Some results of weighted quasi-arithmetic mean of continuous triangular norms
Let \(T_1\) and \(T_2\) be two continuous t-norms, \(f:[0,1]\to[-\infty,\infty]\) be strictly monotone. Then the function \(T(x,y)=f^{-1}(pf(T_1(x,y))+(1-p)f(T_2(x,y)))\), where \(p\in(0,1)\), is called the weighted quasi-arithmetic mean of \(T_1\) and \(T_2\) with respect to \(f\). This function satisfies all properties of t-norms except associativity. The authors investigate the associativity of the weighted quasi-arithmetic mean for several classes of continuous t-norms. Some results on convex combination of continuous t-norms are generalized. Several methods coming from functional equations are applied and various open problems are stated.
scientific article; zbMATH DE number 5368817
scientific article