Rough sets based on Galois connections

Alcalde, C., Burusco, A., Díaz-Moreno, J.C. and Medina, J. (2017). Fuzzy concept lattices and fuzzy relation equations in the retrieval processing of images and signals, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems25(Supplement-1): 99–120.10.1142/s0218488517400050Search in Google Scholar

Benítez-Caballero, M.J., Medina, J., Ramírez-Poussa, E. and Ślęzak, D. (2020). A computational procedure for variable selection preserving different initial conditions, International Journal of Computer Mathematics97(1–2): 387–404.10.1080/00207160.2019.1613530Search in Google Scholar

Birkhoff, G. (1967). Lattice Theory, 3rd Edn, American Mathematical Society, Providence, RI.Search in Google Scholar

Bloch, I. (2000). On links between mathematical morphology and rough sets, Pattern Recognition33(9): 1487–1496.10.1016/S0031-3203(99)00129-6Search in Google Scholar

Bustince, H., Madrid, N. and Ojeda-Aciego, M. (2015). The notion of weak-contradiction: Definition and measures, IEEE Transactions on Fuzzy Systems23(4): 1057–1069.10.1109/TFUZZ.2014.2337934Search in Google Scholar

Cornejo, M.E., Díaz-Moreno, J.C. and Medina, J. (2017a). Multi-adjoint relation equations: A decision support system for fuzzy logic, International Journal of Intelligent Systems32(8): 778–800.10.1002/int.21889Search in Google Scholar

Cornejo, M.E., Lobo, D. and Medina, J. (2018a). Syntax and semantics of multi-adjoint normal logic programming, Fuzzy Sets and Systems345: 41–62, DOI: 10.1016/j.fss.2017.12.009.10.1016/j.fss.2017.12.009Search in Google Scholar

Cornejo, M.E., Medina, J. and Ramírez-Poussa, E. (2017b). Attribute and size reduction mechanisms in multi-adjoint concept lattices, Journal of Computational and Applied Mathematics318: 388–402.10.1016/j.cam.2016.07.012Search in Google Scholar

Cornejo, M.E., Medina, J. and Ramírez-Poussa, E. (2018b). Characterizing reducts in multi-adjoint concept lattices, Information Sciences422: 364–376.10.1016/j.ins.2017.08.099Search in Google Scholar

Cornelis, C., Medina, J. and Verbiest, N. (2014). Multi-adjoint fuzzy rough sets: Definition, properties and attribute selection, International Journal of Approximate Reasoning55(1): 412–426.10.1016/j.ijar.2013.09.007Search in Google Scholar

Couso, I. and Dubois, D. (2011). Rough sets, coverings and incomplete information, Fundamenta Informaticae108(3–4): 223–247.10.3233/FI-2011-421Search in Google Scholar

Davey, B. and Priestley, H. (2002). Introduction to Lattices and Order, 2nd Edn, Cambridge University Press, Cambridge.10.1017/CBO9780511809088Search in Google Scholar

Denecke, K., Erné, M. and Wismath, S.L. (Eds) (2004). Galois Connections and Applications, Kluwer Academic Publishers, Dordrecht.10.1007/978-1-4020-1898-5Search in Google Scholar

Di Nola, A., Sanchez, E., Pedrycz, W. and Sessa, S. (1989). Fuzzy Relation Equations and Their Applications to Knowledge Engineering, Kluwer Academic Publishers, Norwell, MA.10.1007/978-94-017-1650-5Search in Google Scholar

Díaz-Moreno, J.C. and Medina, J. (2013). Multi-adjoint relation equations: Definition, properties and solutions using concept lattices, Information Sciences253: 100–109.10.1016/j.ins.2013.07.024Search in Google Scholar

Díaz-Moreno, J.C., Medina, J. and Turunen, E. (2017). Minimal solutions of general fuzzy relation equations on linear carriers. An algebraic characterization, Fuzzy Sets and Systems311: 112–123.Search in Google Scholar

