Hutton Uniformity in the Context of Fuzzy Soft Sets

Vol. 9, No. 4, 2016, 419-433

ISSN 1307-5543 – www.ejpam.com

Hutton Uniformity in the Context of Fuzzy Soft Sets

Vildan Çetkin

, Halis Aygün

Department of Mathematics, Kocaeli University, Umuttepe Campus, 41380, Kocaeli, TURKEY

Abstract. In this paper, we introduce the concept of fuzzy soft uniformity in Hutton’s sense. We de-fine topological fuzzy soft remote neighborhood system and use this for investigating the relationship between fuzzy soft cotopology and fuzzy soft (quasi-)uniformity. We show the existence of the initial structure of fuzzy soft uniformities and also we prove the category of fuzzy soft uniform spaces is a topological category overSET3_.

2010 Mathematics Subject Classifications: 54A05,54A40,54E15

Key Words and Phrases: Fuzzy soft set, Fuzzy soft topology, Fuzzy soft remote neighborhood, Fuzzy soft uniformity

1. Introduction

In 1999, Molodtsov[12]proposed a completely new concept called soft set theory to model uncertainty, which associates a set with a set of parameters. Later, Majiet al. [11]introduced the concept of fuzzy soft set which combines fuzzy sets and soft sets. Soft set and fuzzy soft set theories have a rich potential for applications in several directions. Up till now there are many spectacular and creative works about the theories of soft set and fuzzy soft set in the literature (see [2, 3, 8, 9, 11, 13, 17]). Furthermore, Aygüno˘glu et al. [4] studied the topological structure of fuzzy soft sets based on the sense of Šostak[16].

It is well-known that uniformity is a very important concept close to topology and a con-venient tool for investigating topology. Fuzzy versions of (quasi-)uniformity theory were es-tablished by Hutton [7], Lowen [10], Höhle[6] and Shi[14, 15]. Fuzzy (quasi-)uniformity in Hutton’s sense has been accepted by many authors and has attracted wide attention in the literature, despite this.

In this paper, we give an approach to the concept of fuzzy soft uniformity in the sense of Hutton which is compatible with the fuzzy soft topology. The structure of this paper is orga-nized as follows. In Section 2, we give some preliminary concepts and properties. In Section 3, we give the definition of fuzzy soft remote neighborhood system and investigate relations

∗_{Corresponding author.}

Email addresses:[email protected](V. Çetkin),[email protected](H. Aygün)

between fuzzy soft cotopological space and fuzzy soft remote neighborhood system. In Sec-tion 4, we define fuzzy soft uniformity in the sense of Hutton and we study the relaSec-tionship between fuzzy soft cotopology and fuzzy soft uniformity by using fuzzy soft remote neighbor-hood system. In the last section, we introduce and characterize the initial structure of fuzzy soft uniform spaces.

2. Preliminaries

Throughout this paper, L is a complete lattice, M is a completely distributive lattice and there is an order-reversing involution0on L. Leta,bbe elements in L. An elementain Lis said to be coprime if a ≤ b∨c implies thata ≤ b or a ≤ c. The set of all coprimes of L is denoted byc(L). We sayais way below (wedge below)b, in symbols,ab(aÃb) orba

(bÂa), if for every directed (arbitrary) subsetD⊆ L,∨D≥bimpliesa≤d for somed∈D.

Clearly if a∈ L is coprime, then a b if and only ifa à b. A complete lattice L is said to

be continuous (completely distributive) if every element in Lis the supremum of all elements way below (wedge below) it.

Proposition 1. [5]Let L be a complete lattice. The following conditions are equivalent: (i) L is completely distributive.

(ii) L is distributive continuous lattice with enough coprimes.

(iii) The operator∨:Low(L)→L sending every lower set to its supremum has a left adjointβ, and in this caseβ(a) ={b| bÃa}.

From (iii) in the above proposition it is easy to see that the wedge below relation has the interpolation property in a completely distributive lattice, this is to say,aà bimplies there is

somec∈Lsuch thataÃcà b.

Let EandKbe arbitrary nonempty sets viewed on the sets of parameters. A fuzzy soft set

f on X, is a mapping from Einto LX, i.e., f_e := f(e) is an L-fuzzy set onX, for each e_∈E

(see Figure 1). The family of all L-fuzzy soft sets onX is denoted by(LX)E. By 0_X and 1_X, we denote respectively the null fuzzy soft set and absolute fuzzy soft set. The complement of an

L-fuzzy soft set f is denoted by f0, where f_e0(x) = (f_e(x))0. The set of all coprimes of(LX)E is denoted byc((LX)E).

Definition 1([1, 11]). (i) We say that f is a fuzzy soft subset of g and write f _vg if f_e≤g_e, for each e∈E.

(ii) Union of f and g is the fuzzy soft set h= f _tg, where h_e= f_e∨g_e, for each e_∈E.

(iii) Intersection of f and g is the fuzzy soft set h= f ug, where h_e= f_e∧g_e, for each e∈E.

