A new proof of Nguyen's compatibility theorem in a more general context

Let f((R) over tilde, (S) over tilde) be the image of a pair of fuzzy subsets constructed by applying Zadeh's (1975) extension principle to a function of two variables. Nguyen (1978) gave a necessary and sufficient condition for the alpha-cuts of f((R) over tilde, (S>$) over bar over tilde>) to be equal to the crisp images of the alpha-cuts of (R) over tilde, (S) over tilde. Here we give a simplified proof of this theorem which also holds in a more general context: particularly for second-order fuzzy subsets. (C) 1998 Elsevier Science B.V.