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The structures and the connections on four types of covering rough sets

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Abstract

Covering rough set model is an important extension of Pawlak rough set model, and its structure is the foundation of covering rough set theory. This paper considers four covering approximations and studies the structures of the families of their covering upper (or lower) definable sets by means of lattice theory. We provide some conditions under which the families of covering upper (or lower) definable sets with respect to these covering approximations are lattices of sets, or distributive lattices, or geometric lattices, or Boolean lattices. Furthermore, based on these results, we give the relationship among the four covering approximations and establish the connection between matroids and covering rough sets from the viewpoint of lattice theory.

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References

  • Bonikowski Z, Bryniarski E, Wybraniec-Skardowska U (1998) Extensions and intentions in the rough set theory. Inf Sci 107(1):149–167

    Article  MathSciNet  MATH  Google Scholar 

  • Birkhoff G (1935) Abstract linear dependence and lattices. Am J Math 57(4):800–804

    Article  MathSciNet  MATH  Google Scholar 

  • 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 176:1829–1848

    Article  MathSciNet  MATH  Google Scholar 

  • 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):3500–3518

    Article  MathSciNet  MATH  Google Scholar 

  • Davey BA, Priestley HA (1990) Introduction to lattices and order. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • D’eer 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 191–201

  • D’eer L, Restrepo M, Cornelis C, Gómez J (2016) Neighborhood operators for covering-based rough sets. Inf Sci 336:21–44

    Article  MATH  Google Scholar 

  • D’eer L, Cornelis C, Yao Y (2016) A semantically sound approach to Pawlak rough sets and covering-based rough sets. Int J Approx Reason 78:62–72

    Article  MathSciNet  MATH  Google Scholar 

  • Grätzer G (2011) Lattice theory: foundation. Birkhäuser, Basel

    Book  MATH  Google Scholar 

  • Ge X, Wang P, Yun Z (2017) The rough membership functions on four types of covering-based rough sets and their applications. Inf Sci 390:1–14

    Article  MathSciNet  Google Scholar 

  • Huang A, Zhao H, Zhu W (2014) Nullity-based matroid of rough sets and its application to attribute reduction. Inf Sci 263:153–165

    Article  MathSciNet  MATH  Google Scholar 

  • Kondo M (2005) On the structure of generalized rough sets. Inf Sci 176:589–600

    Article  MathSciNet  MATH  Google Scholar 

  • Liu G, Zhu W (2008) The algebraic structures of generalized rough set theory. Inf Sci 178:4105–4113

    Article  MathSciNet  MATH  Google Scholar 

  • Liu G, Zhu K (2014) The relationship among three types of rough approximation pairs. Knowl-Based Syst 60:28–34

    Article  Google Scholar 

  • Liu G, Sai Y (2009) A comparison of two types of rough sets induced by coverings. Int J Approx Reason 50(3):521–528

    Article  MathSciNet  MATH  Google Scholar 

  • Li T, Wu W (2011) Attribute reduction in formal contexts: a covering rough set approach. Fund Inf 111:15–32

    MathSciNet  MATH  Google Scholar 

  • Li X, Yi H, Liu S (2016) Rough sets and matroids from a lattice-theoretic viewpoint. Inf Sci 342:37–52

    Article  MathSciNet  MATH  Google Scholar 

  • Li X, Liu S (2012) Matroidal approaches to rough sets via closure operators. Int J Approx Reason 53:513–527

    Article  MathSciNet  MATH  Google Scholar 

  • Lin G, Liang J, Qian Y (2013) Multigranulation rough sets: from partition to covering. Inf Sci 241:101–118

    Article  MathSciNet  MATH  Google Scholar 

  • Liang J, Wang F, Dang C, Qian Y (2012) An efficient rough feature selection algorithm with a multi-granulation view. Int J Approx Reason 53:912–926

    Article  MathSciNet  Google Scholar 

  • Medina J (2012) Multi-adjoint property-oriented and object-oriented concept lattices. Inf Sci 190:95–106

    Article  MathSciNet  MATH  Google Scholar 

  • Oxley JG (1992) Matroid theory. Oxford University Press, New York

    MATH  Google Scholar 

  • Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356

    Article  MATH  Google Scholar 

  • Pawlak Z, Skowron A (2007) Rough sets and Boolean reasoning. Inf Sci 177(1):41–73

    Article  MathSciNet  MATH  Google Scholar 

  • Pei Z, Pei D, Zheng L (2011) Topology vs generalized rough sets. Int J Approx Reason 52:231–239

    Article  MathSciNet  MATH  Google Scholar 

  • Pomykala J (1987) Approximation operators in approximation space. Bull Polish Acad Sci Math 35:653–662

    MathSciNet  MATH  Google Scholar 

  • Pomykala J (1988) On definability in the nondeterministic information system. Bull Pol Acad Sci Math 36(3–4):193–210

    MathSciNet  MATH  Google Scholar 

  • Qin K, Gao Y, Pei Z (2007) On covering rough sets. In: Rough sets and knowledge technology. Springer, Berlin, 34–41

  • Qin K, Yang J, Pei Z (2008) Generalized rough sets based on reflexive and transitive relations. Inf Sci 178:4138–4141

