The notation of fuzzy set field is introduced. A fuzzy metric is redefined on fuzzy set field and on arbitrary fuzzy set in a field. The metric redefined is between fuzzy points and constitutes both fuzziness and crisp property of vector. In addition, a fuzzy magnitude of a fuzzy point in a field is defined.

Different researchers introduced the concept of fuzzy field and notion of fuzzy metric on fuzzy sets. How to define a fuzzy metric on a fuzzy set is still active research topic in fuzzy set theory which is very applicable in fuzzy optimization and pattern recognition. The notion of fuzzy sets has been applied in recent years for studying sequence spaces by Tripathy and Baruah [

Wenxiang and Tu [

Many authors introduced different notion of fuzzy metric on a fuzzy set from different points of view. Kaleva and Seikkala [

Wong [

This paper is an attempt to define a fuzzy set field in a field which is assumed to be the generalization of a fuzzy field introduced by [

Let

Let

Let

Provided that

Let

If

In this section the concept of fuzzy set field is introduced which is assumed to generalize a fuzzy field defined by [

The following notations might be used

Every mapping from a a non-empty set

Let

Let

In Definition

Let

Two fuzzy points

Using Definition

Let

Let

Let

for each

for each

Let

Let

It follows from Definitions

Let

for each

given

The results follow from Lemma

According to Theorem

With motivation of Theorem

Let

there exists unique

for each

there exists

given

We call

Let

Let

Let

Let

Let

The proof of (iv) follows from (iii). Therefore, the theorem is proved.

If

Let

If there exists a

If there exists a

For ongoing discussion, we keep the notations

If

Let

Let

Definition

Using (i) to (iv) and definition of

Let

Let

Now, we shall verify Definition

Let

Similar to proof of (1), we can easily verify that

Suppose

Similar to the proof of (3), we show that

Let

Let

Thus,

Similar to the proof of (3) above, we show that

Since

Let

Let

Let

(1)–(9) of Definition

Let

First we show that

Now we introduce definition of a fuzzy magnitude of a fuzzy point and fuzzy metric on a fuzzy set in a fuzzy set field,

Let

Let

From Definition

Let

Let

Let

If

Let

Let

Let

Let

Let

By Theorem

In Section

Let

Let

From Definition

If

Let

Therefore, the theorem is proved.

Let

A function

Let

Let

In the preceding theorem,

Let

The result follows from Theorem

Let

Proof is similar to Theorem

In this paper, fuzzy set field in a field is defined and some of its properties are discussed. We believe that our results will be helpful to develop similar notion for fuzzy linear space. The definition of the fuzzy metric introduced in this paper resembles distance between two fuzzy points and it involves both fuzziness and crisp property of fuzzy points that constitute classical metric properties. The definition is more general and it can be applied to define a fuzzy metric on linguistic variables also. We believe that our metric might be applied in fuzzy-decision theory, pattern recognition, and image processing.

The authors declare that there is no conflict of interests regarding the publication of this paper.