alexa (α, β)-fuzzy Lie algebras over an (α, β)-fuzzy field | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
Make the best use of Scientific Research and information from our 700+ peer reviewed, Open Access Journals that operates with the help of 50,000+ Editorial Board Members and esteemed reviewers and 1000+ Scientific associations in Medical, Clinical, Pharmaceutical, Engineering, Technology and Management Fields.
Meet Inspiring Speakers and Experts at our 3000+ Global Conferenceseries Events with over 600+ Conferences, 1200+ Symposiums and 1200+ Workshops on
Medical, Pharma, Engineering, Science, Technology and Business

(α, β)-fuzzy Lie algebras over an (α, β)-fuzzy field

P. L. ANTONY1 and P. L. LILLY2

1Department of Mathematics, St. Thomas College, Thrissur, Kerala, India, E-mail: [email protected],

2Department of Mathematics, St. Joseph's College, Irinjalakuda, Kerala, India, E-mail: [email protected]

Received July 14, 2009 Revised September 15, 2009

Visit for more related articles at Journal of Generalized Lie Theory and Applications

Abstract

The concept of ( ; )-fuzzy Lie algebras over an ( ; )-fuzzy eld is introduced. We provide characterizations of an (2;2 _q)-fuzzy Lie algebra over an (2;2 _q)-fuzzy eld.

Introduction

Zadeh [12] formulated the notion of fuzzy sets and after that many scholars developed fuzzy system of di erent algebraic structures. The idea of quasi-coincidence of a fuzzy point with a fuzzy set, which is mentioned in [10], has played a vital role in generating some di erent types of fuzzy subgroups. Using the belong-to relation (∈) and quasi-coincidence with relation (q) between fuzzy points and fuzzy sets, the concept of (α, β)-fuzzy subgroup was introduced by Bhakat and Das [4]. Akram [1] introduced (α, β)-fuzzy Lie subalgebras and investigated some of its properties. Nanda [9] introduced fuzzy algebra over fuzzy field. It is natural to investigate similar types of generalization of the existing fuzzy subsystem. In [3], we introduced fuzzy Lie algebra over a fuzzy field and some properties were discussed.

In this paper, we introduce the concept of (α, β)-fuzzy Lie algebra over an (α, β)-fuzzy eld and investigate some of its properties.

Preliminaries

In this section, we present some definitions needed for our study. We denote a complete distributive lattice with the smallest element 0 and the largest element 1 by I. By a fuzzy subset of a nonempty set X, we mean a function from X to I.

Definition 2.1 (see [5]). Let X be a field and let F be a fuzzy subset of X. Then F is called a fuzzy field of X if

image

Remark 2.2. It is seen that if F is a fuzzy field of X, then

image

Definition 2.3. Let A be a fuzzy subset of a Lie algebra L. Then A is called a fuzzy Lie algebra of L over a fuzzy field F, if for all image

image

The relations belong to and quasi-coincidence with

Let L be a Lie algebra over a field X, let A : L → [0; 1] be a fuzzy set on L, and let F : X → [0; 1] be a fuzzy set on X. The support of fuzzy set A is the crisp set that contains all elements of L that have nonzero membership grades in A.

Definition 3.1 (see [10]). A fuzzy set A : L → [0; 1] of the form

image

is said to be a fuzzy point with support x and value t and is denoted by xt.

For a fuzzy point xt and a fuzzy set A in a set L, Pu and Liu [10] gave meaning to the symbol image

A fuzzy point xt is said to belong to a fuzzy set A, written as image A fuzzy point xt is said to be quasi-coincident with a fuzzy set A, denoted by image

For a fuzzy set image we denote image

The following notations are used in this paper.

1. image means that either belong to or quasi-coincident with,

2. image means that α does not hold.

Let min image be denoted by m(t; s) and let maxft; sg be denoted by M(t; s). Take I = [0; 1] and ^ = min, V = max with respect to the usual order in Definitions 2.1 and 2.3.

Lemma 3.2. A fuzzy subset F of a field X is a fuzzy field if and only if it satisfies the following conditions:

image

Lemma 3.3. Let L be a Lie algebra over a field X. Then a fuzzy subset A of Lie algebra L is a fuzzy Lie algebra over a fuzzy field F of X if and only if it satisfies the following conditions:

image

(α, β)-fuzzy Lie algebras over an (α, β)-fuzzy field

Let α and β denote any one of image unless otherwise specified.

Definition 4.1. Let X be a field and let image be a fuzzy subset of X. Then F is called an (α, β)-fuzzy field of X, if it satisfies the following conditions:

image

Definition 4.2. Let L be a Lie algebra over a field X, and let image be animage fuzzy field of X. Then a fuzzy subset image is called an (α, β)-fuzzy Lie algebra of L over an (α, β)fi)-fuzzy field F of X, if it satisfies the following conditions:

image

Example 4.3. In the real vector space image whereimage is cross product of vectors for all image Then image is a Lie algebra over the fieldimage

Define image

image

and define image

image

(i) Then by actual computation, it follows that F is an image -fuzzy field of image and A is an image-fuzzy Lie algebra of image over the image -fuzzy field F of image. Also it can be verified that A is an image -fuzzy Lie algebra of imageover an image-fuzzy field F of image.

