Galois connections among fuzzy groups, similarities, distances.
In a previous paper [\textit{F. Formato}, \textit{G. Gerla} and \textit{L. Scarpati}, Fuzzy subgroups and similarities, Soft Comput. 3, No. 1, 1-6 (1999)], the authors have shown that a fuzzy subgroup (in the original sense of \textit{A. Rosenfeld} [J. Math. Anal. Appl. 35, 512-517 (1971; Zbl 0194.05501)]) of transformations in a given universe \(X\) can be associated with a similarity relation on \(X\) (i.e., a reflexive, symmetric and sup-min transitive fuzzy relation on \(X)\) and vice versa. In this paper it is shown that the previous association can be seen as a Galois connection between the complete lattice of fuzzy subgroups of transformations in a universe on one hand and the complete lattice of similarity relations on this universe. This result turns out to be a fuzzified version of the crisp Galois correspondence between the equivalence relations on a set and the subgroups of transformations in this set. The fuzzy generalisation is based on the extension principle proposed by \textit{G. Gerla} and \textit{L. Scarpati} [Inf. Sci. 106, No. 1-2, 49-69 (1998; Zbl 0929.03055)]. It is a pity that the authors did not link their notion of a *-similarity to the notion of a \((g,d)\)-resemblance relation introduced by \textit{M. De Cock} and \textit{E. Kerre} [Fuzzy Sets Syst. 133, No. 2, 137-153 (2003; Zbl 1020.03049)].
scientific article; zbMATH DE number 2081780
scientific article