Abstract
In this paper, we use one axiom to characterize a series of upper (lower) approximations of L-fuzzy rough sets based on fuzzy lattices. The most interesting result of this paper is the investigation of two operators associated with an abstract operator defined on the fuzzy powerset of a universal set. These two operators will paly an essential role in this study. This work can be regarded as a continuation and generalization of Liu’s work in 2013. In which, he characterized the upper approximations of rough sets and fuzzy rough sets by one axiom.
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References
Chen, D.G., Kwong, S., He, Q., et al.: Geometrical interpretation and applications of membership functions with fuzzy rough sets. Fuzzy Sets Syst. 193, 122–135 (2012)
Chen, X., Li, Q.: Construction of rough approximations in fuzzy setting. Fuzzy Sets Syst. 158, 2641–2653 (2007)
Comer, S.: An algebraic approach to the approximation of information. Fundam. Inform. 14, 492–502 (1991)
Dai, J.H., Tian, H.W.: Fuzzy rough set model for set-valued data. Fuzzy Sets Syst. 229, 54–68 (2013)
Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. Int. J. Gen. Syst. 17, 191–208 (1990)
Dubois, D., Prade, H.: Putting fuzzy sets and rough sets together. In: Slowinski, R. (ed.) Intelligent Decision Support, pp. 203–232. Kluwer Academic, Dordrecht (2004)
Guan, L., Wang, G.: Generalized approximations defined by non-equivalence relations. Inf. Sci. 193, 163–179 (2012)
Hao, J., Li, Q.G.: The relationship between \(L\)-fuzzy rough set and \(L\)-topology. Fuzzy Sets and Systems 178, 74–8 (2011)
Jin, Q., Li, L.Q.: On the second type of \(L\)-fuzzy covering rough sets. Inf. Int. Interdiscip. J. 16(2), 1101–1106 (2013)
Kelley, J.L.: General Topology. Graduate Texts in Mathematics, vol. 27, Springer (1955)
Li, Y., Tang, J., Chin, K., Luo, X., Han, Y.: Rough set-based approach for modeling relationship measures in product planning. Inf. Sci. 193, 199–217 (2012)
Lin, T.Y., Liu, Q.: Rough approximate operators: axiomatic rough set theory. In: Ziarko, W. (ed.) Rough Sets, Fuzzy Sets and Knowledge Discovery, pp. 256–260. Springer, Berlin (1994)
Liu, G.: Generalized rough set over fuzzy lattices. Inf. Sci. 178, 1651–1662 (2008)
Liu, G.: Axiomatic systems for rough sets and fuzzy rough sets. Int. J. Approx. Reason. 48, 857–867 (2008)
Liu, G., Sai, Y.: A comparison of two types of rough sets induced by coverings. Int. J. Approx. Reason. 50, 521–528 (2009)
Liu, G.: Using one axiom to characterize rough set and fuzzy rough set approximations. Inf. Sci. 223, 285–296 (2013)
Liu, G.: The relationship among different covering approximations. Inf. Sci. 250, 178–183 (2013)
Ma, Z.M., Hu, B.Q.: Topological and lattice structures of \(L\)-fuzzy rough sets determined by lower and upper sets. Inf. Sci. 218, 194–204 (2013)
Mi, J., Zhang, W.: An axiomatic characterization of a fuzzy generalization of rough sets. Inf. Sci. 160, 235–249 (2004)
Morsi, N.N., Yakout, M.M.: Axiomatics for fuzzy rough set. Fuzzy Sets Syst. 100, 327–342 (1998)
Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)
Pawlak, Z.: Rough Sets-Theoretical Aspects of Reasoning About Data. Kluwer Academic Publishers, Boston (1991)
Pawlak, Z., Skowron, A.: Rough sets: some extensions. Inf. Sci. 177, 28–40 (2007)
Pei, D.W.: A generalized model of fuzzy rough sets. Int. J. Gen. Syst. 34, 603–613 (2005)
Qin, K., Pei, Z., Yang, J.L.: Approximation operators on complete completely distributive lattices. Inf. Sci. 247, 123–130 (2013)
Radzikowska, A.M., Kerre, E.E.: Fuzzy rough sets based on residuated lattices. In: Transactions on Rough Sets II. LNCS, vol. 3135, pp. 278–296 (2004)
She, Y.H., Wang, G.J.: An axiomatic approach of fuzzy rough sets based on residuated lattices. Comput. Math. Appl. 58, 189–201 (2009)
Thiele, H.: On axiomatic characterization of fuzzy approximation operators. I, the fuzzy rough set based case. In: Conference Proceedings RSCTC 2000, Banff Park Lodge, Bariff, Canada, 19 October, pp. 239–247 (2000)
Thiele, H.: On axiomatic characterization of fuzzy approximation operators II, The rough fuzzy set based case. In: Proceedings of 31st IEEE International Symposium on Multiple-Valued Logic, pp. 330–335 (2001)
Thiele, H.: On axiomatic characterization of fuzzy approximation operators III, The fuzzy diamond and fuzzy box cases. In: The 10th IEEE International Conference on Fuzzy Systems, vol. 2, pp. 1148–1151 (2001)
Tiwari, S.P., Srivastava, A.K.: Fuzzy rough sets, fuzzy preorders and fuzzy topologies. Fuzzy Sets Syst. 210, 63–68 (2013)
Wang, G.J.: Theory of \(L\)-fuzzy Topological Space. Shaanxi Normal University Press, Xi’an (1988). (in Chinese)
Wu, W., Zhang, W.: Constructive and axiomatic approaches of fuzzy approximation operators. Inf. Sci. 159, 233–254 (2004)
Yao, Y.: Two views of the theory of rough sets in finite universes. Int. J. Approx. Reason. 15, 291–317 (1996)
Yao, Y.: Relational interpretations of neighborhood operators and rough set approximation operators. Inf. Sci. 111, 239–259 (1998)
Yao, Y.: Constructive and algebraic methods of the theory of rough sets. Inf. Sci. 109, 21–47 (1998)
Yao, Y., Yao, B.: Covering based rough set approximations. Inf. Sci. 200, 91–107 (2012)
Zhang, Y., Li, J., Wu, W.: On axiomatic characterizations of three pairs of covering based approximation operators. Inf. Sci. 180, 274–287 (2010)
Zhang, Y., Luo, M.: On minimization of axiom sets characterizing covering-based approximation operators. Inf. Sci. 181, 3032–3042 (2011)
Zhu, W.: Topological approaches to covering rough sets. Inf. Sci. 177, 1499–1508 (2007)
Acknowledgements
Thanks to the support by National Natural Science Foundation of China (No. 11501278 and No. 11471152) and Shandong Provincial Natural Science Foundation, China (ZR2014AQ011),and Project Science Foundation of Liaocheng University.
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Hu, K., Jin, Q., Li, Lq. (2019). On the Characterizations of L-fuzzy Rough Sets Based on Fuzzy Lattices. In: Cao, BY., Zhong, YB. (eds) Fuzzy Sets and Operations Research. ICFIE 2017. Advances in Intelligent Systems and Computing, vol 872. Springer, Cham. https://doi.org/10.1007/978-3-030-02777-3_23
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DOI: https://doi.org/10.1007/978-3-030-02777-3_23
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