Advances in algebra and analysis i by v madhu, a manimaran, d easwaramoorthy, d kalpanapriya, m mubashir unnissa

Trends in Mathematics

V. Madhu
A. Manimaran
D. Easwaramoorthy
D. Kalpanapriya
M. Mubashir Unnissa
Editors

Advances in
Algebra and
Analysis
International Conference on
Advances in Mathematical Sciences,
Vellore, India, December 2017 Volume I

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Trends in Mathematics
Trends in Mathematics is a series devoted to the publication of volumes arising
from conferences and lecture series focusing on a particular topic from any area of
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V. Madhu • A. Manimaran • D. Easwaramoorthy
D. Kalpanapriya • M. Mubashir Unnissa
Editors

Advances in Algebra
and Analysis
International Conference on Advances
in Mathematical Sciences, Vellore, India,
December 2017 - Volume I

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Editors
V. Madhu

Department of Mathematics
School of Advanced Sciences
Vellore Institute of Technology
Vellore, Tamil Nadu, India

A. Manimaran
Department of Mathematics
School of Advanced Sciences
Vellore Institute of Technology
Vellore, Tamil Nadu, India

D. Easwaramoorthy
Department of Mathematics
School of Advanced Sciences
Vellore Institute of Technology
Vellore, Tamil Nadu, India

D. Kalpanapriya
Department of Mathematics
School of Advanced Sciences
Vellore Institute of Technology
Vellore, Tamil Nadu, India

M. Mubashir Unnissa
Department of Mathematics
School of Advanced Sciences
Vellore Institute of Technology
Vellore, Tamil Nadu, India

ISSN 2297-0215

ISSN 2297-024X (electronic)
Trends in Mathematics
ISBN 978-3-030-01119-2
ISBN 978-3-030-01120-8 (eBook)
/>Library of Congress Control Number: 2018966815
© Springer Nature Switzerland AG 2018
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This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered
company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

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Preface

The Department of Mathematics, School of Advanced Sciences, Vellore Institute of
Technology (Deemed to be University), Vellore, Tamil Nadu, India, organized the

International Conference on Advances in Mathematical Sciences—2017 (ICAMS
2017) in association with the Society for Industrial and Applied Mathematics
VIT Chapter from December 1, 2017 to December 3, 2017. The major objective
of ICAMS 2017 was to promote scientific and educational activities toward the
advancement of common man’s life by improving the theory and practice of
various disciplines of Mathematics. This prestigious conference was partially
financially supported by the Council of Scientific and Industrial Research (CSIR),
India. The Department of Mathematics has 90 qualified faculty members and 30
research scholars, and all were delicately involved in organizing ICAMS 2017
grandly. In addition, 30 leading researchers from around the world served as an
advisory committee for this conference. Overall, more than 450 participants (professors/scholars/students) enriched their knowledge in the wings of Mathematics.
There were 9 eminent speakers from overseas and 33 experts from various states
of India who delivered the keynote address and invited talks in this conference.
Many leading scientists and researchers worldwide submitted their quality research
articles to ICAMS. Moreover, 305 original research articles were shortlisted for
ICAMS 2017 oral presentations that were authored by dynamic researchers from
25 states in India and 20 countries around the world. We hope that ICAMS will
further stimulate research in Mathematics, share research interest and information,
and create a forum of collaboration and build a trust relationship. We feel honored
and privileged to serve the best of recent developments in the field of Mathematics
to the reader.
A basic premise of this book is that quality assurance is effectively achieved
through the selection of quality research articles by a scientific committee consisting
of more than 100 reviewers from all over the world. This book comprises the
contribution of several dynamic researchers in 52 chapters. Each chapter identifies
the existing challenges in the areas of Algebra, Analysis, Operations Research,
and Statistics and emphasizes the importance of establishing new methods and
algorithms to address the challenges. Each chapter presents a research problem, the
v

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vi

Preface

technique suitable for solving the problem with sufficient mathematical background,
and discussions on the obtained results with physical interruptions to understand
the domain of applicability. This book also provides a comprehensive literature
survey which reveals the challenges, outcomes, and developments of higher level
mathematics in this decade. The theoretical coverage of this book is relatively at a
higher level to meet the global orientation of mathematics and its applications in
science and engineering.
The target audience of this book is postgraduate students, researchers, and
industrialists. This book promotes a vision of pure and applied mathematics as
integral to modern science and engineering. Each chapter contains important
information emphasizing core Mathematics, intended for the professional who
already possesses a basic understanding. In this book, theoretically oriented readers
will find an overview of Mathematics and its applications. Industrialists will find a
variety of techniques with sufficient discussion in terms of physical point of view
to adapt for solving the particular application based on mathematical models. The
reader can make use of the literature survey of this book to identify the current
trends in Mathematics. It is our hope and expectation that this book will provide an
effective learning experience and referenced resource for all young mathematicians.
As Editors, we would like to express our sincere thanks to all the administrative
authorities of Vellore Institute of Technology, Vellore, for their motivation and
support. We also extend our profound thanks to all faculty members and research
scholars of the Department of Mathematics and all staff members of our institute.
We especially thank all the members of the organizing committee of ICAMS 2017

who worked as a team by investing their time to make the conference a great
success. We thank the national funding agency, Council of Scientific and Industrial
Research (CSIR), Government of India, for the financial support they contributed
toward the successful completion of this international conference. We express our
sincere gratitude to all the referees for spending their valuable time to review the
manuscripts, which led to substantial improvements and selection of the research
papers for publication. The organizing committee is grateful to Mr. Christopher
Tominich, Editor at Birkhäuser/Springer, for his continuous encouragement and
support toward the publication of this book.
Vellore, India
Vellore, India
Vellore, India
Vellore, India
Vellore, India