Figure 1: A fuzzy soft set f

Letϕ:X1−→X2andψ:E1−→E2be two functions, whereE1andE2 are parameter sets

for the crisp setsX₁ andX₂, respectively. DefineL-fuzzy soft mapping ϕ_ψ→ _{: (L}X1)E1 _→ (_LX2)E2 _{and its L-fuzzy soft inverse mapping} ϕ

ψ← : (LX2)E2 → (LX1)E1

by (ϕ_ψ→(f))_e2(y) = W_ϕ(x)=yW_ψ(e1)=e

2 fe1(x), for all f ∈ (L

X1)E1_{, y ∈ X}

2, e2 ∈ E2 and

(ϕ_ψ←_(g))

e1(x) =gψ(e1)(ϕ(x)), for alle1∈E1, x∈X1and g∈(L X2)E2_.

We refer to[3, 4, 9, 11]for all the basic definitions and notations related to fuzzy soft sets and fuzzy soft mappings.

Definition 2([4]). A mappingτ:K_→M(LX)E is called an(L,M)-fuzzy(E,K)-soft topology on X if it satisfies the following conditions for each k∈K,

(T1) τk(0X) =τk(1X) =1M.

(T2) τk(f ug)≥τk(f)∧τk(g)for each f,g∈(LX)E.

(T3) τ_k(F_i∈Λf_i)_≥V_i∈Λτ_k(f_i)for each_{f_i}i∈Λ⊆(LX)E.

The pair(X,τ)is called an(L,M)-fuzzy(E,K)-soft topological space

Example 1. Let E be a parameter set, I = [0, 1], K = N be the set of natural numbers and

τ:K→I(IX)E be defined as follows: for all k∈K,

τk(f) =

¨

1, if f =0_X, 1_X,

1

k, otherwise.

(1)

It is easy to testify thatτis a fuzzy soft topology on X .

Definition 3([4]). A mapping_T :K_→M(LX)E is called an(L,M)-fuzzy(E,K)-soft cotopology on X if it satisfies the following conditions for each k∈K,

(C1) Tk(0X) =Tk(1X) =1M.

(C3) Tk(ui∈Λfi)≥

V

i∈ΛTk(fi)for all{fi}i∈Λ⊆(LX)E.

The pair(X,T)is called an(L,M)-fuzzy(E,K)-soft cotopological space.

Let (X1,T1) and (X2,T2) be an (L,M)-fuzzy (E1,K1)-soft cotopological space and an

(L,M)-fuzzy(E2,K2)-soft cotopological space, respectively. A fuzzy soft mapping

ϕψ,η:(X1,T1)→(X2,T2)is said to be continuous ifTk1(ϕ←ψ(g))≥ Tη2(k)(g) for each g∈(LX2)E2_{, k∈K}

1, whereϕ :X1→X2,ψ:E1→ E2 andη:K1 →K2 are classical

functions.

Let FSCTOP(L,M) denote the category of(L,M)-fuzzy (E,K)-soft cotopological spaces and continuous mappings.

If T is an(L,M)-fuzzy (E,K)-soft cotopology onX, then τis an(L,M)-fuzzy(E,K)-soft topology onX, whereτ:K_→M(LX)E is defined byτ_k(f) =_Tk(f0), for eachk_∈K.

Example 2. Let X = {x,y} be a classical set, E = {e1,e2}, K = {k1,k2} be parameter sets, L = M = I = [0, 1]. Define h_∈(IX)E as follows: h_e1(x) = 0.6, h_e1(y) = 0.5, h_e2(x) = 0.8

and he2(y) =0.6. Then the mappingT :K →I

(IX)E _{which is defined as follows is a fuzzy soft}

cotopology on X :

Tk(f) =



 

 

1, if f =0_X, 1_X,k_∈K

0.7, if f =h,k=k1,

0, otherwise.

(2)

3. Fuzzy Soft Remote Neighborhood System

In this section, we define fuzzy soft remote neighborhood system and give the relationships between fuzzy soft remote neighborhood system and fuzzy soft cotopological space. If the parameter setsEandKare both one pointed, then we obtain the results given in the paper of

[18].

Definition 4. A topological fuzzy soft remote neighborhood system is a set

R={Rp_|p_∈c((LX)E)_}of mappings Rp:K_→M(LX)E such that for each k_∈K: (RN1) Rp_k(1_X) =0_M, Rp_k(0_X) =1_M.

(RN2) Rp_k(f)6=0_M implies p6v f . (RN3) Rp_k(f tg) =Rp_k(f)∧Rp_k(g).

(RN4) Rp_k(f) =W_g∈p|f V_r6vgRr_k(g).

(1) _RT = _{Rp_{T |} p_∈c((LX)E)_} is a topological fuzzy soft remote neighborhood system, where Rp_T is defined by for all k∈K, p∈c((LX)E)and f ∈(LX)E, as

(Rp_T)_k(f) =

¨W

g∈p|f Tk(g), if p6v f;

0_M, otherwise. (3)

(2) If_T and_S are two(L,M)-fuzzy(E,K)-soft cotopologies which determine the same topolog-ical fuzzy soft remote neighborhood system, thenT =S.

