# The characterizations of upper approximation operators based on coverings

- 8 Downloads

## Abstract

In this paper, We propose a condition of symmetry for the covering \(\mathscr {C}\) in a covering-based approximation space \((U,\mathscr {C})\). By using this condition, we obtain general, topological and intuitive characterizations of the covering \({\mathscr {C}}\) for two types of covering-based upper approximation operators being closure operators. We investigate axiomatic systems for \(\overline{apr}_{S}\) and discuss the relationships among upper approximation operators. We also give a description of \((U,{\mathscr {C}})\) in terms of information exchange systems when these operators are closure ones. We also solve an open problem raised by Ge et al.

## Keywords

Closure operator Covering-based upper approximation operator Partition Third condition of symmetry## Notes

### Acknowledgements

This work is supported by the Natural Science Foundation of China (No. 11371130) and the Natural Science Foundation of Guangxi (Nos. 2014GXNSFBA118015, 2016CSOBDP0004, 2016JC2014 and KY2015YB244).

## Compliance with ethical standards

## Conflict of interest

The authors declare that there is no conflict of interest.

## Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors.

## References

- Bian X, Wang P, Yu Z, Bai X, Chen B (2015) Characterizations of coverings for upper approximation operators being closure operators. Inf Sci 314:41–54MathSciNetCrossRefGoogle Scholar
- Bonikowski Z, Bryniarski E, Skardowska UW (1998) Extensions and intentions in the rough set theory. Inf Sci 107:149–167MathSciNetCrossRefMATHGoogle Scholar
- Bryniarski E (1989) A calculus of rough sets of the first order. Bull Pol Acad Sci Math 16:71–78MathSciNetMATHGoogle Scholar
- Cattaneo G (1998) Abstract approximation spaces for rough theory. In: Rough sets in knowledge discovery 1: methodology and applications, pp 59–98Google Scholar
- Cattaneo G, Ciucci D (2004) Algebraic structures for rough sets. In: LNCS, vol 3135, pp 208–252Google Scholar
- Chen D, Wang C (2007) A new aooproach to arrtibute reduction of consistent and inconsistent covering decision systems with covering rough sets. Inf Sci 176:3500–3518MATHGoogle Scholar
- Chen J, Li J, Lin Y, Lin G, Ma Z (2015) Relations of reduction between covering generalized rough sets and concept lattices. Inf Sci 304:16–27MathSciNetCrossRefMATHGoogle Scholar
- Deer L, Restrepo M, Cornelis C, Gmez J (2016) Neighborhood operators for covering-based rough sets. Inf Sci 336:21–44CrossRefGoogle Scholar
- Engelking R (1989) General topology. Heldermann Verlag, BerlinMATHGoogle Scholar
- Fan N, Hu G, Liu H (2011) Study of definable subsets in covering approximation space of rough sets. In: Proceedings of the 2011 IEEE international conference on information reuse and integration, vol 1, pp 21–24Google Scholar
- Fan N, Hu G, Zhang W (2012) Study on conditions of neighborhoods forming a partition. In: Fuzzy systems and knowledge discovery (FSKD), pp 256–259Google Scholar
- Gao JS et al (2008) A new conflict analysis model based on rough set theory. Chin J Manag 5:813–818 (
**in Chinese**)Google Scholar - Ge X (2010) An application of covering approximation spaces on network security. Comput Math Appl 60:1191–1199MathSciNetCrossRefMATHGoogle Scholar
- Ge X (2014) Connectivity of covering approximation spaces and its applications onepidemiological issue. Appl Soft Comput 25:445–451CrossRefGoogle Scholar
- Ge X, Li Z (2011) Definable subset in covering approximation spaces. Int J Comput Math Sci 5:31–34MathSciNetGoogle Scholar
