Characterization of measures based on strict triangular norms

As a natural generalization of a measure space, Butnariu and Klement introduced T-tribes of fuzzy sets with T-measures. They gave a complete characterization of T-measures for a Frank triangular norm T. Here we characterize T-measures with respect to non-Frank strict triangular norms. We show the speciic roles of Frank triangular norms and a newly introduced family, nearly Frank triangular norms. 1 The notion of T-measure We start with the basic deenitions from 4]. Let X be a set and B a-algebra of subsets of X. The B-generated tribe is the collection T of all functions A: X ! 0; 1] (fuzzy subsets of X) which are B-measurable. In order to deene measures on T , we x a t-norm T (fuzzy conjunction), i.e., a binary operation T: 0; 1] 2 ! 0; 1] which is commutative, associative, nondecreasing, and satisses the boundary condition T(a; 1) = a for all a 2 0; 1] (see 15]). For the other necessary fuzzy logical operations, we take the standard fuzzy negation 0 : 0; 1] ! 0; 1] deened by a 0 := 1 ? a, and the t-conorm S: 0; 1] 2 ! 0; 1] dual to T, i.e., S(a; b) := T(a 0 ; b 0) 0. We extend the operations T, 0 , S to operations T, c , S (fuzzy intersection, fuzzy complement and fuzzy union) on T pointwise: