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Algebraic Operations on Fuzzy Valued Fuzzy Sets

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Advanced Concepts in Fuzzy Logic and Systems with Membership Uncertainty

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 284))

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Abstract

In this chapter, new analytical formulae for membership functions of extended t-norms are derived. We consider the following cases: extended minimum, minimum-based extensions of continuous t-norms, extended continuous t-norms based on drastic-product, and extended Łukasiewicz t-norm based on continuous Archimedean t-norms. As a dual concept to extended t-norms, extended t-conorms and their formulae are considered. These cases cover almost all practical engineering situations when we implement type-2 fuzzy logic systems. In many cases, we get formulae that preserve shapes, which enable us to derive adaptive network fuzzy inference systems of type- 2. Otherwise, some approximations are needful, or more general notion of a triangular norm on fuzzy truth values (t-norm of type-2 for short) is needed, whose axiomatics we provide briefly. Finally, implications on fuzzy truth values, especially their family called simplicatoins, are derived in order to prepare foundations for structures of uncertain fuzzy logic systems.

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Authors and Affiliations

  1. Department of Computer Engineering, Czestochowa University of Technology, Al. Armii Krajowej 36, 42-200, Czestochowa, Poland

    Janusz T. Starczewski

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  1. Janusz T. Starczewski
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Correspondence to Janusz T. Starczewski .

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Starczewski, J.T. (2013). Algebraic Operations on Fuzzy Valued Fuzzy Sets. In: Advanced Concepts in Fuzzy Logic and Systems with Membership Uncertainty. Studies in Fuzziness and Soft Computing, vol 284. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29520-1_2

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  • DOI: https://doi.org/10.1007/978-3-642-29520-1_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-29519-5

  • Online ISBN: 978-3-642-29520-1

  • eBook Packages: EngineeringEngineering (R0)

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