Document Type: Research Paper


1 KERMIT, Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure links 653, B-9000, Gent, Belgium

2 Department of Mathematics, Faculty of Mathematics and Informatics, Med Boudiaf University of Msila, P.O. Box 166 Ichbilia, Msila 28000, Algeria


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 diff erent types of pairs of clones. More specifi 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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