Implication functions in intervalvalued fuzzy set theory
(2013)
Advances in fuzzy implication functions.
In Studies in Fuzziness and Soft Computing
300.
p.7399
 Author
 Glad Deschrijver (UGent)
 Organization
 Abstract
 Intervalvalued 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 intervalvalued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in intervalvalued 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 intervalvalued fuzzy set theory which satisfy the SmetsMagrez axioms, we discuss the solutions of a particular distributivity equation involving strict tnorms, we extend monoidal logic to the intervalvalued 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 intervalvalued fuzzy set theory.
 Keywords
 implication, intervalvalued fuzzy set
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU3196768
 Chicago
 Deschrijver, Glad. 2013. “Implication Functions in Intervalvalued 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 intervalvalued 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 intervalvalued 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 Intervalvalued 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 = {Intervalvalued 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 intervalvalued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in intervalvalued 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 intervalvalued fuzzy set theory which satisfy the SmetsMagrez axioms, we discuss the solutions of a particular distributivity equation involving strict tnorms, we extend monoidal logic to the intervalvalued 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 intervalvalued 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 = {14349922}, language = {eng}, pages = {7399}, publisher = {Springer}, series = {Studies in Fuzziness and Soft Computing}, title = {Implication functions in intervalvalued fuzzy set theory}, url = {http://dx.doi.org/10.1007/9783642356773\_4}, volume = {300}, year = {2013}, }
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