Klement, Erich Peter; Navara, Mirko A characterization of tribes with respect to the Łukasiewicz \(t\)-norm. (English) Zbl 0902.28015 Czech. Math. J. 47, No. 4, 689-700 (1997). Authors’ summary: “We give a characterization of tribes with respect to the Łukasiewicz \(t\)-norm, i.e., of systems of fuzzy sets which are closed with respect to the complement of fuzzy sets and with respect to countably many applications of the Łukasiewicz \(t\)-norm. We also characterize all operations with respect to which all such tribes are closed. This generalizes the characterizations obtained so far for other fundamental \(t\)-norms, e.g., for the product \(t\)-norm”. Reviewer: A.Kufner (Praha) Cited in 6 Documents MSC: 28E10 Fuzzy measure theory 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis 03E72 Theory of fuzzy sets, etc. Keywords:fuzzy sets; crisp set; tribe; triangular norm PDFBibTeX XMLCite \textit{E. P. Klement} and \textit{M. Navara}, Czech. Math. J. 47, No. 4, 689--700 (1997; Zbl 0902.28015) Full Text: DOI EuDML References: [1] Butnariu, D., Klement, E.P.: Triangular norm-based measures and games with fuzzy coalitions. Kluwer, Dordrecht, 1993. · Zbl 0804.90145 [2] Frank, M.J.: On the simultaneous associativity of \(F(x,y)\) and \(x+y-F(x,y)\). Aequationes Math. 19 (1979), 194-226. · Zbl 0444.39003 · doi:10.1007/BF02189866 [3] Klement, E.P.: Construction of fuzzy \(\sigma \)-algebras using triangular norms. J. Math. Anal Appl. 85 (1982), 543-565. · Zbl 0491.28003 · doi:10.1016/0022-247X(82)90015-4 [4] Mesiar, R.: Fundamental triangular norm based tribes and measures. J. Math. Anal. Appl. 177 (1993), 633-640. · Zbl 0816.28014 · doi:10.1006/jmaa.1993.1283 [5] Mesiar, R.: On the structure of \(T_s\)-tribes. Tatra Mountains Math. Publ. 3 (1993), 167-172. · Zbl 0804.28010 [6] Mesiar, R., Navara, M.: \(T_s\)-tribes and \(T_s\)-measures. J. Math. Anal. Appl · Zbl 0852.28009 · doi:10.1006/jmaa.1996.0243 [7] Navara, M.: A characterization of triangular norm based tribes. Tatra Mountains Math Publ. 3 (1993), 161-166. · Zbl 0799.28013 [8] Pykacz, J.: Fuzzy set ideas in quantum logics. Int. J. Theor. Physics 31 (1992), 1765-1781. · Zbl 0789.03049 · doi:10.1007/BF00671785 [9] Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, New York, 1983. · Zbl 0546.60010 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.