QL-implications versus D-implications

Margarita Mas; Miquel Monserrat; Joan Torrens

Kybernetika (2006)

  • Volume: 42, Issue: 3, page 351-366
  • ISSN: 0023-5954

Abstract

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This paper deals with two kinds of fuzzy implications: QL and Dishkant implications. That is, those defined through the expressions I ( x , y ) = S ( N ( x ) , T ( x , y ) ) and I ( x , y ) = S ( T ( N ( x ) , N ( y ) ) , y ) respectively, where T is a t-norm, S is a t-conorm and N is a strong negation. Special attention is due to the relation between both kinds of implications. In the continuous case, the study of these implications is focused in some of their properties (mainly the contrapositive symmetry and the exchange principle). Finally, the case of non continuous t-norms or non continuous t-conorms is studied, deriving new implications of both kinds and showing that they remain strongly connected.

How to cite

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Mas, Margarita, Monserrat, Miquel, and Torrens, Joan. "QL-implications versus D-implications." Kybernetika 42.3 (2006): 351-366. <http://eudml.org/doc/33810>.

@article{Mas2006,
abstract = {This paper deals with two kinds of fuzzy implications: QL and Dishkant implications. That is, those defined through the expressions $I(x,y) = S(N(x),T(x,y))$ and $I(x,y) = S(T(N(x),N(y)),y)$ respectively, where $T$ is a t-norm, $S$ is a t-conorm and $N$ is a strong negation. Special attention is due to the relation between both kinds of implications. In the continuous case, the study of these implications is focused in some of their properties (mainly the contrapositive symmetry and the exchange principle). Finally, the case of non continuous t-norms or non continuous t-conorms is studied, deriving new implications of both kinds and showing that they remain strongly connected.},
author = {Mas, Margarita, Monserrat, Miquel, Torrens, Joan},
journal = {Kybernetika},
keywords = {t-norm; T-conorm; implication operator; QL-implication; D-implication; t-norm; t-conorm; implication operator; QL-implication; D-implication},
language = {eng},
number = {3},
pages = {351-366},
publisher = {Institute of Information Theory and Automation AS CR},
title = {QL-implications versus D-implications},
url = {http://eudml.org/doc/33810},
volume = {42},
year = {2006},
}

TY - JOUR
AU - Mas, Margarita
AU - Monserrat, Miquel
AU - Torrens, Joan
TI - QL-implications versus D-implications
JO - Kybernetika
PY - 2006
PB - Institute of Information Theory and Automation AS CR
VL - 42
IS - 3
SP - 351
EP - 366
AB - This paper deals with two kinds of fuzzy implications: QL and Dishkant implications. That is, those defined through the expressions $I(x,y) = S(N(x),T(x,y))$ and $I(x,y) = S(T(N(x),N(y)),y)$ respectively, where $T$ is a t-norm, $S$ is a t-conorm and $N$ is a strong negation. Special attention is due to the relation between both kinds of implications. In the continuous case, the study of these implications is focused in some of their properties (mainly the contrapositive symmetry and the exchange principle). Finally, the case of non continuous t-norms or non continuous t-conorms is studied, deriving new implications of both kinds and showing that they remain strongly connected.
LA - eng
KW - t-norm; T-conorm; implication operator; QL-implication; D-implication; t-norm; t-conorm; implication operator; QL-implication; D-implication
UR - http://eudml.org/doc/33810
ER -

References

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  9. Mas M., Monserrat, M., Torrens J., QL-Implications on a finite chain, In: Proc. Eusflat-2003, Zittau 2003, pp. 281–284 
  10. Mas M., Monserrat, M., Torrens J., 10.1016/j.ijar.2005.05.001, Internat. J. Approx. Reason. 40 (2005), 262–279 Zbl1084.03021MR2193766DOI10.1016/j.ijar.2005.05.001
  11. Nachtegael M., Kerre E., Classical and fuzzy approaches towards mathematical morphology, In: Fuzzy Techniques in Image Processing (E. Kerre and M. Nachtegael, eds., Studies in Fuzziness and Soft Computing, Vol. 52), Physica–Verlag, Heidelberg 2000, pp. 3–57 
  12. Pei D., R 0 implication: characteristics and applications, Fuzzy Sets and Systems 131 (2002), 297–302 Zbl1015.03034MR1939842
  13. Trillas E., Campo, C. del, Cubillo S., 10.1002/(SICI)1098-111X(200007)15:7<647::AID-INT5>3.0.CO;2-T, Internat. J. Intelligent Systems 15 (2000), 647–655 Zbl0953.03031DOI10.1002/(SICI)1098-111X(200007)15:7<647::AID-INT5>3.0.CO;2-T
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