A characterization of triangular norm based tribes. (English) Zbl 0799.28013

Navara, Mirko

Tatra Mt. Math. Publ. 3, 161-166 (1993).

Summary: We study collections of fuzzy sets which are closed under complementation and under countable products (interpreted as “fuzzy intersections”). We characterize them as certain spaces of functions measurable with respect to the \(\sigma\)-algebra of crisp elements. This solves the problem stated by R. Mesiar [J. Math. Anal. Appl. 177, No. 2, 633-640 (1993) and Tatra Mt. Math. Publ. 1, 105-123 (1992; Zbl 0790.60005)] and enables generalizations of results of D. Butnariu and E. P. Klement [J. Math. Anal. Appl. 162, No. 1, 111-143 (1991; Zbl 0751.60003)], E. P. Klement [J. Math. Anal. Appl. 85, 543-565 (1982; Zbl 0491.28003)], E. P. Klement, W. Schwyhla and R. Lowen [Fuzzy Sets Syst. 5, 21-30 (1981; Zbl 0447.28005)] and R. Mesiar (loc. cit.).

MSC:

28E10	Fuzzy measure theory
46S10	Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
28A20	Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence

Keywords:

fuzzy \(\sigma\)-algebra; generated tribe; \(t\)-norm; \(T\)-norm; measurable function; triangular norm based tribes; fuzzy intersections; fuzzy sets

Citations:

Zbl 0790.60005; Zbl 0751.60003; Zbl 0491.28003; Zbl 0447.28005