AFS Advances in Fuzzy Systems 1687-711X 1687-7101 Hindawi Publishing Corporation 10.1155/2014/968405 968405 Research Article Fuzzy Set Field and Fuzzy Metric Gebray Gebru Reddy B. Krishna Honda Katsuhiro Department of Mathematics UCS Osmania University Hyderabad 500007 India osmania.ac.in 2014 192014 2014 21 04 2014 12 08 2014 2 9 2014 2014 Copyright © 2014 Gebru Gebray and B. Krishna Reddy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 , Tripathy and Sarma , Tripathy and Borgohain , and others.

Wenxiang and Tu  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. ).

Many authors introduced different notion of fuzzy metric on a fuzzy set from different points of view. Kaleva and Seikkala  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 , Quan Xia and Guo , and others.

Wong  defined fuzzy point and discussed its topological properties and there after Deng  defined Pseudo-metric spaces where metric was defined between fuzzy points rather than between fuzzy sets. Hsu  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 , Shi and Zheng , Shi  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 . 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 [<xref ref-type="bibr" rid="B18">18</xref>]).

Let X be a nonempty set and μ:X[0,1] be a mapping. Then Aμ={(α,μ(α)):αX} is said to be a fuzzy set in X with membership function μ.

Definition 2.

Let Aμ={(α,μ(α)):αK} be a fuzzy in K. Then

Aμ is called normal if there is at least one point xR(K=R) with μ(x)=1 (see ).

A fuzzy set Aμ is convex if for any x,yR(K=R) and any λ[0,1], we have μ(λx+(1-λ)y)minμ(x),μ(y) (see ).

A fuzzy number is a fuzzy set on the real line that satisfies the conditions of normality and convexity (see ).

Aμ is said to be fuzzy point if {αK:μ(α)0} = singleton  set. It is usually denoted by xλ, (see [12, 18]), where (1)xλ(y)={λify=x0ifyx.

Support of Aμ is the crisp set, sprt(Aμ)={αK:μ(α)0} (see [12, 18]).

The fuzzy point xλ is said to be contained in a fuzzy set, Aμ, or to belong to Aμ, denoted by xλAμ, if and only if λμ(x)  (see ).

Definition 3 (see [<xref ref-type="bibr" rid="B20">20</xref>]).

Let X 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)={supinf(x1,,xn)f-1(y){μ1(x1),,μn(xn)}if    f-1(y)0otherwise.

Lemma 4 (see [<xref ref-type="bibr" rid="B21">21</xref>]).

Provided that M is either a positive or a negative fuzzy number and that N and P are together either positive or negative fuzzy numbers, then M(NP)=(MN)(MP).

Definition 5 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

Let F be a field and let K be a fuzzy set in F with membership function μ. Suppose the following conditions hold:

μ(x+y)min{τ(x),μ(y)},

μK(-x)τ(x),

μ(xy)min{τ(x),μ(y)},

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 [<xref ref-type="bibr" rid="B4">4</xref>, <xref ref-type="bibr" rid="B8">8</xref>]).

If K is a fuzzy field in F with membership function μ, then

μ(0)τ(x), xF,

μ(1)τ(x), x0,

τ(x)=μ(-x), xF,

μ(x-1)=τ(x), x0.

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 . 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

Every mapping from a a non-empty set K into the unit interval, [0;1] is denoted by small Greek letters;

I=[0;1];  IK={τ:τ:K[0;1]};K=;

xτ=(x,τ(x)), we mean a fuzzy point x with membership value τ(x)  0;

Aγ={xτ:xK,τ(x)0,  τγIK};

xτ=γη, we mean x=y (in usual sense) and τ(x)=η(y).

Definition 7.

Let K 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.

Let γIK.  Aγ={xν:xKforsome,νγ,ν(x)0} is said to be a fuzzy set in K.

In Definition 8, if γ=singletonset, then Aγ is said to be ordinary fuzzy set on K.

Notation 2.

Let Aγ be a fuzzy set in a field F. A fuzzy point x belongs to a fuzzy Aγ (or xAγ); we mean that there exists τγ such that xτAγ. Therefore, we will use xAγ and xτAγ alternatively.

