Negations on type-2 fuzzy sets

So far, the negation that usually has been considered within the type-2 fuzzy sets (T2FSs) framework, and hence T2FS truth values M (set of all functions from [0,1][0,1] to [0,1][0,1]), was obtained by means of Zadeh's extension principle and calculated from standard negation in [0,1][0,1]. But there has been no comparative analysis of the properties that hold for the above operation and the axioms that any negation in M should satisfy. This suggests that negations should be studied more thoroughly in this context. Following on from this, we introduce in this paper the axioms that an operation in M must satisfy to qualify as a negation and then prove that the usual negation on T2FSs, in particular, is antimonotonic in L (set of normal and convex functions of M) but not in M. We propose a family of operations calculated from any suprajective negation in [0,1][0,1] and prove that they are negations in L. Finally, we examine De Morgan's laws for some operations with respect to these negations.