# Fuzzy Set Field and Fuzzy Metric

Authors: Gebru Gebray; B. Krishna Reddy
Journal: Advances in Fuzzy Systems
 (2014)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1], Tripathy and Sarma [2], Tripathy and Borgohain [3], and others.Wenxiang and Tu [4] introduced the concept of fuzzy field in field and fuzzy linear spaces over fuzzy field. Furthermore, different authors are attempting to define fuzzy normed linear spaces, fuzzy inner product space, fuzzy Hilbert space, fuzzy Banach spaces, and so forth (cf. [5–8]).Many authors introduced different notion of fuzzy metric on a fuzzy set from different points of view. Kaleva and Seikkala [9] introduced the notion of a fuzzy metric space where metric was defined between fuzzy sets. The idea behind this notion was to fuzzify the classical metric by replacing real values of a metric by fuzzy values (fuzzy numbers). For the further research work and the properties of this type of fuzzy metric space see for instance Fang [10], Quan Xia and Guo [11], and others.Wong [12] defined fuzzy point and discussed its topological properties and there after Deng [13] defined Pseudo-metric spaces where metric was defined between fuzzy points rather than between fuzzy sets. Hsu [14] introduced fuzzy metric space with metric defined between fuzzy points and examined the completion of fuzzy metric space. For different notions of fuzzy metric space and for further research work see for instance Shi [15], Shi and Zheng [16], Shi [17] and others.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 [4]. We restate fuzzy set in more general form by allowing a particular fuzzy set to consist a family of membership functions. A fuzzy metric on fuzzy set and on fuzzy set field is reintroduced in such way that the classical metric is considered as a special type of fuzzy metric. In the sequel, a notion of magnitude of a fuzzy point in a field is introduced for the first time (up to our knowledge) and some of its properties are investigated.


## 2. Brief Summary of Fuzzy Set, Fuzzy Point, and Fuzzy Field

Definition 1 (see [18]).
LetX be a nonempty set and μ:X→[0,1] be a mapping. Then Aμ={(α,μ(α)):α∈X} is said to be a fuzzy set in X with membership function μ.Definition 2.
LetAμ={(α,μ(α)):α∈K} be a fuzzy in K. Then (i)
Aμ is called normal if there is at least one point x∈R(K=R) with μ(x)=1 (see [19]).(ii)
A fuzzy set Aμ is convex if for any x,y∈R(K=R) and any λ∈[0,1], we have μ(λx+(1-λ)y)≥min⁡μ(x),μ(y) (see [19]).(iii)
A fuzzy number is a fuzzy set on the real line that satisfies the conditions of normality and convexity (see [19]).(iv)
Aμ is said to be fuzzy point if {α∈K:μ(α)≠0} = singleton  set. It is usually denoted by xλ, (see [12, 18]), where
(1)xλ(y)={λify=x0ify≠x.(v)
Support of Aμ is the crisp set, sprt(Aμ)={α∈K:μ(α)≠0} (see [12, 18]).(vi)
The fuzzy point xλ is said to be contained in a fuzzy set, Aμ, or to belong to Aμ, denoted by xλ∈Aμ, if and only if λ≤μ(x)  (see [18]).Definition 3 (see [20]).
LetX be a Cartesian product of universes X=X1×⋯×Xn, Aμ1,…,Aμn be n fuzzy sets in X1,…,Xn, respectively, and f be a mapping from X to a universe Y, y=f(x1,…,xn). Then the extension principle allows us to define a fuzzy set Bγ in Y given by Bγ={(y,γ(y)):y=f(x1,…,xn)}, where f-1 is inverse of f and
(2)γ(y)={sup⁡inf⁡(x1,…,xn)∈f-1(y){μ1(x1),…,μn(xn)}iff-1(y)≠∅0otherwise.Lemma 4 (see [21]).
Provided thatM is either a positive or a negative fuzzy number and that N and P are together either positive or negative fuzzy numbers, then M⊙(N⊕P)=(M⊙N)⊕(M⊙P).Definition 5 (see [4]).
LetF be a field and let K be a fuzzy set in F with membership function μ. Suppose the following conditions hold: (1)
μ(x+y)≥min⁡{τ(x),μ(y)},(2)
μK(-x)≥τ(x),(3)
μ(xy)≥min⁡{τ(x),μ(y)},(4)
μ(x-1)≥τ(x)then K is said to be a fuzzy field in F and is denoted by (Kμ,F).Lemma 6 (see [4, 8]).
IfK is a fuzzy field in F with membership function μ, then (1)
μ(0)≥τ(x), x∈F,(2)
μ(1)≥τ(x), x≠0,(3)
τ(x)=μ(-x), x∈F,(4)
μ(x-1)=τ(x), x≠0.


