On (L,M)-fuzzy quasi-uniform spaces - Science Direct

Fuzzy Sets and Systems 158 (2007) 1472 – 1485 www.elsevier.com/locate/fss

On (L, M)-fuzzy quasi-uniform spaces Yueli Yuea,∗ , Fu-Gui Shib a Department of Mathematics, Ocean University of China, Qingdao 266071, PR China b Department of Mathematics, Beijing Institute of Technology, Beijing 100081, PR China

Received 9 May 2006; received in revised form 29 January 2007; accepted 29 January 2007 Available online 22 February 2007

Abstract An (L, M)-fuzzy topology is a graded extension of topological spaces handling M-valued families of L-fuzzy subsets of a referential, where L and M are completely distributive lattices. When M reduces to the set 2 = {0, 1}, a (2, M)-fuzzy topology is called a fuzzifying topology after Ying. Šostak introduced the notion (L, M)-fuzzy uniform spaces. The aim of this paper is to study the relationship between (2, M)-fuzzy quasi-uniform spaces and (L, M)-fuzzy quasi-uniform spaces as well as the relationship between (2, M)-fuzzy quasi-uniform spaces and pointwise (L, M)-fuzzy quasi-uniform spaces—the extension of Shi’s L-quasiuniform space in a Kubiak–Šostak sense. It is shown that the category of (2, M)-fuzzy quasi-uniform spaces can be embedded in the category of stratified (L, M)-fuzzy quasi-uniform spaces as a both reflective and coreflective full subcategory; and the former category can also be embedded in the category of pointwise (L, M)-fuzzy quasi-uniform spaces. © 2007 Elsevier B.V. All rights reserved. Keywords: (L, M)-fuzzy topology; (L, M)-fuzzy quasi-uniformity; Pointwise (L, M)-fuzzy quasi-uniformity

1. Introduction It is well-known that (quasi-)uniformity is a very important concept close to topology and a convenient tool for investigating topology (see [4,13,15]). L-(quasi-)uniformity in Hutton’s sense (see [9]) has been accepted by many authors and has attracted wide attention in the literature. Up till now there are many works about the theory of Hutton uniformities (see [6,11,26]). Rodabaugh [16] gave a theory of fuzzy uniformities with applications to the fuzzy real lines. It also needs to point out that Shi [19,20] introduced the theory of pointwise L-quasi-uniformities on fuzzy sets and Shi’s theory is simpler and more direct for studying the relationship between pointwise L-quasi-uniformities and pointwise L-topologies. An (L, M)-fuzzy topology is a graded extension of topological spaces handling M-valued families of L-fuzzy subsets of a referential, where L and M are completely distributive lattices. When M reduces to the set 2 = {0, 1}, a (2, M)-fuzzy topology is called a fuzzifying topology after Ying. Extension of Hutton’s quasi-uniformities—[0, 1]-fuzzy uniformity—was considered in [2]. Later, in [22] fuzzy uniformities for lattices more general than [0, 1], namely, the so called (L, M)-fuzzy uniformities were considered. Finally, in [6], a paper specially devoted to the analysis of different approaches to the theory of fuzzy uniformities, an essentially more general concept of an L-valued uniformity was studied using a filter approach. Further, in [18], ∗ Corresponding author.

E-mail address: [email protected] (Y. Yue). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.01.018

