The normal form of a granular fuzzy function

• Authors: Sukhamay Kundu
• Venue: Fuzzy Sets and Systems
• Year: 2001

We consider here a particular kind of fuzzy relations f(x, y) on X × Y such that the α-cut fα is an ordinary function for each 0 ≤ α ≤ 1 from the underlying α-level granular structure on X to that on Y, which are determined by a given pair of similarity relations µX(x,x') and µY(y, y') on X and Y, respectively. This type of fuzzy relations have a close connection with rough sets and functions between them, and are called here granular fuzzy functions. We show that each granular fuzzy function f(x, y) has a unique normal form f'(x, y) ≥ f(x, y) which preserves the values fα([x]α) for each 0 ≤α≤1, where [x]α is the equivalence class of x at the level α for µx. The normal form functions f'(x, y) are in many ways more well-behaved than the non-normal forms in terms of function composition, unique inverse function, etc.