## 3. Fuzzy Set Field and Fuzzy Metric on a Fuzzy Set in Fuzzy Set Field

### 3.1. Fuzzy Set Field

In this section the concept of fuzzy set field is introduced which is assumed to generalize a fuzzy field defined by [4]. Since we are dealing with a subset of original field, the axioms imposed on it are similar to that of the ordinary field. However, the operations (fuzzy sum and fuzzy product) are performed according to the extension principle.Notation 1.
The following notations might be used(1)
Every mapping from a a non-empty setK into the unit interval, [0;1] is denoted by small Greek letters;(2)
I=[0;1];  IK={τ:τ:K[0;1]};K=∅;(3)
xτ=(x,τ(x)), we mean a fuzzy point x with membership value τ(x)≠0;(4)
Aγ={xτ:x∈K,τ(x)≠0,τ∈γ⊂IK};(5)
xτ=γη, we mean x=y (in usual sense) and τ(x)=η(y).Definition 7.
LetK be a field and let μ:K→[0,1] be mapping. A point x is said to be a fuzzy point of K if τ(x)≠0 and τ(x) is said to be membership value of a fuzzy point x.Definition 8.
Letγ⊂IK.Aγ={xν:x∈Kforsome,ν∈γ,ν(x)≠0} is said to be a fuzzy set in K.In Definition8, if γ=singletonset, then Aγ is said to be ordinary fuzzy set on K.Notation 2.
LetAγ be a fuzzy set in a field F. A fuzzy point x belongs to a fuzzy Aγ (or x∈Aγ); we mean that there exists τ∈γ such that xτ∈Aγ. Therefore, we will use x∈Aγ and xτ∈Aγ alternatively.Definition 9.
Two fuzzy pointsxμ and yν are said to be equal if and only if x=y (in usual sense) and μ(x)=ν(y) and we denote it by xμ=yν.Using Definition3 (that is extension principle), we define operations fuzzy product, fuzzy sum, and fuzzy difference between fuzzy points according to the following definition.Definition 10.
Letxτ and yη be fuzzy points in a field K. Then the fuzzy sum, fuzzy product, and fuzzy difference of fuzzy points are defined by(1)
xτ⊕yη:=(x+y,τ(x)⊕η(y)), where τ(x)⊕η(y)=sup⁡inf⁡a=x+y{τ(x),η(y)}.(2)
xτ⊙yη:=(xy,τ(x)⊙η(y)), where τ(x)⊙η(y)=sup⁡inf⁡a=xy{τ(x),η(y)}.(3)
xτ⊖yη:=(x-y,τ(x)⊖η(y)), where τ(x)⊖η(y)=sup⁡inf⁡a=xy{τ(x),η(y)}.Remark 11.
LetK be a fuzzy set in a field K and let xμ;yν∈Ky. Then (1)
τ(x±y)≥min⁡{μ(x),ν(y)};(2)
τ(xy)≥min⁡{μ(x),ν(y)};(3)
τ(x-y)=τ(y-x) for all x, y∈K.Definition 12.
LetAγ be a fuzzy set in K. Fuzzy points w and l in Aγ with respective membership values  ρ(w) and ψ(l) are said to be fuzzy additive and fuzzy multiplicative points, respectively, in Aγ if they satisfy the following conditions:(1)
for eachx∈Aγ, xτ⊕wρ=xτ=wρ⊕xτ;(2)
for eachx∈Aγ, xτ⊙lψ=xτ=lψ⊙xτ.Definition 13.
LetAμ be an ordinary fuzzy set in a field F with membership function μ. Let γμ={ν:ν=μorcombinationsofμcombinedby,⊕orboth}. A fuzzy set, Kγμ={xν:ν∈γμ,x∈F}, is said to be a generated fuzzy set in F with a generating membership function μ denoted  by (Kγμ,F). Alternatively, we call (Kγμ,F) the collection of all fuzzy points generated by μ.Theorem 14.
LetKγμ be a generated fuzzy set in F. Then for all x,y,z∈Kγμ(i)
xτ⊕yη∈Kγμ, xτ⊕yη=yη⊕xτ, and (xτ⊕yη)⊕zκ=xτ⊕(yη⊕zκ);(ii)
xτ⊙yη∈Kγμ, xτ⊙yη=yη⊙xτ, and (xτ⊙yη)⊙zκ=xτ⊙(yη⊙kκ).Proof.
It follows from Definitions10 and 13.Theorem 15.
Let(Kμ,F) be a fuzzy field
(3)ρ(x)={μ(0)ifx=00ifx≠0,ψ(x)={μ(1)ifx=10ifx≠1,γ′=γμ∪{ρ,ψ},
where γμ is as in Definition 13. If x, y, z∈Kγ′ with respective membership values  λ(x), η(y), and κ(z), then (1)
xτ⊕yη=yη⊕xτ;(2)
xτ⊙yη=yη⊙xτ;(3)
xτ⊕(yη⊕zκ)=(xτ⊕yη)⊕zκ;(4)
xτ⊙(yη⊙zκ)=(xτ⊙yη)⊙zκ;(5)
0ρ⊕xτ=xτ=xτ⊕0ρ;(6)
1ψ⊙xτ=xτ=xτ⊙1ψ;(7)
for eachx∈Kγ′, there exists x′∈Kγ′ such that xτ′′⊕xτ=0ρ=xτ⊕xτ′′;(8)
given0≠x∈Kγ′, there exists an x′∈Kγ′ such that xτ′′⊙xτ=1μl=xτ⊙xτ′′.Proof.
The results follow from Lemma6 and Definition 10.According to Theorem15, if (Kμ,F) is a fuzzy field, γμ′={η:η∈γμorη∈{ρ,μl}}, then (Kγμ′,F) represents almost a field over binary operations (·) and (⊕). It is clear that if τ(x)⊙(μ(y)⊕μ(z))=(τ(x)⊙μ(y))⊕(τ(x)⊙μ(y)) for all x,y,z∈Kγμ′, then (Kγμ′,F) satisfies most axioms of a field under the binary operations ⊕ and ⊙.With motivation of Theorem15 and discussion that follows it, we have the following definition.Definition 16.
LetK be a field and let γ⊂IK. A fuzzy set Kγ is said to be a fuzzy set field in K if the binary operators ⊕:Kγ×Kγ→Kγ and ⊙:Kγ×Kγ→Kγ satisfy the following conditions: (1)
xτ⊕yη=yη⊕xτ, ∀x,y∈Kγ, and for all τ, η∈γ;(2)
xτ⊕(yη⊕zκ)=(xτ⊕yη)⊕zκ, ∀x,y,z∈Kγ, and for all τ,η,κ∈γ;(3)
xτ⊙yη=yη⊙xτ, ∀x,y∈Kγ, and for all τ,η∈γ;(4)
xτ⊙(yη⊕zk)=(xτ⊙yη)⊙zκ, ∀x,y,z∈Kγ, and for all τ, η, κ∈γ;(5)
there exists uniqueθ∈Kγ with θρ such that θρ⊕xτ=xτ=xτ⊕θρ, ∀x∈Kγ, and for all τ∈γ;(6)
for eachx∈Kγ, there exists x′∈Kγ such that xτ′⊕xτ=θρ=xτ⊕xτ′′ and for some τ, τ′∈γ;(7)
xτ⊙(yη⊕zk)=(xτ⊙yη)⊕(xτ⊙zk), ∀x,y,z∈Kγ;(8)
there existsl∈Kγ with lψ such that lψ⊙xτ=xτ=xτ⊙lψ;(9)
givenx≠θ and x∈Kγ, there exists an x′∈Kγ such that xτ′′⊙xτ=lμ=xτ⊙xτ′′.We callθ, l∈Kγ the fuzzy additive identity and fuzzy unit points of Kγ, respectively.Theorem 17.
LetKγ be a fuzzy set field in K. Then (i)
θ∈K is the fuzzy additive identity point of Kγ if and only if θ=0, there exists ρ∈γ such that λ(x)=sup⁡inf⁡x=a+b{λ(a),ρ(b)}, ∀x∈K, ∀λ∈γ;(ii)
l∈K is the fuzzy multiplicative identity point of Kγ if and only if l=1 and there exists ψ∈γ such that ∀x∈Kγ∀λ∈γ, λ(x)=sup⁡inf⁡x=ab{λ(a),μl(b)};(iii)
0ρ⊙xτ=0ρ=xτ⊙0ρ, where ρ is as in (i);(iv)
xτ⊙yη=0ρ⇒xτ=0ρ or yη=0ρ.Proof.
LetK be a fuzzy set field in a field K, let τ,η,λ∈γ be arbitrary and let ρ,ψ∈γ be as in Definitions 16 and 21 respectively. We need to show that (i) to (iv) of the theorem.(i)
Letw be the fuzzy additive identity point of Kγ with membership ρ(w). Then we claim that w=0. For if w≠0, then wρ⊕xλ=(x+w,η(x+w))≠(x,λ(x)), hence the claim. Now let x∈Kγμ be arbitrary. Then xλ⊕wρ = (w+x,η(w+x))=(x,λ(x))⇒η(0+x) = λ(x)=sup⁡inf⁡x=a+b{λ(a),ρ(b)}. The converse is clear.(ii)
Letl be as stated in Definition 16 with membership value ψ(l). Let x∈Kγ with corresponding membership value τ(x). Then xτ⊙1ψ=(bx,η(bx))=(x,λ(x)). Thus l=1 and xτ⊙1ψ=(x,η(x))=(x,λ(x))⇒η(x)=λ(x)=sup⁡inf⁡x=ab{λ(a),ψ(b)}. The converse holds trivially.(iii)
Letx,y∈Kγ be arbitrary with membership values λ(x) and ν(y), respectively. Then by (5) and (7), we have xτ⊙yη=xτ⊙(yη⊕0ρ)=(xτ⊙yη)⊕(xτ⊙0ρ)=(xη⊙0ρ)⊕(xτ⊙yη)⇒xτ⊙0ρ=0ρ by uniqueness of 0 in Kγ.(iv)
The proof of (iv) follows from (iii). Therefore, the theorem is proved.Remark 18.
IfKγ is a fuzzy field set in a field K and ρ is as stated in Theorem 17, then for each τ∈γ, either τ(0)=ρ(0) or τ(0)=0.Theorem 19.
LetKγ be a fuzzy set field in a field K. (1)
If there exists aρ∈γ such that ρ(0)≥τ(x) for all x∈K and sup⁡inf⁡x=a+b,b≠0{τ(a),ρ(b)}≤min⁡{τ(x),ρ(0)} for all τ∈γ, then 0∈Kγ with membership value ρ(0) is the additive element of Kγ.(2)
