Convex linear T–S functions: A generalization of Frankʹs equation

The main purpose of this paper is to solve the functional equation T + λ S = Min + λ Max , for 0 ≤ λ ≤ ∞ and a pair ( T , S ) of a t-norm T and a t-conorm S. This equation arises when we consider a convex linear combination of a t-norm and a t-conorm, and set out the problem of finding the intersection of the segments determined by the pairs ( Min , Max ) and ( T , S ) .