ON SOME PROPERTIES OF alpha-PLANES OF TYPE-2 FUZZY SETS
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.
alpha-plane type-2 fuzzy sets fuzzy sets intersection of type-2 fuzzy sets union of type-2 fuzzy sets