A generalization of a contraction principle in probabilistic metric spaces. Part II

1.1. t-norms. A triangular norm (shortly t-norm) is a binary operationT : [0,1]×[0,1]→ [0,1] := I which is commutative, associative, monotone in each place, and has 1 as the unit element. Basic examples are TL : I × I → I , TL(a,b) =Max(a+ b− 1,0) (Łukasiewicz t-norm), TP(a,b) = ab, and TM(a,b) = Min{a,b}. We also mention the following families of tnorms: (i) Sugeno-Weber family (T λ )λ∈(−1,∞), defined by T SW λ =max(0,(x + y− 1 + λxy)/ (1+ λ)), (ii) Domby family (T λ )λ∈(0,∞), defined by T D λ = (1 + (((1 − x)/x)λ + ((1 − y)/ y)λ)1/λ)−1, (iii) Aczel-Alsina family (T λ )λ∈(0,∞), defined by T AA λ = e−(|logx|+|log y|λ)1/λ .