Definition 9.

Two fuzzy points xμ 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 Definition 3 (that is extension principle), we define operations fuzzy product, fuzzy sum, and fuzzy difference between fuzzy points according to the following definition.

Definition 10.

Let xτ and yη be fuzzy points in a field K. Then the fuzzy sum, fuzzy product, and fuzzy difference of fuzzy points are defined by

xτyη:=(x+y,τ(x)η(y)), where τ(x)η(y)=supinfa=x+y{τ(x),η(y)}.

xτyη:=(xy,τ(x)η(y)), where τ(x)η(y)=supinfa=xy{τ(x),η(y)}.

xτyη:=(x-y,τ(x)η(y)), where τ(x)η(y)=supinfa=xy{τ(x),η(y)}.

Remark 11.

Let K be a fuzzy set in a field K and let xμ;yνKy. Then

τ(x±y)min{μ(x),ν(y)};

τ(xy)min{μ(x),ν(y)};

τ(x-y)=τ(y-x) for all x, yK.

Definition 12.

Let Aγ 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:

for each xAγ, xτwρ=xτ=wρxτ;

for each xAγ, xτlψ=xτ=lψxτ.

Definition 13.

Let Aμ be an ordinary fuzzy set in a field F with membership function μ. Let γμ={ν:ν=μorcombinationsofμcombinedby,orboth}. A fuzzy set, Kγμ={xν:νγμ,xF}, 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.

Let Kγμ be a generated fuzzy set in F. Then for all x,y,zKγμ

xτyηKγμ, xτyη=yηxτ, and (xτyη)zκ=xτ(yηzκ);

xτyηKγμ, xτyη=yηxτ, and (xτyη)zκ=xτ(yηkκ).

Proof.

It follows from Definitions 10 and 13.

Theorem 15.

Let (Kμ,F) be a fuzzy field (3)ρ(x)={μ(0)ifx=00ifx0,ψ(x)={μ(1)ifx=10ifx1,γ=γμ{ρ,ψ}, where γμ is as in Definition 13. If x, y, zKγ with respective membership values  λ(x), η(y), and κ(z), then

xτyη=yηxτ;

xτyη=yηxτ;

xτ(yηzκ)=(xτyη)zκ;

xτ(yηzκ)=(xτyη)zκ;

0ρxτ=xτ=xτ0ρ;

1ψxτ=xτ=xτ1ψ;

for each xKγ, there exists xKγ such that xτxτ=0ρ=xτxτ;

given 0xKγ, there exists an xKγ such that xτxτ=1μl=xτxτ.

Proof.

The results follow from Lemma 6 and Definition 10.

According to Theorem 15, if (Kμ,F) is a fuzzy field, γμ={η:ηγμorη{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,zKγμ, then (Kγμ,F) satisfies most axioms of a field under the binary operations and .

With motivation of Theorem 15 and discussion that follows it, we have the following definition.

Definition 16.

Let K be a field and let γIK. A fuzzy set Kγ is said to be a fuzzy set field in K if the binary operators :Kγ×KγKγ and :Kγ×KγKγ satisfy the following conditions:

xτyη=yηxτ, x,yKγ, and for all τ, ηγ;

xτ(yηzκ)=(xτyη)zκ, x,y,zKγ, and for all τ,η,κγ;

xτyη=yηxτ, x,yKγ, and for all τ,ηγ;

xτ(yηzk)=(xτyη)zκ, x,y,zKγ, and for all τ, η, κγ;

there exists unique θKγ with θρ such that θρxτ=xτ=xτθρ, xKγ, and for all τγ;

for each xKγ, there exists xKγ such that xτxτ=θρ=xτxτ and for some τ, τγ;

xτ(yηzk)=(xτyη)(xτzk), x,y,zKγ;

there exists lKγ with lψ such that lψxτ=xτ=xτlψ;

given xθ and xKγ, there exists an xKγ such that xτxτ=lμ=xτxτ.

We call θ, lKγ the fuzzy additive identity and fuzzy unit points of Kγ, respectively.

