{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,8]],"date-time":"2025-04-08T19:45:40Z","timestamp":1744141540375},"reference-count":57,"publisher":"Springer Science and Business Media LLC","issue":"16","license":[{"start":{"date-parts":[[2018,11,8]],"date-time":"2018-11-08T00:00:00Z","timestamp":1541635200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2019,8]]},"DOI":"10.1007\/s00500-018-3616-9","type":"journal-article","created":{"date-parts":[[2018,11,8]],"date-time":"2018-11-08T11:10:27Z","timestamp":1541675427000},"page":"6727-6741","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The structures and the connections on four types of covering rough sets"],"prefix":"10.1007","volume":"23","author":[{"given":"Zhaohao","family":"Wang","sequence":"first","affiliation":[]},{"given":"Hong","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Qinrong","family":"Feng","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,11,8]]},"reference":[{"issue":"1","key":"3616_CR1","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1016\/S0020-0255(97)10046-9","volume":"107","author":"Z Bonikowski","year":"1998","unstructured":"Bonikowski Z, Bryniarski E, Wybraniec-Skardowska U (1998) Extensions and intentions in the rough set theory. Inf Sci 107(1):149\u2013167","journal-title":"Inf Sci"},{"issue":"4","key":"3616_CR2","doi-asserted-by":"publisher","first-page":"800","DOI":"10.2307\/2371015","volume":"57","author":"G Birkhoff","year":"1935","unstructured":"Birkhoff G (1935) Abstract linear dependence and lattices. Am J Math 57(4):800\u2013804","journal-title":"Am J Math"},{"key":"3616_CR3","doi-asserted-by":"publisher","first-page":"1829","DOI":"10.1016\/j.ins.2005.05.009","volume":"176","author":"D Chen","year":"2006","unstructured":"Chen D, Zhang W, Yeung D, Tsang E (2006) Rough approximations on a complete completely distributive lattice with applications to generalized rough sets. Inf Sci 176:1829\u20131848","journal-title":"Inf Sci"},{"issue":"17","key":"3616_CR4","doi-asserted-by":"publisher","first-page":"3500","DOI":"10.1016\/j.ins.2007.02.041","volume":"177","author":"D Chen","year":"2007","unstructured":"Chen D, Wang C, Hu Q (2007) A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets. Inf Sci 177(17):3500\u20133518","journal-title":"Inf Sci"},{"key":"3616_CR5","volume-title":"Introduction to lattices and order","author":"BA Davey","year":"1990","unstructured":"Davey BA, Priestley HA (1990) Introduction to lattices and order. Cambridge University Press, Cambridge"},{"key":"3616_CR6","unstructured":"D\u2019eer L, Cornelis C (2015) New neighborhood based rough sets. In: Proceedings of 10th International Conference on Rough Sets and Knowledge Technology (RSKT2015), LNAI 9436, pp 191\u2013201"},{"key":"3616_CR7","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/j.ins.2015.12.007","volume":"336","author":"L D\u2019eer","year":"2016","unstructured":"D\u2019eer L, Restrepo M, Cornelis C, G\u00f3mez J (2016) Neighborhood operators for covering-based rough sets. Inf Sci 336:21\u201344","journal-title":"Inf Sci"},{"key":"3616_CR8","doi-asserted-by":"publisher","first-page":"62","DOI":"10.1016\/j.ijar.2016.06.013","volume":"78","author":"L D\u2019eer","year":"2016","unstructured":"D\u2019eer L, Cornelis C, Yao Y (2016) A semantically sound approach to Pawlak rough sets and covering-based rough sets. Int J Approx Reason 78:62\u201372","journal-title":"Int J Approx Reason"},{"key":"3616_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-0018-1","volume-title":"Lattice theory: foundation","author":"G Gr\u00e4tzer","year":"2011","unstructured":"Gr\u00e4tzer G (2011) Lattice theory: foundation. Birkh\u00e4user, Basel"},{"key":"3616_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.ins.2017.01.032","volume":"390","author":"X Ge","year":"2017","unstructured":"Ge