Soft Computing

, Volume 23, Issue 20, pp 9885–9897 | Cite as

Bipolar fuzzy matrices

  • Madhumangal Pal
  • Sanjib MondalEmail author


In this article, bipolar fuzzy algebra and bipolar fuzzy relation are defined, and then, the bipolar fuzzy matrix is introduced. Also, an order relation (\(\le \)) is defined and it is proved that the bipolar fuzzy set is a poset and a lattice. Some results on transitive closure and power-convergent of bipolar fuzzy matrices are investigated. Some applications of bipolar fuzzy sets and bipolar fuzzy matrix are included in this study.


Bipolar fuzzy set Bipolar fuzzy relation Bipolar fuzzy matrix Transitive closer Power-convergent 



Authors have no funding support to study and introduce the article.

Compliance with ethical standards

Conflict 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.Department of Applied Mathematics with Oceanology and Computer ProgrammingVidyasagar UniversityMidnaporeIndia

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