Some topological properties of topological rough groups


Let (UR) be an approximation space with U being non-empty set and R being an equivalence relation on U, and let \({\overline{G}}\) and \({\underline{G}}\) be the upper approximation and the lower approximation of subset G of U. A topological rough group G is a rough group \(G=({\underline{G}}, {\overline{G}})\) endowed with a topology, which is induced from the upper approximation space \({\overline{G}}\), such that the product mapping \(f: G\times G\rightarrow {\overline{G}}\) and the inverse mapping are continuous. In the class of topological rough groups, the relations of some separation axioms are obtained; some basic properties of the neighborhoods of the rough identity element and topological rough subgroups are investigated. In particular, some examples of topological rough groups are provided to clarify some facts about topological rough groups. Moreover, the version of open mapping theorem in the class of topological rough group is obtained. Further, some interesting open questions are posed.

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  1. Alharbi N, Altassan A, Aydi H, Özel C (2019) On topological rough groups. arXiv:1909.02500v1

  2. Altassan A, Alharbi N, Aydi H, Özel C (2020) Rough action on topological rough groups. Appl Gen Topol 21(2):295–304

    MathSciNet  MATH  Article  Google Scholar 

  3. Arhangel’ skiǐ AV, Tkachenko M (2008) Topological groups and related structures. Atlantis Press and World Sci, Paris

  4. Baǧırmaz N, Içen Ì, Özcan AF (2016) Topological rough groups. Topol Algebra Appl 4:31–38

    MathSciNet  MATH  Google Scholar 

  5. Baǧırmaz N, Özcan AF, Içen Ì (2016) Rough approximations in a topological group. Gen Math Notes 36:1–18

    Google Scholar 

  6. Biswas R, Nanda S (1994) Rough groups and rough subgroups. Bull Polish Acad Sci Math 42:251–254

    MathSciNet  MATH  Google Scholar 

  7. Bonikowaski Z (1995) Algebraic structures of rough sets. In: Ziarko WP (ed) Rough sets, fuzzy sets and knowledge discovery. Springer, Berlin, pp 242–C247

    Google Scholar 

  8. Cheng W, Mo Z, Wang J (2007) Notes on “the lower and upper approximations in a fuzzy group” and “rough ideals in semigroups.”. Inf Sci 177:5134–5140

    MATH  Article  Google Scholar 

  9. Davvaz B (2004) Roughness in rings. Inf Sci 164:147–163

    MathSciNet  MATH  Article  Google Scholar 

  10. D’eer L, Cornelis C, (2018) A comprehensive study of fuzzy covering-based rough set models: definitions, properties and interrelationships. Fuzzy Sets Syst 336:1–26

  11. Engelking R (1989) General Topology (revised and completed). Heldermann Verlag, Berlin

    Google Scholar 

  12. Frolík Z (1967) Homogeneity problems for extremally disconnected spaces. Comment Math Univ Carolin 8:757–763

    MathSciNet  MATH  Google Scholar 

  13. Iwinski T (1987) Algebraic approach to rough sets. Bull Polish Acad Sci Math 35:673–683

    MathSciNet  MATH  Google Scholar 

  14. Kondo M (2006) On the structure of generalized rough sets. Inf Sci 176:589–600

    MathSciNet  MATH  Article  Google Scholar 

  15. Kuroki N (1997) Rough ideals in semigroups. Inf Sci 100:139–C163

    MathSciNet  MATH  Article  Google Scholar 

  16. Kuroki N, Wang PP (1996) The lower and upper approximations in a fuzzy group. Inf Sci 90:203–220

    MathSciNet  MATH  Article  Google Scholar 

  17. Li F, Zhang Z (2014) The homomorphisms and operations of rough groups. Sci World J 507972:6

    Google Scholar 

  18. Li Z, Xie T, Li Q (2012) Topological structure of generalized rough sets. Comput Math Appl 63:1066–1071

    MathSciNet  MATH  Article  Google Scholar 

  19. Miao D, Han S, Li D, Sun L (2005) Rough group, rough subgroup and their properties. Ślkezak D et al. (eds.), RSFDGrC 2005, LNAI 3641, pp. 104–C113,

  20. Neelima CA, Isaac P (2014) Rough anti-homomorphism on a rough group. Global Math Sci 6:79–C80 Theory and Practical

    MATH  Google Scholar 

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

    MATH  Article  Google Scholar 

  22. Pomykala J, Pomykala JA (1998) The stone algebra of rough sets. Bull Polish Acad Sci Math 36:495–508

    MathSciNet  MATH  Google Scholar 

  23. Radzikowska AM, Kerre EE (2002) A comparative study of fuzzy rough sets. Fuzzy Sets Syst 126:137–155

    MathSciNet  MATH  Article  Google Scholar 

  24. Salama AS, Abd El-Monsef MME (2011) New topological approach of rough set generalizations. Int J Comput Math 8(7):1347–1357

    MathSciNet  MATH  Article  Google Scholar 

  25. Wang CZ, Chen DG (2010) A short note on some properties of rough groups. Comput Math Appl 59:431–436

    MathSciNet  MATH  Article  Google Scholar 

  26. Wang CZ, Chen DG (2013) On rough approximations of groups. Int J Mach Learn Cyber 4:445–449

    Article  Google Scholar 

  27. Wu GB, Huang B (2011) Defect and revision of definition of rough group. J Nanjing Normal Univ 34:39–42 Natural Sci. Edition

    MathSciNet  Google Scholar 

  28. Wu WZ, Mi JS (2019) The Mathematical Structure of Rough Set. Science Press, China

    Google Scholar 

  29. Vlach M (2008) Algebraic and topological aspects of rough set theory. In: Hiroshima University, Fourth International Workshop on Computational Intelligence Applications, Japan, IEEE SCM Hiroshima Chapter

  30. Zhang YL, Li CQ, Li JJ (2019) On characterizations of a pair of covering-based approximation operators. Soft Comput 23:3965–3972

    MATH  Article  Google Scholar 

  31. Zhang YL, Li JJ, Li CQ (2016) Topological structure of relation-based generalized rough sets. Fund Inf 147:477–491

    MathSciNet  MATH  Google Scholar 

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The first author is supported by the Key Program of the Natural Science Foundation of Fujian Province (No: 2020J02043), the NSFC (No. 11571158), the Institute of Meteorological Big Data-Digital Fujian and Fujian Key Laboratory of Data Science and Statistics. The second author is supported by the Young and middle-aged project in Fujian Province (No. JAT190397). The fourth author is supported by the NSFC (No. 11871259).

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Correspondence to Fucai Lin.

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This paper is dedicated to professor Shou Lin on the occasion of his 60th birthday.

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Lin, F., Sun, Q., Lin, Y. et al. Some topological properties of topological rough groups. Soft Comput (2021).

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  • Rough group
  • Topological rough group
  • Topological rough subgroup
  • Strongly topological rough group
  • Connected; extremally disconnected
  • Separation axiom
  • Rough homomorphism
  • Rough kernel
  • Topological group
  • Lower approximation
  • Upper approximation