On the compositional characterization of complete fuzzy pre-orders

Díaz, Susana; Baets, Bernard De; Montes, Susana

doi:10.1016/j.fss.2007.11.017

Cited by 24 publications

(10 citation statements)

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“…a t-norm T , the transitivity of the generated P and I cannot always be expressed w.r.t. a t-norm [17,18,19]. On the other hand, the results presented in the following sections also hold when R is transitive w.r.t.…”

Section: Generalizing T -Transitivitymentioning

confidence: 60%

General results on the decomposition of transitive fuzzy relations

Díaz

Baets

Montes

2010

Fuzzy Optim Decis Making

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We study the transitivity of fuzzy preference relations, often considered as a fundamental property providing coherence to a decision process. We consider the transitivity of fuzzy relations w.r.t. conjunctors, a general class of binary operations on the unit interval encompassing the class of triangular norms usually considered for this purpose. Having fixed the transitivity of a large preference relation w.r.t. such a conjunctor, we investigate the transitivity of the strict preference and indifference relations of any fuzzy preference structure generated from this large preference relation by means of an (indifference) generator. This study leads to the discovery of two families of conjunctors providing a full characterization of this transitivity. Although the expressions of these conjunctors appear to be quite cumbersome, they reduce to more readily used analytical expressions when we focus our attention on the particular case when the transitivity of the large preference relation is expressed w.r.t. one of the three basic triangular norms (the minimum, the product and the Lukasiewicz triangular norm) while at the same time the generator used for decomposing this large preference relation is also one of these triangular norms. During our discourse, we pay ample attention to the Frank family of triangular norms/copulas.keywords Fuzzy preference relation Transitivity Conjunctor Indifference generator Frank family of t-norms

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Section: Generalizing T -Transitivitymentioning

confidence: 60%

General results on the decomposition of transitive fuzzy relations

Díaz

Baets

Montes

2010

Fuzzy Optim Decis Making

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“…Extensive results on the propagation of transitivity can be found in Díaz et al. 19,20,21,22,23 It is easy to verify that I is a T L -equivalence, R is a S L -complete T L -I-ordering and P is a fuzzy strict preference relation compatible with (R, I, T L , S L ). However, P (a, c).…”

Section: The Lukasiewicz T-normmentioning

confidence: 99%

Fuzzy Strict Preference Relations Compatible With Fuzzy Orderings

Llamazares

Baets

2010

Int. J. Unc. Fuzz. Knowl. Based Syst.

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One of the most important issues in the field of fuzzy preference modelling is the construction of a fuzzy strict preference relation and a fuzzy indifference relation from a fuzzy weak preference relation. Here, we focus on a particular class of fuzzy weak preference relations, the so-called fuzzy orderings. The definition of a fuzzy ordering involves a fuzzy equivalence relation and, in this paper, the latter will be considered as the corresponding fuzzy indifference relation. We search for fuzzy strict preference relations compatible with a given fuzzy ordering and its fuzzy indifference relation. In many situations, depending on the t-norm and t-conorm used, this quest results in a unique fuzzy strict preference relation. Our aim is to characterize these fuzzy strict preference relations and to study their transitivity.

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“…In order to define the concept of transitivity for a reciprocal preference relation, since any relation R 2 L n ðAÞ assumes values in the finite set L n , we could consider the idea of t-norm on a finite set (see [23]), as it is usually generalized in the fuzzy context. However, for this purpose, associativity, commutativity and general boundary conditions are not required, as it is the case for additive fuzzy preference structures (see [3,4,6,7]). Without these properties, we are going to work only with the monotonicity condition and a specific boundary condition.…”

Section: Transitivitymentioning

confidence: 99%