If there exists aψ∈γ such that ψ(1)≥τ(x) for all 0≠x∈K and sup⁡inf⁡x=ab,b≠1{τ(a),ψ(b)}≤min⁡{τ(x),ψ(1)} for all τ∈γ, then 1∈Kγ with membership value ψ(1) is the multiplicative element of Kγ.For ongoing discussion, we keep the notationsρ and ψ as they are stated in Theorems 17 and 19.Remark 20.
IfKγ is a fuzzy set field in K, μ, τ, k∈γ and ρ,ψ∈γ be as stated in Theorem 19, then (1)
ρ(0)≥ρ(x), x∈K.(2)
ρ(0)=sup⁡x∈Kτ(x) for each τ∈γ.(3)
xτ⊖xτ=0ρ for each x∈Kγ.(4)
ψ(1)≥ψ(x),x∈K.(5)
ψ(1)=sup⁡0≠x∈Kτ(x) for each τ∈γ.(6)
τ(x)≠0⇒η(x-1)≠0, x≠0 for some η∈γ.(7)
τ(x)⊙(μ(y)⊕k(z))=(τ(x)⊙k(y))⊕(τ(x)⊙k(z)) for all x,y,z∈K.Definition 21.
LetF be a field. A function μ:F→[0,1] is said to have property (Du) if it satisfies the following conditions: (i)
μ(0)≥μ(x)∀x∈F,0≠μ(1)≥μ(x)∀x∈F, x≠0, and {x∈F:τ(x)≠0}, is a subfield.(ii)
τ(x)⊙(μ(y)⊕μ(z))=(τ(x)⊙μ(y))⊕(τ(x)⊙μ(z)), ∀x,y,z∈F.Theorem 22.
LetAμ be an ordinary fuzzy set in a field F, let γμ be as in Definition 13. If μ has property (Du),
(4)ρ(x)={μ(0)ifx=00ifx≠0,ψ(x)={μ(1)ifx=10ifx≠1;
then Fω={xv:v∈ω} is a fuzzy set field.Proof.
Definition16 ((1)–(4)) follows from Theorem 14 and Definition 16 (7) follows from Definition 21. Therefore, we need to verify (5), (6), (8), and (9) of Definition 16. From Definition 21 and definition of Fω, for all τ∈ω, we can easily verify that(i)
ρ(0)≥τ(x), x∈F,(ii)
ρ(0)=τ(0),(iii)
τ(1)≥τ(x), x∈F,x≠0,(iv)
ψ(1)=τ(1).
Using (i) to (iv) and definition ofψ and ρ, we have for each λ∈ω,λ(x)≤ρ(0), we have λ(x)≤ψ(1), ρ(x)≤λ(x), x≠0, and ψ(x)≤λ(x), x≠1. Consequently, sup⁡inf⁡x=a+b,b≠0{λ(a),ρ(b)}≤min⁡{λ(x),ρ(0)}, and  sup⁡inf⁡x=ab,b≠1{λ(a),ψ(b)}≤min⁡{λ(x),ψ(1)},∀x∈K,∀λ∈γ. Therefore, by Theorem 19, θ=0 and l=1 with respective membership values  ρ(0) and ψ(1) are fuzzy zero and fuzzy unit of Kω, respectively. Hence ((5) and (8)) of Definition 16 are proved. Since K={x∈F:τ(x)≠0} is subfield, given x∈K, x≠0, there exists y∈K such that xy=1 and xτ⊙yλ=(1,η(1)). Using ((iii) and (iv)) of this theorem, η(1)=sup⁡inf⁡1=xy{τ(x),ν(y)}=ψ(1). Thus xτ⊙yν=lψ=yν⊙xτ. Similarly, given x∈K, there is a unique y∈K such that xτ⊕yη=0ρ=yη⊕xτ, hence the theorem.Theorem 23.
LetF and K be fields and let Fγ be fuzzy set field of F. If there exists ψ,ρ∈γ as stated in Theorem 19 and f:F→K is a homomorphism, then f(Fγ) is a fuzzy set field in K.Proof.
Let1 and 1′ be units (multiplicative) of F and K, respectively, 0 and 0′ be neutral elements with respect to addition (+) of F and K, respectively and M=f(F). Since f is homomorphism, we have f(1)=1′ and f(0)=0. Since Fγ is fuzzy set in F, Mω is a corresponding fuzzy set in K with membership function(s) given by ωτ(y)=sup⁡{τ(x):y=f(x),τ∈γ} [20].
Now, we shall verify Definition16 ((1) to (9)). (1)
Leta,b∈Mω with membership values ωμ(a) and ων(a), respectively. Then aωμ⊕bων = (a+b,ωτ(a+b)) and there are x,y∈F such that f(x) = a,f(y)=b. But
(5)ωτ(a+b)=sup⁡⁡{τ(x+y):a+b=f(x+y)}=sup⁡{τ(y+x):b+a=f(y+x)}=ωτ(a+b).
Thus aωμ⊕bων=bων⊕aωμ.(2)
Similar to proof of (1), we can easily verify that(aωμ⊕bων)⊕cωκ=aωμ⊕(bων⊕cωκ).(3)
Supposea,b∈Mω. Since a,b∈M, then there are fuzzy points x,y∈F such that f(x) = a and f(y)=b. Since f is a homomorphism, ab=f(xy)∈M. So aωμ⊙bων = (ab,ω(ab))=(ab,sup⁡{τ(xy):ab=f(xy)})∈Mω. But sup⁡{τ(xy):ab = f(xy)}=sup⁡{τ(yx):ba = f(yx)}=ω(ba). Thus, aωμ⊙bων=bων⊙aωμ.(4)
Similar to the proof of (3), we show that(aωμ⊙bων)⊙cωκ=aωμ⊙(bων⊙cωκ).(5)
Lety∈Mω. Since f(0)=0′,0′∈M and by Theorem 17, for every x∈Fγ with corresponding membership value τ(x), there exists ρ∈γ such that τ(x) = sup⁡inf⁡x=a+b{τ(a),ρ(b)}. Since ωτ(y) = sup⁡{τ(x):y=f(x)}, ∀y∈M, we have sup⁡inf⁡y=y1+y2(ωτ(y1),ωρ(y2)) = sup⁡inf⁡y=y1+y2(sup⁡y1=f(a)τ(a),sup⁡y2=f(b)ρ(b)) = sup⁡inf⁡y=y1+y2{sup⁡x=a+b,y=f(x)(τ(a),ρ(b))} = sup⁡x,y=f(x){sup⁡inf⁡x=a+b,y1=f(a),y2=f(b)(τ(a),ρ(b))}=sup⁡x,y=f(x)τ(x) = ωτ(y). Therefore, by Theorem 17, 0′=0 with membership value ωρ(0′) is the fuzzy additive identity point of Mω and for every y∈Mω, 0ωρ′⊕yωτ=yωτ⊕0ωρ′=yωτ.(6)
Leta∈Mω. Since a∈M, there exist x,y∈Fγ such that f(x)=a and xτ⊕yν=0ρ, by Definition 16. But f(x+y)=f(x)+f(y)=a+b=0′, where f(y)=b, and aωμ⊕bωv=(0,ωτ(0′)). Since x⊕y = 0ρ and ρ(0)=ωρ(0) (be extension principle followed by Remak 20), it follows that ωτ(0=a+b) = sup⁡{τ(x+y):0=f(x+y)}=ρ(0)=ωρ(0).Thus,aωμ⊕bων=0ωρ′=bων⊕aωμ. Since a∈Mω was arbitrary, so given y∈Mω there exists z such that yων⊕zωκ=0ωρ′=zωκ⊕yων.(7)
Similar to the proof of (3) above, we show thataωμ⊙(bων⊕cωκ)=(aωμ⊙bων)⊕(aωμ⊙cωκ).(8)
SinceFγ is a fuzzy field set, by Theorem 17, for every x∈Fγ with corresponding membership value τ(x), there exists ψ∈γ such that τ(x)=sup⁡inf⁡x=ab{τ(a),ψ(b)}. Since ωτ(y)=sup⁡{τ(x):y=f(x)},∀y∈M, we have sup⁡inf⁡y=y1y2(ωτ(y1),ωψ(y2)) = sup⁡inf⁡y=y1y2(sup⁡y1=f(a)τ(a),sup⁡y2=f(b)ψ(b)) = sup⁡inf⁡y=y1y2{sup⁡x=ab,y=f(x)(τ(a),ψ(b))} = sup⁡x,y=f(x){sup⁡inf⁡x=ab,y1=f(a),y2=f(b)(τ(a),ψ(b))} = sup⁡x,y=f(x)τ(x)=ωτ(y). Therefore, by Theorem 17, l′=1′ with membership value ωψ(1′) is the fuzzy multiplicative identity point of Mω and for every y∈Mω,1ωψ′⊕yωτ=yωτ⊕1ωψ′=yωτ.(9)
Leta∈Mω,a≠0′. Since a∈M, there exist x,y∈Fγ such that f(x)=a and xτ⊙yη=1ψ and f(xy) = f(x)f(y)=ab=1′, where f(y)=b. Thus a⊙b=(1′,ωτ(1′)). Since xt⊙yη=lψ and ψ(1)=ωψ(1′) (by extension prnciple followed by Remark 20), it follows that ωτ(1′) = sup⁡{τ(xy):1′=f(xy)}=ψ(1)=ωψ(1′). Thus, aωμ⊙bων=1ωψ′=bων⊙aωμ.Corollary 24.
LetF, K be fields, and let Kω be a fuzzy set field in K. If f:F→K is injective homomorphism, then f-1(Kω) is a fuzzy set field in F.Example 25.
LetF be a field and let τ:F→[0,1] be given by τ(x)=1∀x∈F. Then Kτ={xτ:τ(x)≠0}=F×{1} is a fuzzy set field in F.Proof.
(1)–(9) of Definition16 hold trivially, with 0ρ=(0,1)=0τ and 1ψ=(1,1)=1τ.Example 26.
LetF=R and let μ:R→[0,1] be given by
(6)τ(x)={11+|x|if-12≤x≤1211+|x+1|ifx≤-1211+|x-1|ifx≥12,ρ(x)={μ(0)=1ifx=00ifx≠0,ψ(x)={μ(1)=1ifx=10ifx≠1.
Let γμ be as in Definition 13 and ω={η:η∈γμ or η∈{ρ,ψ}}, then Fω={xv:v∈ω} is a fuzzy set field in R.Proof.
First we show thatμ has property (Du). Clearly, μ satisfies Definition 21(i). To verify Definition 21(ii), Lemma 4 will be applied. Since τ(x)=μ(-x) for all x∈R, we can consider only for x≥0. Moreover, Aμ=Aμ1∪Aμ2, where
(7)μ1(x)={11+xifx≥00ifx<0,μ2(x)={11+|x-1|ifx≥00ifx<0,
and τ(x)=max⁡{μ1(x),μ2(x)} [20]. Therefore, μ1 and μ2 are both positive fuzzy numbers. Hence, by Lemma 4, μj⊙(μi⊕μj) = (μj⊙μi)⊕(μj⊙μj)i,j=1,2,…. Thus, τ(x)⊙(μ(y)⊕μ(z)) = (τ(x)⊙μ(y))⊕(τ(x)⊙μ(z))  ∀x,y,z∈F. Therefore, applying Theorem 22 the result follows.


