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

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  • 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)

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 parametrizedt-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) fort-norms of this class and strong ∨-definability (resp., divisibility) for their counterparts on [0, 1].

Suggested Citation

  • Bart Van Gasse & Chris Cornelis & Glad Deschrijver & Etienne E. Kerre, 2006. "On The Properties Of A Generalized Class Of T-Norms In Interval-Valued Fuzzy Logics," New Mathematics and Natural Computation (NMNC), World Scientific Publishing Co. Pte. Ltd., vol. 2(01), pages 29-41.
    • Handle: RePEc:wsi:nmncxx:v:02:y:2006:i:01:n:s1793005706000361
    DOI: 10.1142/S1793005706000361
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