X, Wang P, Yun Z (2017) The rough membership functions on four types of covering-based rough sets and their applications. Inf Sci 390:1\u201314","journal-title":"Inf Sci"},{"key":"3616_CR11","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1016\/j.ins.2013.11.014","volume":"263","author":"A Huang","year":"2014","unstructured":"Huang A, Zhao H, Zhu W (2014) Nullity-based matroid of rough sets and its application to attribute reduction. Inf Sci 263:153\u2013165","journal-title":"Inf Sci"},{"key":"3616_CR12","doi-asserted-by":"publisher","first-page":"589","DOI":"10.1016\/j.ins.2005.01.001","volume":"176","author":"M Kondo","year":"2005","unstructured":"Kondo M (2005) On the structure of generalized rough sets. Inf Sci 176:589\u2013600","journal-title":"Inf Sci"},{"key":"3616_CR13","doi-asserted-by":"publisher","first-page":"4105","DOI":"10.1016\/j.ins.2008.06.021","volume":"178","author":"G Liu","year":"2008","unstructured":"Liu G, Zhu W (2008) The algebraic structures of generalized rough set theory. Inf Sci 178:4105\u20134113","journal-title":"Inf Sci"},{"key":"3616_CR14","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/j.knosys.2014.01.001","volume":"60","author":"G Liu","year":"2014","unstructured":"Liu G, Zhu K (2014) The relationship among three types of rough approximation pairs. Knowl-Based Syst 60:28\u201334","journal-title":"Knowl-Based Syst"},{"issue":"3","key":"3616_CR15","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1016\/j.ijar.2008.11.001","volume":"50","author":"G Liu","year":"2009","unstructured":"Liu G, Sai Y (2009) A comparison of two types of rough sets induced by coverings. Int J Approx Reason 50(3):521\u2013528","journal-title":"Int J Approx Reason"},{"key":"3616_CR16","first-page":"15","volume":"111","author":"T Li","year":"2011","unstructured":"Li T, Wu W (2011) Attribute reduction in formal contexts: a covering rough set approach. Fund Inf 111:15\u201332","journal-title":"Fund Inf"},{"key":"3616_CR17","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/j.ins.2016.01.029","volume":"342","author":"X Li","year":"2016","unstructured":"Li X, Yi H, Liu S (2016) Rough sets and matroids from a lattice-theoretic viewpoint. Inf Sci 342:37\u201352","journal-title":"Inf Sci"},{"key":"3616_CR18","doi-asserted-by":"publisher","first-page":"513","DOI":"10.1016\/j.ijar.2011.12.005","volume":"53","author":"X Li","year":"2012","unstructured":"Li X, Liu S (2012) Matroidal approaches to rough sets via closure operators. Int J Approx Reason 53:513\u2013527","journal-title":"Int J Approx Reason"},{"key":"3616_CR19","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/j.ins.2013.03.046","volume":"241","author":"G Lin","year":"2013","unstructured":"Lin G, Liang J, Qian Y (2013) Multigranulation rough sets: from partition to covering. Inf Sci 241:101\u2013118","journal-title":"Inf Sci"},{"key":"3616_CR20","doi-asserted-by":"publisher","first-page":"912","DOI":"10.1016\/j.ijar.2012.02.004","volume":"53","author":"J Liang","year":"2012","unstructured":"Liang J, Wang F, Dang C, Qian Y (2012) An efficient rough feature selection algorithm with a multi-granulation view. Int J Approx Reason 53:912\u2013926","journal-title":"Int J Approx Reason"},{"key":"3616_CR21","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1016\/j.ins.2011.11.016","volume":"190","author":"J Medina","year":"2012","unstructured":"Medina J (2012) Multi-adjoint property-oriented and object-oriented concept lattices. Inf Sci 190:95\u2013106","journal-title":"Inf Sci"},{"key":"3616_CR22","volume-title":"Matroid theory","author":"JG Oxley","year":"1992","unstructured":"Oxley JG (1992) Matroid theory. Oxford University Press, New York"},{"key":"3616_CR23","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1007\/BF01001956","volume":"11","author":"Z