Theorem 17.

Let Kγ be a fuzzy set field in K. Then

θK is the fuzzy additive identity point of Kγ if and only if θ=0, there exists ργ such that λ(x)=supinfx=a+b{λ(a),ρ(b)}, xK, λγ;

lK is the fuzzy multiplicative identity point of Kγ if and only if l=1 and there exists ψγ such that xKγλγ, λ(x)=supinfx=ab{λ(a),μl(b)};

0ρxτ=0ρ=xτ0ρ, where ρ is as in (i);

xτyη=0ρxτ=0ρ or yη=0ρ.

Proof.

Let K be a fuzzy set field in a field K, let τ,η,λγ be arbitrary and let ρ,ψγ be as in Definitions 16 and 21 respectively. We need to show that (i) to (iv) of the theorem.

Let w be the fuzzy additive identity point of Kγ with membership ρ(w). Then we claim that w=0. For if w0, then wρxλ=(x+w,η(x+w))(x,λ(x)), hence the claim. Now let xKγμ be arbitrary. Then xλwρ = (w+x,η(w+x))=(x,λ(x))η(0+x) = λ(x)=supinfx=a+b{λ(a),ρ(b)}. The converse is clear.

Let l be as stated in Definition 16 with membership value ψ(l). Let xKγ with corresponding membership value τ(x). Then xτ1ψ=(bx,η(bx))=(x,λ(x)). Thus l=1 and xτ1ψ=(x,η(x))=(x,λ(x))η(x)=λ(x)=supinfx=ab{λ(a),ψ(b)}. The converse holds trivially.

Let x,yKγ be arbitrary with membership values λ(x) and ν(y), respectively. Then by (5) and (7), we have xτyη=xτ(yη0ρ)=(xτyη)(xτ0ρ)=(xη0ρ)(xτyη)xτ0ρ=0ρ by uniqueness of 0 in Kγ.

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

Remark 18.

If Kγ is a fuzzy field set in a field K and ρ is as stated in Theorem 17, then for each τγ, either τ(0)=ρ(0) or τ(0)=0.

Theorem 19.

Let Kγ be a fuzzy set field in a field K.

If there exists a ργ such that ρ(0)τ(x) for all xK and supinfx=a+b,b0{τ(a),ρ(b)}min{τ(x),ρ(0)} for all τγ, then 0Kγ with membership value ρ(0) is the additive element of Kγ.

If there exists a ψγ such that ψ(1)τ(x) for all 0xK and supinfx=ab,b1{τ(a),ψ(b)}min{τ(x),ψ(1)} for all τγ, then 1Kγ with membership value ψ(1) is the multiplicative element of Kγ.

For ongoing discussion, we keep the notations ρ and ψ as they are stated in Theorems 17 and 19.

Remark 20.

If Kγ is a fuzzy set field in K, μ, τ, kγ and ρ,  ψγ be as stated in Theorem 19, then

ρ(0)ρ(x), xK.

ρ(0)=supxKτ(x) for each τγ.

xτxτ=0ρ for each xKγ.

ψ(1)ψ(x),  xK.

ψ(1)=sup0xKτ(x) for each τγ.

τ(x)0η(x-1)0, x0 for some ηγ.

τ(x)(μ(y)k(z))=(τ(x)k(y))(τ(x)k(z)) for all x,y,zK.

Definition 21.

Let F be a field. A function μ:F[0,1] is said to have property (Du) if it satisfies the following conditions:

μ(0)μ(x)xF,0μ(1)μ(x)xF, x0, and {xF:τ(x)0}, is a subfield.

τ(x)(μ(y)μ(z))=(τ(x)μ(y))(τ(x)μ(z)), x,y,zF.

Theorem 22.

Let Aμ 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=00ifx0,ψ(x)={μ(1)ifx=10ifx1; then Fω={xv:vω} is a fuzzy set field.

Proof.

Definition 16 ((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 τω, we can easily verify that

ρ(0)τ(x), xF,

ρ(0)=τ(0),

τ(1)τ(x), xF,x0,

ψ(1)=τ(1).

