ON THE PROPERTIES OF A GENERALIZED CLASS OF T-NORMS IN INTERVAL-VALUED FUZZY LOGICS

Bart van Gasse (), Chris Cornelis (), Glad Deschrijver () and Etienne E. Kerre ()
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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

New Mathematics and Natural Computation (NMNC), 2006, vol. 02, issue 01, 29-41

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].

Keywords: Formal fuzzy logic; interval-valued fuzzy sets; triangular norms; residuated lattices (search for similar items in EconPapers)
Date: 2006
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DOI: 10.1142/S1793005706000361

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