Soft Computing

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

On the category of rough sets

Foundations

Abstract

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.

Keywords

Approximation space Product Coproduct Lower (Upper) natural transformation 

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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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