Bipolar \(\delta\)-equal complex fuzzy concept lattice with its application

  • Prem Kumar SinghEmail author
Original Article


Recently, bipolar as well as vague concept lattice visualization is introduced for precise representation of inconsistency and incompleteness in data sets based on its acceptation and rejection part simultaneously. In this process, a problem is addressed while measuring the periodic fluctuation in bipolar information at the given phase of time. This changes in human cognition used coexist often in our daily life where the sentiments (i.e., love or hatred) for anyone may change several times from morning to evening office time. In this case precise representation of this type of bipolar information and measuring its pattern is a major issue for the researchers. To deal with this problem, the current paper proposes three methods for adequate representation of bipolar complex data set using the calculus of complex fuzzy matrix, \(\delta\)-equality and the calculus of granular computing, respectively. Hence, the proposed method provides an umbrella way to navigate or decompose the bipolar complex data sets and their semantics using an illustrative example. The results obtained from the proposed methods are also compared to validate the results.


Bipolar complex fuzzy set Bipolar complex fuzzy graph Bipolar concept lattice Bipolar fuzzy graph Formal concept analysis (FCA) Granulation 

List of symbols

(X, Y, \(\tilde{R}\))







Set of universe





\(\hbox {Av}_d\)

Average distance


Bipolar fuzzy set of vertices


Bipolar fuzzy set of edges


Complex fuzzy set


Bipolar graph


Total number of attributes


Total number of objects










A map from \(X \times Y\) to L


Scale of truth degree


Residuated lattice

a, b, c

Elements in L


Set of objects


Set of attributes




Complex granules

\(\mu ^{\mathrm{P}}(z)\)

Position information

\(\mu ^{\mathrm{N}}(z)\)

Negative information

(\(\uparrow , \downarrow\))

Galois connection


Projection operator

\(L^{\tiny {\textit{X}}}\)

L-set of objects

\(L^{\tiny {\textit{Y}}}\)

L-set of attributes







The author is grateful to the anonymous reviewers and the Editor-in-Chief for their valuable remarks on partial improvement of this paper.

Compliance with ethical standards

Conflict of interest

Author declares that he has no competing interests.


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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Amity Institute of Information TechnologyAmity UniversityNoidaIndia

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