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THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 25, Number 2, June 2017

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An Introduction to NP-completeness in Fuzzy Context

Ajith S. Kurup

Lecturer in Mathematics, Government Polytechnic College, Adoor, Kerala, India. E-mail: ajithskurup@gmail.com

Mathews M. George

Prof. in Mathematics, Providence College of Engineering, Chengannur, Kerala, India. E-mail: mathewjas@gmail.com

Abstract: The traditional set theory models the world as black or white and makes no provision for sets of grey. Life is not black and white. This two valued logic has proved effective and successful in solving well defined problems. However, a class of problems exists which are typically complex in nature, and are often left to human beings to deal with. It is known that no polynomial algorithm could be found for the problems in NP or NP-hard problems like traveling salesman Problem, the Hamiltonian Cycle Problem etc. While a method for computing the solutions to NP-complete problems using a reasonable amount of time remains undiscovered, computer scientists and programmers still frequently encounter NP-complete problems. NP-complete problems are often addressed by using algorithms. The objective of this paper is to study more about NP-completeness by using the notion of fuzzy algorithms.

Key words: Polynomial time, deterministic algorithm, non-deterministic algorithm, the classes P and NP, NP-completeness, NP-hardness, fuzzy sets, fuzzy operations, fuzzy algorithm, fuzzy Turing machine.

Characterization of Anti Fuzzy Ideal of Near-ring Group

G. K. Barthakur

Research Scholar, Department of Mathematical Science Bodoland University, Kokrajhar, Assam, India. E-mail: gopik2003@gmail.com

S. Basak

Department of Mathematics Kokrajhar Govt. College, BTAD, Assam, India. E-mail: bshibu.math@gmail.com

Abstract: In this paper, we shall study anti fuzzy N-subgroup, anti fuzzy ideal of a near-ring group or N-group and investigate some of there related properties. we are also discuss that, the sum of an anti fuzzy N-subgroup and an anti fuzzy ideal of an N-group is again an anti fuzzy N-subgroup and sum of two anti fuzzy ideal is an anti fuzzy ideal.

Key words and phrases: Near-ring, N-group, anti fuzzy N-subgroup, anti fuzzy ideal

Covering Dimension of Fuzzy Normal Spaces

B. Amudhambigai and N. Krithika

Department of Mathematics, Sri Sarada College for Women, Salem, Tamilnadu, India E-mail: rbamudha@yahoo.co.in, krithikasureshkumar.idp@gmail.com

R. Dhavaseelan

Department of Mathematics, Sona College of Technology, Salem, Tamilnadu, India. E-mail: dhavaseelan.r@gmail.com

Abstract: In this paper, we study some interesting properties of covering dimension of fuzzy normal spaces. First we establish a kind of overlapping relation on any fuzzy set and construct a kind of fuzzy normal space. Then we find the covering dimension of fuzzy normal spaces. Furthermore, a quantity of appealing properties and characterizations of them are studied.

Key words: Fuzzy swelling, fuzzy shrinking, covering dimension of fuzzy normal space.

Fuzzy Generalized Closed Sets in A Fuzzy Topological Space

Anjana Bhattacharyya

Department of Mathematics Victoria Institution (College) 78 B, A.P.C.Road Kolkata - 700009, INDIA. E-mail: anjanabhattacharyya@hotmail.com

Abstract: By introducing fb*-set in a fuzzy topological space, in this paper a new class of fuzzy sets, viz., fb*-closed sets is introduced and studied here. It has been shown that fb*-closed sets is stronger than fg-closed [3] sets and weaker than fuzzy closed sets. Also two new types of fuzzy separation axioms are introduced here as applications of fb*g-closed sets. Again fb*g-continuity and fb*g-irresoluteness are also introduced.

Key words: fb*-set, fb*g-closed set, fb*g-continuity.