- Ge X, Bai X, Yun Z (2012) Topological characterizations of covering for special covering-based upper approximation operators. Inf Sci 204:70–81MathSciNetCrossRefMATHGoogle Scholar
- Ge X, Bai X, Yun Z (2012) Topological characterizations of covering for special covering-based upper approximation operators. Inf Sci 204:70–81MathSciNetCrossRefMATHGoogle Scholar
- Kondo M (2005) On the structure of generalized rough sets. Inf Sci 176:589–600MathSciNetCrossRefMATHGoogle Scholar
- Lin TY (1997) Neighborhood systems C application to qualitative fuzzy and rough sets. In: Wang PP (ed) Advances in machine intelligence and soft computing IV. Department of Electrical Engineering, Durham, pp 132–155Google Scholar
- Liu J et al (2003) Application study of rough comprehensive evaluation method in green manufacturing evaluation. Chongqing Environ Sci 12:65–67 (
**in Chinese**)Google Scholar - Liu G (2006) The axiomatization of the rough set upper approximation operations. Fundam Inf 69:331–342MathSciNetMATHGoogle Scholar
- Liu G (2008) Axiomatic systems for rough sets and fuzzy rough sets. Int J Approx Reason 48:857–867MathSciNetCrossRefMATHGoogle Scholar
- Liu G (2013) Using one axiom to characterize rough set and fuzzy rough set approximations. Inf Sci 223:285–296MathSciNetCrossRefMATHGoogle Scholar
- Liu G, Sai Y (2009) A comparison of two types of rough sets induced by coverings. Int J Approx Reason 50:521–528MathSciNetCrossRefMATHGoogle Scholar
- Liu G, Zhu W (2008) The algebraic structures of generalized rough set theory. Inf Sci 178:4105–4113MathSciNetCrossRefMATHGoogle Scholar
- Mrozek A (1996) Methodology of rough controller synthesis. In: Proceedings of the IEEE international conference on fuzzy systems, pp 1135–1139Google Scholar
- Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356CrossRefMATHGoogle Scholar
- Pawlak Z (1991) Rough sets: theoretical aspects of reasoning about data. Kluwer Academic Publishers, BostonCrossRefMATHGoogle Scholar
- Pawlak Z, Skowron A (2007) Rudiments of rough sets. Inf Sci 177:3–27MathSciNetCrossRefMATHGoogle Scholar
- Pawlak Z, Skowron A (2007) Rough sets: some extensions. Inf Sci 177:28–40MathSciNetCrossRefMATHGoogle Scholar
- Pawlak Z, Skowron A (2007) Rough sets and boolean reasoning. Inf Sci 177:41–73MathSciNetCrossRefMATHGoogle Scholar
- Pomykala JA (1987) Approximation operations in approximation space. Bull Pol Acad Sci Math 35:653–662MathSciNetMATHGoogle Scholar
- Qin K, Pei Z (2005) On the topological properties of fuzzy rough sets. Fuzzy Sets Syst 151:601–613MathSciNetCrossRefMATHGoogle Scholar
- Qin K, Gao Y, Pei Z (2007) On covering rough sets. In: Lecture notes in artificial intelligence, vol 4481, pp 34–41Google Scholar
- Samanta P, Chakraborty MK (2009) Covering based approaches to rough sets and implication lattices. In: RSFDGRC 2009 LANI 5908, pp 127–134Google Scholar
- Skowron A (1989) The relationship between the rough sets and evidence theory. Bull Pol Acad Sci Math 37:160–173Google Scholar
- Skowron A, Stepaniuk J (1996) Tolerance approximation spaces. Fundam Inf 272:45–253MathSciNetMATHGoogle Scholar
- Slowinski R, Vanderpooten D (2000) A generalized definition of rough approximations based on similarity. IEEE Trans Knowl Data Eng 12:331–336CrossRefGoogle Scholar
- Tang XJ et al (2015) Application of rough set theory in item cognitive attribute identification. Acta Psychol Sin 47:950–957 (
