Soft Computing

, Volume 23, Issue 1, pp 27–38 | Cite as

On injectivity in category of rough sets

  • A. A. Estaji
  • M. MobiniEmail author


The main purpose of this paper is to verify injectivity in two categories of approximation spaces. We show that (Wr) is \(\mathcal {M}_u\)-injective if and only if there exists an element \(x\in W\) such that \(|[x]_r|=1\); also we prove that \(\underline{\mathbf{Apr }}{} \mathbf S \) does not have any \(\mathcal {M}_l\)-injective object. We introduce the concept of language of an approximation space and show that an approximation space (Ut) is isomorphic to an approximation space (Vs) in \(\overline{\mathbf{Apr }}{} \mathbf S \) if and only if \(\left| \dfrac{U}{t}\right| =\left| \dfrac{V}{s}\right| =\alpha \), and they have the same language. Also, we introduce the concepts of upper weakly monomorphisms and cover for an approximation space, and then characterize their properties in these categories.


Approximation space Upper monomorphism Lower monomorphism \(\mathcal {M}_u\)-injective \(\mathcal {M}_l\)-injective 



The authors thank the anonymous referees for their valuable comments and suggestions for improving the paper.

Compliance with ethical standards

Conflict of interest

All the authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHakim Sabzevari UniversitySabzevarIran
  2. 2.Department of Mathematics, Faculty of MathematicsUniversity of Sistan and BaluchestanZahedanIran

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