## Abstract

The main purpose of this paper is to verify injectivity in two categories of approximation spaces. We show that (*W*, *r*) 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 (*U*, *t*) is isomorphic to an approximation space (*V*, *s*) 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.

## Keywords

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

### Acknowledgements

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.

## References

- Adámek J, Herrlich H, Strecker GE (1990) Abstract and concrete categories. Wiley, New YorkzbMATHGoogle Scholar
- Asperti A, Longo G (1991) Categories, types, and structures: an introduction to category theory for the working computer scientist. Foundations of computing series. M.I.T Press, CambridgezbMATHGoogle Scholar
- Banerjee M, Chakraborty MK (1993) A category for rough sets. Found Comput Decis Sci 18(3–4):167–180MathSciNetzbMATHGoogle Scholar
- Bonikowski Z (1994) Algebraic structures of rough sets. In: Ziarco W (ed) Rough sets. Fuzzy sets and knowledge discovery. Springer, Berlin, pp 243–247Google Scholar
- Borzooei RA, Estaji AA, Mobini M (2017) On the category of rough sets. Soft Comput 21:2201–2214CrossRefzbMATHGoogle Scholar
- Bryniaski E (1989) A calculus of rough sets of the first order. Bull Polish Acad Sci 36(16):71–77MathSciNetGoogle Scholar
- Chen DG, Zhang W-X, Yeung D, Tsang ECC (2006) Rough approximations on a complete completely distributive lattice with applications to generalized rough sets. Inf Sci 176:1829–1848MathSciNetCrossRefzbMATHGoogle Scholar
- Devlin K (1993) The joy of sets: fundamentals of contemporary set theory. Springer, BerlinCrossRefzbMATHGoogle Scholar
- Estaji AA, Hooshmandasl MR, Davvaz B (2012) Rough set theory applied to lattice theory. Inf Sci 200:108–122MathSciNetCrossRefzbMATHGoogle Scholar
- Greco S, Matarazzo B, Slowinski R (2001) Rough sets theory for multicriteria decision analysis. Eur J Oper Res 129:1–47CrossRefzbMATHGoogle Scholar
- Haruna T, Gunji YP (2009) Double approximation and complete lattices. Rough sets and knowledge technology, lecture notes in computer science 5589:52–59CrossRefzbMATHGoogle Scholar
- Liu GL, Sai Y (2004) Invertible approximation operators of generalized rough sets and fuzzy rough sets. Inf Sci 180:2221–2229MathSciNetCrossRefzbMATHGoogle Scholar
- Pagliani P (2001) Rough set approach on lattice. J Uncertain Syst 5:72–80Google Scholar
- Pagliani P (2004) Pretopologies and dynamic spaces. Fundam Inf 59(2–3):221–239MathSciNetzbMATHGoogle Scholar
- Pagliani P, Chakraborty M (2008) A geometry of approximation. Springer, BerlinCrossRefzbMATHGoogle Scholar
- Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356CrossRefzbMATHGoogle Scholar
- Pawlak Z (2002) Rough set theory and its applications. J Telecommun Inf Technol 3(3):7–10Google Scholar
- Pawlak Z, Skowron A (2007) Rudiments of rough sets. Inf Sci 177:3–27MathSciNetCrossRefzbMATHGoogle Scholar
- Polkowski L (2008) Rough sets. Mathematical foundations. Springer, BerlinzbMATHGoogle Scholar
- Roy SK, Bera S (2015) Approximation of rough soft set and its application to lattice. Fuzzy Inf Eng 7:379–387MathSciNetCrossRefGoogle Scholar
- Xu Z, Wang Q (2005) On the properties of covering rough sets model. J Henan Norm Univ (Nat Sci) 33(1):130–132MathSciNetzbMATHGoogle Scholar
- Yao YY (1998) Constructive and algebraic methods of the theory of rough sets. Inf Sci 109:21–47MathSciNetCrossRefzbMATHGoogle Scholar
- Yao Y-Y (2008) Probabilistic rough set approximations. Int J Approx Reason 49:255–271CrossRefzbMATHGoogle Scholar
- Zhu W (2009) Relationship among basic concepts in covering based rough sets. Inf Sci 179:2478–2686MathSciNetCrossRefzbMATHGoogle Scholar