Categories and Algebras from Rough Sets: New Facets
Anuj Kumar More, Mohua Banerjee Β· Fundamenta Informaticae Β· 2016
Rough sets are investigated from the viewpoint of topos theory. Two categories RSC and ROUGH of rough sets and a subcategory ΞΎ-RSC are focussed upon. It is shown that RSC and ROUGH are equivalent. Generalizations RSC(π) and ΞΎ-RSC(π) are proposed over an arbitrary topos π. RSC(π) is shown to be a qu asitopos, while ΞΎ-RSC(π) forms a topos in the special case when π is Boolean. An example of RSC(π) is given, through which one is able to define monoid actions on rough sets. Next, the algebra of strong subobjects of an object in RSC is studied using the notion of relative rough complementation. A class of contrapositionally complemented βc. β¨ c.β lattices is obtained as a result, from the object class of RSC. Moreover, it is shown that such a class can also be obtained if the construction is generalized over an arbitrary Boolean algebra.