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

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.. Also, for a given IT-2 fuzz

International Conference on

Advances in Mathematical Sciences, Vellore, India, December 2017 -

Volume I

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from conferences and lecture series focusing on a particular topic from any area ofmathematics Its aim is to make current developments available to the community asrapidly as possible without compromise to quality and to archive these for reference.Proposals for volumes can be submitted using the Online Book Project SubmissionForm at our website www.birkhauser-science.com

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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

V Madhu

Department of Mathematics

School of Advanced Sciences

Vellore Institute of Technology

Vellore, Tamil Nadu, India

A ManimaranDepartment of MathematicsSchool of Advanced SciencesVellore Institute of TechnologyVellore, Tamil Nadu, India

D Easwaramoorthy

Department of Mathematics

School of Advanced Sciences

Vellore Institute of Technology

Vellore, Tamil Nadu, India

D KalpanapriyaDepartment of MathematicsSchool of Advanced SciencesVellore Institute of TechnologyVellore, Tamil Nadu, India

M Mubashir Unnissa

Department of Mathematics

School of Advanced Sciences

Vellore Institute of Technology

Vellore, Tamil Nadu, India

Trends in Mathematics

ISBN 978-3-030-01119-2 ISBN 978-3-030-01120-8 (eBook)

https://doi.org/10.1007/978-3-030-01120-8

Library of Congress Control Number: 2018966815

© Springer Nature Switzerland AG 2018

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The Department of Mathematics, School of Advanced Sciences, Vellore Institute ofTechnology (Deemed to be University), Vellore, Tamil Nadu, India, organized theInternational Conference on Advances in Mathematical Sciences—2017 (ICAMS2017) in association with the Society for Industrial and Applied MathematicsVIT Chapter from December 1, 2017 to December 3, 2017 The major objective

of ICAMS 2017 was to promote scientific and educational activities toward theadvancement of common man’s life by improving the theory and practice ofvarious disciplines of Mathematics This prestigious conference was partiallyfinancially supported by the Council of Scientific and Industrial Research (CSIR),India The Department of Mathematics has 90 qualified faculty members and 30research scholars, and all were delicately involved in organizing ICAMS 2017grandly In addition, 30 leading researchers from around the world served as anadvisory committee for this conference Overall, more than 450 participants (pro-fessors/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 researcharticles to ICAMS Moreover, 305 original research articles were shortlisted forICAMS 2017 oral presentations that were authored by dynamic researchers from

25 states in India and 20 countries around the world We hope that ICAMS willfurther stimulate research in Mathematics, share research interest and information,and create a forum of collaboration and build a trust relationship We feel honoredand privileged to serve the best of recent developments in the field of Mathematics

v

technique suitable for solving the problem with sufficient mathematical background,and discussions on the obtained results with physical interruptions to understandthe domain of applicability This book also provides a comprehensive literaturesurvey which reveals the challenges, outcomes, and developments of higher levelmathematics in this decade The theoretical coverage of this book is relatively at ahigher level to meet the global orientation of mathematics and its applications inscience and engineering.

The target audience of this book is postgraduate students, researchers, andindustrialists This book promotes a vision of pure and applied mathematics asintegral to modern science and engineering Each chapter contains importantinformation emphasizing core Mathematics, intended for the professional whoalready possesses a basic understanding In this book, theoretically oriented readerswill find an overview of Mathematics and its applications Industrialists will find avariety of techniques with sufficient discussion in terms of physical point of view

to adapt for solving the particular application based on mathematical models Thereader can make use of the literature survey of this book to identify the currenttrends in Mathematics It is our hope and expectation that this book will provide aneffective learning experience and referenced resource for all young mathematicians

As Editors, we would like to express our sincere thanks to all the administrativeauthorities of Vellore Institute of Technology, Vellore, for their motivation andsupport We also extend our profound thanks to all faculty members and researchscholars of the Department of Mathematics and all staff members of our institute

We especially thank all the members of the organizing committee of ICAMS 2017who worked as a team by investing their time to make the conference a greatsuccess We thank the national funding agency, Council of Scientific and IndustrialResearch (CSIR), Government of India, for the financial support they contributedtoward the successful completion of this international conference We express oursincere gratitude to all the referees for spending their valuable time to review themanuscripts, which led to substantial improvements and selection of the researchpapers for publication The organizing committee is grateful to Mr ChristopherTominich, Editor at Birkhäuser/Springer, for his continuous encouragement andsupport toward the publication of this book