### 3.2. Fuzzy Metric of a Fuzzy Set and Fuzzy Magnitude of a Fuzzy Point in a Fuzzy Set Field

Now we introduce definition of a fuzzy magnitude of a fuzzy point and fuzzy metric on a fuzzy set in a fuzzy set field,Kγ, such that ρ∈γ satisfies (1) in Theorem 19. For a fuzzy set field Kγ, such that ρ∈γ satisfies (1) in Theorem 19, it follows that sup⁡{τ(x):x∈K}=ρ(0) (Remark 20). That is τ(xo)≤ρ(x) for all τ∈γ and for each x∈K. Therefore, we assume that any fuzzy point, xτ with a membership value τ(x)=ρ(0) (that is τ(x)=ρ(0)) as a nonfuzzy point of a particular fuzzy set field under discussion. Hence we have the following definition.Definition 27.
LetKγ be a fuzzy field set of a field K as in Theorem 19, and let  τ∈γ. A fuzzy point x with membership value τ(x) is said to be nonfuzzy point of a fuzzy set field Kγ if and only if τ(x)=ρ(0).Definition 28.
LetKγ be a fuzzy set field of a field K, μ∈γ, and let x be a fuzzy point with membership value μ(x). Then the fuzzy magnitude of xτ, denoted by |xτ|F, is given by |xτ|F=|x|+ρ(0)-τ(x). Where ρ is as in Theorem 19.From Definition28, |x|F=|x| if and only x is nonfuzzy in Kγ provided that ρ∈γ is as in Theorem 19. Therefore one can redefine the magnitude of given vector in a field as a fuzzy magnitude. Moreover, |x|F=0 if and only if x=0 and nonfuzzy and |x-y|F=0 if and only if x=y and x-y is nonfuzzy. Furthermore, |xτ|F→|x| whenever τ(x)→ρ(0).Theorem 29.
Letxτ, yν, and zk be fuzzy points in a fuzzy set field Kγ. Then (1)
|xλ⊕yη|F≤|xτ|F+|yν|F,(2)
|xτ⊖yν|≤|x|+|y|F,(3)
|xτ⊖yν|F≤|xτ⊖zκ|F+|zκ⊖yν|F.Proof.
Letxτ, yν, and zκ be fuzzy points. Then (1)
|xτ⊕yν|F=|x+y|+ρ(0)-η(x+y)≤|x+y|+2ρ(0)-τ(x)-ν(y)≤|xτ|F+|yν|F,(2)
|xτ⊖yν|F=|x-y|+ρ(0)-ι(x-y)≤|x-y|+2ρ(0)-τ(x)-ν(y)≤|xτ|F+|yν|F,(3)
|xτ⊖yν|F=|x-y|+ρ(0)-η(x+y)=|x-y|+ρ(0)-η(x-z)+(z-y)≤|x-y|+2ρ(0) − ι(x-z)-ι(y-z)≤|xτ⊖zκ|F+|zκ⊖yν|F.Hence the result is proved.Definition 30.
LetKγ be a fuzzy set field of a field K such that ρ∈γ is as in Theorem 19, μ⊂γ and Aμ be a nonempty fuzzy set in field K. A fuzzy metric on A is a mapping dF:Aμ×Aμ→[0,∞) satisfying the following conditions: (1)
dF(xτ,yη)≥ρ(0)-τ(x-y) for all x,y∈Aμ,(2)
dF(xτ,yη)=0 if and only if x=y and both are nonfuzzy in Kγ,(3)
dF(yη,xτ)=dF(yη,xτ)forallx,y∈Aμ,(4)
dF(xτ,yη)≤dF(xτ,zκ)+dF(zκ,yη)forallx,y,z∈Aμ.A pair (Aμ,dF) is said to be fuzzy metric spaces.Ifd is a metric on a field K and A is any subset of K, then by considering characteristic function on A as its membership function, d defines a fuzzy metric on A. Thus, one can consider an ordinary metric as fuzzy metric.Theorem 31.
LetKγ be a fuzzy set field of a field K such that ρ∈γ is as in Theorem 19, μ⊂γ and Aμ be a nonempty fuzzy set in field K. A mapping dF:Aμ×Aμ→[0,∞) given by dF(xτ,yη)=d(x,y)+ρ(0)-min⁡{τ(x),η(y)} defines a fuzzy metric on Aμ.Proof.
Letxτ,yη,zκ∈Aμ. We need to verify (1) to (4) of Definition 30.(1)
dF(xτ,yη)=d(x,y)+ρ(0)-min⁡{τ(x),η(y)}≥ρ(0)-min⁡{τ(x),η(y)}≥ρ(0)-ϕ(x-y) (by Remark 11),(2)
dF(xτ,yη)=0⇔d(x,y)+ρ(0)-min⁡{τ(x),η(y)}=0⇔d(x,y)=0, and min⁡{τ(x),η(y)} = ρ(0)⇔x=y,ρ(0)=τ(x)=η(y) by Definition 27 both are nonfuzzy,(3)
dF(xτ,yη)=d(x,y)+ρ(0)-min⁡{τ(x),η(y)}=d(y,x)+ρ(0) − min⁡{η(y),τ(x)},(4)
dF(xτ,yη)=d(x,y)+ρ(0)-min⁡{τ(x),η(y)}≤d(x,y)+ρ(0) − min⁡{τ(x),η(y),κ(z)}≤d(x,y) + ρ(0)-min⁡{τ(x),κ(z)}+ρ(0) − min⁡{κ(z),η(y)}≤dF(xτ,zκ)+dF(zκ,yη). Hence, the theorem is proved.Corollary 32.
LetKγ be a fuzzy set field of a field K such that ρ∈γ is as in Theorem 19, v⊂γ and Aν be a non empty fuzzy set in field K. A mapping dF:Aν×Aν→[0,∞) defined by dF(xτ,yη)=|x-y|+ρ(0)-min⁡{τ(x),η(y)} is fuzzy metric on Aν.Theorem 33.
LetKγ be a fuzzy set of a field K such that ρ∈γ is as in Theorem 19, μ⊂γ and Aμ be a nonempty fuzzy set in field K. A mapping dF:Aμ×Aμ→[0,∞) defined by dF(xτ,yη)=|xτ⊖yη|F+ρ(0)-min⁡{τ(x),η(y)} is fuzzy metric on Aμ.Proof.
Letxτ,yη,zκ∈Aν. We need to verify (1)–(4) of Definition 30. (1)
dF(xτ,yη)≥ρ(0)-min⁡{τ(x),η(y)}≥ρ(0)-ϕ(x-y), (By Remark 11 ),(2)
dF(xτ,yη)=0⇔|xτ⊖yη|F+ρ(0)-min⁡{τ(x),η(y)}=0⇔|xτ⊖yη|F = 0 and  min⁡{τ(x),η(y)} = ρ(0)⇔x=y,ρ(0)=min⁡{τ(x),η(y)}⇔μ(y)=ρ(0),(3)
By Theorem29, |xτ⊖yη|F=|yη⊖xτ|F⇒dF(xτ,yη)=dF(yη,xτ),(4)
dF(xτ,yη)=|yη⊖xτ|F+ρ(0)-min⁡{τ(x),η(y)}≤|xτ⊖zκ|F+|zκ⊖yη|F+ρ(0) − min⁡{τ(x),η(y)} ≤ |xτ⊖zκ|F+|zκ⊖yη|F+2ρ(0)-min⁡{τ(x),κ(z)} − min⁡{κ(z),η(y)} = dF(xτ,zκ)+dF(zκ,yη).Hence, the theorem is proved.