V. Madhu
A. Manimaran
D. Easwaramoorthy
D. Kalpanapriya
M. Mubashir Unnissa

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Contents

Part I Algebra
IT-2 Fuzzy Automata and IT-2 Fuzzy Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . .
M. K. Dubey, P. Pal, and S. P. Tiwari

3

Level Sets of i_v_Fuzzy β-Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
P. Hemavathi and K. Palanivel

13

Interval-Valued Fuzzy Subalgebra and Fuzzy
INK-Ideal in INK-Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
M. Kaviyarasu, K. Indhira, V. M. Chandrasekaran, and Jacob Kavikumar

19

On Dendrites Generated by Symmetric Polygonal Systems: The
Case of Regular Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mary Samuel, Dmitry Mekhontsev, and Andrey Tetenov

27

Efficient Authentication Scheme Based on the Twisted Near-Ring
Root Extraction Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
V. Muthukumaran, D. Ezhilmaran, and G. S. G. N. Anjaneyulu

37

Dimensionality Reduction Technique to Solve E-Crime Motives . . . . . . . . . . .
R. Aarthee and D. Ezhilmaran

43

Partially Ordered Gamma Near-Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
T. Nagaiah

49

Novel Digital Signature Scheme with Multiple Private Keys on
Non-commutative Division Semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
G. S. G. N. Anjaneyulu and B. Davvaz
Cozero Divisor Graph of a Commutative Rough Semiring . . . . . . . . . . . . . . . . .
B. Praba, A. Manimaran, V. M. Chandrasekaran, and B. Davvaz

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Contents

Gorenstein F I -Flat Complexes and (Pre)envelopes . . . . . . . . . . . . . . . . . . . . . . . . . .
V. Biju

77

Bounds of Extreme Energy of an Intuitionistic Fuzzy Directed Graph . . . .
B. Praba, G. Deepa, V. M. Chandrasekaran, Krishnamoorthy Venkatesan,

and K. Rajakumar

85

Part II Analysis
On Ultra Separation Axioms via αω-Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
M. Parimala, Cenap Ozel, and R. Udhayakumar

97

Common Fixed Point Theorems in 2-Metric Spaces Using
Composition of mappings via A-Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
J. Suresh Goud, P. Rama Bhadra Murthy, Ch. Achi Reddy,
and K. Madhusudhan Reddy
Coefficient Bounds for a Subclass of m-Fold Symmetric λ-Pseudo
Bi-starlike Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Jay M. Jahangiri, G. Murugusundaramoorthy, K. Vijaya, and K. Uma
Laplacian and Effective Resistance Metric in Sierpinski
Gasket Fractal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
P. Uthayakumar and G. Jayalalitha
Some Properties of Certain Class of Uniformly Convex Functions
Defined by Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
V. Srinivas, P. Thirupathi Reddy, and H. Niranjan
A New Subclass of Uniformly Convex Functions Defined by Linear
Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A. Narasimha Murthy, P. Thirupathi Reddy, and H. Niranjan
Coefficient Bounds of Bi-univalent Functions Using Faber Polynomial . . . 151
T. Janani and S. Yalcin
Convexity of Polynomials Using Positivity of Trigonometric Sums. . . . . . . . . 161
Priyanka Sangal and A. Swaminathan

Local Countable Iterated Function Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
A. Gowrisankar and D. Easwaramoorthy
On Intuitionistic Fuzzy C -Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
T. Yogalakshmi and Oscar Castillo
Generalized Absolute Riesz Summability of Orthogonal Series . . . . . . . . . . . . 185
K. Kalaivani and C. Monica
Holder’s Inequalities for Analytic Functions Defined by
Ruscheweyh-Type q-Difference Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
N. Mustafa, K. Vijaya, K. Thilagavathi, and K. Uma

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Fuzzy Cut Set-Based Filter for Fixed-Value Impulse Noise Reduction . . . . 205
P. S. Eliahim Jeevaraj, P. Shanmugavadivu, and D. Easwaramoorthy
On (p, q)-Quantum Calculus Involving Quasi-Subordination . . . . . . . . . . . . . 215
S. Kavitha, Nak Eun Cho, and G. Murugusundaramoorthy
Part III Operations Research
Sensitivity Analysis for Spanning Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
K. Kavitha and D. Anuradha
On Solving Bi-objective Fuzzy Transportation Problem . . . . . . . . . . . . . . . . . . . . 233
V. E. Sobana and D. Anuradha
Nonlinear Programming Problem for an M-Design Multi-Skill Call
Center with Impatience Based on Queueing Model Method . . . . . . . . . . . . . . . . 243
K. Banu Priya and P. Rajendran
Optimizing a Production Inventory Model with Exponential

