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Erschienen in:
30.08.2018 | Foundations
On injectivity in category of rough sets
Erschienen in: Soft Computing | Ausgabe 1/2019
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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.