On Fuzzy g*m-continuity in Fuzzy *Matroids

Dr. M. Sudha

Department of Mathematics(Associate Professor), Sri Sarada College For Women, Salem (T.N.) 636 016, India. E-mail: sudhaslm05@yahoo.com

J. Mahalakshmi

Department of Mathematics(Research Scholar), Sri Sarada College For Women, Salem (T.N.) 636 016, India. E-mail: paapumaha13@gmail.com

Abstract: In this paper, fuzzy *matroids are introduced via fuzzy *flat axioms. The concepts of fuzzy *m-structure, fuzzy *m-continuous function, fuzzy generalized *m-structure, fuzzy generalized *m-continuous function are introduced. Besides discussing interesting properties, several characterizations of the concepts introduced are investigated.

Key words: Fuzzy *m_E open set; Fuzzy *m_E flat; fuzzy *m-continuous function; fuzzy generaliz-ed *m_E flat; fuzzy generalized *m_E open set; fuzzy generalized *m-continuous function.

On Soft M-continuous Mappings

S. S. Thakur

Department of Applied Mathematics, Jabalpur Engineering College, Jabalpur (M. P.) 482011 India. E-mail:samjh_singh@rediffmail.com

Alpa Singh Rajput

Department of Applied Mathematics, Jabalpur Engineering College, Jabalpur (M. P.) 482011 India. E-mail: alpasinghrajput09@gmail.com

Abstract: The present paper introduces the concepts of soft -structure and soft -continuity and studied some of their properties and characterizations.

Key words: Soft m-structure, Soft M-continuous mappings, Soft m-compact and Soft m-connected.

Fuzzy Soft Ideal Theory Fuzzy Soft Local Function and Generated Fuzzy Soft Topological Spaces

A. Kandil

Mathematics Department, Faculty of Science, Helwan University, Helwan, Egypt.

O. A. E. Tantawy

Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt.

S. A. El-Sheikh and Sawsan S. S. El-Sayed

Mathematics Department, Faculty of Education, Ain Shams University, Cairo, Egypt. E-mail:sawsan_809@yahoo.com E-mail:s.elsayed@mu.edu.sa

Abstract: The purpose of this paper is to introduce the notion of fuzzy soft ideal in fuzzy soft set theory. The concept of fuzzy soft local function is also introduced. These concepts are discussed with a view to find new fuzzy soft topologies from the original one. The basic structure, especially a basis for such generated fuzzy soft topologies also studied here. Finally, the notion of compatibility of fuzzy soft ideals with fuzzy soft topologies is introduced and some equivalent conditions concerning this topic are established here.

Key words: Fuzzy soft set, fuzzy soft topological space, fuzzy soft interior, fuzzy soft closure, fuzzy soft open, fuzzy soft closed, fuzzy soft ideal, fuzzy soft local function, -fuzzy soft topology, compatible fuzzy soft ideal.

Embedding Lemma in Soft Topological Spaces

Moumita Chiney and S. K. Samanta

Department of Mathematics, Visa-Bharati University, Santiniketan 731 235, India. E-mail:syamal_123@yahoo.co.in

Abstract: The aim of this paper is to extend the embedding lemma in soft topological spaces and to use it to characterise a soft weak Tychonoff space as an embedded subspace of a soft cube.

Key words: Soft topology; soft mappings; continuous soft mappings; soft completely regular spaces; soft weak Tychonoff spaces; embedding; soft cube.

Slightly Regular and Somewhat Slightly Generalized Regular Double Fuzzy Continuous Functions

A. Vadivel

Department of Mathematics, Annamalai University, Annamalainagar, Tamil Nadu-608002. E-mail: avmaths@gmail.com

E. Elavarasan

E. Elavarasan Research Scholar, Department of Mathematics, Annamalai University Annamalainagar, Tamil Nadu-608002.. E-mail: maths.aras@gmail.com

Abstract: In this paper, we introduce the concept of slightly regular and somewhat slightly generalized regular double fuzzy continuous functions in double fuzzy topological spaces. Several interesting properties and characterizations are introduced and discussed. Furthermore, the relationship among the new concepts are introduced and established with some interesting counter examples.

Key words: Double fuzzy topology, slightly regular double fuzzy continuous, slightly generalized regular double fuzzy continuous, somewhat slightly generalized regular double fuzzy continuous functions.

