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By Li G., Datta S.

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University of Turku: Turku Center for Computer Science. Järvinen, J. (2006). Lattice theory of rough sets. Unpublished manuscript. Katriniak, T. (1973). The structure of distributive double p-algebras. Regularity and Congruences. 3, 238-246.. Keefe, R. (2000). vagueness. Cambridge, UK: Cambridge University Press. , & Pawlak, Z. (1981).. Knowledge representation systems. Technical Report 433, Institute of Computer Science, Polish Academy of Sciences. Koppelberg, S. (1989). General theory of Boolean algebras.

It was proved that in such a case, the family of rough sets of any approximation space is a regular double Stone algebra. Let us recall the definition of regular double Stone algebras. Definition 15 An algebra S , ∧, ∨, ¬, ,0,1 is a double. Stone. algebra if S , ∧, ∨, ¬,0,1 is a Stone algebra and, additionally, ° is a unary operation that satisfies for any x ∈ S : S6: x ∨ x = 1 S7: x ∨ y = 1 implies  x ≤ y S8: x ∧   x = 0 Operation ° is called a dual pseudocomplementation. A double Stone algebra S , ∨, ∧, ¬, ,0,1 is regular if for any x, y ∈ S the following implication holds: if ¬x = ¬y and  x =  y, then x = y S9: It follows from this definition that  x is a dual pseudocomplement of x, that is  x is the least element in the set { y : x ∨ y = 1}.

Sciences,. Mathematics, 35, 673-683. Järvinen, J. (1999). sets. Doctoral dissertation. University of Turku: Turku Center for Computer Science. Järvinen, J. (2006). Lattice theory of rough sets. Unpublished manuscript. Katriniak, T. (1973). The structure of distributive double p-algebras. Regularity and Congruences. 3, 238-246.. Keefe, R. (2000). vagueness. Cambridge, UK: Cambridge University Press. , & Pawlak, Z. (1981).. Knowledge representation systems. Technical Report 433, Institute of Computer Science, Polish Academy of Sciences.

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