Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application

Cornelis, Chris; Deschrijver, Glad; Kerre, Etienne

doi:10.1016/s0888-613x(03)00072-0

Cited by 384 publications

(189 citation statements)

References 34 publications

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“…Furthermore, T W , T P and S W are natural extensions of T W , T P and S W respectively. In [2,4,7] it is shown that T W inherits more interesting properties from the Lukasiewicz t-norm on the unit interval than the t-representable extension of T W to L I .…”

Section: Then T Is Called the Representant Of T And T Is Denoted Bymentioning

confidence: 99%

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Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory

Deschrijver

2009

Fuzzy Sets and Systems

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In this paper we study extensions of the arithmetic operators +, −, ·, ÷ to the lattice L I of closed subintervals of the unit interval. Starting from a minimal set of axioms that these operators must fulfill, we investigate which properties they satisfy. We also investigate some classes of t-norms on L I which can be generated using these operators; these classes provide natural extensions of the Lukasiewicz, product, Frank, Schweizer-Sklar and Yager t-norms to L I .

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Section: Then T Is Called the Representant Of T And T Is Denoted Bymentioning

confidence: 99%

“…It is easy to see that these operators satisfy (add-1)-(add-5), (mul-1)-(mul-5), (add-5'), (mul-5'), (1), (2), (3) and (4).…”

Section: Examplementioning

confidence: 99%

Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory

Deschrijver

2009

Fuzzy Sets and Systems

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“…A consequence of Theorem 3 is the following corollary (see [22,23,43,51,52,60,[64][65][66][68][69][70][71][72][73][74][75][76][77]). This corollary provides a construction method for IV t-norms and IV t-conorms, which was presented in the first papers on this topic.…”

Section: S Is An IV T-conorm If and Only If Kmentioning

confidence: 99%

“…In [23,43,[72][73][74][75][76][77], it is proved that some important representation theorems can be also shown for IV t-norms, but not for t-representable t-norms. This shows that IVFS theory cannot be reduced to an approach in which all operators are t-representable.…”

Section: We Denote By T Ta T B the T-representable IV T-norms Obtainmentioning

confidence: 99%

“…This definition (Definition 6) extends classical two-valued implication; that is We must point out that the characterization and construction theorems for the implicators on L([0, 1]) (see [73]) are a generalization of the construction and characterization theorems for fuzzy S-implications and fuzzy R-implications (see [20]). In this connection, we see the following open problem.…”

Section: Interval-valued Fuzzy Implication Operatorsmentioning

confidence: 99%

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A Survey of Interval‐Valued Fuzzy Sets

Bustince

Montero

Pagola

et al. 2008

Handbook of Granular Computing

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From fuzzy sets to shadowed sets: Interpretation and computing

Pedrycz

2009

Int. J. Intell. Syst.

120

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In this study, we discuss a concept of shadowed sets and present their applications. To establish some sound compromise between the qualitative Boolean (two-valued) description of data and quantitative membership grades, we introduce an interpretation framework of shadowed sets. Shadowed sets are discussed as three-valued constructs induced by fuzzy sets assuming three values (that could be interpreted as full membership, full exclusion, and uncertain membership). The algorithm of converting membership functions into this quantification is a result of a certain optimization problem guided by the principle of uncertainty localization. We revisit fundamental ideas of relational calculus in the setting of shadowed sets. We demonstrate how shadowed sets help in problems in data interpretation in fuzzy clustering by leading to the three-valued quantification of data structure that consists of core, shadowed, and uncertain structure. C 2008 Wiley Periodicals, Inc.

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Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application

Cited by 384 publications

References 34 publications

Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory

Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory

A Survey of Interval‐Valued Fuzzy Sets

From fuzzy sets to shadowed sets: Interpretation and computing

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