Using (i) to (iv) and definition of ψ and ρ, we have for each λω,  λ(x)ρ(0), we have λ(x)ψ(1), ρ(x)λ(x), x0, and ψ(x)λ(x), x1. Consequently, supinfx=a+b,b0{λ(a),ρ(b)}min{λ(x),ρ(0)}, and  supinfx=ab,b1{λ(a),ψ(b)}min{λ(x),ψ(1)},xK,λγ. 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={xF:τ(x)0} is subfield, given xK, x0, there exists yK such that xy=1 and xτyλ=(1,η(1)). Using ((iii) and (iv)) of this theorem, η(1)=supinf1=xy{τ(x),ν(y)}=ψ(1). Thus xτyν=lψ=yνxτ. Similarly, given xK, there is a unique yK such that xτyη=0ρ=yηxτ, hence the theorem.

Theorem 23.

Let F and K be fields and let Fγ be fuzzy set field of F. If there exists ψ,ργ as stated in Theorem 19 and f:FK is a homomorphism, then f(Fγ) is a fuzzy set field in K.

Proof.

Let 1 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),  τγ} .

Now, we shall verify Definition 16 ((1) to (9)).

Let a,bMω with membership values ωμ(a) and ων(a), respectively. Then aωμbων = (a+b,ωτ(a+b)) and there are x,yF 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 aωμbων=bωνaωμ.

Similar to proof of (1), we can easily verify that (aωμbων)cωκ=  aωμ(bωνcωκ).

Suppose a,  bMω. Since a,  bM, then there are fuzzy points x,yF such that f(x) = a and f(y)=b. Since f is a homomorphism, ab=f(xy)M. So aωμbων = (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, aωμbων=bωνaωμ.

Similar to the proof of (3), we show that (aωμbων)cωκ=    aωμ(bωνcωκ).

Let yMω. Since f(0)=0,  0M and by Theorem 17, for every xFγ with corresponding membership value τ(x), there exists ργ such that τ(x) = supinfx=a+b{τ(a),ρ(b)}. Since ωτ(y) = sup{τ(x):y=f(x)}, yM, we have supinfy=y1+y2(ωτ(y1),ωρ(y2)) = supinfy=y1+y2(supy1=f(a)τ(a),supy2=f(b)ρ(b)) = supinfy=y1+y2{supx=a+b,y=f(x)(τ(a),ρ(b))} = supx,y=f(x){supinfx=a+b,y1=f(a),y2=f(b)(τ(a),ρ(b))}=supx,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 yMω, 0ωρyωτ=yωτ0ωρ=yωτ.

Let aMω. Since aM, there exist x,yFγ such that f(x)=a and xτyν=0ρ, by Definition 16. But f(x+y)=f(x)+f(y)=a+b=0, where f(y)=b, and aωμbωv=(0,ωτ(0)). Since xy = 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, aωμbων=0ωρ=bωνaωμ. Since aMω was arbitrary, so given yMω there exists z such that yωνzωκ=0ωρ=zωκyων.

Similar to the proof of (3) above, we show that aωμ(bωνcωκ)=(aωμbων)(aωμcωκ).

Since Fγ is a fuzzy field set, by Theorem 17, for every xFγ with corresponding membership value τ(x), there exists ψγ such that τ(x)=supinfx=ab{τ(a),ψ(b)}. Since ωτ(y)=sup{τ(x):y=f(x)},  yM, we have supinfy=y1y2(ωτ(y1),ωψ(y2)) = supinfy=y1y2(supy1=f(a)τ(a),supy2=f(b)ψ(b)) = supinfy=y1y2{supx=ab,y=f(x)(τ(a),ψ(b))} = supx,y=f(x){supinfx=ab,y1=f(a),y2=f(b)(τ(a),ψ(b))} = supx,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 yMω,  1ωψyωτ=yωτ1ωψ=yωτ.

Let aMω,  a0. Since aM, there exist x,yFγ such that f(x)=a and xτyη=1ψ and f(xy) = f(x)f(y)=ab=1, where f(y)=b. Thus ab=(1,ωτ(1)). Since xtyη=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, aωμbων=1ωψ=bωνaωμ.