Proof. By Definition 4, we need to show (RN4) in the following. First of all, (RN1)-(RN2) are trivial.

(RN3): Let k_∈ K and f,g _∈(LX)E. From the definition of R_Tp, we have f _v g implies

(Rp_T)k(f) ≥ (R p

T)k(g). This is to say (R p

T)k(f tg) ≤ (R p

T)k(f)∧(R p

T)k(g). Suppose that αÃ((R_Tp)k(f)∧(R

p

T)k(g)), whereα∈c(M). Then αÃ(R p

T)k(f)andαà (R p

T)k(g). Then there existu_∈p_| f andv_∈p_|gsuch thatα_{≤ Tk}(u)andα_{≤ Tk}(v). Therefore,

α≤ Tk(u)∧ Tk(v)≤ Tk(utv).

It is clear thatp6v(utv), f tgvutv. Hence by the definition ofRp_T, we have α≤(Rp_T)_k(f _tg).

From the arbitrariness ofα, we get for eachk∈K,(R_Tp)k(f tg)≥(R p

T)k(f)∧(R p T)k(g). (RN4): For each g_∈p_| f andk_∈K, we have

Tk(g)≤

^

p6vg

(Rp_T)_k(g)_≤(Rp_T)_k(g)_≤(Rp_T)_k(f).

Therefore,(Rp_T)k(f) =

W

g∈p|f Tk(g)≤

W

g∈p|f

V

r6vg(RrT)k(g)≤(R p T)k(f). This means that for eachk∈K,(Rp_T)k(f) =

W

g∈p|f

V

r6vg(R r T)k(g).

(2) For the proof of the second claim of Lemma 1, it is sufficient to show the validity of the following equality;_Tk(f) =V_p6vf(Rp_T)k(f)for all f ∈(LX)E andk∈K.

Obviously, _Tk(f)≤V

p6vf(R p

T)k(f)for all f ∈(LX)E andk∈K. So it is enough to prove

Tk(f)≥

V

p6vf(R p

T)k(f). In fact, we have

^

p6vf

(Rp_T)_k(f) = ^

p6vf

_

g∈p|f

Tk(g) =

_

A∈Πp6vfp|f

^

p6vf

Tk(A(p))≤

_

A∈Πp6vfp|f

Tk(up6vfA(p)) =Tk(f).

The last equality is due to_up6vfA(p) = f for everyA∈Πp6vfp| f.

Lemma 2. LetR={Rp|p∈c((LX)E)}be a topological fuzzy soft remote neighborhood system andT :K→M(LX)E be defined by for all k∈K and f ∈(LX)E,

Tk(f) =

^

p6vf

Rp_k(f).

Proof. By Definition 3, (C1) is trivial.

(C2) is proved by the following equations: for eachk∈K,

Tk(f tg) =

^

p6v(ftg)

Rp_k(f _tg)_≥(^

p6vf

Rp_k(f))_∧(^

p6vg

Rp_k(g)) =_Tk(f)_{∧ Tk}(g).

Finally, (C3) is shown by the following computation: for eachk∈K,

Tk(uj∈Jfj) =

^

p6vuj∈Jfj

Rp_k(uj∈Jfj) =

^

j∈J

^

p6vfj

Rp_k(uj∈Jfj)≥

^

j∈J

^

p6vfj

Rp_k(fj) =

^

j∈J

Tk(fj).

This completes the proof.

Moreover, it is obvious that _R = _P if _R and_P are two topological fuzzy soft remote neighborhood systems which determine the same fuzzy soft cotopology.

Lemma 3. LetR ={Rp | p ∈c((LX)E)} be a set satisfying (RN1)-(RN3). Then the following statements are equivalent:

(RN4) Rp_k(f) =W_g∈p|f V_r6vgRr_k(g). (RN4*) Rp_k(f) =W_g∈p|f(Rp_k(g)∧V

r6vgRrk(f)).

Proof. Suppose (RN4*) holds, i.e.,Rp_k(f) =W_g∈p|f(Rp_k(g)∧V

r6vgR r

k(f)). Letα∈ c(M) such thatαÃRp_k(f) =W_g∈p|f(Rp_k(g)∧V_r6vgRr_k(f)). Then there exists some g∈ p| f such

that

(1) αÃRp_k(g);

(2) αÃRr_k(f), for eachr6vg.

It is clear that the meet of fuzzy soft sets containing f and fulfilling (1) and (2) is still of such kind. So we can defineg∗to be the minimal fuzzy soft set containingf and fulfilling (1), (2), i.e.,αÃRp_k(g∗)andαÃRr_k(f)for allr6vg∗. Thus, for eachr6vg∗, it follows fromαÃRr_k(f)

that there existshr_∈r_| f such that (3) Rr_k(hr)Âα;

(4) Rr_k(f)Âα, for eachg6vhr.

