EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 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 over SET3. 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, Maji et 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, Aygünoğlu 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: vcetkin@gmail.com (V. Çetkin), halis@kocaeli.edu.tr (H. Aygün) http://www.ejpam.com 419 c© 2016 EJPAM All rights reserved. V. Çetkin, H. Aygün / Eur. J. Pure Appl. Math, 9 (2016), 419-433 420 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 relationship 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 involution ′ on L. Let a, b be elements in L. An element a in L is said to be coprime if a ≤ b ∨ c implies that a ≤ b or a ≤ c. The set of all coprimes of L is denoted by c(L). We say a is way below (wedge below) b, in symbols, a� b (a à b) or b� a (b  a), if for every directed (arbitrary) subset D ⊆ L, ∨D ≥ b implies a ≤ d for some d ∈ D. Clearly if a ∈ L is coprime, then a � b if and only if a à b. A complete lattice L is said to be continuous (completely distributive) if every element in L is 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 adjoint β , and in this case β(a) = {b | b à a}. 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, a à b implies there is some c ∈ L such that a à c à b. Let E and K be arbitrary nonempty sets viewed on the sets of parameters. A fuzzy soft set f on X , is a mapping from E into LX , i.e., fe := f (e) is an L-fuzzy set on X , for each e ∈ E (see Figure 1). The family of all L-fuzzy soft sets on X is denoted by (LX )E . By 0X and 1X , 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 f ′, where f ′e (x) = ( fe(x))′. The set of all coprimes of (LX )E is denoted by c((LX )E). Definition 1 ([1, 11]). (i) We say that f is a fuzzy soft subset of g and write f v g if fe ≤ ge, for each e ∈ E. (ii) Union of f and g is the fuzzy soft set h= f t g, where he = fe ∨ ge, for each e ∈ E. (iii) Intersection of f and g is the fuzzy soft set h= f u g, where he = fe ∧ ge, for each e ∈ E. Let p | f denote the set {g ∈ (LX )E | p 6v g v f } for p ∈ c((LX )E) and f ∈ (LX )E . V. Çetkin, H. Aygün / Eur. J. Pure Appl. Math, 9 (2016), 419-433 421 Figure 1: A fuzzy soft set f Let ϕ : X1 −→ X2 and ψ : E1 −→ E2 be two functions, where E1 and E2 are parameter sets for the crisp sets X1 and X2, respectively. Define L-fuzzy soft mapping ϕψ → : (LX1)E1 → (LX2)E2 and its L-fuzzy soft inverse mapping ϕψ ← : (LX2)E2 → (LX1)E1 by (ϕψ→( f ))e2 (y) = ∨ ϕ(x)=y ∨ ψ(e1)=e2 fe1 (x), for