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Lattice-type fuzzy order is uniquely given by its 1-cut: Proof and consequences. (English) Zbl 1044.06002

Summary: A 1-cut of a fuzzy relation (sometimes called a core) does not contain all the information that is represented by the fuzzy relation. Particularly, a fuzzy order \(\preccurlyeq\) on a universe \(X\) equipped with an fuzzy equality \(\approx\) is not uniquely determined by its 1-cut \({}^1\!\preccurlyeq\;=\{ \langle x,y\rangle | (x \preccurlyeq y)=1\}\). That is, there are in general several fuzzy orders with a common 1-cut. We show that, if the fuzzy order obeys in addition the lattice structure (which many natural examples of fuzzy orders do), it is uniquely determined by its 1-cut. Moreover, we discuss consequences of this result for the so-called fuzzy concept lattices and formal concept analysis.

MSC:

06A99 Ordered sets
06B99 Lattices
03E72 Theory of fuzzy sets, etc.
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