Corollary 24.

Let F, K be fields, and let Kω be a fuzzy set field in K. If f:FK is injective homomorphism, then f-1(Kω) is a fuzzy set field in F.

Example 25.

Let F be a field and let τ:F[0,1] be given by τ(x)=1xF. Then Kτ={xτ:τ(x)0}=F×{1} is a fuzzy set field in F.

Proof.

(1)–(9) of Definition 16 hold trivially, with 0ρ=(0,1)=0τ and 1ψ=(1,1)=1τ.

Example 26.

Let F=R and let μ:R[0,1] be given by (6)τ(x)={11+|x|if-12x1211+|x+1|ifx-1211+|x-1|ifx12,ρ(x)={μ(0)=1ifx=00ifx0,ψ(x)={μ(1)=1ifx=10ifx1. Let γμ be as in Definition 13 and ω={η:ηγμ or η{ρ,ψ}}, then Fω={xv:vω} is a fuzzy set field in R.

Proof.

First we show that μ 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 xR, we can consider only for x0. Moreover, Aμ=Aμ1Aμ2, where (7)μ1(x)={11+xifx00ifx<0,μ2(x)={11+|x-1|ifx00ifx<0, and τ(x)=max{μ1(x),μ2(x)} . Therefore, μ1 and μ2 are both positive fuzzy numbers. Hence, by Lemma 4, μj(μiμj) = (μjμi)(μjμj)i,  j=1,2,. Thus, τ(x)(μ(y)μ(z)) = (τ(x)μ(y))(τ(x)μ(z))  x,y,zF. 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 ργ satisfies (1) in Theorem 19. For a fuzzy set field Kγ, such that ργ satisfies (1) in Theorem 19, it follows that sup{τ(x):xK}=ρ(0) (Remark 20). That is τ(xo)ρ(x) for all τγ and for each xK. 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.

Let Kγ be a fuzzy field set of a field K as in Theorem 19, and let  τγ. 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.

Let Kγ be a fuzzy set field of a field K, μγ, 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 Definition 28, |x|F=|x| if and only x is nonfuzzy in Kγ provided that ργ 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.

Let xτ, yν, and zk be fuzzy points in a fuzzy set field Kγ. Then

|xλyη|F|xτ|F+|yν|F,

|xτyν||x|+|y|F,

|xτyν|F|xτzκ|F+|zκyν|F.

Proof.

Let xτ, yν, and zκ be fuzzy points. Then

|xτyν|F=|x+y|+ρ(0)-η(x+y)|x+y|+2ρ(0)-τ(x)-ν(y)|xτ|F+|yν|F,

|xτyν|F=|x-y|+ρ(0)-ι(x-y)|x-y|+2ρ(0)-τ(x)-ν(y)|xτ|F+|yν|F,

|xτyν|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)|xτzκ|F+|zκyν|F.

Hence the result is proved.

Definition 30.

Let Kγ be a fuzzy set field of a field K such that ργ is as in Theorem 19, μγ 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:

dF(xτ,yη)ρ(0)-τ(x-y) for all x,yAμ,

dF(xτ,yη)=0 if and only if x=y and both are nonfuzzy in Kγ,

dF(yη,xτ)=dF(yη,xτ)  for  all  x,yAμ,

dF(xτ,yη)dF(xτ,zκ)+dF(zκ,yη)  for  all  x,y,zAμ.

A pair (Aμ,dF) is said to be fuzzy metric spaces.

If d 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.

Let Kγ be a fuzzy set field of a field K such that ργ is as in Theorem 19, μγ 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.

Let xτ,yη,zκAμ. We need to verify (1) to (4) of Definition 30.

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),

dF(xτ,yη)=0d(x,y)+ρ(0)-min{τ(x),η(y)}=0d(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,

dF(xτ,yη)=d(x,y)+ρ(0)-min{τ(x),η(y)}=d(y,x)+ρ(0)min{η(y),τ(x)},

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.

Let Kγ be a fuzzy set field of a field K such that ργ is as in Theorem 19, vγ 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.

