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# Implication functions in interval-valued fuzzy set theory

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Abstract
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.
Keywords
implication, interval-valued fuzzy set

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## Citation

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Chicago
Deschrijver, Glad. 2013. “Implication Functions in Interval-valued Fuzzy Set Theory.” In Advances in Fuzzy Implication Functions, ed. Michał Baczyński, Gleb Beliakov, Humberto Bustince, and Ana Pradera, 300:73–99. Berlin, Germany: Springer.
APA
Deschrijver, Glad. (2013). Implication functions in interval-valued fuzzy set theory. In M. Baczyński, G. Beliakov, H. Bustince, & A. Pradera (Eds.), Advances in fuzzy implication functions (Vol. 300, pp. 73–99). Berlin, Germany: Springer.
Vancouver
1.
Deschrijver G. Implication functions in interval-valued fuzzy set theory. In: Baczyński M, Beliakov G, Bustince H, Pradera A, editors. Advances in fuzzy implication functions. Berlin, Germany: Springer; 2013. p. 73–99.
MLA
Deschrijver, Glad. “Implication Functions in Interval-valued Fuzzy Set Theory.” Advances in Fuzzy Implication Functions. Ed. Michał Baczyński et al. Vol. 300. Berlin, Germany: Springer, 2013. 73–99. Print.
```@incollection{3196768,
abstract     = {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.},
author       = {Deschrijver, Glad},
booktitle    = {Advances in fuzzy implication functions},
editor       = {Baczy\'{n}ski, Micha\unmatched{0142} and Beliakov, Gleb and Bustince, Humberto and Pradera, Ana},
isbn         = {9783642356773},
issn         = {1434-9922},
language     = {eng},
pages        = {73--99},
publisher    = {Springer},
series       = {Studies in Fuzziness and Soft Computing},
title        = {Implication functions in interval-valued fuzzy set theory},
url          = {http://dx.doi.org/10.1007/978-3-642-35677-3\_4},
volume       = {300},
year         = {2013},
}

```
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