Rough sets in terms of discrete dynamical systems

Abstract

In the paper we consider a topological approximation space (U, τ ) (induced by a given information system I) as a discrete dynamical system; that is, we are concerned with a finite approximation space U whose topology τ is induced by a function f : U → U. Our aim is to characterise these type of approximation spaces by means of orbits which represent the evolution of points of U with respect to the process f. Apart from topological considerations we also provide some algebraic characterisation of orbits. Due to the finiteness condition imposed by I, any point a ∈ U is eventually cyclic. In consequence, as we demonstrate, orbits are algebraically close to rough sets, e.g. they induce a Łukasiewicz algebra of order two, where the lower approximation operator may be interpreted as the action of retriving a cycle from a given orbit and the upper approximation operator may be interpreted as the action of making a given orbit cyclic. © 2010 Springer-Verlag Berlin Heidelberg.