Demand Rate, Exponential Deterioration and Shortages . . . . . . . . . . . . . . . . . . . 253
M. Dhivya Lakshmi and P. Pandian
Analysis of Batch Arrival Bulk Service Queueing System with
Breakdown, Different Vacation Policies and Multiphase Repair . . . . . . . . . . . 261
M. Thangaraj and P. Rajendran
An Improvement to One’s BCM for the Balanced and Unbalanced
Transshipment Problems by Using Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 271
Kirtiwant P. Ghadle, Priyanka A. Pathade, and Ahmed A. Hamoud
An Articulation Point-Based Approximation Algorithm for
Minimum Vertex Cover Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Jayanth Kumar Thenepalle and Purusotham Singamsetty
On Bottleneck-Rough Cost Interval Integer Transportation Problems . . . . 291
A. Akilbasha, G. Natarajan, and P. Pandian
Direct Solving Method of Fully Fuzzy Linear Programming
Problems with Equality Constraints Having Positive Fuzzy Numbers. . . . . 301
C. Muralidaran and B. Venkateswarlu
An Optimal Deterministic Two-Warehouse Inventory Model for
Deteriorating Items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
K. Rangarajan and K. Karthikeyan
Analysis on Time to Recruitment in a Three-Grade Marketing
Organization Having Classified Sources of Depletion of Two Types
with an Extended Threshold and Shortage in Manpower Forms
Geometric Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
S. Poornima and B. Esther Clara

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Contents

Neutrosophic Assignment Problem via BnB Algorithm . . . . . . . . . . . . . . . . . . . . . 323
S. Krishna Prabha and S. Vimala
Part IV

Statistics

An Approach to Segment the Hippocampus from T 2-Weighted
MRI of Human Head Scans for the Diagnosis of Alzheimer’s
Disease Using Fuzzy C-Means Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
T. Genish, K. Prathapchandran and S. P. Gayathri
Analysis of M[X] /Gk /1 Retrial Queueing Model and Standby . . . . . . . . . . . . . . 343
J. Radha, K. Indhira and V. M. Chandrasekaran
μ-Statistically Convergent Multiple Sequences in Probabilistic
Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
Rupam Haloi and Mausumi Sen
A Retrial Queuing Model with Unreliable Server in K Policy . . . . . . . . . . . . . . 361
M. Seenivasan and M. Indumathi
Two-Level Control Policy of an Unreliable Queueing System with
Queue Size-Dependent Vacation and Vacation Disruption . . . . . . . . . . . . . . . . . . 373
S. P. Niranjan, V. M. Chandrasekaran, and K. Indhira
Analysis of M/G/1 Priority Retrial G-Queue with Bernoulli
Working Vacations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
P. Rajadurai, M. Sundararaman, Sherif I. Ammar, and D. Narasimhan
Time to Recruitment for Organisations having n Types of Policy
Decisions with Lag Period for Non-identical Wastages . . . . . . . . . . . . . . . . . . . . . . 393
Manju Ramalingam and B. Esther Clara
A Novice’s Application of Soft Expert Set: A Case Study on
Students’ Course Registration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

Selva Rani B and Ananda Kumar S
Dynamics of Stochastic SIRS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
R. Rajaji
Steady-State Analysis of Unreliable Preemptive Priority Retrial
Queue with Feedback and Two-Phase Service Under
Bernoulli Vacation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
S. Yuvarani and M. C. Saravanarajan
An Unreliable Optional Stage M X /G/1 Retrial Queue
with Immediate Feedbacks and at most J Vacations. . . . . . . . . . . . . . . . . . . . . . . . . 437
M. Varalakshmi, P. Rajadurai, M. C. Saravanarajan,
and V. M. Chandrasekaran
Weibull Estimates in Reliability: An Order Statistics Approach . . . . . . . . . . . 447
V. Sujatha, S. Prasanna Devi, V. Dharanidharan,
and Krishnamoorthy Venkatesan

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Intuitionistic Fuzzy ANOVA and Its Application Using Different
Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
D. Kalpanapriya and M. M. Unnissa
A Resolution to Stock Prices Prediction by Developing ANN-Based
Models Using PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
Jitendra Kumar Jaiswal and Raja Das
A Novel Method of Solving a Quadratic Programming Problem
Under Stochastic Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

S. Sathish and S. K. Khadar Babu

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Volume II Contents

Part V Differential Equations
Numerical Solution to Singularly Perturbed Differential Equation
of Reaction-Diffusion Type in MAGDM Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
P. John Robinson, M. Indhumathi, and M. Manjumari

3

Application of Integrodifferential Equations Using Sumudu
Transform in Intuitionistic Trapezoidal Fuzzy MAGDM Problems. . . . . . . .
P. John Robinson and S. Jeeva

13

Existence of Meromorphic Solution of Riccati-Abel Differential
Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
P. G. Siddheshwar and A. Tanuja

21

Expansion of Function with Uncertain Parameters in Higher
Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Priyanka Roy and Geetanjali Panda

29

Analytical Solutions of the Bloch Equation via Fractional Operators
with Non-singular Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A. S. V. Ravi Kanth and Neetu Garg

37

Solution of the Lorenz Model with Help from the Corresponding
Ginzburg-Landau Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
P. G. Siddheshwar, S. Manjunath, and T. S. Sushma

47

Estimation of Upper Bounds for Initial Coefficients and
Fekete-Szegö Inequality for a Subclass of Analytic Bi-univalent
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
G. Saravanan and K. Muthunagai
An Adaptive Mesh Selection Strategy for Solving Singularly
Perturbed Parabolic Partial Differential Equations with a Small Delay. . .
Kamalesh Kumar, Trun Gupta, P. Pramod Chakravarthy,
and R. Nageshwar Rao

