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

In this paper we investigate additive generators in Atanassov's intuitionistic fuzzy and interval-valued fuzzy set theory. Starting from generalized arithmetic operators satisfying some axioms we define additive generators and we characterize continuous generators which map exact elements to exact elements in terms of generators on the unit interval. We give a necessary and sufficient condition under which a generator actually generates a t-norm and we show that the generated t-norm belongs to particular classes of t-norms depending on the arithmetic operators involved in the definition of the generator.

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“…Theorem 3 [22] The mapping satisfies the following properties, for all α, β in R and a, b, c inL I ,…”

Section: Arithmetic Operators Onl Imentioning

confidence: 99%

Additive Generators Based on Generalized Arithmetic Operators in Interval-Valued Fuzzy and Atanassov’s Intuitionistic Fuzzy Set Theory

Deschrijver

Kerre

2015

Imprecision and Uncertainty in Information Representation and Processing

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“…We start from two arithmetic operators ⊕ : (L I ) 2 →L I and ⊗ : (L I + ) 2 → L I satisfying the following properties (see [20]), The mapping is defined in [20] by, for all x, y inL I ,…”

Section: Implications Defined Using Arithmetic Operators On Lmentioning

confidence: 99%

Implication Functions in Interval-Valued Fuzzy Set Theory

Deschrijver

2013

Studies in Fuzziness and Soft Computing

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Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory.

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“…Definitions 3.1 and 4.1 are equivalent to the definitions of t-representable tnorm, t-conorm and negation on L I (U in this paper) which are used by the "Fuzziness and Uncertainty Modelling Research Unit", at Ghent University, see [13].…”

Section: Interval Fuzzy Negationsmentioning

confidence: 99%

Analyzing the Relationship between Interval-valued D-Implications and Interval-valued QL-Implications

Reiser¹,

Bedregal²,

Santiago³

et al. 2010

TEMA - Tendências Em Matemática Aplicada E Computacional

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Abstract.The aim of this work is to analyze the relationship between interval QLimplications and their contrapositions named interval D-implications.In order to achieve this aim, the commutative classes relating to these concepts are studied.We also analyze under which conditions the main properties corresponding to punctual D-implications and QL-implications are still valid when an interval-based fuzzy approach,on the best interval representation, is considered.

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

Cited by 61 publications

References 18 publications

Additive Generators Based on Generalized Arithmetic Operators in Interval-Valued Fuzzy and Atanassov’s Intuitionistic Fuzzy Set Theory

Additive Generators Based on Generalized Arithmetic Operators in Interval-Valued Fuzzy and Atanassov’s Intuitionistic Fuzzy Set Theory

Implication Functions in Interval-Valued Fuzzy Set Theory

Analyzing the Relationship between Interval-valued D-Implications and Interval-valued QL-Implications

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