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Algebraic rough sets via algebraic relations

  • Foundation, algebraic, and analytical methods in soft computing
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Abstract

The aim of this paper is to discuss algebraic rough set and its relationships with convex space, rough set and generalized neighborhood space. Specifically, the notion of algebraic relations is introduced and a pair of lower approximation operator and upper approximation operator are presented. Then, several conditions of algebraic relations such as seriality, reflexivity, (resp., weak, primitive) symmetry and (resp., strong) transitivity are characterized by algebraic approximation operators. Based on this, relationships among algebraic rough sets, convex structures and generalized neighborhood systems are investigated. It is proved that the category of reflexive and transitive algebraic rough spaces is isomorphic to the category of convex spaces. In particular, the category of reflexive, weakly symmetric and transitive algebraic rough spaces is isomorphic to the category of convex matroids and the category of reflexive, weakly symmetric and transitive algebraic generalized neighborhood spaces.

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Acknowledgements

The authors would like to thank the anonymous reviewers and the area editor for their careful reading and valuable comments. This work is supported by Doctoral Scientific Research Foundation of Anhui Normal University (751966) and Provincial Postgraduate Academic Innovation Projects (2023xscx037 and 2022xscx041).

Funding

This work is supported by Doctoral Scientific Research Foundation of Anhui Normal University (751966) and Provincial Postgraduate Academic Innovation Projects (2023xscx037 and 2022xscx041).

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Authors and Affiliations

  1. Anhui Normal University, Wuhu, China

    Xiu-Yun Wu & Hui-Min Zhang

  2. School of Mathematics, Hunan University of Science and Engineering, Yongzhou, 425100, China

    Chun-Yan Liao

Authors
  1. Xiu-Yun Wu
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  2. Chun-Yan Liao
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  3. Hui-Min Zhang
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Contributions

The first author provides the main thought and purpose of the paper. He introduces the notion of algebraic relation and characterizes seriality, reflexivity, (resp., weak, primitive) symmetry and (resp., strong) of algebraic relation by algebraic approximation operators. In addition, he also discusses relations between algebraic approximation spaces and generalized neighborhood spaces. The second the third authors discuss relationships among algebraic approximation spaces, approximation spaces and convex spaces. They also provides counterexamples in this paper.

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Correspondence to Xiu-Yun Wu.

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Wu, XY., Liao, CY. & Zhang, HM. Algebraic rough sets via algebraic relations. Soft Comput 28, 9157–9177 (2024). https://doi.org/10.1007/s00500-024-09820-x

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