Rough Sets Congruences and Diagrams
Résumé
The study of congruences in topological spaces is a very old and difficult problem (McKinsey, Tarski 1942). It has been re-discovered recently in connection to a rough equality of sets (Pawlak 1989). The rough equality is a congruence with respect to rough operations of lower and upper approximation and a set complement, but fails to be the congruence with respect to set union and intersection. We define here new operators corresponding to the union and intersection of sets so that rough equality becomes a congruence with respect to them. In order to motivate their definition we introduce a new technique called Rough Diagrams. Rough Diagrams correspond to classical notion of Ven Diagrams and can also serve as a new and very intuitive way of contructing counter models for formulas that are not tautologies of modal logic S5. Finally, we use an automated theorem proving system daTac, to prove that the quotient algebra with respect to the rough congruence is a topological quasi-Boolean algebra.