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Part VI

Volume II Contents

Fluid Dynamics

Steady Finite-Amplitude Rayleigh-Bénard-Taylor Convection
of Newtonian Nanoliquid in a High-Porosity Medium . . . . . . . . . . . . . . . . . . . . . . .
P. G. Siddheshwar and T. N. Sakshath

79

MHD Three Dimensional Darcy-Forchheimer Flow of a Nanofluid
with Nonlinear Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nainaru Tarakaramu, P. V. Satya Narayana, and B. Venkateswarlu

87

Effect of Electromagnetohydrodynamic on Chemically Reacting
Nanofluid Flow over a Cone and Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H. Thameem Basha, I. L. Animasaun, O. D. Makinde, and R. Sivaraj

99

Effect of Non-linear Radiation on 3D Unsteady MHD Nanoliquid
Flow over a Stretching Surface with Double Stratification . . . . . . . . . . . . . . . . . . 109

K. Jagan, S. Sivasankaran, M. Bhuvaneswari, and S. Rajan
Chemical Reaction and Nonuniform Heat Source/Sink Effects on
Casson Fluid Flow over a Vertical Cone and Flat Plate Saturated
with Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
P. Vijayalakshmi, S. Rao Gunakala, I. L. Animasaun, and R. Sivaraj
An Analytic Solution of the Unsteady Flow Between Two Coaxial
Rotating Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Abhijit Das and Bikash Sahoo
Cross Diffusion Effects on MHD Convection of Casson-Williamson
Fluid over a Stretching Surface with Radiation and Chemical
Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
M. Bhuvaneswari, S. Sivasankaran, H. Niranjan, and S. Eswaramoorthi
Study of Steady, Two-Dimensional, Unicellular Convection in a
Water-Copper Nanoliquid-Saturated Porous Enclosure Using
Single-Phase Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
P. G. Siddheshwar and B. N. Veena
The Effects of Homo-/Heterogeneous Chemical Reactions on
Williamson MHD Stagnation Point Slip Flow: A Numerical Study . . . . . . . . 157
T. Poornima, P. Sreenivasulu, N. Bhaskar Reddy, and S. Rao Gunakala
The Influence of Wall Properties on the Peristaltic Pumping
of a Casson Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
P. Devaki, A. Kavitha, D. Venkateswarlu Naidu, and S. Sreenadh
Peristaltic Flow of a Jeffrey Fluid in Contact with a Newtonian
Fluid in a Vertical Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
R. Sivaiah, R. Hemadri Reddy, and R. Saravana

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xv

MHD and Cross Diffusion Effects on Peristaltic Flow of a Casson
Nanofluid in a Duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
G. Sucharitha, P. Lakshminarayana, and N. Sandeep
Axisymmetric Vibration in a Submerged Piezoelectric Rod Coated
with Thin Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Rajendran Selvamani and Farzad Ebrahimi
Numerical Exploration of 3D Steady-State Flow Under the Effect
of Thermal Radiation as Well as Heat Generation/Absorption over
a Nonlinearly Stretching Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
R. Jayakar and B. Rushi Kumar
Radiated Slip Flow of Williamson Unsteady MHD Fluid over a
Chemically Reacting Sheet with Variable Conductivity and Heat
Source or Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Narsu Siva Kumar and B. Rushi Kumar
Approximate Analytical Solution of a HIV/AIDS Dynamic Model
During Primary Infection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Ajoy Dutta and Praveen Kumar Gupta
Stratification and Cross Diffusion Effects on Magneto-Convection
Stagnation-Point Flow in a Porous Medium with Chemical
Reaction, Radiation, and Slip Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
M. Bhuvaneswari, S. Sivasankaran, S. Karthikeyan, and S. Rajan
Natural Convection of Newtonian Liquids and Nanoliquids
Confined in Low-Porosity Enclosures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
P. G. Siddheshwar and K. M. Lakshmi
Study of Viscous Fluid Flow Past an Impervious Cylinder in Porous
Region with Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
D. V. Jayalakshmamma, P. A. Dinesh, N. Nalinakshi, and T. C. Sushma

Numerical Solution of Steady Powell-Eyring Fluid over a Stretching
Cylinder with Binary Chemical Reaction and Arrhenius Activation
Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Seethi Reddy Reddisekhar Reddy and P. Bala Anki Reddy
Effect of Homogeneous-Heterogeneous Reactions in MHD
Stagnation Point Nanofluid Flow Toward a Cylinder with
Nonuniform Heat Source or Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
T. Sravan Kumar and B. Rushi Kumar
Effects of Thermal Radiation on Peristaltic Flow of Nanofluid
in a Channel with Joule Heating and Hall Current . . . . . . . . . . . . . . . . . . . . . . . . . . 301
R. Latha and B. Rushi Kumar

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Chemically Reactive 3D Nonlinear Magneto Hydrodynamic
Rotating Flow of Nanofluids over a Deformable Surface with Joule
Heating Through Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
E. Kumaresan and A. G. Vijaya Kumar
MHD Carreau Fluid Flow Past a Melting Surface with
Cattaneo-Christov Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
K. Anantha Kumar, Janke V. Ramana Reddy, V. Sugunamma,
and N. Sandeep
Effect of Porous Uneven Seabed on a Water-Wave Diffraction
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
Manas Ranjan Sarangi and Smrutiranjan Mohapatra

