Soft Computing

, Volume 21, Issue 9, pp 2201–2214 | Cite as

On the category of rough sets



We consider the class of approximation spaces in the present paper. In this class we define the concept of lower natural transformations, upper natural transformations and natural transformations. We prove that the class of approximation spaces with the lower natural transformations, upper natural transformations and natural transformations form categories which are denoted by \(\underline{\mathbf{Apr }}{} \mathbf S \), \(\overline{\mathbf{Apr }}{} \mathbf S \) and \(\mathbf Apr {} \mathbf S \), respectively. We characterize a lower (upper) natural transformation through equivalence classes in an approximation space. We prove that two categories \(\mathbf Apr {} \mathbf S \) and \(\underline{\mathbf{Apr }}{} \mathbf S \) are the same. We characterize several kinds of epimorphisms and monomorphisms. In addition, we show that \(\underline{\mathbf{Apr }}{} \mathbf S \) is a (ExtrEpiExtrMono)-structured.


Approximation space Product Coproduct Lower (Upper) natural transformation 



The authors would like to express their sincere thanks to the referees for their valuable comments and suggestions which helped a lot to improve the presentation of this paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsShahid Beheshti University, G.C.TehranIran
  2. 2.Department of MathematicsHakim Sabzevari UniversitySabzevarIran

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