NEAR GROUPS ON NEARNESS APPROXIMATION SPACES

Abstract

Near set theory provides a formal basis for observation, comparison and classification of perceptual granules. In the near set approach, every perceptual granule is a set of objects that have their origin in the physical world. Objects that have, in some degree, affinities are considered perceptually near each other, i.e., objects with similar descriptions. In this paper, firstly we introduce the concept of near groups, near subgroups, near cosets, near invariant sub-groups, homomorphisms and isomorphisms of near groups in nearness approximation spaces. Then we give some properties of these near structures.

Keywords

• Near set,Rough set,Approximation space,Nearness approximation space,Near group,2000 AMS Classification: 03 E 75

NEAR GROUPS ON NEARNESS APPROXIMATION SPACES

Abstract

References

1. Biswas, R. and Nanda, S. Rough groups and rough subgroups, Bull. Pol. AC. Math. 42, 251–254, 1994. [2] Davvaz, B. Rough sets in a fundamental ring, Bull. Iranian Math. Soc. 24 (2), 49–61, 1998. [3] Davvaz, B. Roughness in rings, Inform. Sci. 164, 147–163, 2004.
2. Davvaz, B. and Mahdavipour, M. Roughness in modules, Inform. Sci. 176 (24), 3658–3674, 2006.
3. Hassanien, A., Abraham, A., Peters, J., Schaefer, G. and Henry, C. Rough sets and near sets in medical imaging: A review, IEEE Trans Info. Tech. in Biomedicine 13 (6), 955–968, 2009.
4. Henry, C. Near Sets: Theory and Applications (Ph.D. Thesis (supervisor J. F. Peters), Department of Electrical & Computer Engineering, University of Manitoba, 2010)
5. Iwinski, T. B. Algebraic approach to rough sets, Bull. Pol. AC. Math. 35, 673–683, 1987.
6. Kuroki, N. Rough ideals in semigroups, Inform. Sci. 100, 139–163, 1997.
7. Kuroki, N. and Wang, P. P. The lower and upper approximations in a fuzzy group, Inform. Sci. 90, 203–220, 1996.
8. Miao, D., Han, S., Li, D. and Sun, L. Rough Group, Rough Subgroup and their Properties (Springer-Verlag, Heidelberg, 2005), 104–113.

9. Naimpally, S. and Peters, J. Topology with applications. Topological spaces via near and far, World Scientific, Singapore, 2012 (to appear).
10. Pawlak, Z. Classification of Objects by means of Attributes, Institute for Computer Science, Polish Academy of Sciences, Report 429, 1981.
11. Pawlak, Z. Rough sets, Int. J. Comput. Inform. Sci. 11 (5), 341–356, 1982.
12. Pawlak, Z., Peters, J. F. and Blisko, J. How near, Systemy Wspomagania Decyzji I 57 (109), 2002-2007.
13. Pawlak, Z. Rough Sets-Theoretical Aspects of : Reasoning about Data (Kluwer Academic Puplishers, Boston, London, Dordrecht, 1991).
14. Peters, J. F. Near sets. General theory about nearness of objects, Applied Mathematical Sciences 1 (53-56), 2609–2629, 2007.
15. Peters, J. F. Near sets, special theory about nearness of objects, Fund. Inform. 75 (1-4), 407–433, 2007. [18] Peters, J. F. Classification of perceptual objects by means of features, Int. J. Info. Technol. Intell. Comput. 3 (2), 1–35, 2008.
16. Peters, J. F. Sufficiently near sets of neighbourhoods, in: J. Yao, S. Ramanna, G. Wang, Z. Suraj (eds.), Rough Sets and Knowledge Technology, LNCS 6954, Springer, Berlin, 17–24, 2011.
17. Peters, J. F. and Naimpally, S. Approach spaces for near filters, Gen. Math. Notes 2 (1), 159–164, 2011. [21] Peters, J. F. and Tiwari, S. Approach merotopies and near filters, Gen. Math. Notes 3 (1), 1–15, 2011.
18. Polkowski, L. Rough Sets (Mathematical Foundations, Springer-Verlag, Heidelberg, 2002). [23] Skowron, A. and Stepaniuk, J. Tolerance approximation spaces, Fund. Inform. 27 (2-3), 245–253, 1996. [24] Wolski, M. Perception and classification. A note on near sets and rough sets, Fundamenta Informaticae 101, 143–155, 2010.
19. Wolski, M. Gauges, pregauges and completions: Some theoretical aspects of near and rough set approaches to data, in: J. Yao, S. Ramanna, G. Wang, Z. Suraj (eds.), Rough Sets ad Knowledge Technology, LNCS 6954, Springer, Berlin, 559–568, 2011.
20. Yao, Y. Y. On generalizing Pawlak approximation operators, Lecture Notes in Artificial Intelligence 1424, 298–307, 1994.