Geodesic
On some properties of $\alpha $-planes of type-2 fuzzy sets
Kybernetika, Tome 49 (2013) no. 1, pp. 149-163

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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.
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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     author = {Tak\'a\v{c}, Zdenko},
     title = {On some properties of $\alpha $-planes of type-2 fuzzy sets},
     journal = {Kybernetika},
     pages = {149--163},
     publisher = {mathdoc},
     volume = {49},
     number = {1},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2013__49_1_a10/}
}
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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/