Skip to main content
Springer Nature Link
Log in

Two Kinds of Rough Algebras and Brouwer-Zadeh Lattices

  • Conference paper
Rough Sets and Current Trends in Computing (RSCTC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4259))

Included in the following conference series:

Abstract

Many researchers study rough sets from the point of view of description of the rough set pairs(a rough set pair is also called a rough set), i.e. <lower approximation set, upper approximation set>. Comer [4] showed that all the rough sets in an approximation space constructed a regular double Stone algebra. The constructed algebra is called the rough double Stone algebra in this paper. Pagliani [19] interpreted Rough Set System (all the rough sets in an approximation space in disjoint representation) as a Nelson algebra. The constructed Nelson algebra from an approximation space is called the rough Nelson algebra in this paper. It is showed that a rough double Stone algebra is a Brouwer-Zadeh lattice, and a rough Nelson algebra is a Brouwer-Zadeh lattice also.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Banerjee, M., Chakraborty, M.K.: Rough sets through algebraic logic. Fundamenta Informaticae 28, 211–221 (1996)

    MATH  MathSciNet  Google Scholar 

  2. Cattaneo, G., Ciucci, D.: Heyting Wajsberg Algebras as an Abstract Environment Linking Fuzzy and Rough Sets. In: Alpigini, J.J., Peters, J.F., Skowron, A., Zhong, N. (eds.) RSCTC 2002. LNCS (LNAI), vol. 2475, pp. 77–84. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  3. Cattaneo, G., Giuntini, R., Pilla, R.: BZMVdM algebras and stonian MV-algebras. Fuzzy Sets and Systems 108, 201–222 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Comer, S.: On connections between information systems, rough sets and algebraic logic. In: Algebraic methods in logic and computer science, pp. 117–124. Banach Center Publications (1993)

    Google Scholar 

  5. Dai, J.H.: Generalization of rough set theory using molecular lattices. Chinese Journal of Computers (in Chinese) 27, 1436–1440 (2004)

    Google Scholar 

  6. Dai, J.-H.: Structure of rough approximations based on molecular lattices. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymała-Busse, J.W. (eds.) RSCTC 2004. LNCS, vol. 3066, pp. 69–77. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  7. Dai, J.-H.: Logic for Rough Sets with Rough Double Stone Algebraic Semantics. In: Ślęzak, D., Wang, G., Szczuka, M.S., Düntsch, I., Yao, Y. (eds.) RSFDGrC 2005. LNCS, vol. 3641, pp. 141–148. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  8. Dai, J.H., Chen, W.D., Pan, Y.H.: A minimal axiom group of rough set based on Quasi-ordering. Journal of Zhejiang University SCIENCE 7, 810–815 (2004)

    Article  Google Scholar 

  9. Dai, J.H., Chen, W.D., Pan, Y.H.: Sequent caculus system for rough sets based on rough Stone algebras. In: Proc. of IEEE International Conference on Granular Computing (IEEE GrC 2005), pp. 423–426. IEEE Press, New Jersy (2005)

    Google Scholar 

  10. Dai, J.H., Chen, W.D., Pan, Y.H.: Rough Sets and Brouwer-Zadeh Lattices. In: Wang, G.-Y., Peters, J.F., Skowron, A., Yao, Y. (eds.) RSKT 2006. LNCS, vol. 4062, pp. 200–207. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  11. Dai, J.H., Pan, Y.H.: On rough algebras (in Chinese). Journal of Software 16, 1197–1204 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Düntsch, I.: Rough relation algebras. Fundamenta Informace 21, 321–331 (1994)

    MATH  Google Scholar 

  13. Düntsch, I.: Rough sets and algebra of relations. In: Orlowska, E. (ed.) Incomplete information: Rough set analysis, pp. 95–108. Physica-Verlag, Herdberg (1998)