(ii) Let x = (1; 0; 0), y = (2; 0; 0), t = 0:4, s = 0:3. Then image and m(t, s) = 0.3. image Hence A is not an image-fuzzy Lie algebra.

(iii) Let x = (0; 0; 0), y = (2; 0; 0) be elements in image and t = 0:4, s = 0:6. Then image and image This shows thatimage Hence A is not a (q, q)-fuzzy Lie algebra.

Theorem 4.4. Let X be a field. Then a fuzzy subset image is a fuzzy field if and only if F is an image-fuzzy field of X.

Proof. The result follows immediately from Lemma 3.2.

Theorem 4.5. Let L be a Lie algebra over a field X. Then a fuzzy subset A of L is a fuzzy Lie algebra over a fuzzy field F of X if and only if A is an image-fuzzy Lie algebra of L over an image-fuzzy field F of X.

Proof. The result follows immediately from Lemmas 3.2 and 3.3.

Theorem 4.6. Let X be a field and let image be a fuzzy subset of X. Then F is animage-fuzzy field of X if and only if

image

Proof. Suppose that F is an image-fuzzy field of X. It is clear that

image

We consider two possibilities.

Case 1. Let image Then, image If possible, letimage Let image be such thatimage and soimage Alsoimage shows thatimage shows thatimage Therefore,image a contradiction.

Case 2. Let image Then,image If possible, letimage a contradiction. Therefore, it follows that image Similarly, (ii) is proved.

Conversely, suppose that conditions (i) and (ii) are satisfied by a fuzzy set F of X. Let image and soimage Since F satisfies condition (i),

image

Now consider the possibilities image thenimage thenimage and henceimage Therefore, it follows that ifimage Similarly, ifimage Hence F is an image-fuzzy field of X.

Theorem 4.7. Let L be a Lie algebra over a field X. Then a fuzzy subset A of L is an image-fuzzy Lie algebra of L over an image-fuzzy field F of X if and only if

image

Proof. Suppose that A is an image-fuzzy Lie algebra over an image-fuzzy field F of X. It is clear that image We consider two possibilities.

Case 1. Let image If possible, let image be such thatimage Butimage This shows thatimage a contradiction.

Case 2. Let image If possible, letimage Then we have image a contradiction. Therefore, it follows thatimage Thus, (ii) is proved. Similarly, (i) and (iii) are proved.

Conversely, suppose that the conditions (i), (ii), and (iii) are satisfied by a fuzzy set A of L. Let image and soimage Since A satisfies condition (iii),

image

Now consider the possibilitiesimage then, image then,image and hence image Therefore, it follows that if image Similarly, if imageimage Hence, A is an image-fuzzy Lie algebra of L over an image-fuzzy field F of X.

Proposition 4.8. Let L be a Lie algebra over a field X. Then every image-fuzzy Lie algebra of L over an image-fuzzy field of X is an image-fuzzy Lie algebra of L over an image-fuzzy field of X.

Proof. Suppose A is an image-fuzzy Lie algebra of L over an image-fuzzy field F of X. Let image Since F is an image-fuzzy field of image then image shows thatimageSimilarly, image for all image in X. So F is an image-fuzzy field of X. Since A is an image-fuzzy Lie algebra, forimage Then by definitionimage Similarly, image andimageHence A is an image-fuzzy Lie algebra of L over an image-fuzzy field F of X.

Remark 4.9. The converse of this proposition may not be true as seen in the following example.

Example 4.10. Let image is cross product for all image Then L is a Lie algebra over the field image

image

and define image

image

Then by Theorem 4.7, it follows that A is an image-fuzzy Lie algebra of image over an image-fuzzy field F of image.

But this is not an image-fuzzy Lie algebra of image over an image-fuzzy field of image. Letimage Butimage It is clear thatimage and soimage Therefore A is not an image-fuzzy Lie algebra of imageover an image-fuzzy eld F of image.

Theorem 4.11. Let A be an image-fuzzy Lie algebra of L over an image-fuzzy field F of X such that image for all x ∈ L and for all λ ∈ X. Then A is an image- fuzzy Lie algebra of L over an image-fuzzy field F of X.

Proof. Suppose that A is an image-fuzzy Lie algebra of L over an image-fuzzy field F of X. Letimage be such that image Then,image and soimage It follows from Theorem 4.6 thatimage Given that image for allimage for all image

image

Let image and soimage From Theorem 4.7,image and from the given condition we get imageand so, image be such that image Thenimage By Theorem 4.7,

image

So image Similarly,image Therefore, A is an image-fuzzy Lie algebra of L over an image-fuzzy field F of X.

