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\(\mathcal T\)-partitions of the real line generated by idempotent shapes. (English) Zbl 0919.04004
Summary: The idea of generating fuzzy numbers as equivalence classes of particular \({\mathcal T}\)-equivalences on the real line \(\mathbb{R}\) is fully exploited. Scales (or generators) are used to define certain (pseudo-)metrics on \(\mathbb{R}\). By means of a shape (function), these (pseudo-)metrics are then transformed into binary fuzzy relations on \(\mathbb{R}\). Shapes leading to \({\mathcal T}\)-equivalences, and hence to a class of fuzzy numbers forming a \({\mathcal T}\)-partition of \(\mathbb{R}\), are completely characterized in the case of a continuous generator. This characterization problem is shown to be closely related to determining the idempotents w.r.t. the \({\mathcal T}\)-addition of fuzzy numbers.

MSC:
03E72 Theory of fuzzy sets, etc.
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