Related Fixed Point Theorem on Two Modified Intuitionistic Fuzzy Metric Spaces

Bhavana Deshpande

Department of Mathematics B.S. Govt. P. G.College, Jaora, Dist.- Ratlam (M. P.) India 90-Rajiv Nagar (Near Kasturba Nagar), Ratlam (M. P.), India-457001.E-mail:bhavnadeshpande@yahoo.com

Shamim Ahmad Thoker

Department of Mathematics Govt. P. G. Arts ad Science College Ratlam (M. P.) India.E-mail:shamimthoker@gmail.com

Abstract: We establish a related fixed point theorem for two pairs of mappings on two modified intuitionistic fuzzy metric spaces. We also give an example to validate our results.

Key words: Modified intuitionistic fuzzy metric spaces, Common fixed point, Cauchy sequence, Triangular norm, t-representable.

Fuzzy Anti 2-bounded Linear Functional

Parijat Sinha

Department of Mathematics, V.S.S.D. College, Kanpur.

Divya Mishra and Ghanshyam Lal

Department of Mathematics, M.G.C.G. University, Satna (INDIA). E-mail:dvmishra275@gmail.com

Abstract: In this paper, fuzzy anti 2-bounded linear functional and fuzzy anti 2-dual spaces are defined. Two fundamental theorems, namely Hahn-Banach theorem and open-mapping theorem in fuzzy anti 2-normed linear space are established.

Key words: Fuzzy anti 2-dual space, Hahn-Banach theorem, open-mapping theorem.

Fuzzy Soft Line Graphs

Muhammad Akram and Muhammad Tahir

Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan.E-mail:m.akram@pucit.edu.pkE-mail:muhammad.tahir@pucit.edu.pk

Abstract: Fuzzy sets and soft sets are two different soft computing models for representing vagueness and uncertainty. We apply these soft computing models in combination to study vagueness and uncertainty in line graphs. We define the notions of fuzzy soft line graphs, strong fuzzy soft line graphs, complete fuzzy soft line graphs, regular fuzzy soft line graphs, partially regular fuzzy soft line graphs and explore some of their interesting properties.

Key words: Fuzzy soft line graphs, strong fuzzy soft line graphs, complete fuzzy soft line graphs, regular fuzzy soft line graphs.

Fuzzy -fold Boolean Ideals in MV-algebras

F. Forouzesh

Faculty of Mathematics and Computing Higher Education Complex of Bam, Kerman, Iran. E-mail:frouzesh@bam.ac.ir

E. Eslami

Department of Mathematics, Faculty of Mathematics and Computer shahid Bahonar University of Kerman, Kerman, Iran.. E-mail:esfandiar.eslami@uk.ac.ir

Abstract: In this paper, we introduce the notion of fuzzy n-fold Boolean ideals of an MV-algebra and investigate its properties. Several characterizations of fuzzy n-fold Boolean ideals are given. In addition, we introduce the notions of fuzzy congruences and fuzzy quotient algebras in MV-algebras and prove that there is a bijection between the set of fuzzy ideals and the set of fuzzy congruences. We state that for each fuzzy ideal m, there is an associated algebra A/m. Also, we show that A/m is an MV-algebra. In particular, we prove that the fuzzy quotient algebras induced by fuzzy n-fold Boolean ideals are n+1-bounded MV-algebras.

Key words: Fuzzy congruence, fuzzy n-fold Boolean ideal, fuzzy ideal, n-bounded MV-algebra.

On Infi-topological Spaces

Birojit Das, Apu Kumar Saha and Baby Bhattacharya.

Department of Mathematics National Institute of Technology, Agartala Tripura, India.

Abstract: The aim of this present treatise is to introduce the concept of Infitopological space as a generalization of the topological space. The concept of open sets (infi-open) and continuity ( I-continuity, I*-continuity) are defined on the new space. The properties of this newly introduced space have been studied. Also generalizations of infiopen sets, I-continuous functions are introduced and studied some of the properties relating to these concepts.

Key words: Infi-topological space, Infi-open, I-continuity, I*-continuity, Infi ??-open set, ??-continuous function.

