Abstract
The category theoretic approach of Obtułowicz to Pawlak’s rough sets has been reintroduced in a somewhat modified form. A generalization is rendered to this approach that has been motivated by the notion of rough membership function. Thus, a link is established between rough sets and L-fuzzy sets for some special lattices. It is shown that a notion of indistinguishability lies at the root of vagueness. This observation in turn gives a common ground to the theories of rough sets and fuzzy sets.
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References
Banerjee, M., Chakraborty, M.K.: Foundations of vagueness: a category-theoretic approach. Electronic Notes in Theoretical Computer Science 82(4) (2003)
Banerjee, M., Chakraborty, M.K.: Algebras from rough sets. In: Pal, S.K., Polkowski, L., Skowron, A. (eds.) Rough-Neural Computing, pp. 157–184. Springer, Heidelberg (2004)
Chakraborty, M.K.: Pawlak’s landscaping with rough sets. Transactions on Rough Sets VI, 51–63 (2007)
Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. International Journal of General Systems 17, 191–209 (1990)
Dubois, D., Prade, H.: Putting rough sets and fuzzy sets together. In: Slowiniski, R. (ed.) Intelligent Decision Support, pp. 203–232. Kluwer, Dordrecht (1992)
Dubois, D., Prade, H.: The three semantics of fuzzy sets. Fuzzy Sets and Systems 90(2), 141–150 (1997)
Eytan, M.: Fuzzy sets: a topos-logical point of view. Fuzzy Sets and Systems 5, 47–67 (1981)
Goguen, J.: L-fuzzy sets. Journal of Math. Anal. Appl. 18, 145–174 (1967)
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Acadenic Publishers, Dordrecht (1998)
Higgs, D.: A category approach to Boolean-valued set theory. Preprint, University of Waterloo (1973)
Höhle, U.: Monoidal closed categories, weak topoi and generalized logics. Fuzzy Sets and Systems 42, 15–35 (1991)
Höhle, U., Stout, L.N.: Foundations of fuzzy sets. Fuzzy Sets and Systems 40, 257–296 (1991)
Klawonn, F.: Fuzzy sets and vague environments. Fuzzy Sets and Systems 66, 207–221 (1994)
Klawonn, F., Kruse, R.: Equality relations as a basis for fuzzy control. Fuzzy Sets and Systems 54, 143–156 (1993)
Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, India (2006)
Menger, K.: Probabilistic theory of relations. Nat. Acad. Sci. U.S.A. 37, 178–180 (1951)
Obtułowicz, A.: Rough sets and Heyting algebra valued sets. Bulletin of the Polish Academy of Sciences (Mathematics) 35(9-10), 667–671 (1987)
Parikh, R.: The problem of vague predicates. In: Cohen, R.S., Wartofeny, M.W. (eds.) Language, Logic and Method, pp. 241–261. D. Reidel Publishing Co., Dordrecht (1983)
Pawlak, Z.: Rough sets. International Journal of Computer and Information Sciences 11, 341–356 (1982)
Pawlak, Z.: Rough sets and fuzzy sets. Fuzzy Sets and Systems 17, 99–102 (1985)
Pawlak, Z.: Vagueness and uncertainty: a rough set perspective. Computational Intelligence: An International Journal 11, 217–232 (1995)
Pawlak, Z.: Vagueness - a rough set view. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds.) Structures in Logic and Computer Science. LNCS, vol. 1261, pp. 106–117. Springer, Heidelberg (1997)
Pawlak, Z., Skowron, A.: Rough membership functions. In: Yager, R., Fedrizzi, M., Kacprzyk, J. (eds.) Advances in Dempster-Shafer Theory of Evidences, pp. 251–271. Wiley, New York (1994)
Pultr, A.: Fuzziness and fuzzy equality. In: Skala, H.J., Termini, S., Trillas, E. (eds.) Aspects of Vagueness, pp. 119–135. D. Reidel Publishing Co., Dordrecht (1984)
Radzikowska, A.M., Kerre, E.E.: A comparative study of fuzzy rough sets. Fuzzy Sets and Systems 126, 137–155 (2002)
Ruspini, E.: Recent developments in fuzzy clustering. In: Yager, R.R. (ed.) Fuzzy Set and Probability Theory: Recent Developments, pp. 133–147. Pergamon, Oxford (1982)
Shapiro, S.: Vagueness in Context. Clarendon Press, Oxford (2006)
Valverde, L.: On the structure of F-indistinguishability operators. Fuzzy Sets and Systems 17(3), 313–328 (1985)
Vitoria, A., Maluszynski, J., Szalas, A.: Modeling and reasoning with paraconsistent rough sets. Fundamentae Informaticae 97, 405–438 (2009)
Wygralak, M.: Some remarks on rough and fuzzy sets. BUSEFAL 21, 43–49 (1985)
Yao, Y.Y.: A comparative study of fuzzy sets and rough sets. Information Sciences 109, 227–242 (1998)
Zadeh, L.A.: Fuzzy sets. Information and Control 18, 338–353 (1965)
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Chakraborty, M.K. (2011). On Fuzzy Sets and Rough Sets from the Perspective of Indiscernibility. In: Banerjee, M., Seth, A. (eds) Logic and Its Applications. ICLA 2011. Lecture Notes in Computer Science(), vol 6521. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18026-2_4
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