Fundamental triangular norm based tribes and measures. (English) Zbl 0816.28014

Mesiar, Radko

J. Math. Anal. Appl. 177, No. 2, 633-640 (1993).

Let \(\{T_ s: 0< s< \infty\}\) be the family of Frank’s \(t\)-norms. The author shows that for every \(T_ s\)-tribe \(\tau\) [as defined by D. Butnariu and E. P. Klement, J. Math. Anal. Appl. 162, No. 1, 111- 143 (1991; Zbl 0751.60003)] on a countable set \(X\) there exists a partition \((Y, Z)\) of \(X\) such that the set \(\tau| Z\) (of all restrictions of members of \(\tau\) to \(Z\)) is a \(\sigma\)-algebra of subsets of \(Z\) and \(\tau| Y\) consists of all measurable fuzzy sets on \(Y\) with respect to some \(\sigma\)-algebra of subsets of \(Y\). This result then leads to an integral representation of finite monotone \(T_ s\)- measures in the sense of Butnariu and Klement [op. cit.] on a countable set.

Reviewer: T.Kubiak (Poznań)

MSC:

28E10	Fuzzy measure theory
03E72	Theory of fuzzy sets, etc.

Keywords:

Frank’s \(t\)-norms; \(T_ s\)-tribe; measurable fuzzy sets; \(T_ s\)- measures

Citations:

Zbl 0751.60003