**in Chinese**)Google Scholar - Thomas GB (2003) Thomas calculus, 10th edn. Addison Wesley Publishing Company, ReadingGoogle Scholar
- Tsumoto S (1996) Automated discovery of medical expert system rules from clinical database on rough set. In: Proceedings of the second international conference on knowledge discovery and data mining, vol 32, pp 63–72Google Scholar
- Wang JC et al (2004) Study on the application of rough set theory in substrate feeding control and fault diagnosis in fermentation process. Comput Eng Appl 16:203–205 (
**in Chinese**)Google Scholar - Yang B, Hu B (2016) A fuzzy covering-based rough set model and its generalization over fuzzy lattice. Inf Sci 367–368:463–486CrossRefGoogle Scholar
- Yang T, Li Q, Zhou B (2010) Reduction about approximation spaces of covering generalized rough sets. Int J Approx Reason 51:335–345MathSciNetCrossRefMATHGoogle Scholar
- Yao Y (1998) Relational interpretations of neighborhood operators and rough set approximation operators. Inf Sci 111:239–259MathSciNetCrossRefMATHGoogle Scholar
- Yao Y (1998) A comparative study of fuzzy sets and rough sets. Inf Sci 109:227–242MathSciNetCrossRefMATHGoogle Scholar
- Yao Y (1998) Constructive and algebraic methods of theory of rough sets. Inf Sci 109:21–47MathSciNetCrossRefMATHGoogle Scholar
- Yao Y, Yao B (2012) Covering based rough set approximation. Inf Sci 200:91–107MathSciNetCrossRefMATHGoogle Scholar
- Yu Z, Bai X, Qiu Z (2013) A study of rough sets based on 1-neighborhood systems. Inf Sci 248:103–113MathSciNetCrossRefMATHGoogle Scholar
- Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefMATHGoogle Scholar
- Zadeh L (1996) Fuzzy logic = computing with words. IEEE Trans Fuzzy Syst 4:103–111CrossRefGoogle Scholar
- Zakowski W (1983) Approximations in the space\( (U,\Pi ) \). Demonstr Math 16:761–769MATHGoogle Scholar
- Zhang GH et al (2017) Sensory quality prediction of tobacco based on rough sets and gray system. Comput Appl Chem 34:163–166 (
**in Chinese**)CrossRefGoogle Scholar - Zhang DZH et al (2017) Research on the model of audit opinion prediction based on integration of neighborhood rough sets and neural network. J Chongqing Univ Technol 31:96–99 (
**in Chinese**)Google Scholar - Zhang YL, Luo MK (2013) Relationships beween covering-based rough sets and relation-based rough sets. Inf Sci 225:55–71CrossRefMATHGoogle Scholar
- Zhang Y, Li C, Lin M, Lin Y (2015) Relationships between generalized rough sets based on covering and reflexive neighborhood system. Inf Sci 319:56–67MathSciNetCrossRefGoogle Scholar
- Zhao D et al (2010) Classification of biological data based on rough sets. Comput Mod 7:96–99 (
**in Chinese**)Google Scholar - Zhu W (2007) Topological approaches to covering rough sets. Inf Sci 177:1499–1508MathSciNetCrossRefMATHGoogle Scholar
- Zhu W (2007) Generalized rough sets based on relations. Inf Sci 177:4997–5011MathSciNetCrossRefMATHGoogle Scholar
- Zhu W (2009) Relationship between generalized rough sets based on binary relation and covering. Inf Sci 179:210–225MathSciNetCrossRefMATHGoogle Scholar
- Zhu W (2009) Relationship among basic concepts in covering-based rough sets. Inf Sci 179:2478–2486MathSciNetCrossRefMATHGoogle Scholar
- Zhu W, Wang FY (2007) On three types of covering rough sets. IEEE Trans Knowl Data Eng 19:1131–1144CrossRefGoogle Scholar
- Zhu W, Wang FY (2012) The fourth type of covering-based rough sets. Inf Sci 1016:1–13MathSciNetGoogle Scholar
- Zhu W, Zhang WX (2002) Neighborhood operators systems and approximations. Inf Sci 144:201–217MathSciNetCrossRefGoogle Scholar
- Zhu W, Wang F (2007) Properties of the third type of covering-based rough sets. In: ICMLC07, pp 3746–3751Google Scholar