Categories and Algebras from Rough Sets: New Facets

Abstract

Rough sets are investigated from the viewpoint of topos theory. Two categories RSC and ROU GH of rough sets and a subcategory ξ-RSC are focussed upon. It is shown that RSC and ROU GH 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. ∨ 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.

A Note on Categories about Rough Sets

“…Any pair (X 1 , X 2 ) such that X 1 , X 2 ∈ B and X 1 ⊆ X 2 is an I-rough set of (U, B). Then, the category RSC has all I-rough sets as its objects and an arrow f : [12]. While ROUGH and RSC looks different at first glance, they are shown to be equivalent in [12].…”

“…Then, the category RSC has all I-rough sets as its objects and an arrow f : [12]. While ROUGH and RSC looks different at first glance, they are shown to be equivalent in [12].…”

“…On the other hand, the advantage of using I-rough sets is that more general categorical construction is possible. In [12], by changing the base from Set to an arbitrary topos, a natural generalization of RSC and ξ-RSC is obtained. Recalling that a topos, as a category-theoretic abstraction of Set, is a category C that has all finite limits, a subobject classifier, and all exponentials [2].…”

“…ξ-RSC) to a parameterized class of categories RSC(C ) (resp. ξ-RSC(C )) is proposed and its topos-theoretic properties are explored in [12]. Objects of RSC(C ) and ξ-RSC(C ) are both triples (X, Y, m), where X and Y are objects of C and m :…”

“…and it is also a ξ-RSC(C ) arrow if there exist objects (¬X, Y, ¬m) and (¬X ′ , Y ′ , ¬m ′ ) and arrow f − : ¬X → ¬X ′ in C such that ¬m ′ • f − = g • ¬m. By instantiating C to a particular topos M-Set (the category of actions of a monoid M on sets, see [6]), a possible application of RSC(C ) to monoid actions on rough sets is suggested in [12].…”