An Incidence Algebra Approach to Knowledge Granulation in Pawlak Information Systems

• Autorzy: Wolski M. , Gomolińska A.
• Języki publikacji: EN

Abstrakty

EN

Representation theory is a branch of mathematics whose original purpose was to represent information about abstract algebraic structures by means of methods of linear algebra (usually, by linear transformations and matrices). G.-C. Rota in his famous Foundations defined a representation of a locally finite partially ordered set (locally finite poset) P in terms of a module over a ring \mathbbA, which can further be extended by the addition of a convolution operation to an associative \mathbbA-algebra called an incidence algebra of P. He applied this construction to solve a number of important problems in combinatorics. Our goal in this paper is to discuss the concept of an incidence algebra as a representation of a Pawlak information system. We shall analyse both incidence algebras and information systems in the context of granular computing, a paradigm which has recently received a lot of attention in computer science. We discuss therefore the concept of an incidence algebra on two levels: the level of objects which form a preordered set and the level of information granules which form a poset. Since incidence algebras induced on these two levels are Morita equivalent, we may focus our attention on the incidence algebra of information granules. We take the lattice of closed ideals of this algebra, where the maximal elements serve as a representation of information granules. The poset of maximal closed ideals obtained in this way is isomorphic to the set of information granules of the Pawlak information system equipped with a natural information order.

Słowa kluczowe

EN

granular computing; incidence algebra; Pawlak information system; representation theory

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2013
• Tom: Vol. 128, nr 1-2
• Strony: 223--238
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Wolski M.

• Maria Curie-Skłodowska University, Department of Logic and Philosophy of Science, Pl. Marii Curie-Skłodowskiej 4, 20-031 Lublin, Poland

autor

Gomolińska A.

• Białystok University, Institute of Computer Science, Sosnowa 64, 15-887 Białystok, Poland

Bibliografia

• [1] Abrams, G., Haefner, J., Del Rio, A.: Corrections and addenda to “The isomorphism problem for incidence rings”, Pacific Journal of Mathematics, 207(2), 2002, 497-506.
• [2] Belding, W. R.: Incidence rings of pre-ordered sets, Notre Dame J. Formal Logic 14, 1973, 481-509.
• [3] Doubilet, P., Rota, G.-C., Stanley, R. P.: On the foundations of combinatorial theory, VI. The idea of generating function, Proc. 6th Berkeley Symposium on Mathematical Statistics and Probability, University of California, Berkeley, CA, 1970/1971 (R. Proceed, Ed.), vol. 2: Probability Theory, University of California Press, Berkeley, CA, 1972, 267-318.
• [4] Haack, J.: Isomorphism of incidence rings, Illinois Journal of Mathematics 28(4), 1984, 676-683.
• [5] Pawlak, Z.: Rough sets, Int. J. Computer and Information Sci. 11, 1982, 341-356.
• [6] Pawlak, Z.: Rough logic, Bull. Polish Acad. Sc. (Tech. Sc.) 35(5-6), 1987, 253-258.
• [7] Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Acad. Publ., 1991.
• [8] Peters, J. F.: Near sets. Special theory about nearness of objects, Fundamenta Informaticae 75, 2007, 407433.
• [9] Rota, G.-C.: On the foundations of combinatorial theory, I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie Verwandte Gebiete 2, 1964, 340-368.
• [10] Słowiński, R., Greco, S., Matarazzo, B.: Rough set based decision support, in: Search Methodologies: Introductory Tutorials in Optimization and Decision Support Techniques (E. K. Burke, G. Kendall, Eds.), Springer-Verlag, New York, 2005, 475-527.
• [11] Stanley, R. P.: Enumerative Combinatorics, vol. 1, Cambridge University Press, 1997.
• [12] Wolski, M., Gomolińska, A.: Elements of representation theory for Pawlak information systems, Proc. 21st Workshop on Concurrency, Specification, and Programming (CS&P’2012), Berlin, Germany, Sept. 26-28, 2012 (L. Popova-ZeugmannEd.), CEUR Workshop Proceedings vol. 928, 2012, 404-415.