It is easy to check thatg∗ uhrsatisfies (1) and (2). Hence, by the minimality ofg∗, it follows that g∗ vg∗ uhr. Therefore, g∗ vhr. Then we get thatαÃRr_k(hr)≤Rr_k(g∗)for all r6v g∗.

Thus,α_≤V_r6vg∗Rr_k(g_∗). Therefore,α_≤W_g∈p|f V_r6vgRr_k(g).

4. Fuzzy Soft Uniform Spaces

In this section, we introduce the concept of fuzzy soft uniformity as a parameterized family of Hutton uniformity in the spirit of fuzzy soft topology. Also, we consider the categorical relationship between the fuzzy soft remote neighborhood system and fuzzy soft uniform space.

Let_H(X,E)denote the family of all mappingsλ:(LX)E→(LX)E such that: (1) f _vλ(f)for all f _∈(LX)E.

(2) λ(F

j∈J fj) =

F

j∈Jλ(fj)for all{fj}j∈J⊆(LX)E.

λ∗ _{denotes the biggest element ofH(X,E), i.e., λ}∗_{(f) =0}

X when f = 0X and λ∗(f) =1X otherwise.

Forλ,µ_{∈ H}(X,E), we have thatλ∆µ_{∈ H}(X,E)andλ_◦µ_{∈ H}(X,E), where

λ∆µ(f) =u{λ(g)tµ(h)| f = gth}andλ◦µ(f) =λ(µ(f)). For eachλ∈ H(X,E), let λÃ(_g) =_u{h∈(_LX)E_|λ(_h0)_vg0_}.

Proposition 2.

(i) λÃ_{∈ H}(_X,E)_. (ii) (λÃ)Ã=λ_.

(iii) (λ◦µ)Ã_=µÃ_◦λÃ_.

(iv) λ_≤µimpliesλÃ_≤µÃ_.

(v) (λ∆µ)Ã=λÃ∆µÃ_.

(vi) (W_i∈Γλ_i)Ã=W

i∈ΓλÃi .

(vii) Ifλ1≤λ2 andµ1≤µ2, thenλ1∆µ1≤λ2∆µ2.

Supposeϕ_ψ:(LX)E→(LY)F be a fuzzy soft mapping andλ_{∈ H}(Y,F), define ϕ⇐

ψ(λ):(LX)E→(LX)E byϕ⇐ψ(λ)(f) =ϕψ←◦λ◦ϕψ→(f)for all f ∈(LX)E.

Proposition 3.

(i) ϕ_ψ⇐(λ)∈ H(X,E).

(ii) λ≤µimpliesϕ⇐_ψ(λ)≤ϕ_ψ⇐(µ). (iii) ϕ_ψ⇐(λÃ) = (ϕ⇐

ψ(λ))Ã.

(iv) ϕ_ψ⇐(λ◦µ)≤ϕ_ψ⇐(λ)◦ϕ_ψ⇐(µ).

Definition 5. An (L,M)-fuzzy (E,K)-soft quasi-uniformity is a mapping U : K → MH(X,E) which satisfies the following conditions: for each k_∈K,

(U2) _Uk(λ∆µ)_{≥ Uk}(λ)_{∧ Uk}(µ)for eachλ,µ_{∈ H}(X,E). (U3) Ifλ≥µ, thenUk(λ)≥ Uk(µ).

(U4) _Uk(λ)_≤W_{Uk(µ)_|µ_◦µ_≤λ_}for allλ_{∈ H}(X,E).

The pair(X,_U)is called an(L,M)-fuzzy(E,K)-soft quasi-uniform space. An(L,M)-fuzzy(E,K) -soft quasi-uniform space (X,U) is said to be an (L,M)-fuzzy (E,K)-soft uniform space if U

provides the condition:

(U) Uk(λ)≤

W

{Uk(µ)|µ≤λ/}for each k∈K,λ∈ H(X,E).

Given two U1 andU2 uniformities on X, we sayU1 is finer thanU2 (orU2 is coarser than_U1) iff_Uk1(λ)_{≥ Uk}2(λ)for eachk_∈Kandλ_{∈ H}(X,E).

A fuzzy soft mappingϕ_ψ,η:(X1,U1)→(X2,U2)is called (quasi-) uniformly continuous

if U_k1(ϕ⇐_ψ(µ)) ≥ U_η2_(k)(µ) for all µ ∈ H(X2,E2), k ∈ K1, where (X1,U1) and (X2,U2) is

an(L,M)-fuzzy (E₁,K₁)-soft uniform space and an(L,M)-fuzzy (E₂,K₂)-soft uniform space, respectively.

Theorem 1. Let(X1,U1),(X2,U2)and(X3,U3)be(L,M)-fuzzy(Ei,Ki)-soft uniform spaces,

respectively for i=1, 2, 3. Ifϕ_ψ,η:(X₁,_U1)_→(X₂,_U2)andϕ∗_ψ∗,η∗ :(X₂,_U2)_→(X₃,_U3)are uniformly continuous, then the composition is uniformly continuous.

