On the relationship between some extensions of fuzzy set theory

Deschrijver, Glad; Kerre, Etienne

doi:10.1016/s0165-0114(02)00127-6

Cited by 653 publications

(322 citation statements)

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“…In a future paper we will use the arithmetic operators to define additive and multiplicative generators on L I in a more general way. Similarly as Lemma 2.1 in [9] it can be shown that L I is a complete lattice. An intuitionistic fuzzy set A on U in Atanassov's sense can be represented by the L I -fuzzy set A given by…”

Section: Introductionmentioning

confidence: 59%

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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: Introductionmentioning

confidence: 59%

“…Another extension of fuzzy set theory is intuitionistic fuzzy set theory introduced by Atanassov [1]. In [9] it is shown that Atanassov's intuitionistic fuzzy set theory is equivalent to interval-valued fuzzy set theory and that both are equivalent to L-fuzzy set theory in the sense of Goguen [14] w.r.t. a special lattice L I .…”

Section: Introductionmentioning

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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“…Thus, the exact meaning of AFS is rather obscure, showing the semantics of a B1 framework while formally seeming to be a B2 one. Moreover, as Atanassov and his followers failed to give a clear definition of AFS's underlying logical and semantical structure (see [27]), no solid reasons to separate AFS from IVFS were available when both formalisms were proven [8,9] to be equivalent (through the isomorphism :…”

Section: Type-1 and Type-2 Bipolaritymentioning

confidence: 99%

“…model, and an F2 framework could be associated to either a B1 or B2 setting. Nevertheless, a commonly-used instance of type-2 fuzziness, interval valued fuzzy sets (IVFS, see [18]), actually devised as B1 objects, has been shown (see [8,9]) to be in certain sense equivalent to Atanassov fuzzy sets (AFS, see [1]), which however were originally devised as F1 and B2 objects. In fact, as a consequence of this formal equivalence, a bitter dispute (see [10] and [3]) raised between Atanassov and his followers, on one side, and an important part of the fuzzy community, on the other, about the exact meaning of AFS and their real relevance in the context of bipolarity.…”

Section: Introductionmentioning

confidence: 99%

On the Semantics of Bipolarity and Fuzziness

Rodríguez

Franco

Montero

2011

Advances in Intelligent and Soft Computing

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This paper analyzes the relationship between fuzziness and bipolarity, notions which were devised to address different kinds of uncertainty: linguistic imprecision, in the former, and knowledge relevance and character or polarity, in the latter. Although different types of fuzziness and bipolarity have been defined, these relations are not always clear. This paper proposes the use of four-valued extensions to provide a formal method to rigorously define and compare the semantics and logical structure of diverse combinations of fuzziness and bipolarity types. As a result, this paper claims that these notions and their different types are independent and not semantically equivalent despite its possible formal equivalence.

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“…The relation above is transitive and antisymmetric and it expresses the fact that x links strongly to y, so that (L([0, 1]), ≤ L ) is a complete lattice (see [21][22][23]), where the smallest element is 0 L = [0, 0] and the largest is 1 L = [1,1]. Evidently, it is not a linear lattice, for there exist elements that are not comparable.…”

Section: Preliminary Definitionsmentioning

confidence: 99%