Pawlak","year":"1982","unstructured":"Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341\u2013356","journal-title":"Int J Comput Inf Sci"},{"issue":"1","key":"3616_CR24","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/j.ins.2006.06.007","volume":"177","author":"Z Pawlak","year":"2007","unstructured":"Pawlak Z, Skowron A (2007) Rough sets and Boolean reasoning. Inf Sci 177(1):41\u201373","journal-title":"Inf Sci"},{"key":"3616_CR25","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1016\/j.ijar.2010.07.010","volume":"52","author":"Z Pei","year":"2011","unstructured":"Pei Z, Pei D, Zheng L (2011) Topology vs generalized rough sets. Int J Approx Reason 52:231\u2013239","journal-title":"Int J Approx Reason"},{"key":"3616_CR26","first-page":"653","volume":"35","author":"J Pomykala","year":"1987","unstructured":"Pomykala J (1987) Approximation operators in approximation space. Bull Polish Acad Sci Math 35:653\u2013662","journal-title":"Bull Polish Acad Sci Math"},{"issue":"3\u20134","key":"3616_CR27","first-page":"193","volume":"36","author":"J Pomykala","year":"1988","unstructured":"Pomykala J (1988) On definability in the nondeterministic information system. Bull Pol Acad Sci Math 36(3\u20134):193\u2013210","journal-title":"Bull Pol Acad Sci Math"},{"key":"3616_CR28","doi-asserted-by":"crossref","unstructured":"Qin K, Gao Y, Pei Z (2007) On covering rough sets. In: Rough sets and knowledge technology. Springer, Berlin, 34\u201341","DOI":"10.1007\/978-3-540-72458-2_4"},{"key":"3616_CR29","doi-asserted-by":"publisher","first-page":"4138","DOI":"10.1016\/j.ins.2008.07.002","volume":"178","author":"K Qin","year":"2008","unstructured":"Qin K, Yang J, Pei Z (2008) Generalized rough sets based on reflexive and transitive relations. Inf Sci 178:4138\u20134141","journal-title":"Inf Sci"},{"key":"3616_CR30","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1016\/j.knosys.2012.05.019","volume":"36","author":"Y She","year":"2012","unstructured":"She Y, He X (2012) On the structure of the multigranulation rough set model. Knowl-Based Syst 36:81\u201392","journal-title":"Knowl-Based Syst"},{"key":"3616_CR31","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1007\/s13042-014-0314-5","volume":"8","author":"L Su","year":"2017","unstructured":"Su L, Zhu W (2017) Closed-set lattice and modular matroid induced by covering-based rough sets. Int J Mach Learn Cybern 8:191\u2013201","journal-title":"Int J Mach Learn Cybern"},{"key":"3616_CR32","first-page":"290","volume-title":"Rough sets, fuzzy sets and knowledge discovery","author":"S Tsumoto","year":"1993","unstructured":"Tsumoto S, Tanaka A (1993) Rough sets and matroid theory. In: Ziarko W (ed) Rough sets, fuzzy sets and knowledge discovery. Springer, London, pp 290\u2013297"},{"key":"3616_CR33","first-page":"224","volume":"985","author":"S Tsumoto","year":"1995","unstructured":"Tsumoto S, Tanaka H (1995) Algebraic specification of empirical inductive learning methods based on rough sets and matroid theory. LNCS 985:224\u2013243","journal-title":"LNCS"},{"key":"3616_CR34","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1016\/j.ins.2012.07.030","volume":"220","author":"S Wang","year":"2013","unstructured":"Wang S, Zhu Q, Zhu W, Min F (2013) Quantitative analysis for covering-based rough sets through the upper approximation number. Inf Sci 220:483\u2013491","journal-title":"Inf Sci"},{"key":"3616_CR35","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/j.knosys.2012.06.006","volume":"36","author":"S Wang","year":"2012","unstructured":"Wang S, Zhu Q, Zhu W, Min F (2012) Matroidal structure of rough sets and its characterization to attribute reduction. Knowl-Based Syst 36:155\u2013161","journal-title":"Knowl-Based Syst"},{"key":"3616_CR36","doi-asserted-by":"publisher","first-page":"1107","DOI":"10.1016\/j.ijar.2013.03.017","volume":"54","author":"W