Proposition 4.12. If A is an image-fuzzy Lie algebra of L over an image-fuzzy field F, then

image

Proof. Let image Then, from Theorem 4.7, the following hold.

image

Theorem 4.13. Let A be an image-fuzzy Lie algebra of L over an image-fuzzy field F of X. Then, for t ∈ (0; 0:5], At is a Lie subalgebra over Ft when Ft contains at least two elements.

Proof. For t ∈ (0; 0:5], suppose Ft contains at least two elements.

Let image This shows thatimage Therefore,

image

and hence image Thus,image Similarly, image Therefore, Ft is a subfield of X.

Suppose image So image Thenimage and soimage

Similarly, for image Therefore, At is a Lie subalgebra over the field Ft.

Let imagebe a function. If A and B are fuzzy subsets of L and L', respectively, then image are defined using Zadeh's extension principle [6]. If α is one ofimage it follows thatimage if and only ifimagefor all x ∈ L and for all t ∈ (0; 1].

Theorem 4.14. Let L and L' be Lie algebras over a field X and let image be a homo- morphism. If B is an image-fuzzy Lie algebra of L' over an image-fuzzy field F of X, then f-1(B) is an image-fuzzy Lie algebra of L over the image-fuzzy field F of X.

Proof. Let image be such thatimageThen image Since B is an image-fuzzy Lie algebra of L' over an imagefuzzy field F of X,

image

So we haveimage

Let image and so

image

and hence image

Therefore, imagefuzzy Lie algebra of L over the image-fuzzy field F of X.

Definition 4.15. A fuzzy set μ of a set Y is said to possess sup property if for every nonempty subset S of Y , there exists imagesuch that

image

Theorem 4.16. Let L and L' be Lie algebras over a field X and let image be an onto homomorphism. Let A be an image-fuzzy Lie algebra of L over an image-fuzzy field F of X, which satisfies the sup property. Then f(A) is an image-fuzzy Lie algebra of L' over the image-fuzzy field F of X.

Proof. Let imageand imageand so

image

Since f is onto, imageare nonempty subsets of L and by the sup property of A, there exists imagesuch that

image

then image Since A is an image-fuzzy Lie algebra of L over an image-fuzzy field F of X, we haveimageNow imageTherefore,

image

and so image Alsoimage

Let image Then it follows that image ButimageThis shows that image

Therefore, f(A) is an image -fuzzy Lie algebra of L' over the imagefuzzy field F of X.

References

Select your language of interest to view the total content in your interested language
Post your comment

Share This Article

Relevant Topics

Recommended Conferences

  • 7th International Conference on Biostatistics and Bioinformatics
    September 26-27, 2018 Chicago, USA
  • Conference on Biostatistics and Informatics
    December 05-06-2018 Dubai, UAE
  • Mathematics Congress - From Applied to Derivatives
    December 5-6, 2018 Dubai, UAE

Article Usage

  • Total views: 11750
  • [From(publication date):
    November-2010 - Feb 23, 2018]
  • Breakdown by view type
  • HTML page views : 7956
  • PDF downloads : 3794
 

Post your comment

captcha   Reload  Can't read the image? click here to refresh

Peer Reviewed Journals
 
Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals
International Conferences 2018-19
 
Meet Inspiring Speakers and Experts at our 3000+ Global Annual Meetings

Contact Us

Agri & Aquaculture Journals

Dr. Krish

[email protected]

1-702-714-7001Extn: 9040

Biochemistry Journals

Datta A

[email protected]

1-702-714-7001Extn: 9037

Business & Management Journals

Ronald

[email protected]

1-702-714-7001Extn: 9042

Chemistry Journals

Gabriel Shaw

[email protected]

1-702-714-7001Extn: 9040

Clinical Journals

Datta A

[email protected]

1-702-714-7001Extn: 9037

Engineering Journals

James Franklin

[email protected]

1-702-714-7001Extn: 9042

Food & Nutrition Journals

Katie Wilson

[email protected]

1-702-714-7001Extn: 9042

General Science

Andrea Jason

[email protected]

1-702-714-7001Extn: 9043

Genetics & Molecular Biology Journals

Anna Melissa

[email protected]

1-702-714-7001Extn: 9006

Immunology & Microbiology Journals

David Gorantl

[email protected]

1-702-714-7001Extn: 9014

Materials Science Journals

Rachle Green

[email protected]

1-702-714-7001Extn: 9039

Nursing & Health Care Journals

Stephanie Skinner

[email protected]

1-702-714-7001Extn: 9039

Medical Journals

Nimmi Anna

[email protected]

1-702-714-7001Extn: 9038

Neuroscience & Psychology Journals

Nathan T

[email protected]

1-702-714-7001Extn: 9041

Pharmaceutical Sciences Journals

Ann Jose

[email protected]

1-702-714-7001Extn: 9007

Social & Political Science Journals

Steve Harry

[email protected]

1-702-714-7001Extn: 9042

 
© 2008- 2018 OMICS International - Open Access Publisher. Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version