BICOMPLETABLE STANDARD FUZZY QUASI-METRIC SPACE

In this paper we introduce the definition of standard fuzzy quasi-metric space then we discuss several properties after we give an example to illustrate this notion. Then we showed the existence of a standard fuzzy quasi-metric space which is not bicompletable. Here we prove that every bicompletable standard qussi-metric spaceadmits a unique [ up to F-isometic ] bicompletion.


INTRODUCTION
In [1] Kider started the study of a notion of standard fuzzy metric space that constitutesan interesting modification of the notion of metric fuzziness due to George and Veeramani [2]. In this paper we extend the notion standard fuzzy metric space to a standard fuzzy quasi-metric space. On the other hand, it was presented in [4] an example of a standard fuzzy metric space that is not completable, also it has been obtained an internal characterization of completable standard fuzzy metric spaces. Taking these results into account and the fact that the concept of bicompletion provides a theory of completion to quasi-metric spaces in the classical sense ( for instance see [5] ). It seems natural and interesting to discuss the problem of characterizing standard fuzzy quasi-metric spaces that are bicompletable. The main purpose of this paper is to solve this problem. Following the modern terminology ( for instance see Section 11 of [5] ) by a quasi-metric on a set X we mean a function d:X× X [0, ) such that for all x, y, z X :
We introduce the following definition.

Definition 1.3:[1]
A triple (X,M, ) is said to be standard fuzzy metric space if X is an arbitrary set, is a continuous t-norm and M is a fuzzy set on X 2 satisfying the following conditions:

Remark 1.31:[4]
If X and Y are homeomorphic, the homeomorphism puts their points in one-to-one correspondence in such a way that their open sets also correspond to one another. For standard fuzzy metric space X and Y, let means that X and Y are homeomorphic. It is easily verified that the relation is reflexive, symmetric and transitive.

Definition 1.32:[4]
A mapping f from a standard fuzzy metric space (X, , ) into a standard fuzzy metric space for all x, yX. It is obvious that an F-isometry is one-to -one and uniformly continuous. X and Y are said to be F-isometric if there exists an F-isometry between them that is onto. An F-isometry is necessarily a homeomorphism but the converse is not true.  In [4] it was presented the following example of a standard fuzzy metric space that is not completable.
Since ( ) is a Cauchy sequence in ( , , ), the proof of part (b) shows that there is k j such that ( , )) (1-) Then for n , we obtain (1- We conclude that ( , , ) is bicomplete Definition 2.14: A standard fuzzy quasi-metric space (X,M, ) is called bicompletable if it admits a bicompletion.

Theorem 2.15:
Let (X,M, ) be a standard fuzzy quasi-metric space and let (Y,N, ) be a bicomplete standard fuzzy quasi-metric space.
If there is an F-isometry mapping f from a dense subset A of X to Y then f has a unique extension : X→Y.
Proof: We consider any xX but X = so x then there is a sequence ( ) in A such that ( ) converges to x by Lemma 1.28.Then ( ) is Cauchy.Since f is F-isometry (f( )) is Cauchy in Y but Y is complete hence there is yY such that (f( )) converges to y. Now we define (x) = y.We now show that this definition is independent of the particular choice of the sequence in A converging to x .Suppose that ( ) in A converges to x and ( ) in A converges to x. Then ( ) converges to x where ( ) = ( , , , ,…). Hence (f( )) converges and the two subsequence (f( )) and (f( )) of (f( )) must have the same limit. This prove is uniquely defined at every xX. Clearly (x) = f(x) for every xA so that is an extension of f Theorem 2.16: Let (X,M, ) be a standard fuzzy quasi-metric space and let (Y,N, ) be a bicomplete standard fuzzy quasimetric space.
If f is an F-isometry mapping from a dense subset A of X to Y then the unique extension :X→Y is an F-isometry. Proof: Let (Y, , ) and (Z, , ) be two bicompletions of (X,M, ) then we will prove that (Y, , ) and (Z, , ) are F-isometric. Since (Y, , ) is a bicompletion of (X,M, ) then there is an F-isometry f from (X,M, ) to a dense subset of (Y, , ). By Theorem 2.15 and Theorem 2.16 f admits a unique extension onto (Y, , ) which is also an Fisometry. Similarly is an isometry extension (X,M, ) onto (Z, , ). To prove that and are F-isometric it remains to see that and are onto we will show that is onto. Indeed given yY there is a sequence ( ) in X such that ( ) →y. Since is an F-isometry ( ) is a Cauchy sequence, so it converges to some point xX. Consequently (x) = y. Similarly we can prove that is onto. Hence and are F-isometric. Now (Y, , ) is F-isometric to (X,M, )