Wu","year":"2013","unstructured":"Wu W, Leung Y (2013) Optimal scale selection for multi-scale decision tables. Int J Approx Reason 54:1107\u20131129","journal-title":"Int J Approx Reason"},{"key":"3616_CR37","first-page":"421","volume":"72","author":"M Wolski","year":"2006","unstructured":"Wolski M (2006) Complete orders, categories and lattices of approximations. Fund Inf 72:421\u2013435","journal-title":"Fund Inf"},{"key":"3616_CR38","first-page":"503","volume":"128","author":"Z Wang","year":"2013","unstructured":"Wang Z, Shu L, Ding X (2013) Minimal description and maximal description in covering-based rough sets. Fund Inf 128:503\u2013526","journal-title":"Fund Inf"},{"key":"3616_CR39","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1016\/j.ins.2015.02.002","volume":"305","author":"Z Wang","year":"2015","unstructured":"Wang Z, Wang H, Feng Q, Shu L (2015) The approximation number function and the characterization of covering approximation space. Inf Sci 305:196\u2013207","journal-title":"Inf Sci"},{"issue":"1","key":"3616_CR40","first-page":"130","volume":"33","author":"Z Xu","year":"2005","unstructured":"Xu Z, Wang Q (2005) On the properties of covering rough sets model (in Chinese). J Henan Norm Univ 33(1):130\u2013132","journal-title":"J Henan Norm Univ"},{"key":"3616_CR41","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/S0020-0255(98)00012-7","volume":"109","author":"Y Yao","year":"1998","unstructured":"Yao Y (1998a) Constructive and algebraic methods of theory of rough sets. Inf Sci 109:21\u201347","journal-title":"Inf Sci"},{"key":"3616_CR42","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1016\/S0020-0255(98)10006-3","volume":"111","author":"Y Yao","year":"1998","unstructured":"Yao Y (1998b) Relational interpretations of neighborhood operators and rough set approximation operators. Inf Sci 111:239\u2013259","journal-title":"Inf Sci"},{"key":"3616_CR43","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1016\/j.ins.2012.02.065","volume":"200","author":"Y Yao","year":"2012","unstructured":"Yao Y, Yao B (2012) Covering based rough sets approximations. Inf Sci 200:91\u2013107","journal-title":"Inf Sci"},{"key":"3616_CR44","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1016\/j.ins.2015.08.011","volume":"327","author":"Y Yao","year":"2016","unstructured":"Yao Y, She Y (2016) Rough set models in multigranulation spaces. Inf Sci 327:40\u201356","journal-title":"Inf Sci"},{"key":"3616_CR45","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1016\/j.ijar.2009.11.001","volume":"51","author":"T Yang","year":"2010","unstructured":"Yang T, Li Q (2010) Reduction about approximation spaces of covering generalized rough sets. Int J Approx Reason 51:335\u2013345","journal-title":"Int J Approx Reason"},{"key":"3616_CR46","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1016\/j.ins.2013.06.057","volume":"250","author":"X Yang","year":"2013","unstructured":"Yang X, Qi Y, Song X, Yang J (2013) Test cost sensitive multigranulation rough set: model and minimal cost selection. Inf Sci 250:184\u2013199","journal-title":"Inf Sci"},{"key":"3616_CR47","unstructured":"\u017bakowski W (1983) Axiomatization in the space (U, \n                    \n                      \n                    \n                    $$\\varPi $$\n                    \n                      \n                        \u03a0\n                      \n                    \n                  ). Demonstr Math XVI: 761\u2013769"},{"key":"3616_CR48","doi-asserted-by":"publisher","first-page":"4997","DOI":"10.1016\/j.ins.2007.05.037","volume":"177","author":"W Zhu","year":"2007","unstructured":"Zhu W (2007) Generalized rough sets based on relations. Inf Sci 177:4997\u20135011","journal-title":"Inf Sci"},{"key":"3616_CR49","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1016\/j.ins.2008.09.015","volume":"179","author":"W