IT-2 Fuzzy Automata and IT-2 Fuzzy Languages 3

M K Dubey, P Pal, and S P Tiwari

Level Sets of i_v_Fuzzyβ-Subalgebras 13

P Hemavathi and K Palanivel

Interval-Valued Fuzzy Subalgebra and Fuzzy

INK-Ideal in INK-Algebra 19

M Kaviyarasu, K Indhira, V M Chandrasekaran, and Jacob Kavikumar

On Dendrites Generated by Symmetric Polygonal Systems: The

Case of Regular Polygons 27Mary Samuel, Dmitry Mekhontsev, and Andrey Tetenov

Efficient Authentication Scheme Based on the Twisted Near-Ring

Root Extraction Problem 37

V Muthukumaran, D Ezhilmaran, and G S G N Anjaneyulu

Dimensionality Reduction Technique to Solve E-Crime Motives 43

R Aarthee and D Ezhilmaran

Partially Ordered Gamma Near-Rings 49

T Nagaiah

Novel Digital Signature Scheme with Multiple Private Keys on

Non-commutative Division Semirings 57

G S G N Anjaneyulu and B Davvaz

Cozero Divisor Graph of a Commutative Rough Semiring 67

B Praba, A Manimaran, V M Chandrasekaran, and B Davvaz

vii

Gorenstein F I -Flat Complexes and (Pre)envelopes 77

V Biju

Bounds of Extreme Energy of an Intuitionistic Fuzzy Directed Graph 85

B Praba, G Deepa, V M Chandrasekaran, Krishnamoorthy Venkatesan,

and K Rajakumar

Part II Analysis

On Ultra Separation Axioms viaαω-Open Sets 97

M Parimala, Cenap Ozel, and R Udhayakumar

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 ofm-Fold Symmetric λ-Pseudo

Bi-starlike Functions 111Jay 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 161Priyanka Sangal and A Swaminathan

Local Countable Iterated Function Systems 169

A Gowrisankar and D Easwaramoorthy

On Intuitionistic FuzzyC -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-Typeq-Difference Operator 195

N Mustafa, K Vijaya, K Thilagavathi, and K Uma

Contents ix

Fuzzy Cut Set-Based Filter for Fixed-Value Impulse Noise Reduction 205

P S Eliahim Jeevaraj, P Shanmugavadivu, and D Easwaramoorthy

S Kavitha, Nak Eun Cho, and G Murugusundaramoorthy

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 271Kirtiwant P Ghadle, Priyanka A Pathade, and Ahmed A Hamoud

An Articulation Point-Based Approximation Algorithm for

Minimum Vertex Cover Problem 281Jayanth 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

Neutrosophic Assignment Problem via BnB Algorithm 323

S Krishna Prabha and S Vimala

Part IV Statistics

MRI of Human Head Scans for the Diagnosis of Alzheimer’s

Disease Using FuzzyC-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 353Rupam 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 393Manju Ramalingam and B Esther Clara

A Novice’s Application of Soft Expert Set: A Case Study on

Students’ Course Registration 407Selva 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

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

Contents xi

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 469Jitendra 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

Part V Differential Equations

Numerical Solution to Singularly Perturbed Differential Equation

of Reaction-Diffusion Type in MAGDM Problems 3

P John Robinson, M Indhumathi, and M Manjumari

Application of Integrodifferential Equations Using Sumudu

Transform in Intuitionistic Trapezoidal Fuzzy MAGDM Problems . 13

P John Robinson and S Jeeva

Existence of Meromorphic Solution of Riccati-Abel Differential

Equation . 21

P G Siddheshwar and A Tanuja

Expansion of Function with Uncertain Parameters in Higher

Dimension 29Priyanka Roy and Geetanjali Panda

Analytical Solutions of the Bloch Equation via Fractional Operators

with Non-singular Kernels 37

A S V Ravi Kanth and Neetu Garg

Solution of the Lorenz Model with Help from the Corresponding

Ginzburg-Landau Model 47

P G Siddheshwar, S Manjunath, and T S Sushma

Estimation of Upper Bounds for Initial Coefficients and

Fekete-Szegö Inequality for a Subclass of Analytic Bi-univalent

Functions 57

G Saravanan and K Muthunagai

An Adaptive Mesh Selection Strategy for Solving Singularly

Perturbed Parabolic Partial Differential Equations with a Small Delay . 67Kamalesh Kumar, Trun Gupta, P Pramod Chakravarthy,

and R Nageshwar Rao

xiv Volume II Contents

Part VI Fluid Dynamics

Steady Finite-Amplitude Rayleigh-Bénard-Taylor Convection

of Newtonian Nanoliquid in a High-Porosity Medium 79

P G Siddheshwar and T N Sakshath

MHD Three Dimensional Darcy-Forchheimer Flow of a Nanofluid

with Nonlinear Thermal Radiation 87Nainaru Tarakaramu, P V Satya Narayana, and B Venkateswarlu

Effect of Electromagnetohydrodynamic on Chemically Reacting

Nanofluid Flow over a Cone and Plate 99

H Thameem Basha, I L Animasaun, O D Makinde, and R Sivaraj

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

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

xvi Volume II Contents

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

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

xviii 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

Algebra

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 adeterministic IT-2 fuzzy automaton and prove that it is behavioural equivalent to anIT-2 fuzzy automaton Also, for a given IT-2 fuzzy language, we give certain recipefor constructions of deterministic IT-2 fuzzy automata

