Approximations of bipolar fuzzy \(\Gamma \)-hyperideals of \(\Gamma \)-semihypergroups



This paper studies relationship between bipolar fuzzy sets, rough sets and \(\Gamma \)-semihypergroups. We study the notion of rough bipolar fuzzy \(\Gamma \)-hyperideals in \(\Gamma \)-semihypergroups. Then we prove that the lower and upper approximation of a bipolar fuzzy \(\Gamma \)-hyperideal is a bipolar fuzzy \(\Gamma \)-hyperideal.


Rough sets Bipolar fuzzy sets \(\Gamma \)-Semihypergroups Rough bipolar fuzzy \(\Gamma \)-hyperideals 

Mathematics Subject Classification

20N20 03E72 



The authors are highly grateful to the referees for their valuable comments and suggestions which were helpful in improving this paper.

Compliance with ethical standards

Conflict of interest

The authors of this paper Naveed Yaqoob and Muhammad Aslam 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

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of Science Al-ZulfiMajmaah UniversityAl-ZulfiSaudi Arabia
  2. 2.Department of Mathematics and Science, College of Arts and Applied SciencesDhofar UniversitySalalahOman
  3. 3.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan

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