Ganter, B. and Wille, R. (1999). Formal Concept Analysis: Mathematical Foundation, Springer Verlag, Berlin.10.1007/978-3-642-59830-2Search in Google Scholar

Grant, J. and Hunter, A. (2006). Measuring inconsistency in knowledge bases, Journal of Intelligent Information Systems27(2): 159–184.10.1007/s10844-006-2974-4Search in Google Scholar

Hajek, P. (1998). Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht.10.1007/978-94-011-5300-3Search in Google Scholar

Han, S.-E. (2019). Roughness measures of locally finite covering rough sets, International Journal of Approximate Reasoning105: 368–385.10.1016/j.ijar.2018.12.003Search in Google Scholar

Hassanien, A.E., Abraham, A., Peters, J.F., Schaefer, G. and Henry, C. (2009). Rough sets and near sets in medical imaging: A review, IEEE Transactions on Information Technology in Biomedicine13(6): 955–968.10.1109/TITB.2009.201701719304490Search in Google Scholar

Hassanien, A.E., Schaefer, G. and Darwish, A. (2010). Computational intelligence in speech and audio processing: Recent advances, in X.-Z. Gao et al. (Eds), Soft Computing in Industrial Applications, Springer, Berlin/Heidelberg, pp. 303–311.10.1007/978-3-642-11282-9_32Search in Google Scholar

Järvinen, J., Radeleczki, S. and Veres, L. (2009). Rough sets determined by quasiorders, Order26(4): 337–355.10.1007/s11083-009-9130-zSearch in Google Scholar

Kortelainen, J. (1994). On relationship between modified sets, topological spaces and rough sets, Fuzzy Sets and Systems61(1): 91–95.10.1016/0165-0114(94)90288-7Search in Google Scholar

Luo, C., Li, T., Chen, H., Fujita, H. and Yi, Z. (2018). Incremental rough set approach for hierarchical multicriteria classification, Information Sciences429: 72–87.10.1016/j.ins.2017.11.004Search in Google Scholar

Madrid, N. (2017). An extension of f-transforms to more general data: Potential applications, Soft Computing21(13): 3551–3565.10.1007/s00500-017-2622-7Search in Google Scholar

Madrid, N. and Ojeda-Aciego, M. (2011a). Measuring inconsistency in fuzzy answer set semantics, IEEE Transactions on Fuzzy Systems19(4): 605–622.10.1109/TFUZZ.2011.2114669Search in Google Scholar

Madrid, N. and Ojeda-Aciego, M. (2011b). On the existence and unicity of stable models in normal residuated logic programs, International Journal of Computer Mathematics89(3): 310–324.10.1080/00207160.2011.580842Search in Google Scholar

Madrid, N. and Ojeda-Aciego, M. (2017). A view of f-indexes of inclusion under different axiomatic definitions of fuzzy inclusion, in S. Moral et al. (Eds), Scalable Uncertainty Management, Springer, Cham, pp. 307–318.10.1007/978-3-319-67582-4_22Search in Google Scholar

Madrid, N., Ojeda-Aciego, M., Medina, J. and Perfilieva, I. (2019). L-fuzzy relational mathematical morphology based on adjoint triples, Information Sciences474: 75–89.10.1016/j.ins.2018.09.028Search in Google Scholar

Medina, J. (2012a). Multi-adjoint property-oriented and object-oriented concept lattices, Information Sciences190: 95–106.10.1016/j.ins.2011.11.016Search in Google Scholar

Medina, J. (2012b). Relating attribute reduction in formal, object-oriented and property-oriented concept lattices, Computers & Mathematics with Applications64(6): 1992–2002.10.1016/j.camwa.2012.03.087Search in Google Scholar

Medina, J. (2017). Minimal solutions of generalized fuzzy relational equations: Clarifications and corrections towards a more flexible setting, International Journal of Approximate Reasoning84: 33–38.10.1016/j.ijar.2017.02.002Search in Google Scholar