The notion of type-2 fuzzy sets was introduced by Zadeh [21], who gives the structure to model and abbreviate the impact of uncertainty in fuzzy logic rule-basedsystems The author in [9] has pointed out that the membership function of type-1fuzzy sets is totally crisp and hence not able to model certain uncertainty involved

sub-in the model, whereas sub-in case of type-2 fuzzy sets, it is capable to model suchuncertainty because of their fuzzy membership functions Also, the membershipfunction of type-2 fuzzy sets is three dimensional which gives additional degrees offreedom to model the uncertainty directly in comparison to type-1 fuzzy sets whichhave two-dimensional membership function However, it is not easy to understandand use the concept of type-2 fuzzy sets, which can be seen by the fact that almost allapplications use interval type-2 fuzzy set for the sake of all computations to performeasily [10]

From the commencement of the theory of fuzzy sets, Santos [12], Wee [17] andWee and Fu [18] introduced and studied fuzzy automata and languages, and afterMalik, Mordeson and Sen [11] have further studied and developed In the last fewdecades, 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: maheshdubey6@gmail.com ; mahesh@am.ism.ac.in ; priyankapal2192@gmail.com ;

© Springer Nature Switzerland AG 2018

V Madhu et al (eds.), Advances in Algebra and Analysis, Trends in Mathematics,

https://doi.org/10.1007/978-3-030-01120-8_1

3

and fuzzy languages have obtained not only conversion of classical automata tofuzzy automata but also a broad field of applications[2].

Fuzzy automata and fuzzy languages referred above are either based upon type-1fuzzy sets or on certain lattice structures (cf., [4,5,8,20]) Since we know, type-1fuzzy sets cannot be able to minimize the uncertainty involved in the model, andMendel [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 theconcepts of IT-2 fuzzy automata and languages and give the platform to developthe above model of nonclassical computations In this note, we give a brief look atcertain studies for IT-2 fuzzy automata and languages, which may be carried out indetails In particular, we begin by introducing a deterministic IT-2 fuzzy automatonand 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 ofdeterministic IT-2 fuzzy automata Finally, we give a brief look at an algebraic study

of an IT-2 fuzzy automaton

In this section, we memorize certain notions allied with an IT-2 fuzzy set We initiatewith 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 μ F(y, v) , where y ∈ Y and v ∈ J y ⊆ [0, 1], i.e.:

μ F(y, v)/(y, v), J y ⊆ [0, 1] , in which 0 ≤ μ F(y, v) ≤ 1.

From Definition2.1, it has been observed that when uncertainties disappear, a

type-2 membership function must reduce to a type-1 membership function, and in this

case, the variable v equals μ F (y)and 0≤ μ F(y)≤ 1

Definition 2.2 ([ 10 ]) A type-2 fuzzy set F in Y is called an IT-2 fuzzy set if

μ F(y, v) = 1, ∀y ∈ Y and ∀v ∈ J y An IT-2 fuzzy set F can be expressed as



F =y ∈Y v ∈J y 1/(y, v), J y ⊆ [0, 1].

For an IT-2 fuzzy set, we consider J y = [μ F(y), μ F(y) ] for all y ∈ Y , where μ F(y)

and μ F(y)are, respectively, called the lower membership function (LPF) and uppermembership 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]

IT-2 Fuzzy Automata and IT-2 Fuzzy Languages 5

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 isbehavioural equivalent to an IT-2 fuzzy automaton We initiate with the followingconcept 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,iand f is as follows:

(i) λ : S × X → IT 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

xcan enter

(ii) iand 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 isgiven below

Definition 3.2 Let M = (S, X,λ,i,  f )be an IT-2 fuzzy automaton The transitionmap λcan be extended to λ∗: S × X∗→ IT 2F (S), where

∀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 languageρ ∈ IT 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/[∨{μ i (s) ∧ μ λ∗(s,w) (s) ∧ μ f(s) : s, s∈ S},

∨{μ i (s) ∧ μ λ∗(s,w) (s) ∧ μ f(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; s0is 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.

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 languageρ ∈ IT 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∗,

Proposition 3.1 A ρ∈ IT 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:

, i e ,  f)is a DIT2FA, where the IT-2

fuzzy subset of final states f∈ IT 2F (S)is defined as under:

IT-2 Fuzzy Automata and IT-2 Fuzzy Languages 7

Thusρis accepted by a DIT2FA M.

Similarly, we can show that converse is also true

IT-2 Fuzzy Languages

In this section, we give the recipe to constructions of a DIT2FA for a given IT-2fuzzy language In particular, we give two recipes for such constructions andprove that both the DIT2FA are homomorphic The first such recipe is based onright congruence relation (Myhill-Nerode relation), while the other is based on thederivative of given IT-2 fuzzy language We initiate with the following constructionbased on right congruence relation

Proposition 4.1 Letρ ∈ IT 2F (X∗) Then there exists a deterministic IT-2 fuzzy

automaton, which acceptsρ.

Proof Let us define a relation Rρ on X∗such that uR

ρ(u) Now, it is easy to check that both the maps λ∗

Rρand f Rρare well-defined Thus

( [u] R ρ), μ fR ρ ( [u] Rρ )] = f Rρ ( [u] Rρ )= ρ(u).

Hence DIT2FAM Rρ acceptsρ

Now, we introduce the following concept of derivative of an IT-2 fuzzy language.