Let Kγ be a fuzzy set of a field K such that ργ is as in Theorem 19, μγ and Aμ be a nonempty fuzzy set in field K. A mapping dF:Aμ×Aμ[0,) defined by dF(xτ,yη)=|xτyη|F+ρ(0)-min{τ(x),η(y)} is fuzzy metric on Aμ.

Proof.

Let xτ,yη,zκAν. We need to verify (1)–(4) of Definition 30.

dF(xτ,yη)ρ(0)-min{τ(x),η(y)}ρ(0)-ϕ(x-y), (By Remark 11 ),

dF(xτ,yη)=0|xτyη|F+ρ(0)-min{τ(x),η(y)}=0|xτyη|F = 0 and  min{τ(x),η(y)} = ρ(0)x=y,ρ(0)=min{τ(x),η(y)}μ(y)=ρ(0),

By Theorem 29, |xτyη|F=|yηxτ|FdF(xτ,yη)=dF(yη,xτ),

dF(xτ,yη)=|yηxτ|F+ρ(0)-min{τ(x),η(y)}|xτzκ|F+|zκyη|F+ρ(0)min{τ(x),η(y)}|xτzκ|F+|zκyη|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 Section 3, 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 xK and for all τγ,  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 1ρ(0), where ρ is as in Theorem 19. However, the two definitions coincide whenever ρ(0)=1.

Definition 34.

Let Aγ be fuzzy set in a field. A fuzzy point xτAγ is said to be nonfuzzy point if and only if τ(x)=1.

Definition 35.

Let xτ 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 Definition 34 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.

If xτ,  yη, and zκ are fuzzy points in a field K. Then

|xτyη|F=|yηxτ|F,

|xτyη|F|xτ|F+|yη|F,

|xτyη|F|xτzκ|F+|zκyη|F.

Proof.

Let xτ, γη, and zk be fuzzy points in a field K. Then

|xτyη|F=|x-y|+1-ϕ(x-y)=|x-y|+1-ϕ(y-x) (by Remark 11(3))   =|yηxτ|F,

|xτyη|F=|x+y|+1-ν(x+y)|x+y|+2-τ(x)-η(y)|xτ|F+|y|F,

|xτyη|F=|x-y|+1-ϕ(x-y)=|x-y|+1-ϕ(x-z+z-y)|x-y|+2-ϕ(x-z)ϕ(y-z)|xτzκ|F+|zκyη|F.

Therefore, the theorem is proved.

Definition 37.

Let Aμ 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:

dF(xτ,yη)1-ϕ(x-y),

dF(xτ,yη)=0 if and only if x is nonfuzzy in Aμ and x=y,

dF(xτ,yη)=dF(yη,xτ) for all x,yAμ,

dF(xτ,yη)dF(xτ,zκ)+dF(zκ,yη) for all x,y,zAμ.

A function dF is said to be fuzzy metric on Aμ and a pair (Aμ,dF) is said to be fuzzy metric space.

Theorem 38.

Let K 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.

Let x,y,zAμ with respective membership values  τ(x),η(y)and κ(z). We will verify that (1) to (4) of Definition 37.

dF(xτ,yη)=d(x,y)+1-min{τ(x),η(y)}dF(xτ,yη)1-ϕ(x-y), by Remark 11.

dF(xτ,yη)=0d(x,y)+1-min{τ(x),η(y)}=0d(x,y)=0,  min{τ(x),η(y)}=1x=y, 1=τ(x)=η(y)x=y and by Definition 34 both are nonfuzzy.

dF(xτ,yη)=d(x,y)+1-min{τ(x),η(y)}=d(y,x)+1-min{η(y),τ(x)}.

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,yAμ 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.

Let Aμ 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 Theorem 38.

Theorem 40.

Let Aν be a nonempty fuzzy set in a field K. A mapping dF:Aν×Aν[0,) defined by dF(xτ,yη)=|xτyη|F+1-min{τ(x),η(y)} is fuzzy metric on Aν.

Proof.

Proof is similar to Theorem 33, 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.

Conflict of Interests

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

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