all f ∈ (LX1)E1 , y ∈ X2, e2 ∈ E2 and (ϕψ←(g))e1 (x) = gψ(e1)(ϕ(x)), for all e1 ∈ E1, x ∈ X1 and g ∈ (LX2)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 mapping τ : K → M (L X )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 u g)≥ τk( f )∧τk(g) for each f , g ∈ (LX )E . (T3) τk( ⊔ i∈Λ fi)≥ ∧ i∈Λτk( fi) for each { fi}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 (I X )E be defined as follows: for all k ∈ K, τk( f ) = ¨ 1, if f = 0X , 1X , 1 k , otherwise. (1) It is easy to testify that τ is a fuzzy soft topology on X . Definition 3 ([4]). A mapping T : K → M (L X )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 . (C2) Tk( f t g)≥ Tk( f )∧Tk(g) for all f , g ∈ (LX )E . V. Çetkin, H. Aygün / Eur. J. Pure Appl. Math, 9 (2016), 419-433 422 (C3) Tk(ui∈Λ fi)≥ ∧ 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,T 1) and (X2,T 2) 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,T 1)→ (X2,T 2) is said to be continuous if T 1 k (ϕ ← ψ (g))≥ T 2 η(k)(g) for each g ∈ (LX2)E2 , k ∈ K1, where ϕ : X1 → X2, ψ : E1 → E2 and η : 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 on X , then τ is an (L, M)-fuzzy (E, K)-soft topology on X , where τ : K → M (L X )E is defined by τk( f ) = Tk( f ′), for each k ∈ 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: he1 (x) = 0.6, he1 (y) = 0.5, he2 (x) = 0.8 and he2 (y) = 0.6. Then the mapping T : K → I (I X )E which is defined as follows is a fuzzy soft cotopology on X : Tk( f ) =      1, if f = 0X , 1X , 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 sets E and K are 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 (L X )E such that for each k ∈ K: (RN1) Rp k(1X ) = 0M , Rp k(0X ) = 1M . (RN2) Rp k( f ) 6= 0M implies p 6v f . (RN3) Rp k( f t g) = Rp k( f )∧ Rp k(g). (RN4) Rp k( f ) = ∨ g∈p| f ∧ r 6vg Rr k(g). Lemma 1. Let T : K → M (L X )E be an (L, M)-fuzzy (E, K)-soft cotopology. Then the followings are valid. V. Çetkin, H. Aygün / Eur. J. Pure Appl. Math, 9 (2016), 419-433 423 (1) RT = {R p 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 ) = ¨ ∨ g∈p| f Tk(g), if p 6v f ; 0M , 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, then T = S . Proof. By Definition 4, we need to show (RN1)-(RN4) in the following. First of all, (RN1)- (RN2) are trivial. (RN3): Let k ∈ K and f , g ∈ (LX )E . From the definition of Rp T , we have f v g implies (Rp T )k( f ) ≥ (R p T )k(g). This is to say (Rp T )k( f t g) ≤ (Rp T )k( f ) ∧ (R p T )k(g). Suppose that α à ((Rp T )k( f ) ∧ (R p T )k(g)), where α ∈ c(M). Then α à (Rp T )k( f ) and α à (Rp T )k(g). Then there exist u ∈ p | f and v ∈ p | g such that α≤ Tk(u) and α≤ Tk(v). Therefore, α≤ Tk(u)∧Tk(v)≤ Tk(ut v). It is clear that p 6v (ut v), f t g v ut v. Hence by the definition of Rp T , we have α≤ (Rp T )k( f t g). From the arbitrariness of α, we get for each k ∈ K , (Rp T )k( f t g)≥ (Rp T )k( f )∧ (R p T )k(g). (RN4): For each g ∈ p | f and k ∈ K , we have Tk(g)≤ ∧ p 6vg (Rp T )k(g)≤ (R p T )k(g)≤ (R p T )k( f ). Therefore, (Rp T )k( f ) = ∨ g∈p| f Tk(g)≤ ∨ g∈p| f ∧ r 6vg(R r T )k(g)≤ (R p T )k( f ). This means that for each k ∈ K , (Rp T )k( f ) = ∨ g∈p| f ∧ r 6vg(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 ) = ∧ p 6v f (R p T )k( f ) for all f ∈ (LX )E and k ∈ K . Obviously, Tk( f ) ≤ ∧ p 6v f (R p T )k( f ) for all f ∈ (LX )E and k ∈ K . So it is enough to prove Tk( f )≥ ∧ p 6v f (R p T )k( f ). In fact, we have ∧ p 6v f (Rp T )k( f ) = ∧ p 6v f ∨ g∈p| f Tk(g) = ∨ A∈Πp 6v f p| f ∧ p 6v f Tk(A(p))≤ ∨ A∈Πp 6v f p| f Tk(up 6v f A(p)) = Tk( f ). The last equality is due to up 6v f A(p) = f for every A∈ Πp 6v f p | f . Lemma 2. Let R = {Rp | p ∈ c((LX )E)} be a topological fuzzy soft remote neighborhood system and T : K → M (L X )E be defined by for all k ∈ K and f ∈ (LX )E , Tk( f ) = ∧ p 6v f Rp k( f ). Then T is an (L, M)-fuzzy (E, K)-soft cotopology on X . Furthermore, if R and P are two topo- logical fuzzy soft remote neighborhood systems which determine the same (L, M)-fuzzy (E, K)-soft cotopology,then R =P . V. Çetkin, H. Aygün / Eur. J. Pure Appl. Math, 9 (2016), 419-433 424 Proof. By Definition 3, (C1) is trivial. (C2) is proved by the following equations: for each k ∈ K , Tk( f t g) = ∧ p 6v( f tg) Rp k( f t g)≥ ( ∧ p 6v f Rp k( f ))∧ ( ∧ p 6vg Rp k(g)) = Tk( f )∧Tk(g). Finally, (C3) is shown by the following computation: for each k ∈ K , Tk(u j∈J f j) = ∧ p 6vu j∈J f j Rp k(u j∈J f j) = ∧ j∈J ∧ p 6v f j Rp k(u j∈J f j)≥ ∧ j∈J ∧ p 6v f j Rp k( f j) = ∧ j∈J Tk( f j). 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. Let R = {Rp | p ∈ c((LX )E)} be a set satisfying (RN1)-(RN3). Then the following statements are equivalent: (RN4) Rp k( f ) = ∨ g∈p| f ∧ r 6vg Rr k(g). (RN4*) Rp k( f ) = ∨ g∈p| f (R p k(g)∧ ∧ r 6vg Rr k( f )). Proof. Suppose (RN4*) holds, i.e., Rp k( f ) = ∨ g∈p| f (R p k(g) ∧ ∧ r 6vg Rr k( f )). Let α ∈ c(M) such that α à Rp k( f ) = ∨ g∈p| f (R p k(g) ∧ ∧ r 6vg Rr k( f )). Then there exists some g ∈ p | f such that (1) αà Rp k(g); (2) αà Rr k( f ), for each r 6v g. 