Download PDF by Tomasa Calvo, Anna Kolesárová, Magda Komorníková, Radko: Aggregation Operators: New Trends and Applications

By Tomasa Calvo, Anna Kolesárová, Magda Komorníková, Radko Mesiar (auth.), Professor Tomasa Calvo, Professor Gaspar Mayor, Professor Radko Mesiar (eds.)

ISBN-10: 3662003198

ISBN-13: 9783662003190

ISBN-10: 3790817872

ISBN-13: 9783790817874

1. The expanding variety of learn papers seemed within the final years that both utilize aggregation capabilities or give a contribution to its theoretieal examine asses its becoming significance within the box of Fuzzy Logie and in others the place uncertainty and imprecision play a correct position. due to the fact those papers are pub­ lished in lots of journals, few books and a number of other complaints of meetings, books on aggregation are partieularly welcome. To my wisdom, "Agrega­ tion Operators. New tendencies and purposes" is the 1st e-book aiming at generality , and that i take it as a honour to put in writing this Foreword in line with the light call for of its editors, Radko Mesiar, Tomasa Calvo and Gaspar Mayor. My excitement additionally derives from the truth that two decades aga i used to be one of many first Spaniards attracted to the research of aggregation features, and this e-book comprises paintings by means of a number of Spanish authors. The booklet includes great and suitable unique papers, authored through the most notable researchers within the box, and because it will probably serve, because the editors indicate within the Preface, as a small guide on aggregation, the booklet is especially priceless for these coming into the topic for the 1st time. The booklet additionally comprises aside facing power parts of program, so it may be important in gaining perception at the destiny developments.

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X n ) ~ a ~ max(xl, ... , x n ) :=:::} A(xI, ... , x n ) = a. Note also that an aggregation operator A with annihilator a E]O, 1[ cannot have neutral element e. However, A may have a neutral element e if its annihilator a E {O, I}, and e =I- a. Aggregation operators A with annihilator a E {O, I} can have another special element b E {O, I} \ {a} with almost annihilating property: VnEN,Vxl, ... ,XnE[O,I](bE{XI, ... ,Xn}) Downarrow (45) ° Such element bis called a weak annihilator [69]. As an example we can take the 3 - lI-operator E introduced in (39) with annihilator a = and weak annihilator b = 1.

Suppose that A is an aggregation operator with annihilator a E]O,I[. Then we can introduce two operators acting on inputs from [0, a] and on inputs from [a, 1], respectively, and by appropriate linear transformations of the scales [0, a] and [a,l] into [O,IJ to construct two aggregation operators A[o] and A[l] with annihilators ao = 1 and al = 0, respectively. Applying this construction method to a-medians med a introduced in (43) we obtain (meda) [0] = max, independently of a E]O, 1[. 22 The following observation applies specifically to aggregation operators with annihilator a E]O, 1[.

Applying this transformation to any aggregation operator A acting on [0, 1J as deseribed in (63), we obtain the dual aggregation operator A d . Definition 15. Let A: the operator A d U [0, l]n nEN -+ [O,lJ be an aggregation operator. Then = A

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Aggregation Operators: New Trends and Applications by Tomasa Calvo, Anna Kolesárová, Magda Komorníková, Radko Mesiar (auth.), Professor Tomasa Calvo, Professor Gaspar Mayor, Professor Radko Mesiar (eds.)


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