Modelling a linguistic variable as a hierarchical family of partitions induced by an indistinguishability operator

Soto, Adolfo Rodríguez de; Recasens, J.

doi:10.1016/s0165-0114(99)00134-7

Cited by 21 publications

(12 citation statements)

References 8 publications

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“…As already mentioned, it is also closely related to -partitions obtained from an algebraic (or strict) fuzzy partition, as proposed by De Baets and Mesiar [8]. Links between a linguistic variable and indistinguishability relations, introduced by Valverde and Trillas [27,28], have been studied by De Soto and Recasens [14].…”

Section: Discussionmentioning

confidence: 99%

“…Pedrycz proposed two main interpretations: one is based on equalization, like entropy equalization, while the second one relies on the error-free reconstruction of a fuzzified signal by means of a defuzzification interface [24,25]. In [14], de Soto and Recasens showed that fuzzy partitions based on triangular fuzzy numbers are obtained as compatible partitions associated with indistinguishability operators associated with the Lukasiewicz triangular norm. Equality relations were also applied in fuzzy control to justify the use of triangular or trapezoidal membership functions [20].…”

Section: Introductionmentioning

confidence: 99%

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Building a class of fuzzy equivalence relations

Foulloy

Benoit

2006

Fuzzy Sets and Systems

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In this paper, we propose a practical method, given a strict triangular norm with a convex additive generator, for deriving a fuzzy equivalence relation whose reflexivity condition generalizes Ruspini's definition of fuzzy partitions. The properties of the relations, their comparison, their transitivity, the construction of fuzzy equivalence relations on cartesian products are presented. A large part of the paper is devoted to applications with fuzzy partitions defined on the real line. Several examples, including the fairy-tale problem from De Cock and Kerre [On (un)suitable fuzzy relations to model approximate equality, Fuzzy Sets and Systems 133 (2) (2003) 137-153], the comparison of colored objects and comfort situations are proposed.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Building a class of fuzzy equivalence relations

Foulloy

Benoit

2006

Fuzzy Sets and Systems

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“…[3,14,31,34] and many others. The role of these relations in modeling of linguistic semantics has been raised already in [22] (recently also in [6]) where a related concept of the, so called, indiscernibility relation (see [35]) has been employed. The mathematical theory of the natural language semantics presented there relies on the following hypothesis.…”

Section: Remarkmentioning

confidence: 99%

A comprehensive theory of trichotomous evaluative linguistic expressions

Novák

2008

Fuzzy Sets and Systems

154

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In this paper, a logical theory of the, so called, trichotomous evaluative linguistic expressions (TEv-expressions) is presented. These are frequent expressions of natural language, such as "small, very small, roughly medium, extremely big", etc. The theory is developed using the formal system of higher-order fuzzy logic, namely the fuzzy type theory (generalization of classical type theory). First, we discuss informally what are properties of the meaning of TEv-expressions. Then we construct step by step axioms of a formal logical theory T Ev of TEv-expressions and prove various properties of T Ev . All the proofs are syntactical and so, our theory is very general. We also outline construction of a canonical model of T Ev . The main elegancy of our theory consists in the fact that semantics of all kinds of evaluative expressions is modeled in a unified way. We also prove theorems demonstrating that essential properties of the vagueness phenomenon can be captured within our theory.

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“…Among other results, it will be proved that a t-norm T is ET-Lipschitzian and moreover the maps T(n) The powers E of the natural T-indistinguishability operators have been studied in relation with antonymy and fuzzy partitions in [20]. Proposition 3.3: Let E be a T-indistinguishability operator on a set X. E is a T-indistinguishability operator on X.…”

Section: Preliminariesmentioning

confidence: 99%

Aggregation Operators and the Lipschitzian Condition

Jacas

Recasens

2007

2007 IEEE International Fuzzy Systems Conference

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Abstract-Lipschitzian and kernel aggregation operators with respect to the natural T-indistinguishability operator ET and their powers are studied. A t-norm T is proved to be ETLipschitzian, and is interpreted as a fuzzy point and a fuzzy map as well. Given an Archimedean t-norm T with additive generator t, the quasi-arithmetic mean generated by t is proved to be the most stable aggregation operator with respect to T.

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Modelling a linguistic variable as a hierarchical family of partitions induced by an indistinguishability operator

Cited by 21 publications

References 8 publications

Building a class of fuzzy equivalence relations

Building a class of fuzzy equivalence relations

A comprehensive theory of trichotomous evaluative linguistic expressions

Aggregation Operators and the Lipschitzian Condition

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