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Categories and Algebras from Rough Sets : New Facets

More A. K., Banerjee M.

Fundamenta Informaticae

2016

Vol. 148, nr 1/2

173--190

Rough sets are investigated from the viewpoint of topos theory. Two categories RSC and ROUGH of rough sets and a subcategory ξ-RSC are focussed upon. It is shown that RSC and ROUGH are equivalent. Generalizations RSC(C ) and ξ-RSC(C ) are proposed over an arbitrary topos C. RSC(C ) is shown to be a quasitopos, while ξ-RSC(C ) forms a topos in the special case when C is Boolean. An example of RSC(C ) 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 'c.V.c' 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.