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 define g∗ to be the minimal fuzzy soft set containing f and fulfilling (1), (2), i.e., αà Rp k(g∗) and αà Rr k( f ) for all r 6v g∗. Thus, for each r 6v g∗, it follows from αà Rr k( f ) that there exists hr ∈ r | f such that (3) Rr k(h r) α; (4) Rr k( f ) α, for each g 6v hr . It is easy to check that g ∗uhr satisfies (1) and (2). Hence, by the minimality of g∗, it follows that g∗ v g ∗ uhr . Therefore, g∗ v hr . Then we get that α à Rr k(h r) ≤ Rr k(g∗) for all r 6v g∗. Thus, α≤ ∧ r 6vg∗ Rr k(g∗). Therefore, α≤ ∨ g∈p| f ∧ r 6vg Rr k(g). From the arbitrariness of α, we have Rp k( f ) ≤ ∨ g∈p| f ∧ r 6vg Rr k(g), for each k ∈ K . Since for each k ∈ K , Rp k( f ) ≥ ∨ g∈p| f ∧ r 6vg Rr k(g) is obvious. We have Rp k( f ) = ∨ g∈p| f ∧ r 6vg Rr k(g), as desired. V. Çetkin, H. Aygün / Eur. J. Pure Appl. Math, 9 (2016), 419-433 425 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. LetH (X , E) denote the family of all mappings λ : (LX )E → (LX )E such that: (1) f v λ( f ) for all f ∈ (LX )E . (2) λ( ⊔ j∈J f j) = ⊔ j∈J λ( f j) for all { f j} j∈J ⊆ (LX )E . λ∗ denotes the biggest element of H (X , E), i.e., λ∗( f ) = 0X when f = 0X and λ∗( f ) = 1X otherwise. For λ,µ ∈H (X , E), we have that λ∆µ ∈H (X , E) and λ ◦µ ∈H (X , E), where λ∆µ( f ) = u{λ(g) t µ(h) | f = g t h} and λ ◦ µ( f ) = λ(µ( f )). For each λ ∈ H (X , E), let λÃ(g) = u{h ∈ (LX )E | λ(h′)v g ′}. Proposition 2. (i) λà ∈H (X , E). (ii) (λÃ)à = λ. (iii) (λ ◦µ)à = µÃ ◦λÃ. (iv) λ≤ µ implies λà ≤ µÃ. (v) (λ∆µ)à = λÃ∆µÃ. (vi) ( ∨ i∈Γλi)à = ∨ i∈Γλ à i . (vii) If λ1 ≤ λ2 and µ1 ≤ µ2, then λ1∆µ1 ≤ λ2∆µ2. Suppose ϕψ : (LX )E → (LY )F be a fuzzy soft mapping and λ ∈H (Y, F), define ϕ⇐ ψ (λ) : (LX )E → (LX )E by ϕ⇐ ψ (λ)( f ) = ϕ← ψ ◦λ ◦ϕ→ ψ ( f ) for all f ∈ (LX )E . Proposition 3. (i) ϕ⇐ ψ (λ) ∈H (X , E). (ii) λ≤ µ implies ϕ⇐ 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, (U1) Uk(λ∗) = 1M . V. Çetkin, H. Aygün / Eur. J. Pure Appl. Math, 9 (2016), 419-433 426 (U2) Uk(λ∆µ)≥Uk(λ)∧Uk(µ) for each λ,µ ∈H (X , E). (U3) If λ≥ µ, then Uk(λ)≥Uk(µ). (U4) Uk(λ)≤ ∨ {Uk(µ) | µ ◦µ≤ λ} for all λ ∈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)≤ ∨ {Uk(µ) | µ≤ λ/} for each k ∈ K ,λ ∈H (X , E). Given two U 1 and U 2 uniformities on X , we say U 1 is finer than U 2 (or U 2 is coarser than U 1) iff U 1 k (λ)≥U 2 k (λ) for each k ∈ K and λ ∈H (X , E). A fuzzy soft mapping ϕψ,η : (X1,U 1)→ (X2,U 2) is called (quasi-) uniformly continuous if U 1 k (ϕ ⇐ ψ (µ)) ≥ U 2 η(k)(µ) for all µ ∈ H (X2, E2), k ∈ K1, where (X1,U 1) and (X2,U 2) 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,U 