Nonlinear Wave Propagation Through a Radiating van der Waals
Fluid with Variable Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
Madhumita Gangopadhyay
Effect of Slip and Convective Heating on Unsteady MHD
Chemically Reacting Flow Over a Porous Surface with Suction. . . . . . . . . . . . 357
A. Malarselvi, M. Bhuvaneswari, S. Sivasankaran, B. Ganga,
and A. K. Abdul Hakeem
Solution of Wave Equations and Heat Equations Using HPM . . . . . . . . . . . . . . 367
Nahid Fatima and Sunita Daniel
Nonlinear Radiative Unsteady Flow of a Non-Newtonian Fluid Past
a Stretching Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
P. Krishna Jyothi, G. Sarojamma, K. Sreelakshmi, and K. Vajravelu
Heat Transfer Analysis in a Micropolar Fluid with Non-Linear
Thermal Radiation and Second-Order Velocity Slip . . . . . . . . . . . . . . . . . . . . . . . . . 385
R. Vijaya Lakshmi, G. Sarojamma, K. Sreelakshmi, and K. Vajravelu
Analytical Study on Heat Transfer Behavior of an Orthotropic Pin
Fin with Contact Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
M. A. Vadivelu, C. Ramesh Kumar, and M. M. Rashidi
Numerical Investigation of Developing Laminar Convection
in Vertical Double-Passage Annuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
Girish N, M. Sankar, and Younghae Do
Heat and Mass Transfer on MHD Rotating Flow of Second Grade
Fluid Past an Infinite Vertical Plate Embedded in Uniform Porous
Medium with Hall Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
M. Veera Krishna, M. Gangadhar Reddy, and A. J. Chamkha
High-Power LED Luminous Flux Estimation Using a Mathematical
Model Incorporating the Effects of Heatsink and Fins . . . . . . . . . . . . . . . . . . . . . . 429
A. Rammohan, C. Ramesh Kumar, and M. M. Rashidi

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xvii

Soret and Dufour Effects on Hydromagnetic Marangoni Convection
Boundary Layer Nanofluid Flow Past a Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
D. R. V. S. R. K. Sastry, Peri K. Kameswaran, Precious Sibanda,
and Palani Sudhagar
Part VII Graph Theory
An Algorithm for the Inverse Distance-2 Dominating Set of a Graph . . . . . 453
K. Ameenal Bibi, A. Lakshmi, and R. Jothilakshmi
γ -Chromatic Partition in Planar Graph Characterization . . . . . . . . . . . . . . . . . . 461
M. Yamuna and A. Elakkiya
Coding Through a Two Star and Super Mean Labeling . . . . . . . . . . . . . . . . . . . . . 469
G. Uma Maheswari, G. Margaret Joan Jebarani, and V. Balaji
Computing Status Connectivity Indices and Its Coindices
of Composite Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
K. Pattabiraman and A. Santhakumar
Laplacian Energy of Operations on Intuitionistic Fuzzy Graphs. . . . . . . . . . . 489
E. Kartheek and S. Sharief Basha
Wiener Index of Hypertree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
L. Nirmala Rani, K. Jennifer Rajkumari, and S. Roy
Location-2-Domination for Product of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
G. Rajasekar, A. Venkatesan, and J. Ravi Sankar
Local Distance Pattern Distinguishing Sets in Graphs . . . . . . . . . . . . . . . . . . . . . . . 517
R. Anantha Kumar
Construction of Minimum Power 3-Connected Subgraph with k
Backbone Nodes in Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

D. Pushparaj Shetty and M. Prasanna Lakshmi
Fuzzy Inference System Through Triangular and Hendecagonal
Fuzzy Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
A. Felix, A. D. Dhivya, and T. Antony Alphonnse Ligori
Computation of Narayana Prime Cordial Labeling of Book Graphs . . . . . . 547
B. J. Balamurugan, K. Thirusangu, B. J. Murali, and J. Venkateswara Rao
Quotient-3 Cordial Labeling for Path Related Graphs: Part-II . . . . . . . . . . . . 555
P. Sumathi and A. Mahalakshmi
Relation Between k-DRD and Dominating Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
S. S. Kamath, A. Senthil Thilak, and Rashmi M
The b-Chromatic Number of Some Standard Graphs . . . . . . . . . . . . . . . . . . . . . . . 573
A. Jeeva, R. Selvakumar, and M. Nalliah

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Volume II Contents

Encode-then-Encrypt: A Novel Framework for Reliable and Secure
Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
Rajrupa Singh, C. Pavan Kumar, and R. Selvakumar
New Bounds of Induced Acyclic Graphoidal Decomposition
Number of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
Mayamma Joseph and I. Sahul Hamid
Dominating Laplacian Energy in Products of Intuitionistic Fuzzy
Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
R. Vijayaragavan, A. Kalimulla, and S. Sharief Basha
Power Domination Parameters in Honeycomb-Like Networks . . . . . . . . . . . . . 613

J. Anitha and Indra Rajasingh
Improved Bound for Dilation of an Embedding onto Circulant
Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
R. Sundara Rajan, T. M. Rajalaxmi, Joe Ryan, and Mirka Miller

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Part I

Algebra

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IT-2 Fuzzy Automata and IT-2 Fuzzy
Languages
M. K. Dubey, Priyanka Pal, and S. P. Tiwari

Abstract The objective of this work is to give certain determinization and algebraic
studies for an interval type-2 (IT-2) fuzzy automaton and language. We introduce a
deterministic IT-2 fuzzy automaton and prove that it is behavioural equivalent to an
IT-2 fuzzy automaton. Also, for a given IT-2 fuzzy language, we give certain recipe
for constructions of deterministic IT-2 fuzzy automata.

