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

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Abstract

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.

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Abbreviations

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

Context-K

\(\wedge\) :

Infimum

\(\vee\) :

Supremum

Z :

Set of universe

\(\delta\) :

Granulation

d :

Distance

\(\hbox {Av}_d\) :

Average distance

I :

Bipolar fuzzy set of vertices

J :

Bipolar fuzzy set of edges

Z :

Complex fuzzy set

G :

Bipolar graph

n :

Total number of attributes

n :

Total number of objects

\(\otimes\) :

Multiplication

\(\rightarrow\) :

Residuum

X :

Objects

Y :

Attributes

\(\tilde{R}\) :

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

L :

Scale of truth degree

L :

Residuated lattice

a, b, c :

Elements in L

A :

Set of objects

B :

Set of attributes

\(C_1\) :

Concept

CG:

Complex granules

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

Position information

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

Negative information

(\(\uparrow , \downarrow\)):

Galois connection

\(\prod\) :

Projection operator

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

L-set of objects

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

L-set of attributes

\(\bigcup\) :

Union

\(\bigcap\) :

Intersection

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Acknowledgements

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

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  1. Amity Institute of Information Technology, Amity University, Sector-125, Noida, Uttar Pradesh, 201313, India

    Prem Kumar Singh

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Singh, P.K. Bipolar \(\delta\)-equal complex fuzzy concept lattice with its application. Neural Comput & Applic 32, 2405–2422 (2020). https://doi.org/10.1007/s00521-018-3936-9

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