Soft Computing

, Volume 23, Issue 22, pp 11447–11460 | Cite as

Fixed points of covering upper and lower approximation operators

  • A. A. EstajiEmail author
  • M. Vatandoost
  • R. Pourkhandani


Pawlak’s rough set and its extension, covering-based rough set, are important techniques for reasoning in incomplete information systems. In this paper, by studying some results about the Feynman paths, we show that the family of all fixed points of covering upper and lower approximation operators is an atomic frame and a complete lattice, respectively. Then, we find a relation between some major causal operators of relativity theory and covering approximation operators. As a result of this connection, we introduce a Feynman index to classify space–times.


Covering-based Rough set Causal structure Space–time Frame 



The authors thank the editor and the referee for their valuable comments and suggestions for improving the paper.

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer SciencesHakim Sabzevari UniversitySabzevarIran

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