On some properties of α -planes of type-2 fuzzy sets

Zdenko Takáč

Kybernetika (2013)

  • Volume: 49, Issue: 1, page 149-163
  • ISSN: 0023-5954

Abstract

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Some basic properties of α -planes of type-2 fuzzy sets are investigated and discussed in connection with the similar properties of α -cuts of type-1 fuzzy sets. It is known, that standard intersection and standard union of type-1 fuzzy sets (it means intersection and union under minimum t-norm and maximum t-conorm, respectively) are the only cutworthy operations for type-1 fuzzy sets. Recently, a similar property was declared to be true also for α -planes of type-2 fuzzy sets in a few papers. Thus, we study under which t-norms and which t-conorms are intersection and union of the type-2 fuzzy sets preserved in the α -planes. Note that understanding of the term α -plane is somewhat confusing in recent type-2 fuzzy sets literature. We discuss this problem and show how it relates to obtained results.

How to cite

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Takáč, Zdenko. "On some properties of $\alpha $-planes of type-2 fuzzy sets." Kybernetika 49.1 (2013): 149-163. <http://eudml.org/doc/252501>.

@article{Takáč2013,
abstract = {Some basic properties of $\alpha $-planes of type-2 fuzzy sets are investigated and discussed in connection with the similar properties of $\alpha $-cuts of type-1 fuzzy sets. It is known, that standard intersection and standard union of type-1 fuzzy sets (it means intersection and union under minimum t-norm and maximum t-conorm, respectively) are the only cutworthy operations for type-1 fuzzy sets. Recently, a similar property was declared to be true also for $\alpha $-planes of type-2 fuzzy sets in a few papers. Thus, we study under which t-norms and which t-conorms are intersection and union of the type-2 fuzzy sets preserved in the $\alpha $-planes. Note that understanding of the term $\alpha $-plane is somewhat confusing in recent type-2 fuzzy sets literature. We discuss this problem and show how it relates to obtained results.},
author = {Takáč, Zdenko},
journal = {Kybernetika},
keywords = {type-2 fuzzy sets; $\alpha $-plane; intersection of type-2 fuzzy sets; union of type-2 fuzzy sets; fuzzy sets; type-2 fuzzy sets; -plane; intersection of type-2 fuzzy sets; union of type-2 fuzzy sets},
language = {eng},
number = {1},
pages = {149-163},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On some properties of $\alpha $-planes of type-2 fuzzy sets},
url = {http://eudml.org/doc/252501},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Takáč, Zdenko
TI - On some properties of $\alpha $-planes of type-2 fuzzy sets
JO - Kybernetika
PY - 2013
PB - Institute of Information Theory and Automation AS CR
VL - 49
IS - 1
SP - 149
EP - 163
AB - Some basic properties of $\alpha $-planes of type-2 fuzzy sets are investigated and discussed in connection with the similar properties of $\alpha $-cuts of type-1 fuzzy sets. It is known, that standard intersection and standard union of type-1 fuzzy sets (it means intersection and union under minimum t-norm and maximum t-conorm, respectively) are the only cutworthy operations for type-1 fuzzy sets. Recently, a similar property was declared to be true also for $\alpha $-planes of type-2 fuzzy sets in a few papers. Thus, we study under which t-norms and which t-conorms are intersection and union of the type-2 fuzzy sets preserved in the $\alpha $-planes. Note that understanding of the term $\alpha $-plane is somewhat confusing in recent type-2 fuzzy sets literature. We discuss this problem and show how it relates to obtained results.
LA - eng
KW - type-2 fuzzy sets; $\alpha $-plane; intersection of type-2 fuzzy sets; union of type-2 fuzzy sets; fuzzy sets; type-2 fuzzy sets; -plane; intersection of type-2 fuzzy sets; union of type-2 fuzzy sets
UR - http://eudml.org/doc/252501
ER -

References

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  10. Starczewski, J., Extended triangular norms on gaussian fuzzy sets., In: Proc. EUSFLAT-LFA 2005 (E. Montseny and P. Sobrevilla, eds.), 2005, pp. 872-877. 
  11. Takáč, Z., Intersection and union of type-2 fuzzy sets and connection to ( α 1 , α 2 )-double cuts., In: Proc. EUSFLAT-LFA 2011 (S. Galichet, J. Montero and G. Mauris, eds.), 2011, pp. 1052-1059. Zbl1254.03108
  12. Wagner, C., Hagras, H., zSlices-towards bridging the gap between interval and general type-2 fuzzy logic., In: Proc. IEEE FUZZ Conf, Hong Kong 2008, pp. 489-497. 
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