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

in a recent paper, de baets et al. have characterized the fuzzytolerance and fuzzy equivalence relations that a given strict order relation iscompatible with. in this paper, we generalize this characterization by consideringan arbitrary (crisp) relation instead of a strict order relation, while payingattention to the particular cases of a reflexive or irreflexive relation. the reasoninglargely draws upon the notion of the clone relation of a (crisp) relation,introduced recently by bouremel et al., and the partition of this clone relationin terms of three different types of pairs of clones. more specifically, reflexive related clones and irreflexive unrelated clones turn out to play a key role in thecharacterization of the fuzzy tolerance and fuzzy equivalence relations that agiven (crisp) relation is compatible with.