1 Introduction
The notion of type-2 fuzzy sets was introduced by Zadeh [21], who gives the substructure to model and abbreviate the impact of uncertainty in fuzzy logic rule-based
systems. The author in [9] has pointed out that the membership function of type-1
fuzzy sets is totally crisp and hence not able to model certain uncertainty involved
in the model, whereas in case of type-2 fuzzy sets, it is capable to model such

uncertainty because of their fuzzy membership functions. Also, the membership
function of type-2 fuzzy sets is three dimensional which gives additional degrees of
freedom to model the uncertainty directly in comparison to type-1 fuzzy sets which
have two-dimensional membership function. However, it is not easy to understand
and use the concept of type-2 fuzzy sets, which can be seen by the fact that almost all
applications use interval type-2 fuzzy set for the sake of all computations to perform
easily [10].
From the commencement of the theory of fuzzy sets, Santos [12], Wee [17] and
Wee and Fu [18] introduced and studied fuzzy automata and languages, and after
Malik, Mordeson and Sen [11] have further studied and developed. In the last few
decades, many works on fuzzy automata and languages have been done (cf., [1, 2,
4, 5, 7, 8, 13–16, 20]). During the decades, it has been observed that fuzzy automata

M. K. Dubey ( ) · P. Pal · S. P. Tiwari
Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad, India
e-mail: ; ; ;
; ;
© Springer Nature Switzerland AG 2018
V. Madhu et al. (eds.), Advances in Algebra and Analysis, Trends in Mathematics,
/>
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4

M. K. Dubey et al.

and fuzzy languages have obtained not only conversion of classical automata to

fuzzy automata but also a broad field of applications[2].
Fuzzy automata and fuzzy languages referred above are either based upon type-1
fuzzy sets or on certain lattice structures (cf., [4, 5, 8, 20]). Since we know, type-1
fuzzy sets cannot be able to minimize the uncertainty involved in the model, and
Mendel [10] suggested to use an IT-2 fuzzy set model of a word in the concept
of computing with words. Recently, Jiang and Tang [6] introduced and studied the
concepts of IT-2 fuzzy automata and languages and give the platform to develop
the above model of nonclassical computations. In this note, we give a brief look at
certain studies for IT-2 fuzzy automata and languages, which may be carried out in
details. In particular, we begin by introducing a deterministic IT-2 fuzzy automaton
and prove that it is behavioural equivalent to an IT-2 fuzzy automaton. Further,
for a given IT-2 fuzzy language, we give the certain recipe for constructions of
deterministic IT-2 fuzzy automata. Finally, we give a brief look at an algebraic study
of an IT-2 fuzzy automaton.

2 IT-2 Fuzzy Sets
In this section, we memorize certain notions allied with an IT-2 fuzzy set. We initiate
with the following notion of a type-2 fuzzy set. For more description, we refer to
[9, 10, 19, 21].
Definition 2.1 ([9]) A type-2 fuzzy set F in a nonempty set Y is characterized by
a type-2 membership function μF (y, v), where y ∈ Y and v ∈ Jy ⊆ [0, 1], i.e.:
F =
y∈Y

v∈Jy

μF (y, v)/(y, v), Jy ⊆ [0, 1] , in which 0 ≤ μF (y, v) ≤ 1.

From Definition 2.1, it has been observed that when uncertainties disappear, a type2 membership function must reduce to a type-1 membership function, and in this
case, the variable v equals μF (y) and 0 ≤ μF (y) ≤ 1.

Definition 2.2 ([10]) A type-2 fuzzy set F in Y is called an IT-2 fuzzy set if
μF (y, v) = 1, ∀y ∈ Y and ∀v ∈ Jy . An IT-2 fuzzy set F can be expressed as
F = y∈Y v∈Jy 1/(y, v), Jy ⊆ [0, 1].
For an IT-2 fuzzy set, we consider Jy = [μF (y), μF (y)] for all y ∈ Y , where μF (y)
and μF (y) are, respectively, called the lower membership function (LPF) and upper
membership function (UMF) of F which are two type-1 membership functions that
bound the footprint of uncertainty. We shall denote by I T 2F (Y ), the set of all IT-2
fuzzy sets in Y . For more details on IT-2 fuzzy sets and their operations, we refer
to [10].

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IT-2 Fuzzy Automata and IT-2 Fuzzy Languages

5

3 IT-2 Fuzzy Automata and IT-2 Fuzzy Languages
In this section, we give a brief look to determinization of an IT-2 fuzzy automaton.
In particular, we introduce a deterministic IT-2 fuzzy automaton and prove that it is
behavioural equivalent to an IT-2 fuzzy automaton. We initiate with the following
concept of an IT-2 fuzzy automaton.
Definition 3.1 ([6]) An IT-2 fuzzy automaton (IT2FA) is a five-tuple M =
(S, X, λ, i, f ), where S, X are nonempty sets called set of states and set of inputs
and the characterization of λ, i and f is as follows:
(i) λ : S × X → I T 2F (S), called the transition map, such that for a given s ∈ S
and x ∈ X, λ(s, x) is an IT-2 fuzzy subset of S, and it may be seen as the
possibility distribution of the states that the automaton in state s and with input
x can enter.
(ii) i and f are IT-2 fuzzy subsets of S, called the IT-2 fuzzy set of initial states

and IT-2 fuzzy set of final states, respectively.
Now, we need to extend the transition function for defining the notion of the degree
to which a string of input symbols is accepted by an IT-2 fuzzy automaton, which is
given below.
Definition 3.2 Let M = (S, X, λ, i, f ) be an IT-2 fuzzy automaton. The transition
map λ can be extended to λ∗ : S × X ∗ → I T 2F (S), where
λ∗ (s, e) = 1/ [1, 1] /s,
λ∗ (s, wx) =

