By Tomasa Calvo Sánchez, Joan Torrens Sastre

ISBN-10: 3319304194

ISBN-13: 9783319304199

ISBN-10: 3319304216

ISBN-13: 9783319304212

This ebook deals a well timed record on key theories and purposes of soft-computing. Written in honour of Professor Gaspar Mayor on his seventieth birthday, it essentially makes a speciality of parts with regards to his learn, together with fuzzy binary operators, aggregation capabilities, multi-distances, and fuzzy consensus/decision versions. It additionally discusses a few fascinating purposes comparable to the implementation of fuzzy mathematical morphology in line with Mayor-Torrens t-norms. Importantly, the several chapters, authored via prime specialists, current novel effects and provide new views on assorted points of Mayor’s study. The publication additionally contains an outline of evolutionary fuzzy platforms, an issue that's not considered one of Mayor’s major parts of curiosity, and a last bankruptcy written through the Spanish pioneer in fuzzy common sense, Professor E. Trillas. computing device and choice scientists, wisdom engineers and mathematicians alike will locate right here an authoritative evaluate of key soft-computing thoughts and techniques.

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**Extra resources for Fuzzy Logic and Information Fusion: To commemorate the 70th birthday of Professor Gaspar Mayor**

**Sample text**

As a particular case of a more general result of Aglianò and Montagna in [1], we recall a slightly different notion of ordinal sum for finite linearly-ordered Wajsberg hoops. Actually, a hoop is an algebra A = A, ∗, →, 1 such that A, ∗, 1 is a commutative monoid and for all x, y, z ∈ A the following equations hold: x → x = 1, x ∗ (x → y) = y ∗ (y → x), x → (y → z) = (x ∗ y) → z. A Wajsberg hoop is a hoop satisfying the equation: (x → y) → y = (y → x) → x. A bounded hoop is an algebra A = (A, ∗, →, 1, 0) such that A, ∗, →, 1 is a hoop and 0 ≤ x for all x ∈ A.

J. Frank, B. Schweizer, Problems on associative functions. Aequationes Math. 66, 128–140 (2003) 42. W. Trutschnig, On a strong metric on the space of copulas and its induced dependence measure. J. Math. Anal. Appl. 384, 690–705 (2011) 43. J. Fernández-Sánchez, W. Trutschnig, Conditioning-based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and Iterated Function Systems. J. Theor. Probab. (2015). 1007/s10959-014-0541-4 44. P. D. Taylor, A remark on associative copulas.

Many of the definitions and theorems for bivariate copulas have analogous multivariate versions (see [1, 12]). 3 Associative Copulas: A Survey 35 The main problem in the theory of copulas is to determine which sets of (possible different dimensions) copulas are margins of a higher-dimensional copula. The associativity of n-copulas in the sense of Post (see [70]) is studied in [71]— solving an open problem posed in [72]. Specifically, if n ≥ 2 is a natural number and S is a nonempty set, an n-ary operation f : S n −→ S is associative on S if, for any 1 ≤ i < j ≤ n, the equality j−1 n+j−1 2n−1 = f x1 , f xj f x1i−1 , f xin+i−1 , xn+i 2n−1 , xn+j q holds for all x1 , .

### Fuzzy Logic and Information Fusion: To commemorate the 70th birthday of Professor Gaspar Mayor by Tomasa Calvo Sánchez, Joan Torrens Sastre

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