By Radko Mesiar, Magda Komorníková (auth.), Chris Cornelis, Glad Deschrijver, Mike Nachtegael, Steven Schockaert, Yun Shi (eds.)
This booklet is a tribute to Etienne E. Kerre at the get together of his retirement on October 1st, 2010, after being energetic for 35 years within the box of fuzzy set conception. It gathers contributions from researchers which were with reference to him in a single method or one other in the course of his lengthy and fruitful profession. in addition to a foreword via Lotfi A. Zadeh, it includes thirteen chapters on either theoretical and utilized subject matters in fuzzy set idea, divided in 3 elements: 1) logics and connectives, 2) information research, and three) media purposes. the 1st half offers with fuzzy logics and with operators on (extensions of) fuzzy units. half 2 offers with fuzzy equipment in tough set thought, formal idea research, choice making and category. The final half discusses using fuzzy tools for representing and manipulating media gadgets, similar to photos and textual content files. the variety of the themes which are lined mirror the variety of Etienne's learn pursuits, and certainly, the range of present examine within the zone of fuzzy set theory.
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Extra info for 35 Years of Fuzzy Set Theory: Celebratory Volume Dedicated to the Retirement of Etienne E. Kerre
Two variants of intuitonistc fuzzy propositional calculus. Preprint IM-MFAIS-5-88, Soﬁa (1988) 3. : Intuitionistic Fuzzy Sets. Springer Physica-Verlag, Berlin (1999) On the Intuitionistic Fuzzy Implications and Negations 37 4. : Intuitionistic fuzzy implications and Modus Ponens. Notes on Intuitionistic Fuzzy Sets 11(1), 1–5 (2005) 5. : On some types of intuitionistic fuzzy negations. Notes on Intuitionistic Fuzzy Sets 11(4), 170–172 (2005) 6. : On some intuitionistic fuzzy negations. Notes on Intuitionistic Fuzzy Sets 11(6), 13–20 (2005) 7.
Iii) ⊗-transitivity [m = m ] ⊗ [m = m ] ≤ [m = m ] . for all m, m , m ∈ M where [m = m ] denotes a truth value of m = m . A special case of fuzzy equality on the algebra of truth values is biresiduation. Example of a fuzzy equality on M = R with respect to the standard Łukasiewicz algebra is . [m = n] = 1 − (1 ∧ |m − n|), m, n ∈ R. Syntax of FTT is a generalization of the lambda-calculus that is constructed in a classical way. The main difference from classical type theory is thus in definition of additional special connectives.
We should note that Axiom 8 is checked using the classical intuitionistic fuzzy negation (¬1 ); if it is valid using the respective generated by implication I negation N (x), then the axiom is marked as 8N . We should also note that the validity of Axiom 7 does not imply the validity of Axiom 7*. The forms of the axioms with asterisks are Axiom 3∗ : (∀y)(I(0, y) is an IFT). Axiom 5∗ : (∀x)(I(x, x) is an IFT). Axiom 7∗ : (∀x, y)(I(x, y) is an IFT iﬀ x ≤ y). The validity of each of these assertions can be checked directly.
35 Years of Fuzzy Set Theory: Celebratory Volume Dedicated to the Retirement of Etienne E. Kerre by Radko Mesiar, Magda Komorníková (auth.), Chris Cornelis, Glad Deschrijver, Mike Nachtegael, Steven Schockaert, Yun Shi (eds.)