Medina, J., Ojeda-Aciego, M. and Ruiz-Calviño, J. (2009). Formal concept analysis via multi-adjoint concept lattices, Fuzzy Sets and Systems160(2): 130–144.10.1016/j.fss.2008.05.004Search in Google Scholar

Medina, J., Ojeda-Aciego, M. and Vojtáš, P. (2004). Similarity-based unification: A multi-adjoint approach, Fuzzy Sets and Systems146(1): 43–62.10.1016/j.fss.2003.11.005Search in Google Scholar

Novák, V., Mockor, J. and Perfilieva, I. (1999). Mathematical Principles of Fuzzy Logic, Kluwer, Boston, MA.10.1007/978-1-4615-5217-8Search in Google Scholar

Pagliani, P. (2014). The relational construction of conceptual patterns—Tools, implementation and theory, in M. Kryszkiewicz et al. (Eds), Rough Sets and Intelligent Systems Paradigms, Springer International Publishing, Cham, pp. 14–27.10.1007/978-3-319-08729-0_2Search in Google Scholar

Pagliani, P. (2016). Covering Rough Sets and Formal Topology—A Uniform Approach Through Intensional and Extensional Constructors, Springer, Berlin/Heidelberg, pp. 109–145.Search in Google Scholar

Pagliani, P. and Chakraborty, M. (2008). A Geometry of Approximation: Rough Set Theory Logic, Algebra and Topology of Conceptual Patterns (Trends in Logic), 1st Edn, Springer Publishing Company, Berlin/Heidelberg.10.1007/978-1-4020-8622-9Search in Google Scholar

Pawlak, Z. (1982). Rough sets, International Journal of Computer and Information Science11: 341–356.10.1007/BF01001956Search in Google Scholar

Pawlak, Z. (1991). Rough Sets—Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht.10.1007/978-94-011-3534-4_7Search in Google Scholar

Perfilieva, I. (2006). Fuzzy transforms: Theory and applications, Fuzzy Sets and Systems157(8): 993–1023.10.1016/j.fss.2005.11.012Search in Google Scholar

Perfilieva, I., Singh, A.P. and Tiwari, S.P. (2017). On the relationship among f-transform, fuzzy rough set and fuzzy topology, Soft Computing21(13): 3513–3523.10.1007/s00500-017-2559-xSearch in Google Scholar

Ronse, C. and Heijmans, H.J.A.M. (1991). The algebraic basis of mathematical morphology. II: Openings and closings, CVGIP: Image Understanding54(1): 74–97.Search in Google Scholar

Sanchez, E. (1976). Resolution of composite fuzzy relation equations, Information and Control30(1): 38–48.10.1016/S0019-9958(76)90446-0Search in Google Scholar

Shao, M.-W., Liu, M. and Zhang, W.-X. (2007). Set approximations in fuzzy formal concept analysis, Fuzzy Sets and Systems158(23): 2627–2640.10.1016/j.fss.2007.05.002Search in Google Scholar

Skowron, A., Swiniarski, R. and Synak, P. (2004). Approximation spaces and information granulation, in S. Tsumoto et al. (Eds), Rough Sets and Current Trends in Computing, Springer, Berlin/Heidelberg, pp. 116–126.10.1007/978-3-540-25929-9_13Search in Google Scholar

Slowinski, R. and Vanderpooten, D. (1997). Similarity relation as a basis for rough approximations, in P.P.Wang (Ed.), Advances in Machine Intelligence and Soft Computing, Duke University, Durham, NC, pp. 17–33.Search in Google Scholar

Stell, J.G. (2007). Relations in mathematical morphology with applications to graphs and rough sets, in S. Winter et al. (Eds), Spatial Information Theory, Springer, Berlin, Heidelberg, pp. 438–454.10.1007/978-3-540-74788-8_27Search in Google Scholar

