Abstract
Dominance-based rough sets generalize classical indiscernibility based rough sets by handling ordered value sets of attributes and monotonic relationships between values of condition and decision attributes. Dominance based rough sets permit, in particular, a natural hybridization of fuzziness and roughness, which are complementary concepts of vagueness. In this article, we characterize the Dominance-based Rough Set Approach (DRSA) from the point of view of its mathematical foundations, taking into account algebraic structures and topological properties. We present algebraic representations of DRSA in terms of generalizations of several algebras already used to represent the classical rough set approach, namely: bipolar de Morgan Brouwer-Zadeh distributive lattice, bipolar Nelson algebra, bipolar Heyting algebra, bipolar double Stone algebra, bipolar three-valued Łukasiewicz algebra, bipolar Wajsberg algebra.We also present an algebraic model for ordinal classification. With respect to topological properties, using the concept of a bitopological space, we extend on DRSA the results obtained for classical rough sets.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bezhanishvili, G., Bezhanishvili, N., Gabelaia, D., Kurz, A.: Bitopological Duality for Distributive Lattices and Heyting Algebras. To appear in Mathematical Structures in Computer Science (2009)
Cattaneo, G., Ciucci, D.: Algebraic structures for rough sets. In: Peters, J.F., Skowron, A., Dubois, D., Grzymała-Busse, J.W., Inuiguchi, M., Polkowski, L. (eds.) Transactions on Rough Sets II. LNCS, vol. 3135, pp. 208–252. Springer, Heidelberg (2004)
Cattaneo, G., Nisticó, G.: Brouwer-Zadeh Posets and three valued Łukasiewicz posets. Fuzzy Sets and Systems 33, 165–190 (1989)
Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. Internat. J. General Systems 17, 191–209 (1990)
Dubois, D., Prade, H.: Putting rough sets and fuzzy sets together. In: Słowiński, R. (ed.) Intelligent Decision Support - Handbook of Applications and Advances of the Rough Sets Theory, pp. 203–232. Kluwer, Dordrecht (1992)
Figueira, J., Greco, S., Ehrgott, M. (eds.): Multiple Criteria Decision Analysis: State of the Art Surveys. Springer, Berlin (2005)
Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer, Dordrecht (1994)
Greco, S., Inuiguchi, M., Słowiński, R.: Dominance-based rough set approach using possibility and necessity measures. In: Alpigini, J.J., Peters, J.F., Skowron, A., Zhong, N. (eds.) RSCTC 2002. LNCS (LNAI), vol. 2475, pp. 85–92. Springer, Heidelberg (2002)
Greco, S., Matarazzo, B., Słowiński, R.: A new rough set approach to evaluation of bankruptcy risk. In: Zopounidis, C. (ed.) Operational Tools in the Management of Financial Risks, pp. 121–136. Kluwer, Dordrecht (1998)
Greco, S., Matarazzo, B., Słowiński, R.: The use of rough sets and fuzzy sets in MCDM. In: Gal, T., Stewart, T., Hanne, T. (eds.) Advances in Multiple Criteria Decision Making, vol. 14, pp. 14.1–14.59. Kluwer Academic Publishers, Boston (1999)
Greco, S., Matarazzo, B., Słowiński, R.: Rough set processing of vague information using fuzzy similarity relations. In: Calude, C., Paun, G. (eds.) From Finite to Infinite, pp. 149–173. Springer, Heidelberg (2000)
Greco, S., Matarazzo, B., Słowiński, R.: A fuzzy extension of the rough set approach to multicriteria and multiattribute sorting. In: Fodor, J., De Baets, B., Perny, P. (eds.) Preferences and Decisions under Incomplete Information, pp. 131–154. Physica-Verlag, Heidelberg (2000)
Greco, S., Matarazzo, B., Słowiński, R.: Rough sets theory for multicriteria decision analysis. European Journal of Operational Research 129, 1–47 (2001)
Greco, S., Matarazzo, B., Słowiński, R.: Dominance-Based Rough Set Approach to Knowledge Discovery (I) - General Perspective. In: Zhong, N., Liu, J. (eds.) Intelligent Technologies for Information Analysis, ch. 20, pp. 513–552. Springer, Berlin (2004)