The category of(L,M)-fuzzy(E,K)-soft quasi-uniform spaces and continuous mappings is denoted byHFSU(L,M).

Theorem 2. Let(X,_U)be an(L,M)-fuzzy(E,K)-soft quasi-uniform space and Rp_U :K→M(LX)E be defined by for all f ∈(LX)E,

(Rp_U)_k(f) =_

p6vh

_

λ(h0_)vf0

Uk(λ).

ThenRU ={R p

U |p∈c((L X₎E₎

}is a topological fuzzy soft remote neighborhood system.

Proof. We need to check (RN1)-(RN4). (RN1), (RN2) and (RN3) are straightforward, what remains is to prove.

(RN4): From Lemma 3, we know that it is equivalent to check (RN4*). Since

(Rp_U)_k(f)_≥W_g∈p|f €(Rp_U)_k(g)_∧V_r6vg(Rr_U)_k(f)Š, for allk_∈K, is obvious. It is sufficient to show that(Rp_U)_k(f)≤W

g∈p|f

€

(Rp_U)_k(g)∧V

r6vg(RrU)k(f)

Š

, for each

k∈K. Letk∈K andα_∈c(M)such thatαÃ(Rp_U)k(f), that is, αÃ(Rp_U)k(f) =

_

p6vh

_

λ(h0_)vf0

Uk(λ)≤

_

p6vh

_

λ(h0_)vf0

_

µ◦µ≤λ

Uk(µ).

Then there existh∈(LX)E,λ_{∈ H}(X,E)andµ_{∈ H}(X,E)such that

andα_{≤ Uk}(µ).

Let g= (µ(h0))0. Then g∈p| f. Furthermore, we have

(Rp_U)_k(g) = _

p6vd

_

ν(d0_)vg0

Uk(ν)≥

_

ν(h0_)vg0

Uk(ν)≥ Uk(µ)≥α

and

^

r6vg

(R_Ur )_k(f) = ^

r6vg

_

r6vd

_

ν(d0_)vf0

Uk(ν)≥

^

r6vg

_

ν(g0_)vf0

Uk(ν)≥

^

r6vg

Uk(µ)≥α.

Thenα_≤(Rp_U)_k(g)_∧V_r6vg(R_Ur )_k(f). Therefore,α_≤W_g∈p|f €(R_Up )_k(g)_∧V_r6vg(Rr_U)_k(f)Š. From the arbitrariness of α, we have (R_Up)k(f) ≤

W

g∈p|f

€

(Rp_U)_k(g)∧V

r6vg(RrU)k(f)

Š

.

Theorem 3. Let(X,_U)be an(L,M)-fuzzy(E,K)-soft quasi-uniform space. Then,R_Up can also be written as follows:

(i) (_RUp)_k(f) =W_p6vhW_λ◦λ(h0_)vf0Uk(λ).

(ii) (R_Up)k(f) =

W

h∈(LX₎E

W

p6vλ(h0_)w(λ◦λ(h0₎₎0_wf Uk(λ).

(iii) (R_Up)k(f) =

W

p6vλ/_(f)Uk(λ).

Proof. (i) and (ii) are trivial. (iii) can be obtained by the definition ofλ/.

From Lemma 2, we know that_TU is an(L,M)-fuzzy(E,K)-soft cotopology onX and call it the generated(L,M)-fuzzy(E,K)-soft cotopology by_U.

Theorem 4. Let (X,T) be an(L,M)-fuzzy(E,K)-soft cotopological space. Then there is one

(L,M)-fuzzy(E,K)-soft quasi uniformity_UT on X such that the generated(L,M)-fuzzy(E,K) -soft cotopology byUT is justT, i.e.,T =TUT. This is to say that each(L,M)-fuzzy(E,K)-soft cotopological space is(L,M)-fuzzy(E,K)-soft quasi-uniformizable.

Proof. Let g_∈(LX)E andλ_g:(LX)E _→(LX)E be defined as follows:

λg(f) =



 

 

1_X, if f 6vg;

g, if 0_{X 6}= f v g; 0_X, otherwise.

Thenλ_f ∈ H(X,E)andλ_f ◦λf =λf. DefineUT :K→MH(X,E)by

(UT)k(λ) =

_ {

n

^

i=1

Tk(gi)|λ≥∆in=1λg0_i,n∈N}.

