Affiliations: Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India. [email protected]; [email protected]
Correspondence:
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Address for correspondence: Mohua Banerjee, Department of Mathematics and Statistics, IIT Kanpur, Kanpur 208016, India. This work is supported by the Council of Scientific and Industrial Research (CSIR) India, Research Grant No. 09/092(0875)/2013-EMR-I. Part of the work was presented at RST 2015, Warsaw, during a research visit of the author supported by the Indo-European Research Training Network in Logic (IERTNiL).
Abstract: 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. ∨ 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.
Keywords: Rough sets, Elementary Topos, Quasitopos, Lattices with Complementation
