On the compatibility of a crisp relation with a fuzzy equivalence relation

In a recent paper, De Baets et al. have characterized the fuzzy tolerance and fuzzy equivalence relations that a given strict order relation is compatible with. In this paper, we generalize this characterization by considering an arbitrary (crisp) relation instead of a strict order relation, while paying attention to the particular cases of a reflexive or irreflexive relation. The reasoning largely draws upon the notion of the clone relation of a (crisp) relation, introduced recently by Bouremel et al., and the partition of this clone relation in terms of three di ff erent types of pairs of clones. More speci fi cally, reflexive related clones and irreflexive unrelated clones turn out to play a key role in the characterization of the fuzzy tolerance and fuzzy equivalence relations that a given (crisp) relation is compatible with.

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