Geodesic
On some properties of $\alpha $-planes of type-2 fuzzy sets
Kybernetika, Tome 49 (2013) no. 1, pp. 149-163 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 03E72, 68T37
Keywords: type-2 fuzzy sets; $\alpha $-plane; intersection of type-2 fuzzy sets; union of type-2 fuzzy sets; fuzzy sets 
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Takáč, Zdenko. On some properties of $\alpha $-planes of type-2 fuzzy sets. Kybernetika, Tome 49 (2013) no. 1, pp. 149-163. http://geodesic.mathdoc.fr/item/KYB_2013_49_1_a10/

[1] Karnik, N., Mendel, J.: Operations on type-2 fuzzy sets. Fuzzy Sets and Systems 122 (2001), 327-348. | DOI | MR | Zbl

[2] Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Application. Upper-Saddle River, Prentice-Hall, NJ 1995. | MR

[3] Kolesárová, A., Kováčová, A.: Fuzzy množiny a ich aplikácie. STU, Bratislava 2004.

[4] Liu, F.: An Efficient Centroid Type Reduction Strategy for General Type-2 Fuzzy Logic System. IEEE Comput. Intell. Soc., Walter J. Karplus Summer Research Grant Report 2006.

[5] Liu, F.: An efficient centroid type reduction strategy for general type-2 fuzzy logic system. Inform. Sci. 178 (2008), 2224-2236. | DOI | MR

[6] Mendel, J. M.: Comments on '$\alpha$-plane representation for type-2 fuzzy sets: Theory and applications'. IEEE Trans. Fuzzy Syst. 18 (2010), 229-230. | DOI

[7] Mendel, J. M., John, R. I.: Type-2 fuzzy sets made simple. IEEE Trans. Fuzzy Syst. 10 (2002), 117-127. | DOI

[8] Mendel, J. M., Liu, F., Zhai, D.: $\alpha$-plane representation for type-2 fuzzy sets: Theory and applications. IEEE Trans. Fuzzy Syst. 17 (2009), 1189-1207. | DOI

[9] Mizumoto, M., Tanaka, K.: Some properties of fuzzy sets of type 2. Inform. Control 31 (1976), 312-340. | DOI | MR | Zbl

[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 ($\alpha_1,\alpha_2$)-double cuts. In: Proc. EUSFLAT-LFA 2011 (S. Galichet, J. Montero and G. Mauris, eds.), 2011, pp. 1052-1059. | Zbl

[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.

[13] Zadeh, L. A.: The concept of a linguistic variable and its application to approximate reasoning - I. Inform. Sci. 8 (1975), 199-249. | DOI | MR | Zbl