Zhu","year":"2009","unstructured":"Zhu W (2009) Relationship between generalized rough sets based on binary relation and covering. Inf Sci 179:210\u2013225","journal-title":"Inf Sci"},{"key":"3616_CR50","doi-asserted-by":"publisher","first-page":"1499","DOI":"10.1016\/j.ins.2006.06.009","volume":"177","author":"W Zhu","year":"2007","unstructured":"Zhu W (2007) Topological approaches to covering rough sets. Inf Sci 177:1499\u20131508","journal-title":"Inf Sci"},{"key":"3616_CR51","doi-asserted-by":"publisher","first-page":"2478","DOI":"10.1016\/j.ins.2009.02.013","volume":"179","author":"W Zhu","year":"2009","unstructured":"Zhu W (2009) Relationship among basic concepts in covering-based rough sets. Inf Sci 179:2478\u20132486","journal-title":"Inf Sci"},{"key":"3616_CR52","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1016\/S0020-0255(03)00056-2","volume":"152","author":"W Zhu","year":"2003","unstructured":"Zhu W, Wang F (2003) Reduction and axiomatization of covering generalized rough sets. Inf Sci 152:217\u2013230","journal-title":"Inf Sci"},{"key":"3616_CR53","doi-asserted-by":"crossref","unstructured":"Zhu W, Wang F (2006) Properties of the first type of covering-based rough sets. In: Proceedings of the Sixth IEEE International Conference on Data Mining-Workshops (ICDMW\u201906), pp 407\u2013411","DOI":"10.1109\/ICDMW.2006.136"},{"issue":"8","key":"3616_CR54","doi-asserted-by":"publisher","first-page":"1131","DOI":"10.1109\/TKDE.2007.1044","volume":"19","author":"W Zhu","year":"2007","unstructured":"Zhu W, Wang F (2007) On three types of covering based rough sets. IEEE Trans Knowl Data Eng 19(8):1131\u20131144","journal-title":"IEEE Trans Knowl Data Eng"},{"key":"3616_CR55","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1016\/j.ins.2012.01.026","volume":"201","author":"W Zhu","year":"2012","unstructured":"Zhu W, Wang F (2012) The fourth type of covering-based rough sets. Inf Sci 201:80\u201392","journal-title":"Inf Sci"},{"key":"3616_CR56","doi-asserted-by":"publisher","first-page":"114","DOI":"10.1016\/j.knosys.2015.09.036","volume":"91","author":"X Zhang","year":"2016","unstructured":"Zhang X, Miao D, Liu C, Le M (2016) Constructive methods of rough approximation operators and multigranulation rough sets. Knowl-Based Syst 91:114\u2013125","journal-title":"Knowl-Based Syst"},{"key":"3616_CR57","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.ijar.2015.09.003","volume":"68","author":"Z Zhao","year":"2016","unstructured":"Zhao Z (2016) On some types of covering rough sets from topological points of view. Int J Approx Reason 68:1\u201314","journal-title":"Int J Approx Reason"}],"container-title":["Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-018-3616-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00500-018-3616-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-018-3616-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,11,8]],"date-time":"2019-11-08T00:17:42Z","timestamp":1573172262000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00500-018-3616-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,8]]},"references-count":57,"journal-issue":{"issue":"16","published-print":{"date-parts":[[2019,8]]}},"alternative-id":["3616"],"URL":"https:\/\/doi.org\/10.1007\/s00500-018-3616-9","relation":{},"ISSN":["1432-7643","1433-7479"],"issn-type":[{"value":"1432-7643","type":"print"},{"value":"1433-7479","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,11,8]]},"assertion":[{"value":"8 November 2018","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"Author declares that he has no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"This article does not contain any studies with human participants or animals performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Human and animal rights"}}]}}