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Categories and algebras from rough sets: new facets. (English) Zbl 1388.03049

Summary: Rough sets are investigated from the viewpoint of topos theory. Two categories \(RSC\) and \(ROUGH\) of rough sets and a subcategory \(\xi\text{-}RSC\) are focussed upon. It is shown that \(RSC\) and \(ROUGH\) are equivalent. Generalizations \(RSC(\mathcal{C})\) and \(\xi\text{-}RSC(\mathcal{C})\) are proposed over an arbitrary topos \(\mathcal{C}\). \(RSC(\mathcal{C})\) is shown to be a quasitopos, while \(\xi\text{-}RSC(\mathcal{C})\) forms a topos in the special case when is Boolean. An example of \(RSC(\mathcal{C})\) is given, through which one is able to define monoid actions on rough sets. Next, the algebra of strong subobjects of anobject in \(RSC\) is studied using the notion of relative rough complementation. A class of contrapositionally complemented ‘\(c. \vee c.\)’ 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.

MSC:

03E72 Theory of fuzzy sets, etc.
18B25 Topoi
18B05 Categories of sets, characterizations
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