Definition 4.1 Let ρ ∈ IT 2F (X∗) and u ∈ X∗ An IT-2 fuzzy language ρu,defined byρ u (v)= ρ(uv), ∀v ∈ X∗is called a derivative ofρ with respect to u.

The following recipe is construction of a DIT2FA with the help of derivative ofgiven IT-2 fuzzy language

Letρ ∈ IT 2F (X∗) Now, assume Sρ = {ρ u : u ∈ X∗}, and define λ∗ρ and fρ

as under:

λ∗ρ : Sρ × X∗→ Sρ such t hat λ∗ρ (ρ u , v)= ρ uv , ∀ρ u ∈ Sρ , ∀v ∈ X∗, and

fρ ∈ IT 2F (Sρ ) such t hat fρ (ρ u )= ρ u (e), ∀ρ u ∈ Sρ

Then it can be easily seen that the maps λ∗ρ and fρare well-defined Thus M ρ=

(Sρ , X, λ∗ρ ,ρ e , fρ ) is a deterministic IT-2 fuzzy automaton Now, for all w ∈ X∗



ρ A ρ (w) = fρ (λ∗ρ (ρ e , w)) = fρ (ρ ew ) = fρ (ρ w )= ρ w (e)= ρ(we)= ρ(w), itshows that Mρ acceptsρ

Before starting next, we familiarize the following concept of homomorphismbetween two DIT2FA

(ii) φ(λ(s, u)) = λ(φ(s), u); and

(iii) f (s)= f(φ(s)), ∀s ∈ S and ∀u ∈ X∗.



Mis called the homomorphic image of M if φ is an onto map.

Proposition 4.2 Let ρ ∈ IT 2F (X∗) Then DIT2FA M Rρ = S Rρ , X, λ∗

X∗ Then it is easy to check that φ is well-defined onto map Now, φ(λ∗ρ ( ρu , w))=

φ (λ∗ρ (λ∗ρ (ρ e , u), w)) = φ(λ∗ρ (ρ e , uw)) = [uw] Rρ = λ∗

IT-2 Fuzzy Automata and IT-2 Fuzzy Languages 9

This section is towards the study of algebraic aspects of an IT2FA For a betterunderstanding of the structure of automata and their applications, such type ofstudies has been done for both classical and nonclassical automata (cf., [3,11,14]).Here, we consider an IT-2 fuzzy automaton having no IT-2 fuzzy sets of initial andfinal states In particular, an IT-2 fuzzy automaton is a three-tuple M = (S, X,λ),

where S is nonempty finite set of states, X is an input set and  λis transition maphaving usual meaning as in Definition 3.1 We begin with the following

Definition 5.1 Let M = (S, X,λ) be an IT2FFA and  U ∈ IT 2F (S) The IT-2

fuzzy source and the IT-2 fuzzy successor of Uare, respectively, defined as,

(ii)  U ⊆ IT 2F SO( U ) and  U ⊆ IT 2F SU( U );

(iii) I T 2F SO( U i : i ∈ I) = I T 2F SO(  U i : i ∈ I) and IT 2F SU( U i :

i ∈ I) = I T 2F SU (  U i : i ∈ I);

(iv) I T 2F SO(I T 2F SO( U )) = IT 2F SO( U ) and I T 2F SU (I T 2F SU ( U )) =

I T 2F SU ( U ).

Definition 5.2 Let M = (S, X,λ) be an IT2FFA and  U ∈ IT 2F (S) Then  U is

called an IT-2 fuzzy subsystem of M if for all s ∈ S, IT 2F SU( U )⊆ U, i.e.:

μ U(s) ≥ μ U(s) ∧ μ λ∗(s,w) (s)and

μ U(s) ≤ μ U(s) ∧ μ λ ∗(s,w) (s),

∀s∈ S and ∀w ∈ X∗.

Remark 5.1 From the above, it can be observed that for any IT2FFA, M =

(S, X, λ)and U ∈ IT 2F (S), IT2FSU( U) is always an IT-2 fuzzy subsystem of M

6 Conclusion

In this note, we have tried to introduce deterministic IT-2 fuzzy automata and shownthat it is behavioural equivalent to IT-2 fuzzy automata Also, we have providedcertain recipes to constructions of DTI2FA for a given IT-2 fuzzy language andprove that they are homomorphic Finally, we have given a brief look on algebraicaspects of IT2FFA These studies are beginning to develop the theory of automataand languages based on IT2 fuzzy sets

References

1 Bˇelohlávek, R.: Determinism and fuzzy automata Information Sciences, 143, 205–209 (2002)

2 ´ Ciri´c, M., Ignjatovi´c, J.: Fuzziness in automata theory: why? how? Studies in Fuzziness and

Soft Computing, 298, 109–114 (2013)

3 Holcombe, W M L.: Algebraic Automata Theory Cambridge University Press, (1987)

4 Ignjatovi´c, J., ´ Ciri´c, M., Bogdanovi´c, S.: Determinization of fuzzy automata with membership

values in complete residuated lattices Information Sciences, 178, 164–180 (2008)

5 Ignjatovi´c, J., ´ Ciri´c, M., Bogdanovi´c, S., Petkovi´c, T.: Myhill-Nerode type theory for fuzzy

languages and automata Fuzzy Sets and Systems, 161, 1288–1324 (2010)

6 Jiang, Y., Tang, Y.: An interval type-2 fuzzy model of computing with words Information

Sciences, 281, 418–442 (2014)

7 Jun, Y B.: Intuitionistic fuzzy finite state machines Journal of Applied Mathematics and

Computing, 17, 109–120 (2005)

8 Li, Y., Pedrycz, W.: Fuzzy finite automata and fuzzy regular expressions with membership

values in lattice-ordered monoids Fuzzy Sets and Systems, 156, 68–92 (2005)

9 Mendel, J M., John, R I.: Type-2 fuzzy sets made Simple IEEE Transaction on Fuzzy

Systems, 10, 117–127 (2002)

10 Mendel, J M., John, R I., Liu, F.: Interval Type-2 fuzzy logic Systems made Simple IEEE

Transaction on Fuzzy Systems, 14, 808–821 (2006)

11 Mordeson, J N., Malik, D S.: Fuzzy Automata and Languages, Theory and Applications Chapman and Hall/CRC London/Boca Raton, (2000)

12 Santos, E S.: Max-product machines Journal of Mathematical Analysis and Applications, 37,

677–686 (1972)

13 Tiwari, S.P., Yadav, V K., Singh, A.K.: Construction of a minimal realization and monoid for

a fuzzy language: a categorical approach Journal of Applied Mathematics and Computing, 47,

401–416 (2015)

14 Tiwari, S.P., Yadav, V K., Singh, A K.: On algebraic study of fuzzy automata International

Journal of Machine Learning and Cybernetics, 6, 479–485 (2015)

15 Tiwari, S.P., Yadav, V K., Dubey, M.K.: Minimal realization for fuzzy behaviour: A

bicategory-theoretic approach Journal of Intelligent & Fuzzy Systems, 30, 1057–1065 (2016)

16 Tiwari, S.P., Gautam, V., Dubey, M.K.: On fuzzy multiset automata Journal of Applied

Mathematics and Computing, 51, 643–657 (2016)

17 Wee, W G.: On generalizations of adaptive algorithm and application of the fuzzy sets concept

to pattern classification Ph D Thesis, Purdue University, Lafayette, IN, (1967)

18 Wee, W G., Fu, K.S.: A formulation of fuzzy automata and its application as a model of

learning systems IEEE Transactions on Systems, Man and Cybernetics, 5, 215–223 (1969)

IT-2 Fuzzy Automata and IT-2 Fuzzy Languages 11

19 Wu, D., Mendel, J M.: Uncertainty measures for interval type-2 fuzzy Sets Information

Sciences, 177, 5378–5393 (2007)

20 Wu, L., Qiu, D.: Automata theory based on complete residuated lattice-valued logic: Reduction

and minimization Fuzzy Sets and Systems, 161, 1635–1656 (2010)

21 Zadeh, L A.: The concept of a linguistic variable and its application to approximate reasoning

-1 Information Sciences, 8, 199–249 (1975)

P Hemavathi and K Palanivel

Abstract This paper explores the new idea of level sets of i_v_fuzzy β-subalgebras

and discussed some of their properties

The study of two class of algebras, B-algebras and β-algebras, was initiated by

Neggers and Kim in 2002 [6,7] In 1965, the notion of fuzzy sets has beenintroduced by Zadeh [8] In 1994 Biswas [2] proposed Rosenfeld’s fuzzy subgroupswith i-v membership functions Borumand Saeid et al [3] investigated about the

i-v-f B-algebras in 2006 The concept of fuzzy β-subalgebras in β-algebras was

introduced by Ayub Anasri et al [1] in 2013 Inspired by this, Hemavathi et al [4,5]

introduced the notion of i-v-f β-subalgebra and i-v-f translation and multiplication

of i-v-f subalgebras This paper aspires to define the level subset of i-v-f subalgebras with the help of i-v β-subalgebra in β-algebras, and it deals some of

β-their properties and elegant results

Vellore Institute of Technology, Vellore, India

e-mail: drkpalanivel@gmail.com ; palanivel.k@vit.ac.in

14 P Hemavathi and K Palanivel

Definition 2.1 A β-algebra is a nonempty set X with a constant 0 and two binary

operations ( +) and (−) satisfying the following axioms:

Definition 2.4 Let A be a fuzzy set of X, for α ∈ [0, 1] Then A α = {x ∈ X :

σ (x) ≥ α} is known as a level set of A.

Proposition 2.5 Let A be a fuzzy set of a set X For α1, α2 ∈ [0, 1], if α1 ≤ α2, then A α2 ≤α1where A α1 andA α2 are the corresponding level sets of A.

Definition 2.6 Let A be a fuzzy set of X For α ∈ [0, 1], the set A α = {x ∈ X :

σ (x) ≤ α} is called as a lower level set of A.

Definition 2.7 Let A = {x, α A (x) : x ∈ X} be an interval-valued fuzzy subset in

X Then σ A is said to be an interval-valued fuzzy(i_v_fuzzy) β-sub algebra of X

if

(i) σ A (x + y) ≥ rmin{σ A (x), σ A (y) } ∀ x, y ∈ X.

(ii) σ A (x − y) ≥ rmin{σ A (x), σ A (y) } ∀ x, y ∈ X.

Definition 2.8 Let A be an i_v_ fuzzy subset of X, α ∈ D[0, 1] Then

A α = {x ∈ X : A α (x) ≥ α} is called a i_v_ level set of A.

This section classifies the β-subalgebras by their family of level sets of valued fuzzy(i_v_ fuzzy) β-subalgebras of a β-algebra.

interval-Definition 3.1 Consider A be an i_v_ fuzzy β-subalgebra of X, α ∈ D[0, 1] Then

A α = {x ∈ X : A α (x) ≥ α} is called a i_v_ level β-subalgebra of A.

Theorem 3.2 If A = {x, σ A (x) : x ∈ X} is an i_v_ fuzzy set in X, then A α is a subalgebra of X, for every α ∈ D[0, 1].

Proof For x, y ∈ σ A α & σ A (x) ≥ α & σ A (y) ≥ α

σ A (x + y) ≥ rmin{σ A (x), σ A (y)}

≥ rmin{α, α}

≥ α

⇒ x + y ∈ A α Similarly, t x − y ∈ A α

Hence A α is subalgebra of X.

Theorem 3.3 If A = {x, σ A (x)  : x ∈ X} is an i_v_ fuzzy set in X such that A α

is a subalgebra of X for every α ∈ D[0, 1] Then A is an i_v_ fuzzy β-subalgebra

of X.

Proof Consider A = {x, σ A (x)  : x ∈ X} is an i_v_ fuzzy set in X.

Assume that A α is a subalgebra of X for α ∈ D[0, 1].

Now, α = rmin{σ A (x), σ A (y)}

For x + y and x − y ∈ A α,

⇒ x + y ∈ A α ≥ rmin{σ A (x), σ A (y) } = α

Similarly, x − y.

∴ A is an i_v_ fuzzy β-subalgebra of X.

Theorem 3.4 Any β-subalgebra of X can be realized as a level β-subalgebra for

some i_v_ fuzzy β-subalgebra of X.

Proof Consider A be an i_v_ fuzzy β-subalgebra of X.

We define

σ A (x)=



α x ∈ X [0, 0], elsewhere

16 P Hemavathi and K Palanivel

Theorem 3.6 Let X be any β-algebra If {A i } any sequence of β-subalgebra of X

such that A0 ⊂ A1 ⊂ ⊂ A n = X, then ∃ an i_v_ fuzzy β-subalgebras σ of X

whose i_v_ level β-subalgebras are exactly the β-subalgebra {A i }.

Proof Consider a set of numbers α0> α1> > α n , where each α i ∈ D[0, 1] Let σ be a fuzzy set represented by

σ A0 = α0& σ (A i ∼ A i−1) = α i , 0 < i ≤ n

to prove σ is i_v_ fuzzy β-subalgebra of X.

For this x, y ∈ X.

Case (i)

Let x, y ∈ A i ∼ A i−1

Then x, y ∈ A i ⇒ x + y ∈ A i &x − y ∈ A i Also x, y ∈ A i ∼ A i−1⇒ σ(x) =

s i ⇒ σ(y) ⇒ rmin{σ(x), σ (y)} = α i

Now since A i is subalgebra x +y & x−y ∈ A i ⇒ x+y & x−y ∈ A i ∼ A i−1or

Then σ (x) = α i & σ (y) = α j > α i

Hence rmin{σ(x), σ ( y) } = rmin{α i , α j } = α i

Further, y ∈ A i ∼ A j−1 ⇒ y ∈ A j ⊂ A i ⇒ x, y ∈ A i

Since A i is β-subalgebra of X,x + y & x − y ∈ A i,

∴ σ A i (x − y) ≥ s i = rmin{σ(x), σ (y)} &

σ A i (x + y) ≥ α i = rmin{σ(x), σ (y)}.

Thus in both cases, σ is a i_v_ fuzzy β-subalgebra of X.

From the definition of σ , it follows that I m(σ ) = {α0, α1 s n }.

Hence σ α i = {x ∈ X : σ(x) ≥ α i }, for 0 ≤ i ≤ n are the i_v_ level subalgebras of X by Theorem3.3

β-Then the sequence{σ t i } of i_v_level β-subalgebras of σ is in the form of σ α0 ⊂

σ α1 ⊂ ⊂ σ t n = X

Now, σ α0 = {x ∈ X : σ (x) ≤ α0} = A0

Finally, to prove σ t i = A i for 0≤ i ≤ n

clearly,A i ⊆ σ α i If x ∈ σ α i , then σ (x) ≥ α iwhich implies

σ (x) ∈ {α1, α2, α n}

Here x ∈ A0orA1 orA i It follows that x ∈ A i;

∴ σ α i = A i for 0≤ i ≤ n.

Thus the i_v_ level β-subalgebras of σ are exactly the β-subalgebras of X.

Theorem 3.7 Let A = {x, σ A (x) : x ∈ X} be an i_v_ fuzzy β-subalgebra of X If

I m(A) is finite α0< α1< < α n , then any α i , α j ∈ Im(σ A ),σ α i = σ α j implies

α i = α j

Proof Assume that α i = α j

If x ∈ σ α j , then σ A (x) ≥ α j > α i ⇒ x ∈ σ α i ; there exists x ∈ X such that

α i ≤ σ (x) < α j ⇒ x ∈ σ α i but x ∈ σ α j

Hence σ α j ⊂ σ α i and σ α j = σ α i

which is a contradiction

Theorem 3.8 Let A = {x, σ A (x) : x ∈ X} be an i_v_fuzzy β-subalgebra of X.

Two-level subalgebras A α1 & A α2(with α1< α2) of A are equal if and only if there

is no x ∈ X such that α1≤ σ A (x) < α2.

18 P Hemavathi and K Palanivel

Proof Assume thatA α1 = A α2for α1< α2

Then there exists x ∈ X such that the membership function α1< σ A (x) < α2.

Hence σ α2 is a proper subset of σ s1

1 Aub Ayub Anasri, M., Chandramouleeswaran, M.: Fuzzy β-subalgebras of β-algebras

Interna-tional journal of mathematical sciences and engineering applications 5(7), 239–249 (2013)

2 Biswas, R.: Rosenfeld’s fuzzy subgroups with Interval valued membership functions Fuzzy sets

and systems 63(1), 87–90 (1994)

3 Borumand saeid, A.: Interval valued fuzzy B-algebras Iranian Journal of fuzzy systems.

3(2),63–73 (2006)

4 Hemavathi, P., Muralikrishna, P., Palanivel, K.: A note on interval valued fuzzy β-subalgebras.

Global Journal of Pure and Applied Mathematics 11(4), 2553–2560 (2015)

5 Hemavathi, P., Muralikrishna, P., Palanivel, K.:Study on i-v fuzzy translation and multiplication

of i-v fuzzy β-subalgebras, International Journal of Pure and Applied Mathematics, 109(2),

245–256 (2016)

6 Neggers, J., Kim Hee Sik.: On B-algebras Math Vesnik 54(1–2),21–29 (2002)

7 Neggers, J., Kim Hee Sik.: On β-algebras Math Solvaca 52(5), 517–530 (2002)

8 Zadeh, L.A.: Fuzzy sets Inform Control 8(3), 338–353 (1965)

and Fuzzy INK-Ideal in INK-Algebra

M Kaviyarasu, K Indhira, V M Chandrasekaran, and Jacob Kavikumar

Abstract In this paper we examine IVF INK-ideal in INK-algebras by giving a

couple of definitions and relative hypotheses The image and preimage of IVFINK-ideal become i-v fuzzy INK-ideals in INK-algebras

by Zadeh [7] Xi [6] applied the concept of fuzzy set to BCK-algebra and gave some

of its properties In [8], Zadeh made an extension of the concept of fuzzy set by

an interval-valued fuzzy set (i.e., a fuzzy set with an interval-valued membershipfunction) This interval-valued fuzzy set is referred to as i-v fuzzy set Zadeh alsoconstructed a method of approximate inference using his i-v fuzzy sets In [1],Biswas defined interval-valued fuzzy subgroups and investigated some elementaryproperties

In this paper, using the notion of interval-valued fuzzy set by Zadeh, we introducethe concept of interval-valued fuzzy INK-ideals in INK-algebra (briefly i-v fuzzyINK-ideals in INK-algebra) and study some of their properties We prove that everyINK-ideals of INK-algebra X can be realized as i-v level INK-ideals of a INK-algebra X, and then we obtain some related results which have been mentioned inthe abstract

M Kaviyarasu · K Indhira (  ) · V M Chandrasekaran

Department of Mathematics, VIT University, Vellore, India

e-mail: kavitamilm@gmail.com ; kaviyarasu.m2015@vit.ac.in ; kindhira@vit.ac.in;

20 M Kaviyarasu et al.

Definition 1 An algebra (A, , 0) is approached INK-algebra on the off chance

that it satisfies the accompanying conditions:

INK-I a  a = 0

INK-II a  0 = a

INK-III 0 a = a

INK-IV (b  a)  (b  c) = (a  c), for all 0, a, b, c ∈ A.

Definition 2 Let B nonvoid set be a subset of a INK-algebra A Then B is called a

INK-ideal of A if, for all a, b, c ∈ A.

INKI(1) 0∈ B

INKI(2) (c ∗ a) ∗ (c ∗ b) ∈ B, and b ∈ B imply a ∈ B.

Definition 3 Let A be a INK-algebra and T ⊆ A Then T is said to be subalgebra, if a  b ∈ T , for all a, b ∈ A.

INK-Definition 4 An ideal C of a INK-algebra X is said to be closed if 0∗ a ∈ C for all

a ∈ C.

Definition 5 Let A be a nonvoid set A mapping ξ : a → [0, 1] ξ is called a fuzzy set in X The complement of ξ denoted by ξ−(a) = 1 − ξ(a), for all a ∈ A.

Definition 6 A fuzzy set ξ in a INK-algebra A is called a fuzzy subalgebra of A if,

ξ(a  b) ≥ min {ξ(a), ξ(b)}, for all a, b ∈ A.

