The Ordered Set of Rough Sets

We study the ordered set of rough sets determined by relations which are not necessarily reflexive, symmetric, or transitive. We show that for tolerances and transitive binary relations the set of rough sets is not necessarily even a semilattice. We also prove that the set of rough sets determined by a symmetric and transitive binary relation forms a complete Stone lattice. Furthermore, for the ordered sets of rough sets that are not necessarily lattices we present some possible canonical completions. 1 Different Types of Indiscernibility Relations The rough set theory introduced by Pawlak (1982) deals with situations in which the objects of a certain universe of discourse U can be identified only within the limits determined by the knowledge represented by a given indiscernibility relation. Based on such indiscernibility relation the lower and the upper approximation of subsets of U may be defined. The lower and the upper approximation of a subset X of U can be viewed as the sets of elements which certainly and possibly belong to X , respectively. Usually it is presumed that indiscernibility relations are equivalences. However, some authors, for example, Järvinen (2001), Pomykała (2002), and Skowron and Stepaniuk (1996) have studied approximation operators which are defined by tolerances. Slowinski and Vanderpooten (2000) have studied approximation operators defined by reflexive binary relations, and Greco, Matarazzo, and Slowinski (2000) considered approximations based on reflexive and transitive relations. Yao and Lin (1996) have studied approximations determined by arbitrary binary relations, and in a recent survey Düntsch and Gediga (2003) explored various types of approximation operators based on binary relations. Furthermore, Cattaneo (1998) and Järvinen (2002), for instance, have studied approximation operations in a more general lattice-theoretical setting. The structure of the ordered set of rough sets defined by equivalences was examined by Gehrke and Walker (1992), Iwiński (1987), and J. Pomykała and J.A. Pomykała (1988). In this work we study the structure of the ordered sets of rough sets based on indiscernibility relations which are not necessarily reflexive, symmetric, or transitive. 2 Lattices and Orders Here we recall some basic notions of lattice theory which can be found, for example, in the books by Davey and Priestly (2002) and Grätzer (1998). A binary relation ≤ on a S. Tsumoto et al. (Eds.): RSCTC 2004, LNAI 3066, pp. 49–58, 2004. c © Springer-Verlag Berlin Heidelberg 2004