Cayley Vague Graphs

Muhammad Akram

Department of Mathematics, University of the Punjab, New Campus, Lahore-Pakistan. E-mail:makrammath@yahoo.com E-mail:m.akram@pucit.edu.pk

Sovan Samanta

Department of Mathematics, India Institute of Information Technology,Nagpur-440006, India. E-mail:ssamantavu@gmail.com

Madhumangal Pal

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721 102, India.. E-mail:mmpalvu@gmail.com

Abstract: In this paper, we introduce the notion of Cayley vague graphs and investigate some of their properties. We present some interesting properties of vague graphs in terms of algebraic structures. We discuss connectedness in Cayley vague graphs. We also describe different types of ??-connectedness in Cayley vague graphs.

Key words: Cayley vague graphs, vague subsemigroups, connectedness, ??-connectedness.

On t-fuzzy h-bi-ideals and h-quasi-ideals in G-hemiring Debabrata

Debabrata Mandal

Department of Mathematics Raja Peary Mohan College, Uttarpara, Hooghly-712258.E-mail:dmandaljumath@gmail.com

Abstract: In this paper, t-fuzzy h-bi-ideals and t-fuzzy h-quasi-ideals of a G-hemiring are studied and some related properties are investigated. The notions of h -hemiregularity and h-intra-hemiregularity of a G-hemiring are studied and some of their characterizations in terms of t-fuzzy h -ideals are also obtained.

Key words: G-hemiring, t-fuzzy h-ideal, t-fuzzy h-bi-ideal, t-fuzzy h-quasi-ideal, h-hemiregular, h-intra-hemiregular.

Fuzzy Goal Programming with Linear and Some Nonlinear Membership Functions Approach to Multiobjective Solid Transportation Problems

A. K. Bit

Department of Mathematics, Faculty of Civil Engineering, College of Military Engineering, Pune-411031 (M. S.), India.E-mail: amalkbit@yahoo.com

Abstract: The linear multiobjective solid transportation problem in which the supply, demand and capacity constraints are all equality type and the objectives are equally important, non commensurable and conflicting in nature. The fuzzy goal programming with linear and some nonlinear membership functions for obtaining efficient solutions as well as the best compromise solution of a multiobjective solid transportation problem has been presented in this paper. An example is included to illustrate the methodology. Also this method is compared with one existing fuzzy programming algorithm with linear membership function and nonlinear membership functions. It may be noted that the necessity of the multiobjective solid transportation problem arises when there are heterogeneous conveyances available for the shipment of goods. The multiobjective solid transportation problem is of much use in public distribution systems.

Key words: Multiobjective solid transportation problem, fuzzy goal programming, efficient solution, Optimal compromise solution, linear and nonlinear membership functions.

A Comparison of The Other Two Types of Rough Sets Induced by Coverings

Songtao Shao and Bingxue Yao

Department of Mathematical Science, Liaocheng University. Liaocheng, Shandong, 252059, P.R.China..E-mail: sst0306@163.comE-mail:yaobingxue@163.com

Abstract: Rough set theory is a useful tool for data mining and analysis of uncertain. It is based on equivalence relations and has been extended to covering based rough set. We compare the covering rough sets defined by Xu and Zhang with ones defined by Zhu, and we investigate their properties and structures.

Key words: Rough sets, Generalized rough sets, Approximation operators, Coverings

A Modified GPIU Method for Singular Saddle Point Problems

Peng-Bo Xu, Nai-Min Zhang , Jing Li and Yuan Chen

School of Mathematics and Information Science, Wenzhou University, Wenzhou, 325035, People¡¯s Republic of China.

Abstract: In this paper, for solving singular saddle point problems, we extend the generalized parameterized inexact Uzawa (GPIU) method, that is, we propose a modified version of the GPIU (MGPIU ) method by introducing two new factors. We give the semi-convergence analysis of the MGPIU method and prove the semi-convergence result by verifying two necessary and sufficient conditions. Since the MGPIU method covers the GPIU method, then with some proper factors, the MGPIU method may semi-converges faster than the GPIU method for solving singular saddle point problems.

Key words: Singular linear systems, Saddle point problems, Iterative methods, Semi-convergence