    Google Scholar 

  14. Gehrke, M., Walker, E.: On the structure of rough sets. Bulletin of the Polish Academy of Sciences: Mathematics 40, 235–255 (1992)

    MATH  MathSciNet  Google Scholar 

  15. Iturrioz, L.: Rough sets and 3-valued structures. In: Orlowska, E. (ed.) Logic at work, pp. 596–603. Springer, Heidelberg (1998)

    Google Scholar 

  16. Iwiński, T.B.: Algebraic approach to rough sets. Bulletin of the Polish Academy of Sciences: Mathematics 35, 673–683 (1987)

    MATH  MathSciNet  Google Scholar 

  17. Järvinen, J.: On the structure of rough approximations. In: Alpigini, J.J., Peters, J.F., Skowron, A., Zhong, N. (eds.) RSCTC 2002. LNCS, vol. 2475, pp. 123–130. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  18. Lin, T.Y., Liu, Q.: Rough approximate operators: Axiomatic rough set theory. In: Ziarko, W.P. (ed.) Rough Sets, Fuzzy Sets and Knowledge Discovery, pp. 256–260. Springer, Berlin (1994)

    Google Scholar 

  19. Pagliani, P.: Rough sets and Nelson algebras. Fundamenta Informaticae 27, 205–219 (1996)

    MATH  MathSciNet  Google Scholar 

  20. Pagliani, P.: Rough set theory and logic-algebraic structures. In: Orlowska, E. (ed.) Incomplete information: Rough set analysis, pp. 109–190. Physica-Verlag, Heidelberg (1998)

    Google Scholar 

  21. Pawlak, Z.: Rough Sets-Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)

    MATH  Google Scholar 

  22. Pomykala, J., Pomykala, J.A.: The Stone algebra of rough sets. Bulletin of the Polish Academy of Sciences: Mathematics 36, 495–508 (1988)

    MATH  MathSciNet  Google Scholar 

  23. Yao, Y.Y.: Constructive and algebraic methods of the theory of rough sets. Information Sciences 109, 21–47 (1998)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Artificial Intelligence, Zhejiang University, Hangzhou, 310027, P.R. China

    Jian-Hua Dai, Weidong Chen & Yunhe Pan

  2. Department of Information Management, Zhejiang Police Vocational Academy, Hangzhou, 310018, P.R. China

    Hanfei Lv

Authors
  1. Jian-Hua Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hanfei Lv
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Weidong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yunhe Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Economics, University of Catania, Corso Italia, 55, 95129, Catania, Italy

    Salvatore Greco

  2. Graduate School of Engineering, Department of Electrical Engineering and Computer Sciences, University of Hyogo, 2167 Shosha, 671-2280,, Himeji, Hyogo, Japan

    Yutaka Hata

  3. Department of Medical Informatics, Faculty of Medicine, Shimane University, 89-1 Enya-cho, Izumo, 693-8501, Shimane, Japan

    Shoji Hirano

  4. Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyama, Toyonaka, 560-8531, Osaka, Japan

    Masahiro Inuiguchi

  5. Department of Risk Engineering, School of Systems and Information Engineering, University of Tsukuba, 305-8573, Ibaraki, Japan

    Sadaaki Miyamoto

  6. Institute of Mathematics, Warsaw University, Banacha 2, 02-097, Warsaw, Poland

    Hung Son Nguyen

  7. Systems Research Institute, Polish Academy of Sciences, 01-447, Warsaw, Poland

    Roman Słowiński

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Dai, JH., Lv, H., Chen, W., Pan, Y. (2006). Two Kinds of Rough Algebras and Brouwer-Zadeh Lattices. In: Greco, S., et al. Rough Sets and Current Trends in Computing. RSCTC 2006. Lecture Notes in Computer Science(), vol 4259. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11908029_12

Download citation

  • DOI: https://doi.org/10.1007/11908029_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-47693-1

  • Online ISBN: 978-3-540-49842-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics