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

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Abstract
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.
Keywords
LATTICES, SETS, REPRESENTATIONS, CONSTRUCTIONS, ORDERINGS, ALGEBRA, Crisp relation, Fuzzy relation, Clone relation, Compatibility, Tolerance, relation, Equivalence relation

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Citation

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MLA
De Baets, Bernard, Hassane Bouremel, and Lemnaouar Zedam. “On the Compatibility of a Crisp Relation with a Fuzzy Equivalence Relation.” IRANIAN JOURNAL OF FUZZY SYSTEMS 13.7 (2016): 15–31. Print.
APA
De Baets, B., Bouremel, H., & Zedam, L. (2016). On the compatibility of a crisp relation with a fuzzy equivalence relation. IRANIAN JOURNAL OF FUZZY SYSTEMS, 13(7), 15–31.
Chicago author-date
De Baets, Bernard, Hassane Bouremel, and Lemnaouar Zedam. 2016. “On the Compatibility of a Crisp Relation with a Fuzzy Equivalence Relation.” Iranian Journal of Fuzzy Systems 13 (7): 15–31.
Chicago author-date (all authors)
De Baets, Bernard, Hassane Bouremel, and Lemnaouar Zedam. 2016. “On the Compatibility of a Crisp Relation with a Fuzzy Equivalence Relation.” Iranian Journal of Fuzzy Systems 13 (7): 15–31.
Vancouver
1.
De Baets B, Bouremel H, Zedam L. On the compatibility of a crisp relation with a fuzzy equivalence relation. IRANIAN JOURNAL OF FUZZY SYSTEMS. 2016;13(7):15–31.
IEEE
[1]
B. De Baets, H. Bouremel, and L. Zedam, “On the compatibility of a crisp relation with a fuzzy equivalence relation,” IRANIAN JOURNAL OF FUZZY SYSTEMS, vol. 13, no. 7, pp. 15–31, 2016.
@article{8519521,
  abstract     = {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.},
  author       = {De Baets, Bernard and Bouremel, Hassane and Zedam, Lemnaouar},
  issn         = {1735-0654},
  journal      = {IRANIAN JOURNAL OF FUZZY SYSTEMS},
  keywords     = {LATTICES,SETS,REPRESENTATIONS,CONSTRUCTIONS,ORDERINGS,ALGEBRA,Crisp relation,Fuzzy relation,Clone relation,Compatibility,Tolerance,relation,Equivalence relation},
  language     = {eng},
  number       = {7},
  pages        = {15--31},
  title        = {On the compatibility of a crisp relation with a fuzzy equivalence relation},
  volume       = {13},
  year         = {2016},
}

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