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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:
 3e+72 Theory of fuzzy sets, etc.
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##### References:
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