Abstract
The theory of rough sets deals with the approximation of an arbitrary subset of a universe by two definable or observable subsets called, respectively, the lower and the upper approximation. There are at least two methods for the development of this theory, the constructive and the axiomatic approaches. The rough set axiomatic system is the foundation of rough sets theory. This paper proposes a new matrix view of the theory of rough sets, we start with a binary relation and we redefine a pair of lower and upper approximation operators using the matrix representation. Different classes of rough set algebras are obtained from different types of binary relations. Various classes of rough set algebras are characterized by different sets of axioms. Axioms of upper approximation operations guarantee the existence of certain types of binary relations (or matrices) producing the same operators. The upper approximation of the Pawlak rough sets, rough fuzzy sets and rough sets of vectors over an arbitrary fuzzy lattice are characterized by the same independent axiomatic system.