It is easy to verify that_UT is an(L,M)-fuzzy (E,K)-soft quasi uniformity on X. Now we prove thatT =TUT. Noting thatλ

/

g0

i(

f) = f, from the definition ofTUT, we have fork∈K

(TUT)k(f) =

^

p6vf

_

p6vλ/_(f)

_ {

n

^

i=1

Tk(gi)|λ≥∆in=1λg0_i,n∈N} ≥

^

p6vf

This is to sayTUT ≥ T. On the other hand, we have

(TUT)k(f) =

^

p6vf

_

p6vλ/_(f)

_ {

n

^

i=1

Tk(gi)|λ≥∆in=1λg0_i,n∈N}

≤^

p6vf

_

p6vλ/_(f)

_ {

n

^

i=1

Tk(gi)|λ/≥∆in=1λ/g_i0,n∈N}

≤^

p6vf

_

p6vλ/_(f)

_ {

n

^

i=1

Tk(gi)|λ/(f)≥∆ni=1λg0_i(f),n∈N}

≤^

p6vf

_

{Tk(umj=1gj)|p6v umj=1gjw f,m∈N}

≤^

p6vf

_

{Tk(g)|p6vgw f}=Tk(f).

This completes the proof.

Theorem 5. Ifϕ_ψ,η:(X1,U1)→(X2,U2)is quasi uniformly continuous, then

ϕψ,η:(X1,TU1)→(X₂,T_U2)is fuzzy soft continuous. Proof. Let g∈(LX2)E2_,k∈K

1 andαÃ(TU2)_η(k)(g). Sinceϕ_ψ,η:(X₁,U1)→(X₂,U2)is

quasi uniformly continuous, we have_Uk1(ϕ⇐_ψ(µ))_{≥ Uη}2_(k)(µ)for allµ_{∈ H}(X2,E2), k∈K1.

Hence,

αÃ(TU2)_η(k)(g) =

^

p6vg

_

p6vλ/_(g)

U_η2_(k)(λ)≤ ^

p6vg

_

p6vλ/_(g)

U_k1(ϕ⇐

ψ(λ)).

Noting that ϕ_ψ→(h)6v g whenh6v ϕ_ψ←(g), we can find someλ_{(h) ∈ H}(X2,E2)such that

ϕ→

ψ(h)6v λ(h)(g)andα ≤ Uk1(ϕψ⇐(λ(h))). Now letν(h)=ϕ⇐_ψ(λ(h)). Thenν(h)∈ H(X1,E1)

andh6vν/_(h)(ϕ←

ψ(g)). Hence,

α≤ ^

h6vϕ← ψ(g)

Uk1(ν(h))≤

^

h6vϕ← ψ(g)

_

h6vν/_(ϕ← ψ(g))

Uk1(ν) = (TU1)_k(ϕ←_ψ(g)).

Therefore,(_TU2)_η(k)(g)≤(T_U1)_k(ϕ←_ψ(g))from the arbitrariness ofα. So,

ϕ_ψ,η:(X1,TU1)→(X₂,T_U2)is fuzzy soft continuous.

Theorem 6. Ifϕ_ψ,η:(X1,T1)→(X2,T2)is fuzzy soft continuous, then

ϕ_ψ,η:(X1,UT1)→(X₂,U_T2)is quasi uniformly continuous.

Proof. Letϕ_ψ,η:(X1,T1)→(X2,T2)be continuous. From the definition ofUT1, we know

that for each k ∈K, (UT2)_η(k)(λ) =

W {Vn

i=1T_η2₍k)(gi)| λ≥ ∆ n

i=1λg0_i, n∈N}. Moreover, if λ≥∆ni=1λg0_i, then we have

ϕ⇐

ψ(λ)≥ϕψ⇐(∆ni=1λg0_i) = n

^

i=1

(ϕ⇐

ψ(λ(g0i))) = n

^

i=1 λϕ←

Sinceϕ_ψ,η:(X₁,_T1)_→(X₂,_T2)is continuous, we haveVn_i=1Tη2_(k)(g_i)_≤Vn_i=1Tk1(ϕ_ψ←(g_i)). Hence,(_UT2)_η(k)(λ)≥(U_T1)_k(ϕ⇐_ψ(λ)).

Therefore,ϕ_ψ,η:(X1,UT1)→(X₂,U_T2)is quasi uniformly continuous.

Theorem 7. Let G:FSCTOP(L,M)_→HFSU(L,M)be defined by G((X,_T)) = (X,_UT). Then G is an embedding functor fromFSCTOP(L,M)toHFSU(L,M).

5. Category of Fuzzy Soft Uniform Spaces

In this section, we will show that the category HFSU(L,M) of (L,M)-fuzzy (E,K)-soft uniform spaces and continuous functions is a topological category overSET3.

Theorem 8. Let{(X_i,_Ui_)}

i∈Γbe a family of(L,M)-fuzzy(Ei,Ki)-soft uniform spaces, X be a

set, E,K be the parameter sets and for each i∈Γ,ϕ_i :X→X_i,ψ_i :E→E_i andη_i :K→K_i be a function. We define the mappingU :K→MH(X,E)by:

Uk(λ) =

_ {

n

^

j=1

Uij

η_{i j}(k)(λij)|∆ n

j=1(ϕψ)⇐ij (λij)≤λ}, for each k∈K,

where∨is taken over the finite index set{i1, . . . ,in} ⊆Γ. Then the following items are satisfied.

