On connections between information systems, rough sets and algebraic logic

In this note we remark upon some relationships between the ideas of an approximation space and rough sets due to Pawlak ([9] and [10]) and algebras related to the study of algebraic logic - namely, cylindric algebras, relation algebras, and Stone algebras. The paper consists of three separate observations. The first deals with the family of approximation spaces induced by the indiscernability relation for different sets of attributes of an information system. In [3] the family of closure operators defining these approximation spaces is abstractly characterized as a certain type of Boolean algebra with operators. An alternate formulation in terms of a general class of diagonal-free cylindric algebras is given in 1.6. The second observation concerns the lattice theoretic approach to the study of rough sets suggested by Iwiński [6] and the result by J. Pomykała and J. A. Pomykała [11] that the collection of rough sets of an approximation space forms a Stone algebra. Namely, in 2.4 it is shown that every regular double Stone algebra is embeddable into the algebra of all rough subsets of an approximation space. Finally, a notion of rough relation algebra is formulated in Section 3 and a few connections with the study of ordinary relation algebras are established.