Gödel Form of Fuzzy Transitive Relations
DOI:
https://doi.org/10.5644/SJM.08.1.01Keywords:
Gödel fuzzy implicator, fuzzy transitivity, measure of fuzzy transitivityAbstract
The concepts of fuzzy transitivity of a fuzzy relation on a given universe and the measure of fuzzy transitivity are studied with the use of Gödel fuzzy implicator.
2000 Mathematics Subject Classification. 03E72, 46S40, 68T37
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References
W. Bandler and L. Kohout, Semantics of implicators and fuzzy relational products, Int J. Man-Mach. Stud., 12 (1980), 89–1161.
W. Bandler and L. J. Kohout, Fuzzy power sets and fuzzy implication operators, Fuzzy Sets Syst., 4 (1980), 13–30.
I. Beg and S. Ashraf, Fuzzy equivalence relations, Kuwait J. Sci. Eng., 35 (1A) (2008), 191–206.
I. Beg and S. Ashraf, Fuzzy similarity and measure of similarity with Lukasiewicz implicator, New Math. Nat. Comput., 4 (2) (2008), 191–206.
I. Beg and S. Ashraf, Fuzzy inclusion and fuzzy similarity with G¨odel implication operator, New Math. Nat. Comput., 5 (3) (2009) 617–633.
I. Beg and S. Ashraf, Similarity measures for fuzzy sets, Appl. Comput. Math., 8 (2) (2009), 192–202.
I. Beg and S. Ashraf, Fuzzy transitive relations, Fuzzy Syst. Math., 24 (4) (2010) 162–169.
I. Beg and S. Ashraf, Fuzzy set of inclusion under Kleene’s implicator, Appl. Comput. Math., 10 (1) (2011), 65–77.
J. C. Bezdek and J. O. Harris, Fuzzy partitions and relations. An axiomatic basis for clustering, Fuzzy Sets Syst., 1 (1978), 112–127.
M. De Cock and E. E. Kerre, On (un)suitable fuzzy relations to model approximate equality, Fuzzy Sets Syst., 133 (2) (2003), 137–153.
J. C. Fodor and R. Yager, Fuzzy Set-theoretic Operators and Quantifiers, In Fundamentals of Fuzzy Sets, Dubois D. and Prade H., Eds., The Handbook of Fuzzy Sets Series, Kluwer Academic Publ., Dordrecht, (2000), 125–193.
L. Garmendia, The Evolution of the Concept of Fuzzy Measure, In Intelligent Data Mining, Studies in Computational Intelligence series, Springer Berlin / Heidelberg, (2005), 185–200.
S. Gottwald, Fuzzified fuzzy relations, In R. Lowen and M. Roubens, editors, Proceedings of the fourth IFSA. Congress, volume Mathematics (editor P. Wuyts), Brussels (1991), 82–86.
S. Gottwald, Fuzzy Sets and Fuzzy Logic: Foundations of Application-from a Mathematical Point of View, (1993), Vieweg, Wiesbaden.
J. Jacas and J. Recasens, Fuzzified properties of fuzzy relations, In the Proceedings of the 9th IPMU Conference, Annecy, 1 (2002), 157–161.
E. E. Kerre, Introduction to the Basic Principles of Fuzzy Set, Theory and some of its Applications, (1993), Communication and Cognition.
F. Klawonn, The role of similarity in the fuzzy reasoning, Fuzzy sets, logic and reasoning about knowledge, Appl. Log. Ser., 15, Kluwer Acad. Pub., Dordrecht, (1999), 243–253.
F. Klawonn, Should fuzzy equality and similarity satisfy transitivity? Comments on the paper by M. De Cock and E. Kerre, Fuzzy Sets Syst., 133 (2) (2003), 175–180.
K. Menger, Probabilistic theories of relations, Proceedings of the National Academy of Sciences, 37 (1951), 178–180.
H. Poincar´e, La Science et 1’Hypothese, Flammarion, Paris, (1902).
Ph. Smets and P. Magrez, Implication in fuzzy logic, Int. J. Approx. Reasoning, 1 (4) (1987), 327–347.
E. Trillas and L.Valverde, An inquiry into indistinguishability operators, in Aspects of Vagueness, H. J. Skala, S. Termini y E. Trillas (Eds.), Reidel Pubs. (1984), 231–256.
L. Valverde, On the structure of F-indistinguishability operators, Fuzzy Sets Syst., 17 (1985), 313–328.
R. Willmott, On the transitivity of containment and equivalence in fuzzy power set theory, J. Math. Anal. Appl., 120 (1986), 384–396.
L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (3) (1965), 338–353.
L. A. Zadeh, Similarity relations and fuzzy orderings, Inf. Sci., 3 (1971), 177–200.
