Type-2 fuzzy variables and their arithmetic

This paper proposes an axiomatic framework from which we develop the theory of type-2 (T2) fuzziness, called fuzzy possibility theory. First, we introduce the concept of a fuzzy possibility measure in a fuzzy possibility space (FPS). The fuzzy possibility measure takes on regular fuzzy variable (RFV) values, so it generalizes the scalar possibility measure in the literature. One of the interesting consequences of the FPS is that it leads to a new definition of T2 fuzzy set on the Euclidean space $$\Re^m,$$ which we call T2 fuzzy vector, as a map to the space instead of on the space. More precisely, we define a T2 fuzzy vector as a measurable map from an FPS to the space $$\Re^m$$ of real vectors. In the current development, we are suggesting that T2 fuzzy vector is a more appropriate definition for a T2 fuzzy set on $$\Re^m.$$ In the literature, a T2 fuzzy set is usually defined via its T2 membership function, whereas in this paper, we obtain the T2 possibility distribution function as the transformation of a fuzzy possibility measure from a universe to the space $$\Re^m$$ via T2 fuzzy vector. Second, we develop the product fuzzy possibility theory. In this part, we give a general extension theorem about product fuzzy possibility measure from a class of measurable atom-rectangles to a product ample field, and discuss the relationship between a T2 fuzzy vector and T2 fuzzy variables. We also prove two useful theorems about the existence of an FPS and a T2 fuzzy vector based on the information from a finite number of RFV-valued maps. The two results provide the possible interpretations for the concepts of the FPS and the T2 fuzzy vector, and thus reinforce the credibility of the approach developed in this paper. Finally, we deal with the arithmetic of T2 fuzzy variables in fuzzy possibility theory. We divide our discussion into two cases according to whether T2 fuzzy variables are defined on single FPS or on different FPSs, and obtain two theorems about T2 fuzzy arithmetic.