Tan, A., Wu, W.-Z. and Tao, Y. (2018). A unified framework for characterizing rough sets with evidence theory in various approximation spaces, Information Sciences454–455: 144–160.10.1016/j.ins.2018.04.073Search in Google Scholar

Varma, P.R.K., Kumari, V.V. and Kumar, S.S. (2015). A novel rough set attribute reduction based on ant colony optimisation, International Journal of Intelligent Systems Technologies and Applications14(3–4): 330–353.10.1504/IJISTA.2015.074333Search in Google Scholar

Wille, R. (1982). Restructuring lattice theory: An approach based on hierarchies of concepts, in I. Rival (Ed.), Ordered Sets, Reidel, Dordrecht, pp. 445–470.10.1007/978-94-009-7798-3_15Search in Google Scholar

Wille, R. (2005). Formal concept analysis as mathematical theory of concepts and concept hierarchies, in B. Ganter et al. (Eds), Formal Concept Analysis, Lecture Notes in Computer Science, Vol. 3626, Springer, Berlin/Heidelberg, pp. 1–33.10.1007/11528784_1Search in Google Scholar

Zakowski, W. (1983). Approximations in the space (u, π), Demonstratio Mathematica16(3): 761–769.10.1515/dema-1983-0319Search in Google Scholar

Yang, X., Li, T., Fujita, H., Liu, D. and Yao, Y. (2017). A unified model of sequential three-way decisions and multilevel incremental processing, Knowledge-Based Systems134: 172–188.10.1016/j.knosys.2017.07.031Search in Google Scholar

Yao, Y. (1998a). A comparative study of fuzzy sets and rough sets, Information Sciences109(1): 227–242.10.1016/S0020-0255(98)10023-3Search in Google Scholar

Yao, Y. (1998b). Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences111(1): 239–259.10.1016/S0020-0255(98)10006-3Search in Google Scholar

Yao, Y. (2018). Three-way decision and granular computing, International Journal of Approximate Reasoning103: 107–123.10.1016/j.ijar.2018.09.005Search in Google Scholar

Yao, Y. and Chen, Y. (2006). Rough set approximations in formal concept analysis, in J.F. Peters and A. Skowron (Eds), Transactions on Rough Sets V, Springer, Berlin/Heidelberg, pp. 285–305.10.1007/11847465_14Search in Google Scholar

Yao, Y. and Lingras, P. (1998). Interpretations of belief functions in the theory of rough sets, Information Sciences104(1): 81–106.10.1016/S0020-0255(97)00076-5Search in Google Scholar

Yao, Y.Y. (1996). Two views of the theory of rough sets in finite universes, International Journal of Approximate Reasoning15(4): 291–317.10.1016/S0888-613X(96)00071-0Search in Google Scholar

Yao, Y.Y. (2004). A comparative study of formal concept analysis and rough set theory in data analysis, in S. Tsumoto et al. (Eds), Rough Sets and Current Trends in Computing, Springer, Berlin/Heidelberg, pp. 59–68.10.1007/978-3-540-25929-9_6Search in Google Scholar

Yao, Y. and Yao, B. (2012). Covering based rough set approximations, Information Sciences200: 91–107.10.1016/j.ins.2012.02.065Search in Google Scholar

Zhang, Q., Xie, Q. and Wang, G. (2016). A survey on rough set theory and its applications, CAAI Transactions on Intelligence Technology1(4): 323–333.10.1016/j.trit.2016.11.001Search in Google Scholar

Zhu, W. (2007). Generalized rough sets based on relations, Information Sciences177(22): 4997–5011.10.1016/j.ins.2007.05.037Search in Google Scholar

Ziarko,W. (2008). Probabilistic approach to rough sets, International Journal of Approximate Reasoning49(2): 272–284.10.1016/j.ijar.2007.06.014Search in Google Scholar