1), (X2,U 2) and (X3,U 3) be (L, M)-fuzzy (Ei , Ki)-soft uniform spaces, respectively for i = 1, 2,3. If ϕψ,η : (X1,U 1)→ (X2,U 2) and ϕ∗ ψ∗,η∗ : (X2,U 2)→ (X3,U 3) 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 by HFSU(L, M). Theorem 2. Let (X ,U ) be an (L, M)-fuzzy (E, K)-soft quasi-uniform space and Rp U : K → M (L X )E be defined by for all f ∈ (LX )E , (Rp U )k( f ) = ∨ p 6vh ∨ λ(h′)v f ′ Uk(λ). Then RU = {R p U | p ∈ c((LX )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 )≥ ∨ g∈p| f � (Rp U )k(g)∧ ∧ r 6vg(R r U )k( f ) � , for all k ∈ K , is obvious. It is sufficient to show that (Rp U )k( f )≤ ∨ g∈p| f � (Rp U )k(g)∧ ∧ r 6vg(R r U )k( f ) � , for each k ∈ K . Let k ∈ K and α ∈ c(M) such that αà (Rp U )k( f ), that is, αà (Rp U )k( f ) = ∨ p 6vh ∨ λ(h′)v f ′ Uk(λ)≤ ∨ p 6vh ∨ λ(h′)v f ′ ∨ µ◦µ≤λ Uk(µ). Then there exist h ∈ (LX )E , λ ∈H (X , E) and µ ∈H (X , E) such that p 6v hw (µ(h′))′ w ((µ ◦µ)(h′))′ w (λ(h′))′ w f V. Çetkin, H. Aygün / Eur. J. Pure Appl. Math, 9 (2016), 419-433 427 and α≤Uk(µ). Let g = (µ(h′))′. Then g ∈ p | f . Furthermore, we have (Rp U )k(g) = ∨ p 6vd ∨ ν(d ′)vg ′ Uk(ν)≥ ∨ ν(h′)vg ′ Uk(ν)≥Uk(µ)≥ α and ∧ r 6vg (Rr U )k( f ) = ∧ r 6vg ∨ r 6vd ∨ ν(d ′)v f ′ Uk(ν)≥ ∧ r 6vg ∨ ν(g ′)v f ′ Uk(ν)≥ ∧ r 6vg Uk(µ)≥ α. Then α≤ (Rp U )k(g)∧ ∧ r 6vg(R r U )k( f ). Therefore, α≤ ∨ g∈p| f � (Rp U )k(g)∧ ∧ r 6vg(R r U )k( f ) � . From the arbitrariness of α, we have (R p U )k( f ) ≤ ∨ g∈p| f � (Rp U )k(g)∧ ∧ r 6vg(R r U )k( f ) � . Theorem 3. Let (X ,U ) be an (L, M)-fuzzy (E, K)-soft quasi-uniform space. Then, R p U can also be written as follows: (i) (R p U )k( f ) = ∨ p 6vh ∨ λ◦λ(h′)v f ′Uk(λ). (ii) (R p U )k( f ) = ∨ h∈(LX )E ∨ p 6vλ(h′)w(λ◦λ(h′))′w f Uk(λ). (iii) (R p U )k( f ) = ∨ p 6vλ/( f )Uk(λ). Proof. (i) and (ii) are trivial. (iii) can be obtained by the definition of λ/. From Lemma 2, we know that TU is an (L, M)-fuzzy (E, K)-soft cotopology on X 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 by UT is just T , 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 and λg : (LX )E → (LX )E be defined as follows: λg( f ) =      1X , if f 6v g; g, if 0X 6= f v g; 0X , otherwise. Then λ f ∈H (X , E) and λ f ◦λ f = λ f . Define UT : K → MH (X ,E) by (UT )k(λ) = ∨ { n ∧ i=1 Tk(gi) | λ≥∆n i=1λg ′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 that T = TUT . Noting that λ/ g ′i ( f ) = f , from the definition of TUT , we have for k ∈ K (TUT )k( f ) = ∧ p 6v f ∨ p 6vλ/( f ) ∨ { n ∧ i=1 Tk(gi) | λ≥∆n i=1λg ′i , n ∈ N} ≥ ∧ p 6v f Tk( f ) = Tk( f ). V. Çetkin, H. Aygün / Eur. J. Pure Appl. Math, 9 (2016), 419-433 428 This is to say TUT ≥ T . On the other hand, we have (TUT )k( f ) = ∧ p 6v f ∨ p 6vλ/( f ) ∨ { n ∧ i=1 Tk(gi) | λ≥∆n i=1λg ′i , n ∈ N} ≤ ∧ p 6v f ∨ p 6vλ/( f ) ∨ { n ∧ i=1 Tk(gi) | λ/ ≥∆n i=1λ / g ′i , n ∈ N} ≤ ∧ p 6v f ∨ p 6vλ/( f ) ∨ { n ∧ i=1 Tk(gi) | λ/( f )≥∆n i=1λg ′i ( f ), n ∈ N} ≤ ∧ p 6v f ∨ {Tk(um j=1 g j) | p 6v um j=1 g j w f , m ∈ N} ≤ ∧ p 6v f ∨ {Tk(g) | p 6v g w f }= Tk( f ). This completes the proof. Theorem 5. If ϕψ,η : (X1,U 1)→ (X2,U 2) is quasi uniformly continuous, then ϕψ,η : (X1,TU 1)→ (X2,TU 2) is fuzzy soft continuous. Proof. Let g ∈ (LX2)E2 , k ∈ K1 and α à (TU 2)η(k)(g). Since ϕψ,η : (X1,U 1)→ (X2,U 2) is quasi uniformly continuous, we have U 1 k (ϕ ⇐ ψ (µ)) ≥ U 2 η(k)(µ) for all µ ∈ H (X2, E2), k ∈ K1. Hence, αà (TU 2)η(k)(g) = ∧ p 6vg ∨ p 6vλ/(g) U 2 η(k)(λ)≤ ∧ p 6vg ∨ p 6vλ/(g) U 1 k (ϕ ⇐ ψ (λ)). Noting that ϕ→ ψ (h) 6v g when h 6v ϕ← ψ (g), we can find some λ(h) ∈ H (X2, E2) such that ϕ→ ψ (h) 6v λ(h)(g) and α ≤ U 1 k (ϕ ⇐ ψ (λ(h))). Now let ν(h) = ϕ⇐ψ (λ(h)). Then ν(h) ∈ H (X1, E1) and h 6v ν/(h)(ϕ ← ψ (g)). Hence, α≤ ∧ h 6vϕ← ψ (g) U 1 k (ν(h))≤ ∧ h 6vϕ← ψ (g) ∨ h 6vν/(ϕ← ψ (g)) U 1 k (ν) = (TU 1)k(ϕ ← ψ (g)). Therefore, (TU 2)η(k)(g)≤ (TU 1)k(ϕ←ψ (g)) from the arbitrariness of α. So, ϕψ,η : (X1,TU 1)→ (X2,TU 2) is fuzzy soft continuous. Theorem 6. If ϕψ,η : (X1,T 1)→ (X2,T 2) is fuzzy soft continuous, then ϕψ,η : (X1,UT 1)→ (X2,UT 2) is quasi uniformly continuous. Proof. Let ϕψ,η : (X1,T 1)→ (X2,T 2) be continuous. From the definition ofUT 1 , we know that for each k ∈ K , (UT 2)η(k)(λ) = ∨ { ∧n i=1T 2 η(k)(gi) | λ ≥ ∆n i=1λg ′i , n ∈ N}. Moreover, if λ≥∆n i=1λg ′i , then we have ϕ⇐ψ (λ)≥ ϕ ⇐ ψ (∆ n i=1λg ′i ) = n ∧ i=1 (ϕ⇐ψ (λ(g ′ i))) = n ∧ i=1 λϕ← ψ (gi)′ . V. Çetkin, H. Aygün / Eur. J. Pure Appl. Math, 9 (2016), 419-433 429 Sinceϕψ,η : (X1,T 1)→ (X2,T 2) is continuous, we have ∧n i=1T 2 η(k)(gi)≤ ∧n i=1T 1 k (ϕ ← ψ (gi)). Hence, (UT 2)η(k)(λ)≥ (UT 1)k(ϕ⇐ψ (λ)). Therefore, ϕψ,η : (X1,UT 1)→ (X2,UT 2) 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 from FSCTOP(L, M) to HFSU(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 over SET3. Theorem 8. Let {(X i ,U i)}i∈Γ be 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, ϕi : X → X i ,ψi : E→ Ei and ηi : K → Ki be a function. We define the mapping U : K → MH (X ,E) by: Uk(λ) = ∨ { n ∧ j=1 U i j ηi j (k)(λi j ) |∆n j=1(ϕψ) ⇐ i j (λi j )≤ 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 (ϕψ,η)i is uniformly continuous function. (2) A