λ∗ (s, w)(s ) · λ(s , x) ,
s ∈S

∀w ∈ X∗ and ∀x ∈ X, where 1/ [1, 1] /s is an IT-2 fuzzy subset of S with
membership 1. Also, λ∗ (s, w)(s ) · λ(s , x) stands for the scalar product of IT-2
fuzzy set λ(s , x) with the scalar quantity λ∗ (s, w)(s ).
Definition 3.3 An IT-2 fuzzy language ρ ∈ I T 2F (X∗ ) is said to be accepted by an
IT-2 fuzzy automaton M = (S, X, λ, i, f ), if ∀w ∈ X ∗
ρ(w) = 1/[∨{μi (s) ∧ μλ∗ (s,w) (s ) ∧ μf (s ) : s, s ∈ S},
∨{μi (s) ∧ μλ∗ (s,w) (s ) ∧ μf (s ) : s, s ∈ S}].
The notion of a deterministic IT-2 fuzzy automaton is introduced as follows.
Definition 3.4 A deterministic IT-2 fuzzy automaton (DIT2FA) is a five-tuple
M = (S, X, λ, s0 , f ), where S and X are as in an IT-2 fuzzy automaton; s0 is the
initial state; λ : S × X → S is a map, called state transition map; and f is an IT-2
fuzzy set in S, called the IT-2 fuzzy set of final states.

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M. K. Dubey et al.

Definition 3.5 The transition map λ can be extended to λ∗ : S × X∗ → S, such
that λ∗ (s, e) = s and λ∗ (s, wa) = λ(λ∗ (s, w), a), ∀w ∈ X ∗ and a ∈ X.
Definition 3.6 An IT-2 fuzzy language ρ ∈ I T 2F (X∗ ) is said to be accepted by a
deterministic IT-2 fuzzy automaton M = (S, X, λ, s0 , f ), if for all w ∈ X ∗ ,
ρ(w) = 1/[μf (λ∗ (s0 , w)), μf (λ∗ (s0 , w))].
We shall denote an IT-2 fuzzy language ρ by ρM , if ρ is accepted by a deterministic
IT-2 fuzzy automaton M .
Now, the following result is towards the behavioural equivalent between an IT-2
fuzzy automaton and a deterministic IT-2 fuzzy automaton.
Proposition 3.1 A ρ ∈ I T 2F (X ∗ ) is accepted by an IT-2 fuzzy automaton if and
only if it is accepted by a deterministic IT-2 fuzzy automaton.
Proof Let M = (S, X, λ, i, f ) be an IT-2 fuzzy automaton. Then for all w ∈ X∗
and for all s ∈ S, define an IT-2 fuzzy subset of S as under:
iw (s) = 1/[∨s ∈S {μi (s ) ∧ μλ∗ (s ,w) (s)}, ∨s ∈S {μi (s ) ∧ μλ∗ (s ,w) (s)}],
or that iw (s) = s ∈S i(s ) · λ∗ (s , w)(s) . Now, let S = {iw : w ∈ X∗ } and the
map λ∗ : S × X∗ → S such that λ∗ (iw , w ) = iww , ∀w, w ∈ X∗ . It is clear
that λ∗ is well-defined. Now, M = (S , X, λ∗ , ie , f ) is a DIT2FA, where the IT-2
fuzzy subset of final states f ∈ I T 2F (S ) is defined as under:
f (iw ) =

iw (s) · f (s)
s∈S

{i(s ) · λ∗ (s , w)(s)} · f (s)

=
s∈S

s ∈S

i(s ) · λ∗ (s , w)(s) · f (s) ,

=
s ∈S

or that
f (iw )(s) = 1/[∨s ∈S {μi (s ) ∧ μλ∗ (s ,w) (s) ∧ μf (s)},
∨s ∈S {μi (s ) ∧ μλ∗ (s ,w) (s) ∧ μf (s)}].
Finally, let ρ ∈ I T 2F (X∗ ) be accepted by M . Then for all w ∈ X ∗ ,
ρ(w) = 1/[∨s,s ∈S {μi (s) ∧ μλ∗ (s,w) (s ) ∧ μf (s )},
∨s,s ∈S {μi (s) ∧ μλ∗ (s,w) (s ) ∧ μf (s )}]

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IT-2 Fuzzy Automata and IT-2 Fuzzy Languages

7

= 1/[∨s ∈S {∨s∈S μi (s) ∧ μλ∗ (s,w) (s ) ∧ μf (s )},
∨s ∈S {∨s∈S μi (s) ∧ μλ∗ (s,w) (s ) ∧ μf (s )}]
= 1/[∨s ∈S {μi (s ) ∧ μf (s )}, ∨s ∈S {μiw (s ) ∧ μf (s )}]
w

= 1/[∨s ∈S {μf (iw )(s ), ∨s∈S {μf (iw )(s )}]
= 1/[μf (λ∗ (ie , w)), μf (λ∗ (ie , w)] = ρA (w).
Thus ρ is accepted by a DIT2FA M .
Similarly, we can show that converse is also true.

