Fuzzy groups, fuzzy functions and fuzzy equivalence relations.
The paper applies the notion of fuzzy functions with respect to fuzzy equalities introduced by the first author [cf. Fuzzy Sets Syst. 106, No. 2, 239-246 (1999; Zbl 0934.03067); J. Math. Anal. Appl. 252, No. 1, 495-517 (2000; Zbl 0973.03071); Int. J. Gen. Syst. 32, No. 2, 123-155 (2003; Zbl 1028.03044); ibid. 157-175 (2003; Zbl 1028.03045); ibid. 177-201 (2003; Zbl 1028.03046)]. The main goal of the paper is a comparison between properties of fuzzy groups [cf., e.g., \textit{N. Kuroki}, Inf. Sci. 53, No. 3, 203-236 (1991; Zbl 0714.20052); ibid. 75, No. 3, 201-211 (1993; Zbl 0808.20052)] and vague groups [cf. \textit{M. Demirci}, J. Math. Anal. Appl. 230, No. 1, 142-156 (1999; Zbl 0949.20050)] with applications to fuzzy subgroups and vague groups on the real line. Final results of the paper are very restrictive: reduce a fuzzy case to the crisp one. For example (Theorem 5.4), any admissible fuzzy equivalence relation on the real line reduces to a two-valued function (like the characteristic function of the identity relation). Similarly, fuzzy numbers considered as fuzzy groups reduce to fuzzy points (Corollary 5.5) or to triangular fuzzy numbers (Example 5.10).
scientific article; zbMATH DE number 2093611
scientific article