Greco, S., Matarazzo, B., Słowiński, R.: Dominance-Based Rough Set Approach to Knowledge Discovery (II) - Extensions and Applications. In: Zhong, N., Liu, J. (eds.) Intelligent Technologies for Information Analysis, ch. 21, pp. 553–612. Springer, Berlin (2004)
Greco, S., Matarazzo, B., Słowiński, R.: Decision rule approach. In: Figueira, J., Greco, S., Ehrgott, M. (eds.) Multiple Criteria Decision Analysis: State of the Art Surveys, ch. 13, pp. 507–563. Springer, Berlin (2005)
Greco, S., Matarazzo, B., Słowiński, R.: Dominance-based Rough Set Approach as a proper way of handling graduality in rough set theory. In: Peters, J.F., Skowron, A., Marek, V.W., Orłowska, E., Słowiński, R., Ziarko, W.P. (eds.) Transactions on Rough Sets VII. LNCS, vol. 4400, pp. 36–52. Springer, Heidelberg (2007)
Greco, S., Matarazzo, B., Słowiński, R.: Fuzzy set extensions of the Dominance-based Rough Set Approach. In: Bustince, H., Herrera, F., Montero, J. (eds.) Fuzzy Sets and their Extensions, Representation, Aggregation and Models, pp. 239–261. Springer, Heidelberg (2007)
Greco, S., Matarazzo, B., Słowiński, R.: Algebraic structures for Dominance-based Rough Set Approach. In: Wang, G., Li, T., Grzymala-Busse, J.W., Miao, D., Skowron, A., Yao, Y. (eds.) RSKT 2008. LNCS (LNAI), vol. 5009, pp. 252–259. Springer, Heidelberg (2008)
Greco, S., Matarazzo, B., Słowiński, R.: Dominance-based Rough Set Approach and Bipolar Abstract Approximation Spaces. In: Chan, C.-C., Grzymala-Busse, J.W., Ziarko, W.P. (eds.) RSCTC 2008. LNCS (LNAI), vol. 5306, pp. 31–40. Springer, Heidelberg (2008)
Greco, S., Matarazzo, B., Słowiński, R.: Dominance-based rough set approach to interactive multiobjective optimization. In: Branke, J., Deb, K., Miettinen, K., Słowiński, R. (eds.) Multiobjective Optimization. LNCS, vol. 5252, pp. 121–155. Springer, Heidelberg (2008)
Kelly, J.C.: Bitopological spaces. Proc. London Math. Soc. 13(3), 71–89 (1963)
Michael, E.: Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71, 152–182 (1951)
Pagliani, P.: Rough Set and Nelson algebras. Fundamenta Informaticae 18, 1–25 (1993)
Pagliani, P.: Rough Set theory and logic-algebraic structures. In: Orłowska, E. (ed.) Incomplete Information: Rough Set Analysis, pp. 109–190. Physica, Heidelberg (1998)
Pawlak, Z.: Rough Sets. International Journal of Computer and Information Sciences 11, 341–356 (1982)
Pawlak, Z.: Rough Sets. Kluwer, Dordrecht (1991)
Polkowski, L.: Rough set: mathematical foundations. Physica-Verlag, Heidelberg (2002)
Pomykała, J., Pomykała, J.A.: The Stone algebra of rough sets. Bulletin of the Polish Academy of Sciences, Mathematics 36, 495–508 (1988)
Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2, 186–190 (1970)
Radzikowska, A.M., Kerre, E.E.: A comparative study of fuzzy rough sets. Fuzzy Sets and Systems 126, 137–155 (2002)
Słowiński, R., Greco, S., Matarazzo, B.: Rough set based decision support. In: Burke, E.K., Kendall, G. (eds.) Search Methodologies: Introductory Tutorials in Optimization and Decision Support Techniques, ch. 16, pp. 475–527. Springer, New York (2005)
Vakarelov, D.: Consequence relations and information systems. In: Słowiński, R. (ed.) Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory, pp. 391–399. Kluwer, Dordrecht (1992)
Wiweger, A.: On topological rough sets. Bulletin Polish Academia of Science Mathematics 37, 89–93 (1988)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Greco, S., Matarazzo, B., Słowiński, R. (2010). Algebra and Topology for Dominance-Based Rough Set Approach. In: Ras, Z.W., Tsay, LS. (eds) Advances in Intelligent Information Systems. Studies in Computational Intelligence, vol 265. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-05183-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-05183-8_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05182-1
Online ISBN: 978-3-642-05183-8
eBook Packages: EngineeringEngineering (R0)