Introducing Network Structure on Fuzzy Banach Manifold

Introducing Network Structure on Fuzzy Banach Manifold

1. C. P. Halakatti* Department of Mathematics, Karnatak University Dharwad,

   Karnataka, India.

   Archana Halijol

   Research Scholar, Department of Mathematics,

   Karnatak University Dharwad, Karnataka, India.

   Abstract: we define concept of connectedness on fuzzy Banach manifold with well defined fuzzy norm. Also we show that connectedness is an equivalence relation and induces a network structure on fuzzy Banach manifold.

   Keywords: Fuzzy Banach manifold, local path connectedness, internal path connectedness, maximal path connectedness.

   1. INTRODUCTION

      In our previous paper [4], we have introduced the concept of fuzzy Banach manifold and proved some related results. In this paper we define norm on fuzzy Banach manifold and hence fuzzy Banach space.

      Connectedness on manifold plays very important role in geometry. S. C. P. Halakatti and H. G. Haloli[5] have introduced different forms of connectedness and network structure on differentiable manifold, using their approach we extend different forms of connectedness on fuzzy Banach manifold and show that fuzzy Banach manifold admits network structure by admitting maximal connectedness as an equivalence relation, further it will be shown that network structure on fuzzy Banach manifolds is preserved under smooth map.

   2. PRELIMINARIES

      Definition 2.1: A binary operation is said to be a continuous t-norm if ([0, 1], ) is a topological monoid with unit 1 such that a whenever and

      Definition 2.2: The triplets is said to be a fuzzy metric space if is an arbitrary set, is a continuous t-norm and is a fuzzy set on satisfying the following conditions for all and

      i)

      ii) for all iff

      iii)

      iv) for all

      v) is continuous.

      Definition 2.3: The triplet is said to be a fuzzy normed space if is a vector space, is a continuous t- norm and is a fuzzy set on satisfying following conditions for every and ;

      i) ,

      ii) iff ,

      iii) for 0,

      iv) , v) is continuous, vi) .

      Lemma 2.1: Let is a fuzzy normed linear space. If we define then is a fuzzy metric on , which is called the fuzzy metric induced by the fuzzy norm .

      Definition 2.4: Let be any fuzzy topological space, is a fuzzy subset of such that , and

      is a fuzzy homeomorphism defined on the support of

      , which maps onto an open fuzzy subset in some fuzzy Banach space . Then the pair is called as fuzzy Banach chart.

      Definition 2.5: A fuzzy Banach atlas of class on M is a collection of pairs (i subjected to the following conditions:

      1. that is the domain of fuzzy Banach charts in cover M.

      2. Each fuzzy homeomorphism , defined on the support of which maps onto an open fuzzy subset

         in some fuzzy Banach space , and

         and are open fuzzy subsets in .

      3. The maps which maps

      onto is a fuzzy diffeomorphism of class for each pair of indices .

      The maps and for are called

      fuzzy transition maps.

      Definition 2.6: A fuzzy topological space modelled on fuzzy Banach space is called fuzzy Banach manifold.

   3. NORM ON FUZZY BANACH MANIFOLD

      In this section we define norm on fuzzy Banach manifold and hence fuzzy Banach space.

      Definition 3.1: Let M be a fuzzy Banach manifold, then addition on M is defined as follows:

      for any is a fuzzy Banach manifold).

      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 addition and scalar multiplication is well defined forms a vector space.

      Definition 3.4: Let be a fuzzy Banch manifold which is a vector space and is a continuous t-norm and is a fuzzy set on satisfying the following conditions for any and

      i) with

      ii) iff

      iii) :

      i) if

      ii)

      if .

      iv) .

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

      Definition 3.5: Let

      be a fuzzy Banach manifold

      Definition 4.1: Let be a fuzzy Banach manifold and if there exists a path such that:

      Then is locally path connected to q. If it is true for all

      then is locally path connected.

      Definition 4.2: Let be a fuzzy Banach manifold and if there exists a path such that:

      Then is internally path connected to . If it is true for all

      (i, j I) respectively then M is internally path connected.

      Definition 4.3: Let be a fuzzy Banach manifold and if there exists a path

      such that:

      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 called the limit of the sequence and is denoted by .

      Definition 3.6: A sequence in is said to be a Cauchy sequence if

      and

      Definition 3.7: A fuzzy Banach manifold which is a fuzzy normed linear space is said to be complete if every Cauchys sequence in converges in . Then the triplet is called a fuzzy Banach space.

   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 local, internal and maximal connectivity on fuzzy Banach manifold.

Then is maximally path connected to . If it is true for all

i, j, k I respectively then M is maximally path connected.

Now, we shall show that maximal path connectedness on

is an equivalence relation.

Theorem 4.1: A maximally path connectedness on fuzzy Banach manifold is an equivalence relation.

Proof: Let be a fuzzy Banach manifold. Now we shall show that maximal path connectedness on is an equivalence relation.

We say is maximally path connected to if for

there exists a path such that:

1. Reflexive: Reflexive relation is trivial by considering a constant paths i.e.,

2. Symmetry: Suppose is a path from to then let

   such that:

   Now we shall show that any two fuzzy Banach manifolds preserve network structure under smooth map.

   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 preserves a maximal path connectedness and hence network structure on .

   is continuous since it is the composition of where

   Proof: Let be a Banach manifold admitting network structure and be any fuzzy Banach

   is the map .

   Then is a path from to such that:

   manifold and if is a smooth map.

   Since is a Banach manifold admitting network structure, for there exists a path

   such that:

   Therefore, maximal path connectedness relation is symmetric.

3. Transitivity: Suppose is a path from to and is a path from to .

Let defined as,

Also, we know that and are both continuous and differentiable so the composition is also continuous and differentiable so we can say that for

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

Therefore, is a path from to .

Hence maximal path connectedness relation is transitive.

Therefore, maximal path connectedness is an equivalence relation.

The equivalence relation maximal path connectedness on

induces a network structure.

Definition 4.4: A fuzzy Banach manifold

relation hence forms a network structure on .

Example 1:

Let be a set of all continuous functions on a closed interval [0, 1] be a Banach space with the norm

, is a fuzzy Banach space and hence a fuzzy Banach manifold.

Solution: We know that is a fuzzy Banach space with the norm function given by

Let be linear space. We define

admitting maximal path connectedness relation as an equivalence relation induces a network structure on

as and

where

denoted by .

clearly, is a fuzzy norm defined

by the norm . Hence is a fuzzy normed linear space.

Now, we shall show that every Cauchy sequence in

converges in . For this we need to show that

Consider,

, is fuzzy Banach space and hence a fuzzy Banach manifold.

Solution: We know that with norm

forms a Banach space . First we shall show that is a fuzzy Banach space.

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

Since is a Banach space, we have

Therefore = 1

space and with identity mapping , is a fuzzy Banach manifold.

CONCLUSION

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

i.e.,

Hence, every Cauchy sequence in converges in

.

Therefore, is a fuzzy Banach space.

Let and be fuzzy Banach space . Introduce a chart and be identity mapping. This single chart satisfies all the conditions of charts and atlas hence, with identity mapping is a fuzzy Banach manifold.

Example 2:

Let be collection of all continuous functions defined on fuzzy Banach manifold M be a Banach space with norm

REFERENCE

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3. Robert Geroch, Infinite-Dimensional Manifolds 1975 lecture Notes,Minkowski Institute Press(2013).

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5. S. C. P. Halakatti and H. G. Haloli, SOME TOPOLOGICAL PROPERTIES ON DIFFERENTIABLE MANIFOLD,IJMA-6(1), 2015, 93-101.

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