Elements of Representation Theory for Pawlak Information Systems.
Marcin Wolski, Anna Gomolińska · 2012
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). Rota in his famous “Foundations ” defined a representation of a locally finite partially ordered set (poset) P in terms of a module over a ring A, which can be further extended to an associative A-algebra called incidence algebra of P. He applied this construction to solve a number of important problems in combinatorics. Our goal in this paper is to apply Rota’s construction of incidence algebras to (arbitrary) Pawlak information systems. To be more precise, we analyse both incidence algebras and information systems in the context of granular computing. Therefore, starting from objects and an indiscernibility relation, we focus our attention upon information granules (i.e. equivalence classes) and a corresponding incidence algebra; finally, we discuss a lattice of closed ideals of this algebra (whose maximal elements serve as a representation of information granules). In this way we obtain a (partially ordered) set of maximal (closed) ideals which is isomorphic to the set of information granules of a Pawlak information system (also equipped with a natural information order). 1