    Article  MathSciNet  MATH  Google Scholar 

  • She Y, He X (2012) On the structure of the multigranulation rough set model. Knowl-Based Syst 36:81–92

    Article  Google Scholar 

  • Su L, Zhu W (2017) Closed-set lattice and modular matroid induced by covering-based rough sets. Int J Mach Learn Cybern 8:191–201

    Article  Google Scholar 

  • Tsumoto S, Tanaka A (1993) Rough sets and matroid theory. In: Ziarko W (ed) Rough sets, fuzzy sets and knowledge discovery. Springer, London, pp 290–297

    Google Scholar 

  • Tsumoto S, Tanaka H (1995) Algebraic specification of empirical inductive learning methods based on rough sets and matroid theory. LNCS 985:224–243

    MathSciNet  Google Scholar 

  • Wang S, Zhu Q, Zhu W, Min F (2013) Quantitative analysis for covering-based rough sets through the upper approximation number. Inf Sci 220:483–491

    Article  MathSciNet  MATH  Google Scholar 

  • Wang S, Zhu Q, Zhu W, Min F (2012) Matroidal structure of rough sets and its characterization to attribute reduction. Knowl-Based Syst 36:155–161

    Article  Google Scholar 

  • Wu W, Leung Y (2013) Optimal scale selection for multi-scale decision tables. Int J Approx Reason 54:1107–1129

    Article  MathSciNet  MATH  Google Scholar 

  • Wolski M (2006) Complete orders, categories and lattices of approximations. Fund Inf 72:421–435

    MathSciNet  MATH  Google Scholar 

  • Wang Z, Shu L, Ding X (2013) Minimal description and maximal description in covering-based rough sets. Fund Inf 128:503–526

    MathSciNet  MATH  Google Scholar 

  • Wang Z, Wang H, Feng Q, Shu L (2015) The approximation number function and the characterization of covering approximation space. Inf Sci 305:196–207

    Article  MathSciNet  MATH  Google Scholar 

  • Xu Z, Wang Q (2005) On the properties of covering rough sets model (in Chinese). J Henan Norm Univ 33(1):130–132

    MathSciNet  MATH  Google Scholar 

  • Yao Y (1998a) Constructive and algebraic methods of theory of rough sets. Inf Sci 109:21–47

    Article  MathSciNet  MATH  Google Scholar 

  • Yao Y (1998b) Relational interpretations of neighborhood operators and rough set approximation operators. Inf Sci 111:239–259

    Article  MathSciNet  MATH  Google Scholar 

  • Yao Y, Yao B (2012) Covering based rough sets approximations. Inf Sci 200:91–107

    Article  MathSciNet  MATH  Google Scholar 

  • Yao Y, She Y (2016) Rough set models in multigranulation spaces. Inf Sci 327:40–56

    Article  MathSciNet  MATH  Google Scholar 

  • Yang T, Li Q (2010) Reduction about approximation spaces of covering generalized rough sets. Int J Approx Reason 51:335–345

    Article  MathSciNet  MATH  Google Scholar 

  • Yang X, Qi Y, Song X, Yang J (2013) Test cost sensitive multigranulation rough set: model and minimal cost selection. Inf Sci 250:184–199

    Article  MathSciNet  MATH  Google Scholar 

  • Żakowski W (1983) Axiomatization in the space (U, \(\varPi \)). Demonstr Math XVI: 761–769

  • Zhu W (2007) Generalized rough sets based on relations. Inf Sci 177:4997–5011

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu W (2009) Relationship between generalized rough sets based on binary relation and covering. Inf Sci 179:210–225

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu W (2007) Topological approaches to covering rough sets. Inf Sci 177:1499–1508

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu W (2009) Relationship among basic concepts in covering-based rough sets. Inf Sci 179:2478–2486

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu W, Wang F (2003) Reduction and axiomatization of covering generalized rough sets. Inf Sci 152:217–230

    Article  MATH  Google Scholar 

  • 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’06), pp 407–411

  • Zhu W, Wang F (2007) On three types of covering based rough sets. IEEE Trans Knowl Data Eng 19(8):1131–1144

    Article  Google Scholar 

  • Zhu W, Wang F (2012) The fourth type of covering-based rough sets. Inf Sci 201:80–92

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang X, Miao D, Liu C, Le M (2016) Constructive methods of rough approximation operators and multigranulation rough sets. Knowl-Based Syst 91:114–125

    Article  Google Scholar 

  • Zhao Z (2016) On some types of covering rough sets from topological points of view. Int J Approx Reason 68:1–14

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank all of the editors and reviewers for their constructive comments as well as helpful suggestions, which have substantially improved this paper. This work is supported by the Foundation of Shanxi Normal University (Grant No. 872022).

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi, China

    Zhaohao Wang, Hong Wang & Qinrong Feng

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  1. Zhaohao Wang
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  2. Hong Wang
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  3. Qinrong Feng
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Correspondence to Zhaohao Wang.

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Wang, Z., Wang, H. & Feng, Q. The structures and the connections on four types of covering rough sets. Soft Comput 23, 6727–6741 (2019). https://doi.org/10.1007/s00500-018-3616-9

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