On the distributivity of fuzzy implications over representable uninorms
The classical tautology \((p\land q)\to r\equiv (p\to r) \lor (q\to r)\) rewritten in the language of fuzzy connectives can be understand as a functional equation \[ I(U_1(x,y),z)= U_2(I(x,z), I(y,z)), \] where \(U_1\), \(U_2\) are representable uninorms and \(I\) is a fuzzy implication. The author shows that this functional equation has no solution. On the other hand, the functional equation \[ I(x,U_1(y,z))= U_2(I(x,y), I(x,z)) \] (generalizing the tautologies \(x\to (y\land z)\equiv (x\to y)\land(x\to z)\) and \(x\to (y\lor z)\equiv (x\to y)\lor (x\to z)\)) admits several solutions. The author shows that then the fuzzy implication \(I\) cannot be continuous, and he characterizes the solutions where the vertical sections of \(I\) are continuous. As a by-product, some new results concerning the Cauchy functional equation \(f(x+ y)= f(x)+ f(y)\) on the extended real line \([-\infty,+\infty]\) are shown.
scientific article; zbMATH DE number 5780076