized t-norms on $\mathcal{L}^I$. We derive their corresponding residual implicators, and examine commonly imposed logical properties. Importantly, we formally establish one-to-one correspondences between ∨-definability (respectively, weak divisibility) for t-norms of this class and strong ∨-definability (resp., divisibility) for their counterparts on [0, 1].

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    Bibliographic Info

    Article provided by World Scientific Publishing Co. Pte. Ltd. in its journal New Mathematics and Natural Computation.

    Volume (Year): 02 (2006)
    Issue (Month): 01 ()
    Pages: 29-41

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    Handle: RePEc:wsi:nmncxx:v:02:y:2006:i:01:p:29-41

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    Related research

    Keywords: Formal fuzzy logic; interval-valued fuzzy sets; triangular norms; residuated lattices;

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