(1) U is the coarsest(L,M)-fuzzy(E,K)-soft uniformity on X for which each(ϕ_ψ,η)iis uniformly

continuous function.

(2) A functionϕ_ψ,η:(Z,V)→(X,U)is uniformly continuous iff

(ϕ_ψ,η)i◦ϕψ,η:(Z,V)→(Xi,Ui)is uniformly continuous for all i∈Γ.

Proof. (1) Firstly, we will prove that_U is an(L,M)-fuzzy(E,K)-soft uniformity onX. (U1) and (U3) are clear.

(U2): Suppose there existλ,µ_{∈ H}(X,E)andk_∈Ks.t. _Uk(λ∆µ)_{6≥ Uk}(λ)_{∧ Uk}(µ). By the definition of _U, there exist finite index sets_{i1, . . . ,in},{j1, . . . ,jm} ⊆Γsuch that

Uk(λ∆µ)6≥

Vn

r=1U ir

ηir(k)(λir)

∧

Vm

s=1U js

ηjs(k)(µjs)

where∆n_r=1(ϕ_ψ)⇐_i

r (λir)≤λand

∆m

s=1(ϕψ)⇐js(µjs)≤µ. Since(∆n

r=1(ϕψ)⇐ir (λir))∆(∆ m

s=1(ϕψ)⇐js(µjs))≤λ∆µ. By Proposition 2 (vii), we have

Uk(λ∆µ)≥

Vn

r=1U ir

η_ir(k)(λir)

∧

Vm

s=1U js

η_js(k)(µjs)

.

This is a contradiction. Hence for eachk∈Kandλ,µ_{∈ H}(X,E), we obtain

Uk(λ∆µ)≥ Uk(λ)∧ Uk(µ).

(U4): Suppose there existk∈Kandλ_{∈ H}(X,E)such that

Uk(λ)6≤

W

{Uk(µ)|µ◦µ≤λ}.

By the definition of _Uk(λ), there exists a finite index set J = _{i₁, . . . ,i_{n} ⊆} Γ such that

Vn

j=1U ij

η_{i j}(k)(λij)6≤

W

{Uk(µ)|µ◦µ≤λ}, where∆nj=1(ϕψ)⇐ij (λij)≤λ. Since(Xij,U

(L,M)-fuzzy(E_i

j,Kij)-soft uniformity for eachij∈ {i1, . . . ,in}, by Definition 5,

Uij

η_{i j}(k)(λij)≤

W {Uij

η_{i j}(k)(ν)|ν◦ν≤λij}.

For each ij ∈ {i1, . . . ,in}, there exists νij ∈ H(Xij,Eij) with νij ◦νij ≤ λij such that

Vn

j=1U ij

η_{i j}(k)(νij)6≤

W

{Uk(µ)| µ◦µ≤ λ}. Putν∗ = ∆nj=1(ϕψ)⇐ij (νij). For each ij ∈J, we haveν∗_◦ν∗= (∆n

j=1(ϕψ)i⇐j (νij))◦(∆ n

j=1(ϕψ)⇐ij (νij)). Hence, ν∗_◦ν∗_≤∆n

j=1((ϕψ)⇐ij(νij)◦(ϕψ) ⇐

ij (νij))≤∆ n

j=1((ϕψ)⇐ij (νij◦νij))≤∆ n

j=1(ϕψ)⇐ij(λij)≤λ. Then we haveν∗◦ν∗_≤λandU

k(ν∗)≥

Vn j=1U

ij

η_{i j}(k)(νij). This is a contradiction.

(U4): Let_{(Xi,Ui)}i∈Γbe a family of (L,M)-fuzzy(Ei,Ki)-soft uniform spaces. Suppose there exists λ∈ H(X,E) andk ∈K such that Uk(λ) 6≤

W

{Uk(µ) | µ ≤ λ/}. By using the definition of_Uk(λ), there exists a finite index setJ = _{j₁, . . . ,j_n}ofΓsuch thatW_{Uk(µ)_| µ≤λ/} 6≥Vn

i=1U ji

η_ji(k)(λji), where∆ n

i=1(ϕψ)⇐ji (λji)≤λ. Since_Uji _{is an}(_L,M)_-fuzzy(_E

ji,Kji)-soft uniformity onXji, then

W {Uji

η_ji(k)(ν) | ν ≤ λ

/

ji} ≥ U ji

η_ji(k)(λji). For each ji ∈ J, there exists ν ∗

ji ∈ H(Xji,Eji) with ν∗

ji ≤λ

/

ji such that

W

{Uk(µ)|µ≤λ/} 6≥

Vn i=1U

ji

η_ji(k)(ν∗ji). On the other hand, we have

∆n

i=1(ϕψ)⇐ji (ν ∗

ji)≤∆ n

i=1(ϕψ)⇐ji (λ

/

ji) = ∆ n

i=1((ϕψ)⇐ji (λji))

/_{= (∆}n

i=1(ϕψ)⇐ji (λji))

/_≤λ/_.

