ON A FUNCTIONAL EQUATION CONNECTED TO THE DISTRIBUTIVITY OF FUZZY IMPLICATIONS OVER TRIANGULAR NORMS AND CONORMS

Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see [9, 15] and [4]). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see [5]) f (min(x + y, a)) = min(f (x) f (y), b), where a, b > 0 and f: [0, a] -> [0, b] is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation f (m(1)(x + y)) = m(2)(f(x) f(y)), where m(1), m(2) are functions defined on some intervals of R satisfying additional assumptions. We analyze the cases when m(2) is injective and when m(2) is not injective.