On Fuzzy Orders and Metrics

In this paper we present results regarding fixed point theory using notion of fuzzy orders.We propose an approach based on quasi metrics which in a way unifies the fixed point theory for ordered sets and metric spaces.


Introduction:
Much mathematical effort is expanded on solving fixed point equations i.e.equations of the form where is a map. A solution of such fixed point equations,when one exists,often has to be obtained by process of successive approximations.Order theory plays a role when X carries an order and when solution can be realized as joint of elements which approximates it. At some instances we find that fixed point theorems for ordered sets appear to be insufficient . In such cases , it is helpful to apply fixed point techniques in metric spaces.
In this paper give some results in this direction. By using the notion of quasi metric spaces we prove a theorem simultaneously generalizing fixed point theorems for ordered structures and metric spaces.All results are interpreted in terms of fuzzy set theory.

Preliminaries:
We recall some basic notions, is a complete completely distributive lattice.
Definition 2.1 Let X be a nonempty set,a fuzzy subset  of X is a function :XI.
Given two fuzzy sets , we set provided that for every . Given a fuzzy set is called -cut of .
The notion of fuzzy order and similarity was introduced by L. Zadeh  , for all A reflexive and transitive fuzzy relation is called a fuzzy preordering. Moreover a preordering which is antisymmetric is called a fuzzy order relation.A fuzzy symmetric fuzzy preorder is called a fuzzy equivalence.
A set X equipped with fuzzy order relation , denoted as ,is called a fuzzy ordered set(foset).
The domain of  is the fuzzy set on X, denoted by Dom , whose membership is defined as Similarly, the range of , denoted as Ran  is defined as The height of , denoted by is defined as is also a preorder on X, called the opposite of .  is a fuzzy partial order (fuzzy equivalence) iff is a fuzzy partial order( fuzzy equivalence). Suppose that is a collection of fuzzy preorders on X. Then ,the pointwise intersection is also a fuzzy preorder on X.
If  is a fuzzy order on X then is a fuzzy equivalence on X. This proves that  is transitive. Hence,  is a fuzzy order on X. If  is a fuzzy preorder and 1, then in general the cut is not a preorder .But, is always a preorder ,called the preorder associated with .

Relationship between Ordered structures and Topological Structures
It is well known that crisp topological structures and classical ordered structures are closely related.We observe that if (X,) is a preordered set and 2 X is the corresponding power set with inclusion ordering ,we can define a map which is a homomorphism. If  is an order then is an isomorphism.We extend this idea in the fuzzy setting.

Definition 3.1: A fuzzy set :XI on a preordered set (X, ) is called an upper set if
Dually,  is called a lower set if A fuzzy set  is an upper set in iff it is a lower set in (X, ) In particular, if  is a fuzzy equivalence relation then a fuzzy set  is an upper set in iff it is an lower set in .

Definition 3.2: Let be a fuzzy preordered set and zX then the fuzzy set is an upper set, called the principal upper set generated by z.
Similarly, the fuzzy set is a down set, called the principal down set generated by z.

Proposition 3.5 Let
Then,  is a fuzzy order on I X . Proof: Clearly, . So,  is reflexive. Using definition Thus,  is symmetric.
Thus,  is a fuzzy order on I X . Also, Hence, So, l s a homomorphism. Now, suppose l is a fuzzy order and z,wX such that l(z)=l(w) Then This implies and Hence, l:Xl(X) is an isomorphism.

Orders and Metrics
In this section we show that fuzzy preorders on a set X are closely related to pseudo quasi metric on X.  1)  is a fuzzy preorder iff  is a pseudo quasi ultra metic.
2)  is a fuzzy order iff  is a quasi ultra metic Proof: 1) Suppose  is a fuzzy preorder then Also, Hence,  is a pseudo quasi ultra metic. Converse can be proved similarly.
2) Suppose  is a fuzzy order then by i)  is a pseudo quasi ultra metrics.

Further, if implies
Hence,  is a fuzzy order. Converse can be proved similarly.

Fixed Point Theorem for Fuzzy Ordered Structures
Here we introduce order preserving sequences for fuzzy orders is almost -preserving. It follows from the completeness of  that there is a limit l of the sequence .
By continuity of f , f(l) is the limit of sequence and hence of . So, by uniqueness of limit of almost -preserving sequences we have f(l)=l.
Thus l is a fixed point of f.