## 3.1. Fuzzy Set Field

In this section the concept of fuzzy set field is introduced which is assumed to generalize a fuzzy field defined by [4]. Since we are dealing with a subset of original field, the axioms imposed on it are similar to that of the ordinary field. However, the operations (fuzzy sum and fuzzy product) are performed according to the extension principle.Notation 1.
The following notations might be used(1)
Every mapping from a a non-empty setK into the unit interval, [0;1] is denoted by small Greek letters;(2)
I=[0;1];  IK={τ:τ:K[0;1]};K=∅;(3)
xτ=(x,τ(x)), we mean a fuzzy point x with membership value τ(x)≠0;(4)
Aγ={xτ:x∈K,τ(x)≠0,τ∈γ⊂IK};(5)
xτ=γη, we mean x=y (in usual sense) and τ(x)=η(y).Definition 7.
LetK be a field and let μ:K→[0,1] be mapping. A point x is said to be a fuzzy point of K if τ(x)≠0 and τ(x) is said to be membership value of a fuzzy point x.Definition 8.
Letγ⊂IK.Aγ={xν:x∈Kforsome,ν∈γ,ν(x)≠0} is said to be a fuzzy set in K.In Definition8, if γ=singletonset, then Aγ is said to be ordinary fuzzy set on K.Notation 2.
LetAγ be a fuzzy set in a field F. A fuzzy point x belongs to a fuzzy Aγ (or x∈Aγ); we mean that there exists τ∈γ such that xτ∈Aγ. Therefore, we will use x∈Aγ and xτ∈Aγ alternatively.Definition 9.
Two fuzzy pointsxμ and yν are said to be equal if and only if x=y (in usual sense) and μ(x)=ν(y) and we denote it by xμ=yν.Using Definition3 (that is extension principle), we define operations fuzzy product, fuzzy sum, and fuzzy difference between fuzzy points according to the following definition.Definition 10.
Letxτ and yη be fuzzy points in a field K. Then the fuzzy sum, fuzzy product, and fuzzy difference of fuzzy points are defined by(1)
xτ⊕yη:=(x+y,τ(x)⊕η(y)), where τ(x)⊕η(y)=sup⁡inf⁡a=x+y{τ(x),η(y)}.(2)
xτ⊙yη:=(xy,τ(x)⊙η(y)), where τ(x)⊙η(y)=sup⁡inf⁡a=xy{τ(x),η(y)}.(3)
xτ⊖yη:=(x-y,τ(x)⊖η(y)), where τ(x)⊖η(y)=sup⁡inf⁡a=xy{τ(x),η(y)}.Remark 11.
LetK be a fuzzy set in a field K and let xμ;yν∈Ky. Then (1)
τ(x±y)≥min⁡{μ(x),ν(y)};(2)
τ(xy)≥min⁡{μ(x),ν(y)};(3)
τ(x-y)=τ(y-x) for all x, y∈K.Definition 12.
LetAγ be a fuzzy set in K. Fuzzy points w and l in Aγ with respective membership values  ρ(w) and ψ(l) are said to be fuzzy additive and fuzzy multiplicative points, respectively, in Aγ if they satisfy the following conditions:(1)
for eachx∈Aγ, xτ⊕wρ=xτ=wρ⊕xτ;(2)
for eachx∈Aγ, xτ⊙lψ=xτ=lψ⊙xτ.Definition 13.
LetAμ be an ordinary fuzzy set in a field F with membership function μ. Let γμ={ν:ν=μorcombinationsofμcombinedby,⊕orboth}. A fuzzy set, Kγμ={xν:ν∈γμ,x∈F}, is said to be a generated fuzzy set in F with a generating membership function μ denoted  by (Kγμ,F). Alternatively, we call (Kγμ,F) the collection of all fuzzy points generated by μ.Theorem 14.
LetKγμ be a generated fuzzy set in F. Then for all x,y,z∈Kγμ(i)
xτ⊕yη∈Kγμ, xτ⊕yη=yη⊕xτ, and (xτ⊕yη)⊕zκ=xτ⊕(yη⊕zκ);(ii)
xτ⊙yη∈Kγμ, xτ⊙yη=yη⊙xτ, and (xτ⊙yη)⊙zκ=xτ⊙(yη⊙kκ).Proof.
It follows from Definitions10 and 13.Theorem 15.
Let(Kμ,F) be a fuzzy field
(3)ρ(x)={μ(0)ifx=00ifx≠0,ψ(x)={μ(1)ifx=10ifx≠1,γ′=γμ∪{ρ,ψ},
where γμ is as in Definition 13. If x, y, z∈Kγ′ with respective membership values  λ(x), η(y), and κ(z), then (1)
xτ⊕yη=yη⊕xτ;(2)
xτ⊙yη=yη⊙xτ;(3)
xτ⊕(yη⊕zκ)=(xτ⊕yη)⊕zκ;(4)
xτ⊙(yη⊙zκ)=(xτ⊙yη)⊙zκ;(5)
0ρ⊕xτ=xτ=xτ⊕0ρ;(6)
1ψ⊙xτ=xτ=xτ⊙1ψ;(7)
for eachx∈Kγ′, there exists x′∈Kγ′ such that xτ′′⊕xτ=0ρ=xτ⊕xτ′′;(8)
given0≠x∈Kγ′, there exists an x′∈Kγ′ such that xτ′′⊙xτ=1μl=xτ⊙xτ′′.Proof.
The results follow from Lemma6 and Definition 10.According to Theorem15, if (Kμ,F) is a fuzzy field, γμ′={η:η∈γμorη∈{ρ,μl}}, then (Kγμ′,F) represents almost a field over binary operations (·) and (⊕). It is clear that if τ(x)⊙(μ(y)⊕μ(z))=(τ(x)⊙μ(y))⊕(τ(x)⊙μ(y)) for all x,y,z∈Kγμ′, then (Kγμ′,F) satisfies most axioms of a field under the binary operations ⊕ and ⊙.With motivation of Theorem15 and discussion that follows it, we have the following definition.Definition 16.
LetK be a field and let γ⊂IK. A fuzzy set Kγ is said to be a fuzzy set field in K if the binary operators ⊕:Kγ×Kγ→Kγ and ⊙:Kγ×Kγ→Kγ satisfy the following conditions: (1)
xτ⊕yη=yη⊕xτ, ∀x,y∈Kγ, and for all τ, η∈γ;(2)
xτ⊕(yη⊕zκ)=(xτ⊕yη)⊕zκ, ∀x,y,z∈Kγ, and for all τ,η,κ∈γ;(3)
xτ⊙yη=yη⊙xτ, ∀x,y∈Kγ, and for all τ,η∈γ;(4)
xτ⊙(yη⊕zk)=(xτ⊙yη)⊙zκ, ∀x,y,z∈Kγ, and for all τ, η, κ∈γ;(5)
there exists uniqueθ∈Kγ with θρ such that θρ⊕xτ=xτ=xτ⊕θρ, ∀x∈Kγ, and for all τ∈γ;(6)
for eachx∈Kγ, there exists x′∈Kγ such that xτ′⊕xτ=θρ=xτ⊕xτ′′ and for some τ, τ′∈γ;(7)
xτ⊙(yη⊕zk)=(xτ⊙yη)⊕(xτ⊙zk), ∀x,y,z∈Kγ;(8)
there existsl∈Kγ with lψ such that lψ⊙xτ=xτ=xτ⊙lψ;(9)
givenx≠θ and x∈Kγ, there exists an x′∈Kγ such that xτ′′⊙xτ=lμ=xτ⊙xτ′′.We callθ, l∈Kγ the fuzzy additive identity and fuzzy unit points of Kγ, respectively.Theorem 17.
LetKγ be a fuzzy set field in K. Then (i)
θ∈K is the fuzzy additive identity point of Kγ if and only if θ=0, there exists ρ∈γ such that λ(x)=sup⁡inf⁡x=a+b{λ(a),ρ(b)}, ∀x∈K, ∀λ∈γ;(ii)
l∈K is the fuzzy multiplicative identity point of Kγ if and only if l=1 and there exists ψ∈γ such that ∀x∈Kγ∀λ∈γ, λ(x)=sup⁡inf⁡x=ab{λ(a),μl(b)};(iii)
0ρ⊙xτ=0ρ=xτ⊙0ρ, where ρ is as in (i);(iv)
xτ⊙yη=0ρ⇒xτ=0ρ or yη=0ρ.Proof.
LetK be a fuzzy set field in a field K, let τ,η,λ∈γ be arbitrary and let ρ,ψ∈γ be as in Definitions 16 and 21 respectively. We need to show that (i) to (iv) of the theorem.(i)
Letw be the fuzzy additive identity point of Kγ with membership ρ(w). Then we claim that w=0. For if w≠0, then wρ⊕xλ=(x+w,η(x+w))≠(x,λ(x)), hence the claim. Now let x∈Kγμ be arbitrary. Then xλ⊕wρ = (w+x,η(w+x))=(x,λ(x))⇒η(0+x) = λ(x)=sup⁡inf⁡x=a+b{λ(a),ρ(b)}. The converse is clear.(ii)
Letl be as stated in Definition 16 with membership value ψ(l). Let x∈Kγ with corresponding membership value τ(x). Then xτ⊙1ψ=(bx,η(bx))=(x,λ(x)). Thus l=1 and xτ⊙1ψ=(x,η(x))=(x,λ(x))⇒η(x)=λ(x)=sup⁡inf⁡x=ab{λ(a),ψ(b)}. The converse holds trivially.(iii)
Letx,y∈Kγ be arbitrary with membership values λ(x) and ν(y), respectively. Then by (5) and (7), we have xτ⊙yη=xτ⊙(yη⊕0ρ)=(xτ⊙yη)⊕(xτ⊙0ρ)=(xη⊙0ρ)⊕(xτ⊙yη)⇒xτ⊙0ρ=0ρ by uniqueness of 0 in Kγ.(iv)
The proof of (iv) follows from (iii). Therefore, the theorem is proved.Remark 18.
IfKγ is a fuzzy field set in a field K and ρ is as stated in Theorem 17, then for each τ∈γ, either τ(0)=ρ(0) or τ(0)=0.Theorem 19.
LetKγ be a fuzzy set field in a field K. (1)
If there exists aρ∈γ such that ρ(0)≥τ(x) for all x∈K and sup⁡inf⁡x=a+b,b≠0{τ(a),ρ(b)}≤min⁡{τ(x),ρ(0)} for all τ∈γ, then 0∈Kγ with membership value ρ(0) is the additive element of Kγ.(2)
If there exists aψ∈γ such that ψ(1)≥τ(x) for all 0≠x∈K and sup⁡inf⁡x=ab,b≠1{τ(a),ψ(b)}≤min⁡{τ(x),ψ(1)} for all τ∈γ, then 1∈Kγ with membership value ψ(1) is the multiplicative element of Kγ.For ongoing discussion, we keep the notationsρ and ψ as they are stated in Theorems 17 and 19.Remark 20.
IfKγ is a fuzzy set field in K, μ, τ, k∈γ and ρ,ψ∈γ be as stated in Theorem 19, then (1)
ρ(0)≥ρ(x), x∈K.(2)
ρ(0)=sup⁡x∈Kτ(x) for each τ∈γ.(3)
xτ⊖xτ=0ρ for each x∈Kγ.(4)
ψ(1)≥ψ(x),x∈K.(5)
ψ(1)=sup⁡0≠x∈Kτ(x) for each τ∈γ.(6)
τ(x)≠0⇒η(x-1)≠0, x≠0 for some η∈γ.(7)
τ(x)⊙(μ(y)⊕k(z))=(τ(x)⊙k(y))⊕(τ(x)⊙k(z)) for all x,y,z∈K.Definition 21.
LetF be a field. A function μ:F→[0,1] is said to have property (Du) if it satisfies the following conditions: (i)
μ(0)≥μ(x)∀x∈F,0≠μ(1)≥μ(x)∀x∈F, x≠0, and {x∈F:τ(x)≠0}, is a subfield.(ii)
τ(x)⊙(μ(y)⊕μ(z))=(τ(x)⊙μ(y))⊕(τ(x)⊙μ(z)), ∀x,y,z∈F.Theorem 22.
LetAμ be an ordinary fuzzy set in a field F, let γμ be as in Definition 13. If μ has property (Du),
(4)ρ(x)={μ(0)ifx=00ifx≠0,ψ(x)={μ(1)ifx=10ifx≠1;
then Fω={xv:v∈ω} is a fuzzy set field.Proof.
Definition16 ((1)–(4)) follows from Theorem 14 and Definition 16 (7) follows from Definition 21. Therefore, we need to verify (5), (6), (8), and (9) of Definition 16. From Definition 21 and definition of Fω, for all τ∈ω, we can easily verify that(i)
ρ(0)≥τ(x), x∈F,(ii)
ρ(0)=τ(0),(iii)
τ(1)≥τ(x), x∈F,x≠0,(iv)
ψ(1)=τ(1).
Using (i) to (iv) and definition ofψ and ρ, we have for each λ∈ω,λ(x)≤ρ(0), we have λ(x)≤ψ(1), ρ(x)≤λ(x), x≠0, and ψ(x)≤λ(x), x≠1. Consequently, sup⁡inf⁡x=a+b,b≠0{λ(a),ρ(b)}≤min⁡{λ(x),ρ(0)}, and  sup⁡inf⁡x=ab,b≠1{λ(a),ψ(b)}≤min⁡{λ(x),ψ(1)},∀x∈K,∀λ∈γ. Therefore, by Theorem 19, θ=0 and l=1 with respective membership values  ρ(0) and ψ(1) are fuzzy zero and fuzzy unit of Kω, respectively. Hence ((5) and (8)) of Definition 16 are proved. Since K={x∈F:τ(x)≠0} is subfield, given x∈K, x≠0, there exists y∈K such that xy=1 and xτ⊙yλ=(1,η(1)). Using ((iii) and (iv)) of this theorem, η(1)=sup⁡inf⁡1=xy{τ(x),ν(y)}=ψ(1). Thus xτ⊙yν=lψ=yν⊙xτ. Similarly, given x∈K, there is a unique y∈K such that xτ⊕yη=0ρ=yη⊕xτ, hence the theorem.Theorem 23.
LetF and K be fields and let Fγ be fuzzy set field of F. If there exists ψ,ρ∈γ as stated in Theorem 19 and f:F→K is a homomorphism, then f(Fγ) is a fuzzy set field in K.Proof.
Let1 and 1′ be units (multiplicative) of F and K, respectively, 0 and 0′ be neutral elements with respect to addition (+) of F and K, respectively and M=f(F). Since f is homomorphism, we have f(1)=1′ and f(0)=0. Since Fγ is fuzzy set in F, Mω is a corresponding fuzzy set in K with membership function(s) given by ωτ(y)=sup⁡{τ(x):y=f(x),τ∈γ} [20].
Now, we shall verify Definition16 ((1) to (9)). (1)
Leta,b∈Mω with membership values ωμ(a) and ων(a), respectively. Then aωμ⊕bων = (a+b,ωτ(a+b)) and there are x,y∈F such that f(x) = a,f(y)=b. But
(5)ωτ(a+b)=sup⁡⁡{τ(x+y):a+b=f(x+y)}=sup⁡{τ(y+x):b+a=f(y+x)}=ωτ(a+b).
Thus aωμ⊕bων=bων⊕aωμ.(2)
Similar to proof of (1), we can easily verify that(aωμ⊕bων)⊕cωκ=aωμ⊕(bων⊕cωκ).(3)
Supposea,b∈Mω. Since a,b∈M, then there are fuzzy points x,y∈F such that f(x) = a and f(y)=b. Since f is a homomorphism, ab=f(xy)∈M. So aωμ⊙bων = (ab,ω(ab))=(ab,sup⁡{τ(xy):ab=f(xy)})∈Mω. But sup⁡{τ(xy):ab = f(xy)}=sup⁡{τ(yx):ba = f(yx)}=ω(ba). Thus, aωμ⊙bων=bων⊙aωμ.(4)
Similar to the proof of (3), we show that(aωμ⊙bων)⊙cωκ=aωμ⊙(bων⊙cωκ).(5)
Lety∈Mω. Since f(0)=0′,0′∈M and by Theorem 17, for every x∈Fγ with corresponding membership value τ(x), there exists ρ∈γ such that τ(x) = sup⁡inf⁡x=a+b{τ(a),ρ(b)}. Since ωτ(y) = sup⁡{τ(x):y=f(x)}, ∀y∈M, we have sup⁡inf⁡y=y1+y2(ωτ(y1),ωρ(y2)) = sup⁡inf⁡y=y1+y2(sup⁡y1=f(a)τ(a),sup⁡y2=f(b)ρ(b)) = sup⁡inf⁡y=y1+y2{sup⁡x=a+b,y=f(x)(τ(a),ρ(b))} = sup⁡x,y=f(x){sup⁡inf⁡x=a+b,y1=f(a),y2=f(b)(τ(a),ρ(b))}=sup⁡x,y=f(x)τ(x) = ωτ(y). Therefore, by Theorem 17, 0′=0 with membership value ωρ(0′) is the fuzzy additive identity point of Mω and for every y∈Mω, 0ωρ′⊕yωτ=yωτ⊕0ωρ′=yωτ.(6)
Leta∈Mω. Since a∈M, there exist x,y∈Fγ such that f(x)=a and xτ⊕yν=0ρ, by Definition 16. But f(x+y)=f(x)+f(y)=a+b=0′, where f(y)=b, and aωμ⊕bωv=(0,ωτ(0′)). Since x⊕y = 0ρ and ρ(0)=ωρ(0) (be extension principle followed by Remak 20), it follows that ωτ(0=a+b) = sup⁡{τ(x+y):0=f(x+y)}=ρ(0)=ωρ(0).Thus,aωμ⊕bων=0ωρ′=bων⊕aωμ. Since a∈Mω was arbitrary, so given y∈Mω there exists z such that yων⊕zωκ=0ωρ′=zωκ⊕yων.(7)
Similar to the proof of (3) above, we show thataωμ⊙(bων⊕cωκ)=(aωμ⊙bων)⊕(aωμ⊙cωκ).(8)
SinceFγ is a fuzzy field set, by Theorem 17, for every x∈Fγ with corresponding membership value τ(x), there exists ψ∈γ such that τ(x)=sup⁡inf⁡x=ab{τ(a),ψ(b)}. Since ωτ(y)=sup⁡{τ(x):y=f(x)},∀y∈M, we have sup⁡inf⁡y=y1y2(ωτ(y1),ωψ(y2)) = sup⁡inf⁡y=y1y2(sup⁡y1=f(a)τ(a),sup⁡y2=f(b)ψ(b)) = sup⁡inf⁡y=y1y2{sup⁡x=ab,y=f(x)(τ(a),ψ(b))} = sup⁡x,y=f(x){sup⁡inf⁡x=ab,y1=f(a),y2=f(b)(τ(a),ψ(b))} = sup⁡x,y=f(x)τ(x)=ωτ(y). Therefore, by Theorem 17, l′=1′ with membership value ωψ(1′) is the fuzzy multiplicative identity point of Mω and for every y∈Mω,1ωψ′⊕yωτ=yωτ⊕1ωψ′=yωτ.(9)
Leta∈Mω,a≠0′. Since a∈M, there exist x,y∈Fγ such that f(x)=a and xτ⊙yη=1ψ and f(xy) = f(x)f(y)=ab=1′, where f(y)=b. Thus a⊙b=(1′,ωτ(1′)). Since xt⊙yη=lψ and ψ(1)=ωψ(1′) (by extension prnciple followed by Remark 20), it follows that ωτ(1′) = sup⁡{τ(xy):1′=f(xy)}=ψ(1)=ωψ(1′). Thus, aωμ⊙bων=1ωψ′=bων⊙aωμ.Corollary 24.
LetF, K be fields, and let Kω be a fuzzy set field in K. If f:F→K is injective homomorphism, then f-1(Kω) is a fuzzy set field in F.Example 25.
LetF be a field and let τ:F→[0,1] be given by τ(x)=1∀x∈F. Then Kτ={xτ:τ(x)≠0}=F×{1} is a fuzzy set field in F.Proof.
(1)–(9) of Definition16 hold trivially, with 0ρ=(0,1)=0τ and 1ψ=(1,1)=1τ.Example 26.
LetF=R and let μ:R→[0,1] be given by
(6)τ(x)={11+|x|if-12≤x≤1211+|x+1|ifx≤-1211+|x-1|ifx≥12,ρ(x)={μ(0)=1ifx=00ifx≠0,ψ(x)={μ(1)=1ifx=10ifx≠1.
Let γμ be as in Definition 13 and ω={η:η∈γμ or η∈{ρ,ψ}}, then Fω={xv:v∈ω} is a fuzzy set field in R.Proof.
First we show thatμ has property (Du). Clearly, μ satisfies Definition 21(i). To verify Definition 21(ii), Lemma 4 will be applied. Since τ(x)=μ(-x) for all x∈R, we can consider only for x≥0. Moreover, Aμ=Aμ1∪Aμ2, where
(7)μ1(x)={11+xifx≥00ifx<0,μ2(x)={11+|x-1|ifx≥00ifx<0,
and τ(x)=max⁡{μ1(x),μ2(x)} [20]. Therefore, μ1 and μ2 are both positive fuzzy numbers. Hence, by Lemma 4, μj⊙(μi⊕μj) = (μj⊙μi)⊕(μj⊙μj)i,j=1,2,…. Thus, τ(x)⊙(μ(y)⊕μ(z)) = (τ(x)⊙μ(y))⊕(τ(x)⊙μ(z))  ∀x,y,z∈F. Therefore, applying Theorem 22 the result follows.


## 3.2. Fuzzy Metric of a Fuzzy Set and Fuzzy Magnitude of a Fuzzy Point in a Fuzzy Set Field

Now we introduce definition of a fuzzy magnitude of a fuzzy point and fuzzy metric on a fuzzy set in a fuzzy set field,Kγ, such that ρ∈γ satisfies (1) in Theorem 19. For a fuzzy set field Kγ, such that ρ∈γ satisfies (1) in Theorem 19, it follows that sup⁡{τ(x):x∈K}=ρ(0) (Remark 20). That is τ(xo)≤ρ(x) for all τ∈γ and for each x∈K. Therefore, we assume that any fuzzy point, xτ with a membership value τ(x)=ρ(0) (that is τ(x)=ρ(0)) as a nonfuzzy point of a particular fuzzy set field under discussion. Hence we have the following definition.Definition 27.
LetKγ be a fuzzy field set of a field K as in Theorem 19, and let  τ∈γ. A fuzzy point x with membership value τ(x) is said to be nonfuzzy point of a fuzzy set field Kγ if and only if τ(x)=ρ(0).Definition 28.
LetKγ be a fuzzy set field of a field K, μ∈γ, and let x be a fuzzy point with membership value μ(x). Then the fuzzy magnitude of xτ, denoted by |xτ|F, is given by |xτ|F=|x|+ρ(0)-τ(x). Where ρ is as in Theorem 19.From Definition28, |x|F=|x| if and only x is nonfuzzy in Kγ provided that ρ∈γ is as in Theorem 19. Therefore one can redefine the magnitude of given vector in a field as a fuzzy magnitude. Moreover, |x|F=0 if and only if x=0 and nonfuzzy and |x-y|F=0 if and only if x=y and x-y is nonfuzzy. Furthermore, |xτ|F→|x| whenever τ(x)→ρ(0).Theorem 29.
Letxτ, yν, and zk be fuzzy points in a fuzzy set field Kγ. Then (1)
|xλ⊕yη|F≤|xτ|F+|yν|F,(2)
|xτ⊖yν|≤|x|+|y|F,(3)
|xτ⊖yν|F≤|xτ⊖zκ|F+|zκ⊖yν|F.Proof.
Letxτ, yν, and zκ be fuzzy points. Then (1)
|xτ⊕yν|F=|x+y|+ρ(0)-η(x+y)≤|x+y|+2ρ(0)-τ(x)-ν(y)≤|xτ|F+|yν|F,(2)
|xτ⊖yν|F=|x-y|+ρ(0)-ι(x-y)≤|x-y|+2ρ(0)-τ(x)-ν(y)≤|xτ|F+|yν|F,(3)
|xτ⊖yν|F=|x-y|+ρ(0)-η(x+y)=|x-y|+ρ(0)-η(x-z)+(z-y)≤|x-y|+2ρ(0) − ι(x-z)-ι(y-z)≤|xτ⊖zκ|F+|zκ⊖yν|F.Hence the result is proved.Definition 30.
LetKγ be a fuzzy set field of a field K such that ρ∈γ is as in Theorem 19, μ⊂γ and Aμ be a nonempty fuzzy set in field K. A fuzzy metric on A is a mapping dF:Aμ×Aμ→[0,∞) satisfying the following conditions: (1)
dF(xτ,yη)≥ρ(0)-τ(x-y) for all x,y∈Aμ,(2)
dF(xτ,yη)=0 if and only if x=y and both are nonfuzzy in Kγ,(3)
dF(yη,xτ)=dF(yη,xτ)forallx,y∈Aμ,(4)
dF(xτ,yη)≤dF(xτ,zκ)+dF(zκ,yη)forallx,y,z∈Aμ.A pair (Aμ,dF) is said to be fuzzy metric spaces.Ifd is a metric on a field K and A is any subset of K, then by considering characteristic function on A as its membership function, d defines a fuzzy metric on A. Thus, one can consider an ordinary metric as fuzzy metric.Theorem 31.
LetKγ be a fuzzy set field of a field K such that ρ∈γ is as in Theorem 19, μ⊂γ and Aμ be a nonempty fuzzy set in field K. A mapping dF:Aμ×Aμ→[0,∞) given by dF(xτ,yη)=d(x,y)+ρ(0)-min⁡{τ(x),η(y)} defines a fuzzy metric on Aμ.Proof.
Letxτ,yη,zκ∈Aμ. We need to verify (1) to (4) of Definition 30.(1)
dF(xτ,yη)=d(x,y)+ρ(0)-min⁡{τ(x),η(y)}≥ρ(0)-min⁡{τ(x),η(y)}≥ρ(0)-ϕ(x-y) (by Remark 11),(2)
dF(xτ,yη)=0⇔d(x,y)+ρ(0)-min⁡{τ(x),η(y)}=0⇔d(x,y)=0, and min⁡{τ(x),η(y)} = ρ(0)⇔x=y,ρ(0)=τ(x)=η(y) by Definition 27 both are nonfuzzy,(3)
dF(xτ,yη)=d(x,y)+ρ(0)-min⁡{τ(x),η(y)}=d(y,x)+ρ(0) − min⁡{η(y),τ(x)},(4)
dF(xτ,yη)=d(x,y)+ρ(0)-min⁡{τ(x),η(y)}≤d(x,y)+ρ(0) − min⁡{τ(x),η(y),κ(z)}≤d(x,y) + ρ(0)-min⁡{τ(x),κ(z)}+ρ(0) − min⁡{κ(z),η(y)}≤dF(xτ,zκ)+dF(zκ,yη). Hence, the theorem is proved.Corollary 32.
LetKγ be a fuzzy set field of a field K such that ρ∈γ is as in Theorem 19, v⊂γ and Aν be a non empty fuzzy set in field K. A mapping dF:Aν×Aν→[0,∞) defined by dF(xτ,yη)=|x-y|+ρ(0)-min⁡{τ(x),η(y)} is fuzzy metric on Aν.Theorem 33.
LetKγ be a fuzzy set of a field K such that ρ∈γ is as in Theorem 19, μ⊂γ and Aμ be a nonempty fuzzy set in field K. A mapping dF:Aμ×Aμ→[0,∞) defined by dF(xτ,yη)=|xτ⊖yη|F+ρ(0)-min⁡{τ(x),η(y)} is fuzzy metric on Aμ.Proof.
Letxτ,yη,zκ∈Aν. We need to verify (1)–(4) of Definition 30. (1)
dF(xτ,yη)≥ρ(0)-min⁡{τ(x),η(y)}≥ρ(0)-ϕ(x-y), (By Remark 11 ),(2)
dF(xτ,yη)=0⇔|xτ⊖yη|F+ρ(0)-min⁡{τ(x),η(y)}=0⇔|xτ⊖yη|F = 0 and  min⁡{τ(x),η(y)} = ρ(0)⇔x=y,ρ(0)=min⁡{τ(x),η(y)}⇔μ(y)=ρ(0),(3)
By Theorem29, |xτ⊖yη|F=|yη⊖xτ|F⇒dF(xτ,yη)=dF(yη,xτ),(4)
dF(xτ,yη)=|yη⊖xτ|F+ρ(0)-min⁡{τ(x),η(y)}≤|xτ⊖zκ|F+|zκ⊖yη|F+ρ(0) − min⁡{τ(x),η(y)} ≤ |xτ⊖zκ|F+|zκ⊖yη|F+2ρ(0)-min⁡{τ(x),κ(z)} − min⁡{κ(z),η(y)} = dF(xτ,zκ)+dF(zκ,yη).Hence, the theorem is proved.


