On the transitivity of containment and equivalence in fuzzy power set theory
For fuzzy sets it is natural to consider a fuzzified inclusion relation as done e.g. by \textit{S. Gottwald} [Fuzzy Sets Syst. 2, 125--151 (1979; Zbl 0408.03042)] and \textit{W. Bandler} and \textit{L. Kohout} [ibid. 4, 13--30 (1980; Zbl 0433.03013)] and also a fuzzified equality. In each case a many-valued implication operator is involved. For both types of relations and a series of implication operators of Bandler and Kohout [loc. cit.] a generalized transitivity property is proven which in a more general setting also is given by \textit{S. Gottwald} [Elektron. Informationsverarb. Kybernetik 22, 67--91 (1986; Zbl 0607.03016)] for at least each continuous t-norm. Additionally, other fuzzified inclusion relations are defined by some mean value procedures. For such inclusion relations it is proven that the generalized transitivity not always holds true.
scientific article; zbMATH DE number 3985204