Affiliations: Maria Curie-Skłodowska University, Department of Logic and Philosophy of Science, Pl. Marii Curie-Skłodowskiej 4, 20-031 Lublin, Poland. marcin.wolski@umcs.lublin.pl | Białystok University, Institute of Computer Science, Sosnowa 64, 15-887 Białystok, Poland. anna.gom@math.uwb.edu.pl
Note: [] Address for correspondence: Maria Curie-Skłodowska University, Department of Logic and Philosophy of Science, Pl. Marii Curie-Skłodowskiej 4, 20-031 Lublin, Poland
Abstract: 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 $\mathbb{A}$, which can further be extended by the addition of a convolution operation to an associative $\mathbb{A}$-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.
Keywords: Pawlak information system, granular computing, representation theory, incidence algebra