Definition 7 Let ξ be a fuzzy set of a set A For a fixed t ∈ [0, 1], the set ξ t =

{a ∈ A/ξ(a)} is called an upper level of ξ.

Definition 8 Let ξ be a fuzzy set of INK-algebra A which is said to be fuzzy ideal

of A, if

FI1 ξ(0) ≥ ξ(a)

FI2 ξ(a) ≥ min {ξ(a  b), ξ(b)}, for all a, b ∈ A.

Definition 9 A fuzzy subset ξ in a INK-algebra A is called a fuzzy INK-ideal of A,

if for all a, b, c ∈ A,

FINKI-1 ξ(0) ≥ ξ(a)

FINKI-2 ξ(a) ≥ min {ξ(c  a)  (c  b), ξ(y)}

An i-v fuzzy set τ on A is given by τ = a,

sub-intervals of[0, 1] It is clear that if ξ L

Theorem 1 Let τ be a IVF (INK)-subalgebra of A If there exists a sequence {a n}

in A such that lim n→∞ξ−

Theorem 2 A IVF set τ =ξ τ L , ξ τ U

in A is a IVF INK-algebra of A if and only if

ξ τ L and ξ τ U are fuzzy INK-subalgebra of A.

Proof Let ξ τ L and ξ τ U are fuzzy INK-subalgebra of A and a, b ∈ A.

Hence ξ τ L and ξ τ U are fuzzy INK-ideals of A.

Theorem 4 Let τ1and τ2be IVF (INK)-ideal of a INK-algebra A Then τ1∩ τ2is

Theorem 5 Let A be a INK-algebra and τ be an v fuzzy subset in A Then τ is an

i-v fuzzy INK-ideal of A, if and only if U−(τ ; [ϑ1, ϑ2])=a ∈ A/ξ−

τ (a) ≥ [ϑ1, ϑ2]

is a INK-ideal of τ , for every [ϑ1, ϑ2] ∈ D(0, 1) We call U−(τ ; [ϑ1, ϑ2]) the i-v

level INK-ideal of τ

Proof Assume that τ is an i-v fuzzy INK-ideal of A.

Let [ϑ1, ϑ2]) ∈ θ(0, 1) such that (c  b)  (c  b), b ∈ U−(τ ; [ϑ1, ϑ2])

= min {min [η1, η2] , min [η3, η4]}

So ϑ1< min [η1, η2] and ϑ1< min [η3, η4]

a, b, c ∈ A.

Definition 11 Let g : (X; , 0) → (Y ; , , 0) be a mapping from set X into a set

Y Let τ be an i-v fuzzy subset in Y Then the inverse image of τ , denoted by g−1(τ ),

is an i-v fuzzy subset in X with the membership function given ξ g−1

τ (a) = ξ−(g(a)),

for all x ∈ X.

Theorem 6 An into homomorphic preimage of a fuzzy ideal is also fuzzy

INK-ideal.

Proof Let f : X → Xbe an into homomorphism of INK-algebra, B a fuzzy

INK-ideal of X, and ξ the preimage of B under f; then B(f (x)) = ξ(x), for all

x ∈ X.

24 M Kaviyarasu et al.Then

ξ(x) ≥ min {B((f (z)  f (x))  (f (z)  f (y))), B(f (y))}

ξ(x) = min {B(f (z  x)  (z  y)), B(f (y))}

ξ(x) = min {ξ((z  x)  (z  y)), ξ(y)}

Hence ξ(x) = B(f (x)) = B ◦ f (x) is a fuzzy INK-ideal of X.

Proposition 2 Let g : X → Y Let n =n L , n U

and m=m L , m U

be i-v fuzzy subset in X and Y Then

1 g−1(m)=g−1(m L ), g−1(m U )

2 g(n)=g(n L ), f (n U )

.

Theorem 7 Let g : X → Y be a homomorphism from a algebra X into a

INK-algebra Y If B is an i-v fuzzy INK-ideal of Y, then the inverse image f−1(B) of B is

an i-v fuzzy INK-ideal of X.

Proof Since B = [ξ L

B , ξ B U] is an i-v fuzzy INK-ideal of Y, it follows from Theorem3

that (ξ B L ) and (ξ B U ) are fuzzy INK-ideals of Y Using Theorem 6, we know

is an i-v fuzzy INK-ideal of X

Theorem 8 Let f : X → Y be a homomorphism If A is an i-v fuzzy INK-ideal of

X, then f[A] of A is an i-v fuzzy INK-ideal of Y.

Proof Assume that A is an i-v fuzzy INK-ideal of X Note that A = [ξ L

A , ξ A U] is an

i-v fuzzy INK-ideal of X Let f : X → Y be a homomorphism between INK-algebra

X and Y For every fuzzy INK-ideal ξ in X, f (ξ ) is a fuzzy INK-ideal of Y Then the image f (ξ A L ) and f (ξ A U )are fuzzy INK-ideals of Y Combining Theorem3andProposition2, we conclude that f [A] = [f (ξ L

A ), f (ξ A U )] is an i-v fuzzy INK-ideal