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On approximate-type systems generated by \( L\)-relations. (English) Zbl 1461.03052

Summary: The aim of this work is to study approximate-type systems induced by \( L\)-relations in the framework of the general theory of M-approximate systems introduced in [the last author, Fuzzy Sets Syst. 161, No. 18, 2440–2461 (2010; Zbl 1222.54008)] and its generalizations. Special attention is payed to the structural properties of lattices of such systems and to the study of connections between categories of such systems and the corresponding categories of sets endowed with \(L\)-relations.

MSC:

03E72 Theory of fuzzy sets, etc.
68T37 Reasoning under uncertainty in the context of artificial intelligence

Citations:

Zbl 1222.54008
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References:

[2] Bodenhofer, U., Ordering of fuzzy sets based on fuzzy orderings. I: The basic approach, Mathware Soft Comput., 15, 201-218 (2008) · Zbl 1160.03036
[3] Bodenhofer, U., Ordering of fuzzy sets based on fuzzy orderings. II: Generalizations, Mathware Soft Comput., 15, 219-249 (2008) · Zbl 1170.03338
[4] Brown, L. M.; Ertürk, R.; Dost, Ş., Ditopological texture spaces and fuzzy topology, I. Basic concepts, Fuzzy Sets Syst., 110, 227-236 (2000) · Zbl 0953.54010
[5] Brown, L. M.; Ertürk, R.; Dost, Ş., Ditopological texture spaces and fuzzy topology, II. Topological considerations, Fuzzy Sets Syst., 147, 171-199 (2004) · Zbl 1070.54002
[7] Ciucci, D., Approximation algebra and framework, Fund. Inform., 94, 2, 147-161 (2009) · Zbl 1192.68669
[8] Chen, P.; Zhang, D., Alexandroff L-cotopological spaces, Fuzzy Sets Syst., 161, 2505-2514 (2010) · Zbl 1207.54014
[9] Dubois, D.; Prade, H., Rough fuzzy sets and fuzzy rough sets, Int. J. General Syst., 17, 2-3, 191-209 (1990) · Zbl 0715.04006
[10] Gierz, G.; Hoffman, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M. W.; Scott, D. S., Continuous Lattices and Domains (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1088.06001
[13] Estaji, A. A.; Hooshmandasl, M. R.; Davvaz, B., Rough set theory applied to lattice theory, Inform. Sci., 200, 108-112 (2012) · Zbl 1248.06003
[14] Hao, J.; Li, Q., The relation between L-fuzzy rough sets and L-topology, Fuzzy Sets Syst., 178 (2011) · Zbl 1238.54005
[15] Han, S.-E.; Šostak, A., A compression of digital images derived from a Khalimsky topological structure, Comput. Appl. Math., 32, 521-536 (2013) · Zbl 1300.54012
[16] Järvinen, J., On the structure of rough approximations, Fund. Inform., 53, 135-153 (2002) · Zbl 1012.68200
[17] Järvinen, J.; Kortelainen, J., A unified study between modal-like operators, topologies and fuzzy sets, Fuzzy Sets Syst., 158, 1217-1225 (2007) · Zbl 1119.03054
[18] Kaufman, A., Theory of Fuzzy Subsets (1975), Academic Press: Academic Press New York
[19] Klement, E. P.; Mesiar, R.; Pap, E., Triangular Norms (2000), Kluwer Acad. Publ, S.E. Rodabaugh, E.P. Klement (Eds.), Kluwer Acad. Publ. 2003 · Zbl 0972.03002
[20] Kong, T. Y.; Rosenfeld, A., Digital topology: introduction and survey, Comput. Vis. Graph. Image Process., 48, 357-393 (1989)
[21] (Kong, T. Y.; Rosenfeld, A., Topological Algorithms for Digital Image Processing (1966), Elsevier)
[22] Kortelainen, J., On relationship between modified sets, topological spaces and rough sets, Fuzzy Sets Syst., 61, 91-95 (1994) · Zbl 0828.04002
[24] Mi, J. S.; Hu, B. Q., Topological and lattice structure of L-fuzzy rough sets determined by upper and lower sets, Inform. Sci., 218, 194-204 (2013) · Zbl 1293.03025
