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International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol.4 Issue 7, July-2015 Introducing Network Structure on Fuzzy Banach

S. C. P. Halakatti* Archana Hali jol

Department of , Research Schola r, Karnatak University Dharwad, Department of Mathematic s, Karnataka, India. Karnatak University Dharwad, Karnata ka, India.

Abstract : we define concept of connectedness on fuzzy Banach ii) iff ,

manifold with well defined fuzzy . Also we show that iii) for 0, connectedness is an and induces a network structure on fuzzy Banac h manifold. iv) , v) is continuous,

Keywords: Fuzzy Banach manifold, local path connectedness, vi) . internal path connectedness, maximal path connectedne ss.

Lemma 2.1: Let is a fuzzy normed linear space. If 1. INTRODUCTION we define then is a fuzzy In our previous paper [4], we have introduced the concept metric on , which is called the fuzzy metric induced by the of fuzzy Banach manifold and proved some related results. fuzzy norm . In this paper we define norm on fuzzy Banach manifold and hence fuzzy . Definition 2.4: Let be any fuzzy , is a fuzzy of such that , and Connectedness on manifold plays very important role in is a fuzzy defined on the support of geometry. S. C. P. Halakatti and H. G. Haloli[5] have , which maps onto an open introduced different forms of connectedness and network fuzzy subset in some fuzzy Banach space . Then the structure on , using their approach we pair is called as fuzzy Banach chart. extend different forms of connectedness on fuzzy Banach manifold and show that fuzzy Banach manifold admits Definition 2.5: A fuzzy Banach of class on M is network structure by admitting maximal connectedness as a collection of pairs (i subjected to the an equivalence relation, further it will be shown that following conditions: network structure on fuzzy Banach is preserved i) that is the domain of fuzzy under smooth map. Banach charts in cover M. ii) Each fuzzy homeomorphism , defined on 2. PRE LIMINARIES the support of which maps onto an open fuzzy subset Definition 2.1: A binary operation in some fuzzy Banach space , and is said to be a continuous t-norm if ([0, 1], ) is a and are topological monoid with unit 1 such that a open fuzzy in . whenever and iii) The maps which maps

Definition 2.2: The triplets is said to be a fuzzy onto is a fuzzy of if is an arbitrary , is a continuous t-norm class for each pair of indices . and is a fuzzy set on satisfying the following The maps and for are called conditions for all and fuzzy transition maps. i) ii) for all iff Definition 2.6: A fuzzy topological space modelled on iii) fuzzy Banach space is called fuzzy Banach manifold. iv) for all

3. NORM ON FUZZY BANACH MANIFOLD v) is continuous.

In this section we define norm on fuzzy Banach manifold Definition 2.3: The triplet is said to be a fuzzy and hence fuzzy Banach space. normed space if is a vector space, is a continuous t- Definition 3.1: Let M be a fuzzy Banach manifold, then norm and is a fuzzy set on satisfying addition on M is defined as follows: following conditions for every and ; i) ,

IJERTV4IS070350 http://www.ijert.org 438 (This work is licensed under a Creative Commons Attribution 4.0 International License.) International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol.4 Issue 7, July-2015

for any is Definition 4.1: Let be a fuzzy Banach manifold a fuzzy Banach manifold). and if there exists a path such that: Definition 3.2: Let M be a fuzzy Banach manifold, then scalar multiplication on is defined as follows: for any ( and where if either field of real number or complex numbers) .

Definition 3.3: A fuzzy Banach manifold on which Then is locally path connected to q. If it is true for all addition and scalar multiplication is well defined forms a then is locally path connected. vector space. Definition 4.2: Let be a fuzzy Banach manifold Definition 3.4: Let be a fuzzy Banch manifold which is a and if there exists a path vector space and is a continuous t-norm and is a fuzzy such that: set on satisfying the following conditions for any and

i) with ii) iff Then is internally path connected to . If it is true for all iii) : (i, j I) respectively then M is internally i) if path connected.

ii) Definition 4.3: Let be a fuzzy Banach manifold if . and if there exists a path iv) . such that:

Then the triplet is called as a fuzzy normed linear fuzzy Banach manifold.

Definition 3.5: Let be a fuzzy Banach manifold and is a fuzzy normed linear space. Let be a sequence of elements in . Then is said to be convergent if (i such that: Then is maximally path connected to . If it is true for all i, j, k I respectively then M is

maximally path connected.

Then is called the limit of the sequence and is Now, we shall show that maximal path connectedness on denoted by . is an equivalence relation.