function ϕψ,η : (Z ,V )→ (X ,U ) is uniformly continuous iff (ϕψ,η)i ◦ϕψ,η : (Z ,V )→ (X i ,U i) is uniformly continuous for all i ∈ Γ. Proof. (1) Firstly, we will prove that U is an (L, M)-fuzzy (E, K)-soft uniformity on X . (U1) and (U3) are clear. (U2): Suppose there exist λ,µ ∈H (X , E) and k ∈ K s.t. Uk(λ∆µ) 6≥ Uk(λ)∧Uk(µ). By the definition of U , there exist finite index sets {i1, . . . , in}, { j1, . . . , jm} ⊆ Γ such that Uk(λ∆µ) 6≥ � ∧n r=1U ir ηir (k) (λir ) � ∧ � ∧m s=1U js η js (k) (µ js) � where ∆n r=1(ϕψ) ⇐ ir (λir )≤ λ and ∆m s=1(ϕψ) ⇐ js (µ js)≤ µ. Since (∆n r=1(ϕψ) ⇐ ir (λir ))∆(∆ m s=1(ϕψ) ⇐ js (µ js))≤ λ∆µ. By Proposition 2 (vii), we have Uk(λ∆µ)≥ � ∧n r=1U ir ηir (k) (λir ) � ∧ � ∧m s=1U js η js (k) (µ js) � . This is a contradiction. Hence for each k ∈ K and λ,µ ∈H (X , E), we obtain Uk(λ∆µ)≥Uk(λ)∧Uk(µ). (U4): Suppose there exist k ∈ K and λ ∈H (X , E) such that Uk(λ) 6≤ ∨ {Uk(µ) | µ ◦µ≤ λ}. By the definition of Uk(λ), there exists a finite index set J = {i1, . . . , in} ⊆ Γ such that ∧n j=1U i j ηi j (k)(λi j ) 6≤ ∨ {Uk(µ) | µ ◦ µ ≤ λ}, where ∆n j=1(ϕψ) ⇐ i j (λi j ) ≤ λ. Since (X i j ,U i j ) is an V. Çetkin, H. Aygün / Eur. J. Pure Appl. Math, 9 (2016), 419-433 430 (L, M)-fuzzy (Ei j , Ki j )-soft uniformity for each i j ∈ {i1, . . . , in}, by Definition 5, U i j ηi j (k)(λi j )≤ ∨ {U i j ηi j (k)(ν) | ν ◦ ν≤ λi j }. For each i j ∈ {i1, . . . , in}, there exists νi j ∈ H (X i j , Ei j ) with νi j ◦ νi j ≤ λi j such that ∧n j=1U i j ηi j (k)(νi j ) 6≤ ∨ {Uk(µ) | µ ◦ µ ≤ λ}. Put ν∗ = ∆n j=1(ϕψ) ⇐ i j (νi j ). For each i j ∈ J , we have ν∗ ◦ ν∗ = (∆n j=1(ϕψ) ⇐ i j (νi j )) ◦ (∆n j=1(ϕψ) ⇐ i j (νi j )). Hence, ν∗ ◦ ν∗ ≤∆n j=1((ϕψ) ⇐ i j (νi j ) ◦ (ϕψ)⇐i j (νi j ))≤∆n j=1((ϕψ) ⇐ i j (νi j ◦ νi j ))≤∆n j=1(ϕψ) ⇐ i j (λi j )≤ λ. Then we have ν∗ ◦ ν∗ ≤ λ and Uk(ν∗)≥ ∧n j=1U i j ηi j (k)(νi j ). This is a contradiction. (U4): Let {(X i ,U i)}i∈Γ be a family of (L, M)-fuzzy (Ei , Ki)-soft uniform spaces. Suppose there exists λ ∈ H (X , E) and k ∈ K such that Uk(λ) 6≤ ∨ {Uk(µ) | µ ≤ λ/}. By using the definition of Uk(λ), there exists a finite index set J = { j1, . . . , jn} of Γ such that ∨ {Uk(µ) | µ≤ λ/} 6≥ ∧n i=1U ji η ji (k)(λ ji ), where ∆n i=1(ϕψ) ⇐ ji (λ ji )≤ λ. Since U ji is an (L, M)-fuzzy (E ji , K ji )-soft uniformity on X ji , then ∨ {U ji η ji (k)(ν) | ν ≤ λ / ji } ≥ U ji η ji (k)(λ ji ). For each ji ∈ J , there exists ν∗ji ∈ H (X ji , E ji ) with ν∗ji ≤ λ/ji such that ∨ {Uk(µ) | µ≤ λ/} 6≥ ∧n 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 (λ ji )) / = (∆n i=1(ϕψ) ⇐ ji (λ ji )) / ≤ λ/. Put ν∗ =∆n i=1(ϕψ) ⇐ ji (ν∗ji ). Then there exists ν∗ ∈H (X , E) such that ν∗ ≤ λ/ and Uk(ν∗)≥ ∧n i=1U ji η ji (k)(ν ∗ ji ). Thus n ∧ i=1 U ji η ji (k)(ν ∗ ji )≤Uk(ν ∗)≤ ∨ {Uk(µ) | µ≤ λ/}. This is a contradiction. Hence for each k ∈ K and λ ∈H (X , E), we have Uk(λ) ≤ ∨ {Uk(µ) | µ≤ λ/}. Secondly by using the definition of U , we have Uk((ϕψ)⇐i (λi)) ≥ U i ηi(k) (λi) for each k ∈ K , i ∈ Γ and λi ∈H (X i , Ei). Hence (ϕψ,η)i is uniformly continuous function. Finally, if (ϕψ,η)i : (X ,V )→ (X i ,U i) is uniformly continuous, i.e., Vk((ϕψ)⇐i (λi)) ≥ U i ηi(k) (λi) for each k ∈ K , i ∈ Γ and λi ∈ H (X i , Ei). Then for k ∈ K , we have Uk( f ) = ∨ { n ∧ j=1 U i j ηi j (k)(λi j ) |∆n j=1(ϕψ) ⇐ i j (λi j )≤ λ} ≤ ∨ { n ∧ j=1 Vk((ϕψ) ⇐ i j (λi j )) |∆n j=1(ϕψ) ⇐ i j (λi j )≤ λ} ≤ ∨ {Vk(∆ n j=1(ϕψ) ⇐ i j (λi j )) |∆n j=1(ϕψ) ⇐ i j (λi j )≤ λ} ≤ Vk(λ). V. Çetkin, H. Aygün / Eur. J. Pure Appl. Math, 9 (2016), 419-433 431 (2) Necessity of the composition condition is clear. Suppose that for (L, M)-fuzzy (E∗, K∗) -soft uniform space (Z ,V ), ϕψ,η : (Z ,V )→ (X ,U ) is not uniformly continuous. Then there exist k∗ ∈ K∗ and λ ∈H (X , E) such that Vk∗((ϕψ)⇐(λ)) 6≥ Uη(k∗)(λ). By the definition of U , there exists a finite index set J = { j1, . . . , jn} of Γ such thatVk∗((ϕψ)⇐(λ)) 6≥ ∧n i=1U ji η ji (η(k∗))(λ ji ), where ∆n i=1(ϕψ) ⇐ ji (λ ji )≤ λ. On the other hand, since (ϕψ,η) ji ◦ϕψ,η is uniformly continuous, we have n ∧ i=1 U ji η ji (η(k∗))(λ ji )≤ n ∧ i=1 Vk∗(ϕ ⇐ ψ,η ◦ (ϕψ,η) ⇐ ji (λ ji )) ≤Vk∗(∆ n i=1ϕ ⇐ ψ,η((ϕψ,η) ⇐ ji (λ ji ))) =Vk∗(ϕ ⇐ ψ,η(∆ n i=1(ϕψ,η) ⇐ ji (λ ji ))) ≤Vk∗(ϕ ⇐ ψ,η(λ)). This is a contradiction. Definition 6. Let {(X i ,U i)}i∈Γ be a family of (L, M)-fuzzy (Ei , Ki)-soft uniform spaces, X be a set, E, K be the parameter sets and ϕi : X → X i , 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 , (ϕψ,η)i , (X i ,U i),Γ) is the coarsest (L, M)-fuzzy (E, K)-soft uniform structure on X for which all i ∈ Γ, (ϕψ,η)i are uniformly continuous. From Theorem 8 and Definition 6, we have the following theorem: Theorem 9. The category HFSU(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 (ϕψ,η) = (ϕ,ψ,η). Definition 7. Let X = Πi∈ΓX i , E = Πi∈ΓEi and K = Πi∈ΓKi be the product sets and {(X i ,U i)}i∈Γ be a family of (L, M)-fuzzy (Ei , Ki)-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 {(pq,r)i : X → (X i ,U i)}i∈Γ of all projection functions is called the product of (L, M)-fuzzy (Ei , Ki)-soft uniformity {U i}i∈Γ. 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