A new construction for t-norms and their application to an open problem of Alsina, Frank and Schweizer

Abstract

In this contribution we will investigate triangular norms T1,T2,T3 such that T1<T2<T3 and T3(x,y)−T2(x,y)=T2(x,1−y)−T1(x,1−y). This is an open problem that was published in the monograph by Alsina, Frank and Schweizer. A partial solution to this problem in the case when T2 is the product t-norm, was published already in the mentioned monograph. Further progress in solving the problem for Frank t-norms was done by Havlena at a students' conference. In this contribution, we aim to present another solution to this problem that, in a sense, generalizes the well-known ordinal sum construction for t-norms.