Definition 3.6: A sequence in is said to be a Cauchy sequence if Theorem 4.1: A maximally path connectedness on fuzzy Banach manifold is an equivalence relation.

and Proof: Let be a fuzzy Banach manifold. Now we Definition 3.7: A fuzzy Banach manifold which is shall show that maximal path connectedness on is a fuzzy normed linear space is said to be complete if every an equivalence relation. Cauchy’s sequence in converges in . Then We say is maximally path connected to if for the triplet is called a fuzzy Banach space. there exists a path such that: 4. CONNECTEDNESS ON FUZZY BANACH MANIFOLD

Let , by the property of fuzzy metric defined by fuzzy norm given by intuitively means, there is a triangular relation between elements of . This relation leads to define 1) Reflexive: Reflexive relation is trivial by considering a local, internal and maximal connectivity on fuzzy Banach constant paths i.e., manifold.

IJERTV4IS070350 http://www.ijert.org 439 (This work is licensed under a Creative Commons Attribution 4.0 International License.) International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol.4 Issue 7, July-2015

Now we shall show that any two fuzzy Banach manifolds preserve network structure under smooth map. 2) Symmetry: Suppose is a path from to then let Theorem 4.2: Let be a fuzzy Banach manifold

admitting network structure and be any fuzzy Banach manifold and if is a smooth map then such that: preserves a maximal path connectedness and hence network structure on .

Proof: Let be a Banach manifold admitting is continuous since it is the composition of where network structure and be any fuzzy Banach is the map . manifold and if is a smooth map.

Then is a path from to such that: Since is a Banach manifold admitting network structure, for there exists a path

such that:

Therefore, maximal path connectedness relation is symmetric. Also, we know that and are both continuous and 3) Transitivity: Suppose is a path from to and is a path from to . differentiable so the composition is also continuous and differentiable so we can say that for

Let defined as, we have and where such that for every

path there is a path

such that

Then is well defined since . Also is continuous since its restriction to each of the two closed subsets ] and of is continuous. i.e.,

Since is a smooth map this is true for all and

.

Therefore, is maximally path connected.

Similarly it can be easily shown with the help of Theorem 4.1 that maximal connectedness on is an equivalence relation hence forms a network structure on .

Therefore, is a path from to . Example 1:

Hence maximal path connectedness relation is transitive. Let be a set of all continuous functions on a closed interval [0, 1] be a Banach space with the norm Therefore, maximal path connectedness is an equivalence , is a fuzzy Banach space and relation. hence a fuzzy Banach manifold.

The equivalence relation maximal path connectedness on Solution: We know that is a fuzzy Banach space induces a network structure. with the norm function given by Definition 4.4: A fuzzy Banach manifold Let be linear space. We define admitting maximal path connectedness relation as an as and where equivalence relation induces a network structure on denoted by . clearly, is a fuzzy norm defined

IJERTV4IS070350 http://www.ijert.org 440 (This work is licensed under a Creative Commons Attribution 4.0 International License.) International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol.4 Issue 7, July-2015

by the norm . Hence is a fuzzy normed linear , is fuzzy Banach space and hence a space. fuzzy Banach manifold.

Now, we shall show that every Cauchy sequence in Solution: We know that with norm converges in . For this we need to forms a Banach space . First we shall show that show that is a fuzzy Banach space.

Consider, Let be defined as and

where clearly,

is a fuzzy norm defined by the norm , hence is a fuzzy normed linear space, from above

example, we can say that is a fuzzy Banach

space and with identity mapping , is a fuzzy Banach manifold. Since is a Banach space, we have CONCLUSION

An equivalence relation maximal connectedness on fuzzy Banach manifold induces a network structure on fuzzy

Therefore = 1 Banach manifold.

i.e., REFER ENCE

[1] G. Rano and T. Bag, “Fuzzy Normed Linear Spaces”, International Hence, every Cauchy sequence in converges in Journal Of Mathematics and Scientific Computing, Vol.2(2012), 16- . 19. [2] R. Saadati and M. Vaezpour, “SO ME RESULTS ON FUZZY Therefore, is a fuzzy Banach space. BANACH SPACES”, J.Appl. M ath. & Computing Vol. 17(2005), 475-484. [3] Robert Geroch, “Infinite-Dimensional Manifolds 1975 lecture Let and be fuzzy Banach space . Introduce a Notes”,Minkowski Institute Press(2013). chart and be identity mapping. This [4] S. C. P. Halakatti and Archana Halijol, “FUZZ Y BANACH single chart satisfies all the conditions of charts and atlas MANIFOLD ”, IJMA-6(3), 2015, 1-5. [5] S. C. P. Halakatti and H. G. Haloli, “SOME TOPOLOGICAL hence, with identity mapping is a fuzzy PROPERTIES ON DIFFERENTIABLE MANIFOLD ”,IJMA-6(1), Banach manifold. 2015, 93-101. [6] S. C. P. Halakatti and H. G. Haloli, “Convergence and Example 2: connectedness on complete measure manifo ld”, IJERT-4(03)2015, 225-232. Let be collection of all continuous functions defined on fuzzy Banach manifold M be a Banach space with norm

IJERTV4IS070350 http://www.ijert.org 441 (This work is licensed under a Creative Commons Attribution 4.0 International License.)