Putν∗= ∆n_i=1(ϕ_ψ)⇐_j i (ν

∗

ji). Then there existsν

∗_{∈ H(X,E)such thatν}∗_≤λ/_and

Uk(ν∗)≥

Vn

i=1U ji

η_ji(k)(ν ∗

ji). Thus n

^

i=1

Uji

η_ji(k)(ν ∗

ji)≤ Uk(ν

∗_)≤_

{Uk(µ)|µ≤λ/}.

This is a contradiction. Hence for eachk∈K andλ_{∈ H}(X,E), we have_Uk(λ)_≤W

{Uk(µ)| µ≤λ/}.

Secondly by using the definition of U, we have Uk((ϕψ)⇐i (λi)) ≥ U_ηi_i(k)(λi) for each

k∈K,i∈Γandλ_{i∈ H}(Xi,Ei). Hence(ϕψ,η)i is uniformly continuous function. Finally, if(ϕ_ψ,η)_i :(X,_V)_→(Xi,Ui)is uniformly continuous, i.e.,

Vk((ϕψ)⇐i (λi)) ≥ U_ηi

i(k)(λi) for each k ∈ K,i∈Γ andλi ∈ H(Xi,Ei). Then for k∈ K, we have

Uk(f) =

_ {

n

^

j=1

Uij

η_{i j}(k)(λij)|∆ n

j=1(ϕψ)⇐ij(λij)≤λ}

≤_{

n

^

j=1

Vk((ϕψ)⇐ij (λij))|∆ n

j=1(ϕψ)⇐ij (λij)≤λ}

≤_{Vk(∆nj=1(ϕψ)⇐ij(λij))|∆ n

(2) Necessity of the composition condition is clear. Suppose that for(L,M)-fuzzy(E∗,K∗)

-soft uniform space(Z,_V), ϕ_ψ,η:(Z,_V)_→(X,_U)is not uniformly continuous. Then there existk∗∈K∗andλ_{∈ H}(X,E)such that_Vk∗((ϕ_ψ)⇐(λ))6≥ U_η(k∗₎(λ). By the definition ofU,

there exists a finite index setJ={j1, . . . ,jn}ofΓsuch thatVk∗((ϕ_ψ)⇐(λ))6≥

Vn i=1U

ji

η_ji(η(k∗₎₎(λji), where∆n_i=1(ϕ_ψ)⇐_j

i (λji)≤λ.

On the other hand, since(ϕ_ψ,η)_j

i ◦ϕψ,ηis uniformly continuous, we have n

^

i=1

Uji

η_ji(η(k∗₎₎(λji)≤ n

^

i=1

Vk∗(ϕ_ψ⇐_,η◦(ϕ_ψ,η)⇐_j

i (λji))

≤Vk∗(∆n_i=1ϕ⇐_ψ,η((ϕ_ψ,η)⇐ji (λji)))

=Vk∗(ϕ_ψ⇐_,η(∆n_i=1(ϕ_ψ,η)⇐ji (λji)))

≤Vk∗(ϕ_ψ⇐_,η(λ)).

This is a contradiction.

Definition 6. Let_{(X_i,_Ui)_}i∈Γ be a family of (L,M)-fuzzy(E_i,K_i)-soft uniform spaces, X be a set, E,K be the parameter sets and ϕi : X → Xi, ψi : E → Ei and ηi : K → Ki be

func-tions for each i∈Γ. The initial (L,M)-fuzzy(E,K)-soft uniform structure on X with respect to

(X,(ϕ_ψ,η)_i,(X_i,_Ui),Γ)is the coarsest(L,M)-fuzzy(E,K)-soft uniform structure on X for which all i∈Γ,(ϕ_ψ,η)i are uniformly continuous.

From Theorem 8 and Definition 6, we have the following theorem:

Theorem 9. The categoryHFSU(L,M)of(L,M)-fuzzy(E,K)-soft uniform spaces and uniformly continuous functions is a topological category over the category SET3 with respect to the usual forgetful functor V :HFSU(L,M)→SET3 which is defined by V(X,_U) = (X,E,K)and

V(ϕ_ψ,η) = (ϕ,ψ,η).

Definition 7. Let X= Πi∈ΓXi, E= Πi∈ΓEiand K= Πi∈ΓKibe the product sets and{(Xi,Ui)}i∈Γ be a family of(L,M)-fuzzy(E_i,K_i)-soft uniform spaces, for each i∈Γ. The initial(L,M)-fuzzy

(E,K)-soft uniformity structure_U on X with respect to the family_{(p_q,r)_i:X _→(X_i,_Ui)_}i∈Γof all projection functions is called the product of(L,M)-fuzzy(E_i,K_i)-soft uniformity{Ui_}

i∈Γ. The pair(X,_U)is called the product(L,M)-fuzzy(E,K)-soft uniform space.

6. Conclusion

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