## 4. Metric on Fuzzy Set of a Field

In Section3, we have introduced the definition of fuzzy magnitude and a fuzzy metric of a fuzzy point and a fuzzy set, respectively, in a fuzzy set field, Kγ, in which for all x∈K and for all τ∈γ,τ(x)≤ρ(0). In this section, we try to define a fuzzy magnitude and fuzzy metric of a fuzzy point and fuzzy set in a field, respectively. The definitions introduced differ slightly from that of preceding section due to targeted highest membership value which is assumed to be 1≥ρ(0), where ρ is as in Theorem 19. However, the two definitions coincide whenever ρ(0)=1.Definition 34.
LetAγ be fuzzy set in a field. A fuzzy point xτ∈Aγ is said to be nonfuzzy point if and only if τ(x)=1.Definition 35.
Letxτ be a fuzzy point in a field K. Then the fuzzy magnitude of xτ, denoted by |xτ|F, is given by
(8)|xτ|F=|x|+1-τ(x).From Definition34 a fuzzy magnitude of a nonfuzzy point coincides with that of usual magnitude. Therefore one can redefine the magnitude of a given vector in a field as a fuzzy magnitude.Theorem 36.
Ifxτ,yη, and zκ are fuzzy points in a field K. Then(1)
|xτ⊖yη|F=|yη⊖xτ|F,(2)
|xτ⊕yη|F≤|xτ|F+|yη|F,(3)
|xτ⊖yη|F≤|xτ⊖zκ|F+|zκ⊖yη|F.Proof.
Letxτ, γη, and zk be fuzzy points in a field K. Then(1)
|xτ⊖yη|F=|x-y|+1-ϕ(x-y)=|x-y|+1-ϕ(y-x) (by Remark 11(3))   =|yη⊖xτ|F,(2)
|xτ⊕yη|F=|x+y|+1-ν(x+y)≤|x+y|+2-τ(x)-η(y)≤|xτ|F+|y|F,(3)
|xτ⊖yη|F=|x-y|+1-ϕ(x-y)=|x-y|+1-ϕ(x-z+z-y)≤|x-y|+2-ϕ(x-z) − ϕ(y-z)≤|xτ⊖zκ|F+|zκ⊖yη|F.
Therefore, the theorem is proved.Definition 37.
LetAμ be a nonempty fuzzy set in fuzzy field K. A fuzzy metric on Aμ is a mapping dF:Aμ×Aμ→[0,∞) satisfying the following conditions: (1)
dF(xτ,yη)≥1-ϕ(x-y),(2)
dF(xτ,yη)=0 if and only if x is nonfuzzy in Aμ and x=y,(3)
dF(xτ,yη)=dF(yη,xτ) for all x,y∈Aμ,(4)
dF(xτ,yη)≤dF(xτ,zκ)+dF(zκ,yη) for all x,y,z∈Aμ.A functiondF is said to be fuzzy metric on Aμ and a pair (Aμ,dF) is said to be fuzzy metric space.Theorem 38.
LetK be a field, let d be metric on K and Aμbe a nonempty fuzzy set in K. A mapping dF:Aμ×Aμ→[0,∞) defined by dF(xτ,yη)=d(x,y)+1-min⁡{τ(x),η(y)} is a fuzzy metric on Aμ.Proof.
Letx,y,z∈Aμ with respective membership values  τ(x),η(y)and κ(z). We will verify that (1) to (4) of Definition 37. (1)
dF(xτ,yη)=d(x,y)+1-min⁡{τ(x),η(y)}⇒dF(xτ,yη)≥1-ϕ(x-y), by Remark 11.(2)
dF(xτ,yη)=0⇔d(x,y)+1-min⁡{τ(x),η(y)}=0⇔d(x,y)=0,min⁡{τ(x),η(y)}=1⇔x=y, 1=τ(x)=η(y)⇔x=y and by Definition 34 both are nonfuzzy.(3)
dF(xτ,yη)=d(x,y)+1-min⁡{τ(x),η(y)}=d(y,x)+1-min⁡{η(y),τ(x)}.(4)
dF(xτ,yη)=d(x,y)+1-min⁡{τ(x),η(y)}≤d(x,y)+1-min⁡{τ(x),η(y),η(z)}≤d(x,y) + 1-min⁡{λ(x),κ(z)}+1-min⁡{κ(z),η(y)}≤dF(xτ,zκ)+dF(zκ,yη).Hence the theorem is proved.In the preceding theorem,dF(xτ,yη)=d(x,y) provided that x,y∈Aμ are nonfuzzy in Aμ. Therefore, if μ is characteristic function, then fuzzy metric coincides with usual metric of Aμ. So we call a fuzzy metric given as in Theorem 38 a standard fuzzy metric on a fuzzy set.Corollary 39.
LetAμ be a fuzzy set in a field K. A mapping dF:Aμ×Aμ→[0,∞) given by dF(xτ,yη)=|x-y|+1-min⁡{τ(x),η(y)} defines a fuzzy metric on Aμ.Proof.
The result follows from Theorem38.Theorem 40.
LetAν be a nonempty fuzzy set in a field K. A mapping dF:Aν×Aν→[0,∞) defined by dF(xτ,yη)=|xτ⊖yη|F+1-min⁡{τ(x),η(y)} is fuzzy metric on Aν.Proof.
Proof is similar to Theorem33, so it is omitted.


## 5. Conclusion

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.



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