The algebra of fuzzy truth values

Abstract

The purpose of this paper is to give a straightforward mathematical treatment of algebras of fuzzy truth values for type-2 fuzzy sets.

Examples of Aggregation Operators on Membership Degrees of Type-2 Fuzzy Sets

“…Walker and Walker justify in [19] that the operations on Map(X, M) can be defined naturally from the operations on M and have the same properties. In fact, given the operation * :…”

“…, for all x, where f (x), g(x) ∈ M (see [19], [10]). Therefore, in this paper, we will work on M, as all the results are easily and directly extensible to Map(X, M).…”

“…Definition 6. ( [7], [19], [4], [5]) The operations of ⊔ (generalized maximum), ⊓ (generalized minimum), ¬ and the elements0 y1 are defined on M as follows:…”

“…Note that0 and1 are precisely the characteristic functions of 0 and 1, respectively. We can easily prove that ⊔ and ⊓ satisfy De Morgan's laws with respect to the given operation ¬, but M = (M, ⊔, ⊓, ¬,0,1) does not have a lattice structure, as it does not comply with the absorption law [7], [19]. On the other hand, the operations ⊔ and ⊓ have the properties required to each define a partial order on M.…”

“…Whereas for FSs the membership degree of an element of a set is determined by a value in the interval [0, 1], the membership degree of an element for T2FSs is a fuzzy set in [0,1], that is, a T2FS is determined by a membership function µ : X → M, where M = [0, 1] [0, 1] is the set of functions from [0,1] to [0,1] (see [11], [14], [15], [19]). In this paper we will get results for T2FSs with membership degrees in M = [0, 1] [0,1] (set of functions from [0,1] to [0,1]) and also in the subset L of normal and convex functions of M. Because the membership degree of T2FSs is fuzzy, they are better able to model uncertainty than FSs [12].…”