35 Years of Fuzzy Set Theory: Celebratory Volume Dedicated by Radko Mesiar, Magda Komorníková (auth.), Chris Cornelis,

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 idea. It gathers contributions from researchers which have been on the subject of him in a single approach or one other in the course of his lengthy and fruitful profession. along with a foreword through Lotfi A. Zadeh, it comprises thirteen chapters on either theoretical and utilized issues in fuzzy set thought, divided in 3 components: 1) logics and connectives, 2) facts research, and three) media purposes. the 1st half offers with fuzzy logics and with operators on (extensions of) fuzzy units. half 2 bargains with fuzzy tools in tough set concept, formal thought research, selection making and type. The final half discusses using fuzzy equipment for representing and manipulating media items, comparable to photos and textual content files. the range of the subjects which are lined mirror the range of Etienne's examine pursuits, and certainly, the variety of present learn within the zone of fuzzy set theory.

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1 (ii) and other problems. Let the pair (m, n) denote the expression with m-th implication and n-th negation. Following [12], first, we shall formulate the following Theorem 2. The pairs (4, 1), (5, 1), (7, 1), (12, 1), (13, 1), (15, 1), (20, 2), (22, 4), (23, 5), satisfy the three axioms. We must note that no one pair for 1 ≤ m ≤ 23 and 1 ≤ n ≤ 5 satisfy exactly two axioms and more precisely, axioms (L − CP ) and (R − CP ). In [12], solutions are given for 1 ≤ m ≤ 138 and 1 ≤ n ≤ 34. 1 (iii). 1 (i).

1 (ii), where the assertions are related to (CP ), (L − CP ) and (R − CP ). In [13], it is shown that there are 1322 pairs (m, n) satisfying the three equalities, when 1 ≤ m ≤ 138 and 1 ≤ n ≤ 34. But, no one pair (m, n) for 1 ≤ m ≤ 23 and 1 ≤ n ≤ 5 satisfies exactly two equalities, namely – (L − CP ) and (R − CP ). 1 (iii). 1 (i). Another result of our search is Theorem 5. Only the pair (21, 3) satisfies only equality (R − CP ). The most interesting is the following Theorem 6. The pairs (12, 3), (17, 3) satisfy only equality (L − CP ).

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