By János Fodor, Marc Roubens (auth.)

This booklet offers in-depth insurance of crucial effects approximately fuzzy common sense together with negations, conjunctions, disjunctions, implications and offers the interrelations among these diverse connectives. The paintings brings jointly a number of effects approximately valued binary kin pleasing different transitivity-type stipulations.
The authors suggest the 1st sound advent to valued choice modelling in the course of the systematic use of fuzzy set idea and useful equations and derive the potential foundations for multicriteria choice relief utilizing aggregation, score and selection strategies at the foundation of axiomatic effects. The textual content offers a unified view of varied multicriteria determination making instruments which have been independently derived some time past, facing pairwise comparisons. The monograph is mathematically orientated however the effects should be of the best curiosity for engineers and economists who layout and enforce choice aid platforms in perform. it's also provided with a enough variety of examples to make it appealing to nonspecialists.

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By representation T(x, y) = T(x, y) = z > 0 is equivalnet to f(x) r + f(y) = Iy(x, z) = r f(z). 55), that 1 (max{f(z)- f(x), 0}) = y. • Remarks. 1) Generally speakig, T(x, y) = 0 does not imply y = 1T'(x, 0). 8. 2) In general, the statement of the previous lemma is not true if T is not Archimedean. 4 on ordinal sums). 75. g. Dubois and Prade (1991b), Fodor (1993c), Goguen (1969)). 1 was defined by several authors in a heuristic way. However, the fact that it is an R-, S- and QL-implication based on appropriate t-norms and t-conorms was pointed out by Fodor (1993c).

8 lf a valued binary relation R on A is reflexive then crisp relations R>. \ E (0, 1]. • Proof. Immediate. This statement is not true in general for c-reflexive or weakly reflexive valued binary relations. Irreflexive (sometimes called antireflexive) valued binary relations are introduced now. 8 A valued binary relation R on A is ir-reflexive if R(a, a) =0 for every a E A. We left to the reader the definition of both c-irreflexive and weakly irreflexive valued relations. Characterizations of irreflexivity are given in terms of traces.

AE(O,l]. • Proof. Easy. Clearly, if a valued binary relation R on Ais T-antisymmetric (T-asymmetric) for a certain t-norm T, then R is T'-antisymmetric (T'-asymmetric) for any t-norm T' such that T' ::::; T. Therefore, an antisymmetric relation is T-antisymmetric for any t-norm T. Remarkable new definitions of antisymmetry (asymmetry) (that is, different from the case min) are obtained by using t-norms with zero divisors. In these cases, roughly speaking, values R(a, b) and R(b, a) cannot be too high at the same time.

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