[25] Mi, J. S.; Zhang, W. X., An axiomatic characterization of a fuzzy generalization of a rough set, Inform. Sci., 160, 235-249 (2004) · Zbl 1041.03038
[26] Morillas, S.; Gregori, V.; Herves, A., Fuzzy peer groups for reducing mixed gaussianimpulse noise from color images, IEEE Trans. Image Process., 18, 1452-1466 (2009) · Zbl 1371.94269
[28] Ouyang, Y.; Wang, Z.; Zhang, H., On fuzzy rough sets based on tolerance relations, Inform. Sci., 180, 532-542 (2010) · Zbl 1189.68131
[29] Pawlak, Z., Rough sets, Int. J. Comput. Inform. Sci., 11, 341-356 (1982) · Zbl 0501.68053
[30] Qin, K.; Pei, Z., On the topological properties of fuzzy rough sets, Fuzzy Sets Syst., 151, 601-613 (2005) · Zbl 1070.54006
[31] Qin, K.; Pei, Z., Generalized rough sets based on reflexive and transitive relations, Inform. Sci., 178, 4138-4141 (2008) · Zbl 1153.03316
[32] Radzikowska, A. M.; Kerre, E. E., A comparative study of fuzzy rough sets, Fuzzy Sets Syst., 126, 137-155 (2002) · Zbl 1004.03043
[33] Raney, G. N., A subdirect-union representation for completely distributive complete lattices, Proc. Am. Math. Soc., 4, 518-522 (1953) · Zbl 0053.35201
[34] Rodabaugh, S. E., Powerset operator based foundations for point-set lattice-theoretic (poslat) fuzzy set theories and topologies, Quaest. Math., 20, 463-530 (1997) · Zbl 0911.04003
[35] Rodabaugh, S. E., Powerset operator foundations for poslat fuzzy set theories and topologies, (Höhle, U.; Rodabaugh, S. E., Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory (1999), Kluwer Acad. Publ.), 91-116 · Zbl 0974.03047
[36] Rosenfeld, A., Digital topology, Am. Math. Monthly, 86, 621-630 (1979) · Zbl 0432.68061
[37] Rosenfeld, A., Fuzzy digital topology, Inform. Control, 40, 76-87 (1979) · Zbl 0404.68071
[38] Rosenthal, K. I., Quantales and Their Applications, Pirman Research Notes in Mathematics, vol. 234 (1990), Lungman: Lungman Burnt Mill, Harlow · Zbl 0703.06007
[39] Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), North Holland: North Holland New York · Zbl 0546.60010
[42] Šostak, A., Towards the theory of M-approximate systems: fundamentals and examples, Fuzzy Sets Syst., 161, 2440-2461 (2010) · Zbl 1222.54008
[43] Šostak, A., Towards the theory of approximate systems: variable range categories, (Proceedings of ICTA2011, Islamabad, Pakistan (2012), Cambridge University Publ.), 265-284 · Zbl 1301.03044
[44] Tiwari, S. P.; Srivastava, A. K., Fuzzy rough sets, fuzzy preorders and fuzzy topologies, Fuzzy Sets Syst., 210, 63-68 (2013) · Zbl 1260.54024
[45] Valverde, L., On the structure of F-indistinguishibility operators, Fuzzy Sets Syst., 17, 313-328 (1985) · Zbl 0609.04002
[46] Yao, Y. Y., A comparative study of fuzzy sets and rough sets, Inform. Sci., 109, 227-242 (1998) · Zbl 0932.03064
[47] Yao, Y. Y., On generalizing Pawlak approximation operators, Proceedings of the First International Conference Rough Sets and Current Trends in Computing, 298-307 (1998) · Zbl 0955.68505
[48] Yao, Yiyu, Bingxue Yao covering based rough set approximations, Inform. Sci., 200, 91-107 (2012) · Zbl 1248.68496
[49] Yu, H.; Zhan, W. R., On the topological properties of generalized rough sets, Inform. Sci., 263, 141-152 (2014) · Zbl 1347.68335
[50] Zadeh, L., Similarity relations and fuzzy orderings, Inform. Sci., 3, 177-200 (1971) · Zbl 0218.02058
[51] Zhou, N. L.; Hu, B. Q., Rough sets based on completely distributive lattice, Inform. Sci., 269, 378-387 (2014) · Zbl 1339.68250
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