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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; $\alpha $-plane; intersection of type-2 fuzzy sets; union of type-2 fuzzy sets; fuzzy sets
Summary:
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.
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