4 Construction of Deterministic IT-2 Fuzzy Automata for
IT-2 Fuzzy Languages
In this section, we give the recipe to constructions of a DIT2FA for a given IT-2
fuzzy language. In particular, we give two recipes for such constructions and
prove that both the DIT2FA are homomorphic. The first such recipe is based on
right congruence relation (Myhill-Nerode relation), while the other is based on the
derivative of given IT-2 fuzzy language. We initiate with the following construction
based on right congruence relation.
Proposition 4.1 Let ρ ∈ I T 2F (X∗ ). Then there exists a deterministic IT-2 fuzzy
automaton, which accepts ρ.
Proof Let us define a relation Rρ on X∗ such that uRρ v ⇔ ρ(uw) = ρ(vw), ∀w ∈
X ∗ . Then Rρ is an equivalence relation on X∗ . Now, let SRρ = X∗ /Rρ = {[u]Rρ :
u ∈ X∗ }, where [u]Rρ = {v ∈ X∗ : uRρ v}. Define the maps λ∗Rρ : SRρ × X ∗ → SRρ
such that λ∗Rρ ([u]Rρ , v) = [uv]Rρ and fRρ ∈ I T 2F (SRρ ) such that fRρ ([u]Rρ ) =

ρ(u). Now, it is easy to check that both the maps λ∗Rρ and fRρ are well-defined. Thus
MRρ = (SRρ , X, λ∗Rρ , [e]Rρ , fRρ ) is a deterministic IT-2 fuzzy automaton. Finally,
for all u ∈ X∗ ,
ρMRρ (u) = fRρ (λ∗Rρ ([e]Rρ , u))
= 1/[μf (λ∗Rρ ([e]Rρ , u)), μfR (λ∗Rρ ([e]Rρ , u))]
Rρ

ρ

= 1/[μf ([u]Rρ ), μfR ([u]Rρ )] = fRρ ([u]Rρ ) = ρ(u).
Rρ

ρ

Hence DIT2FA MRρ accepts ρ.

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M. K. Dubey et al.

Now, we introduce the following concept of derivative of an IT-2 fuzzy language.
Definition 4.1 Let ρ ∈ I T 2F (X∗ ) and u ∈ X∗ . An IT-2 fuzzy language ρ u ,
defined by ρ u (v) = ρ(uv), ∀v ∈ X∗ is called a derivative of ρ with respect to u.
The following recipe is construction of a DIT2FA with the help of derivative of
given IT-2 fuzzy language.
Let ρ ∈ I T 2F (X∗ ). Now, assume S ρ = {ρ u : u ∈ X∗ }, and define λ∗ρ and f ρ
as under:
λ∗ρ : S ρ × X∗ → S ρ such that λ∗ρ (ρ u , v) = ρ uv , ∀ρ u ∈ S ρ , ∀v ∈ X ∗ , and
f ρ ∈ I T 2F (S ρ ) such that f ρ (ρ u ) = ρ u (e), ∀ρ u ∈ S ρ .
Then it can be easily seen that the maps λ∗ρ and f ρ are well-defined. Thus M ρ =
(S ρ , X, λ∗ρ , ρ e , f ρ ) is a deterministic IT-2 fuzzy automaton. Now, for all w ∈ X∗
ρA ρ (w) = f ρ (λ∗ρ (ρ e , w)) = f ρ (ρ ew ) = f ρ (ρ w ) = ρ w (e) = ρ(we) = ρ(w), it
shows that M ρ accepts ρ.
Before starting next, we familiarize the following concept of homomorphism
between two DIT2FA.
Definition 4.2 Let M = S, X, λ, s0 , f and M = S , X, λ , s0 , f be two
deterministic IT-2 fuzzy automata. A map φ : S → S is called a homomorphism
from M to M if
(i) φ(s0 ) = s0 ;
(ii) φ(λ(s, u)) = λ (φ(s), u); and
(iii) f (s) = f (φ(s)), ∀s ∈ S and ∀u ∈ X∗ .

M is called the homomorphic image of M if φ is an onto map.
Proposition 4.2 Let ρ ∈ I T 2F (X∗ ). Then DIT2FA MRρ =

SRρ , X, λ∗Rρ ,

[e]Rρ , fRρ is a homomorphic image of DIT2FA M ρ = (S ρ , X, λ∗ρ , ρ e , f ρ ).

Proof Define a map φ : M ρ → MRρ such that φ(ρ u ) = [u]Rρ , ∀ ρ u ∈ S ρ and u ∈
X∗ . Then it is easy to check that φ is well-defined onto map. Now, φ(λ∗ρ (ρ u , w)) =
φ(λ∗ρ (λ∗ρ (ρ e , u), w)) = φ(λ∗ρ (ρ e , uw)) = [uw]Rρ = λ∗Rρ ([u]Rρ , w) =
λ∗Rρ (φ(ρ u ), w). Also, for all ρ u ∈ S ρ , f ρ (ρ u ) = f ρ (λ∗ρ (ρ e , u)) = f ρ (ρ u ) =

ρ u (e) = ρ(u) = fRρ ([u]Rρ ) = fRρ (φ(ρ u )). Hence φ : M ρ → MRρ is a
homomorphism, and MRρ is a homomorphic image of M ρ .

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