Abstract
In this paper, we introduce the notion of generalized algebraic lower (upper) approximation operator and give its characterization theorem. That is, for any atomic complete Boolean algebra \({\mathcal B}\) with the set \({\mathcal A}({\mathcal B})\) of atoms, a map \(L: {\mathcal B} \rightarrow {\mathcal B}\) is an algebraic lower approximation operator if and only if there exists a binary relation R on \({\mathcal A}({\mathcal B})\) such that L = R −, where R − is the lower approximation defined by the binary relation R. This generalizes the results given by Yao.
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References
Birkhoff, G.: Lattice Theory. In: AMS, Providence Rhode Island (1995)
Davey, B.A., Priestly, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (2002)
Gediga, G., Düntsch, I.: Modal-style operators in qualitative data analysis. In: Proceedings of the 2002 IEEE International Conference on Data Mining, pp. 155–162 (2002)
Hughes, G.E., Cresswell, M.J.: A companion to modal logic. Methuen, London (1984)
Iwinski, T.B.: Algebraic approach to rough sets. Bull. Pol. Ac. Math. 35, 673–683 (1987)
Jonsson, B., Tarski, A.: Boolean algebras with operators. Part I. American Jour. of Math. 73, 891–939 (1951)
Jarvinen, J.: Knowledge representation and rough sets. TUCS Dissertations 14, Turku Center for Computer Science, Turku, Finland (1999)
Jarvinen, J.: On the structure of rough approximations. TUCS Technical report 447, Turku Center for Computer Science, Turku, Finland (2002)
Goldblatt, R.: Logics of time and computation. CSLI Lecture Notes 7 (1987)
Kondo, M.: On the structure of generalized rough sets. To appear in Information Sciences
Orlowska, E.: Rough set semantics for non-classical logics. In: Ziarko, W. (ed.) Rough Sets, Fuzzy Sets and Knowledge Discovery, pp. 143–148. Springer, Heidelberg (1994)
Pawlak, Z.: Rough sets. Int. J.Inform. Comp.Sci. 11, 341–356 (1982)
Polkowski, L.: Rough Sets: Mathematical Foundations. Physica/Springer, Heidelberg (2002)
Vakarelov, D.: Modal logics for knowledge representation systems. Theoretical Computer Science 90, 433–456 (1991)
Yao, Y.Y.: Two views of the theory of rough sets in finite universes. Int. J. Approximate Reasoning 15, 291–317 (1996)
Yao, Y.Y.: Constructive and algebraic methods of the theory of rough sets. Information Sciences 109, 21–47 (1998)
Yao, Y.Y., Lin, T.Y.: Generalization of rough sets using modal logic. Intelligent Automation and Soft Computing, An International Journal 2, 103–120 (1996)
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Kondo, M. (2005). Algebraic Approach to Generalized Rough Sets. In: Ślęzak, D., Wang, G., Szczuka, M., Düntsch, I., Yao, Y. (eds) Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing. RSFDGrC 2005. Lecture Notes in Computer Science(), vol 3641. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11548669_14
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DOI: https://doi.org/10.1007/11548669_14
Publisher Name: Springer, Berlin, Heidelberg
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