Y. Yue, F.-G. Shi / Fuzzy Sets and Systems 158 (2007) 1472 – 1485

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there is a significant extension of Hutton approach for quasi-uniformities without using filters explicitly and without any distributivity and with general tensor products generating the intersection axiom. In [24], the relationship between (L, M)-fuzzy topologies and (L, M)-fuzzy quasi-uniformities was investigated. Zhang [25] gave a way to embed the category of uniform spaces in the category of Hutton uniform spaces. There is another way to embed the category of uniform spaces in the category of Hutton uniform spaces first in Katsaras [10] for the valued lattice [0, 1] and in Liu and Liang [14] for the general case. One aim of this paper is to study the relationship between (2, M)-fuzzy quasi-uniform spaces and (L, M)-fuzzy quasi-uniform spaces by Katsaras and Zhang’s approach. It is shown that the category of (2, M)-fuzzy quasi-uniform spaces can be embedded in the category of stratified (L, M)-fuzzy quasi-uniform spaces as a both reflective and coreflective full subcategory. Another aim of this paper is to study the relationship between (2, M)-fuzzy quasi-uniform spaces and pointwise (L, M)-fuzzy quasiuniform spaces—the extension of Shi’s L-quasi-uniform space in a Kubiak–Šostak sense. It is shown that the category of (2, M)-fuzzy quasi-uniform spaces can be embedded in the category of pointwise (L, M)-fuzzy quasi-uniform spaces. 2. Preliminaries Let L be a complete lattice. An element a ∈ L is said to be coprime (resp., prime) if a b ∨ c (resp., a b ∧ c) implies that a b or a c (resp., a b or a c). The set of all coprimes (primes) of L is denoted by c(L)(resp., p(L)). We say a is way below (wedge below) b, in symbols, a>b (ab) or b?a (ba), if for every directed (arbitrary) subset D ⊆ L, D b implies a d for some d ∈ D. Clearly if a ∈ L is a coprime, then a>b if and only if ab. A complete lattice L is said to be continuous (completely distributive) if every element in L is the supremum of all the elements way below (wedge below) it. Proposition 2.1 (Gierz et al. [5]). Let L be a complete lattice. The following conditions are equivalent: (1) L is completely distributive; (2) L is distributive continuous lattice with enough coprimes; (3) The operator : Low(L) → L sending every lower set to its supremum has a left adjoint , and in this case (a) = {b| ba}. From (3) 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, ab implies there is some c ∈ L such that acb. For more detail about completely distributive lattices, please refer to [5]. In the following, L and M are two completely distributive lattices and L possesses an order reversing involution . X L is the set of all L-fuzzy sets on X. A ∈ LX is defined by A (x) = (A(x)) . The set of all coprimes of LX is denoted by c(LX ). U denotes the characteristic function ofU ∈ 2X . Let F : X → Y be an ordinary mapping, define FL→ : LX → LY and FL← : LY → LX by FL→ (A)(y) = {A(x)| x ∈ X, F (x) = y} for A ∈ LX and y ∈ Y , and FL← (B)(x) = B(F (x)) for B ∈ LY and x ∈ X (following the notation in [17]), respectively. Definition 2.2 (Höhle [7], Höhle and Šostak [8], Kubiak [12], Šostak [21]). An (L, M)-fuzzy topology is a mapping : LX → M such that (FCT1) (1X ) = (0X ) = 1; X (FCT2) (A ∧ B)(A) ∧ (B) for all A, B ∈ L ; (FCT3) ( j ∈J Aj ) j ∈J (Aj ) for every family {Aj |j ∈ J } ⊆ LX . The pair (LX , ) is called an (L, M)-fuzzy topological space. A mapping F : (LX , ) → (LY , 1 ) is said to be continuous with respect to and 1 if (FL← (B))1 (B) for all B ∈ LY . Let (L, M)-FTOP denote the category of (L, M)-fuzzy topological spaces and continuous mappings. When L = {0, 1}, Definition 2.2 will reduce to that of M-fuzzifying topology. Let M-FYS denote the category of M-fuzzifying topological spaces.

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The relationship between M-FYS and (L, M)-FTOP was first studied in [23] when L = M = [0, 1] and in Fang [3] for the general case. M-FYS can be embedded in (L, M)-FTOP by and defined in Lemmas 2.3 and 2.4. X Lemma 2.3 (Fang [3]). Let (X, ) be an M-fuzzifying topological space and define () : L → M by ()(A) = X r∈L (r (A)) for A ∈ L , where r (A) = {x|A(x)r}. Then ()(A) = r∈p(L) (r (A)) and () is an (L, M)fuzzy topology on X. Furthermore, is an embedding functor of M-FYS in (L, M)-FTOP.

Lemma 2.4. Let (X, ) be an M-fuzzifying topological space and define () : LX → M as follows: ∀A ∈ LX , (U ), A = U , ()(A) = 0 others. Then () is an (L, M)-fuzzy topology on X and is an embedding functor of M-FYS in (L, M)-FTOP. Now we recall some notions and terminologies about (L, M)-fuzzy quasi-uniform spaces used in this paper. Let H (LX ) denote the family of all mappings d : LX → LX such that: (1) A d(A) for all A ∈ LX ; (2) d( j ∈J Aj ) = j ∈J d(Aj ) for {Aj }j ∈J ⊆ LX . d1 denotes the biggest element of H (LX ), i.e., d1 (A) = 0X when A = 0X and d1 (A) = 1X otherwise. For d, e ∈ H (LX ), we have that d ∧ e ∈ H (LX ) and d ◦ e ∈ H (LX ), where d ∧ e(A) = d(B) ∨ e(C) = d(x ) ∧ e(x ) and d ◦ e(A) = d(e(A)). B∨C=A

x A,x ∈c(LX )

Suppose F : X → Y is a mapping, d ∈ H (LY ), define F ⇐ (d) : LX → LX by F ⇐ (d)(A) = FL← ◦ d ◦ FL→ (A) for all A ∈ LX , then F ⇐ (d) ∈ H (LX ). When f ∈ H (2Y ), similarly, define F (f ) : 2X → 2X by F (f )(U ) = F ← ◦ f ◦ F → (U ) for all U ∈ 2X , then F (f ) ∈ H (2X ). Definition 2.5 (Dzhajanbajev and Šostak [2], Gutierrez Garcia et al. [6], Šostak [22]). An (L, M)-fuzzy quasi-uniformity is a mapping FU : H (LX ) → M such that (FQU1) FU(d1 ) = 1; (FQU2) FU(d ∧ e)= FU(d) ∧ FU(e) for all d, e ∈ H (LX ); (FQU3) FU(d) = e◦e d FU(e) for all d ∈ H (LX ). An (L, M)-fuzzy quasi-uniformity is said to be a stratified (L, M)-fuzzy quasi-uniformity if it also satisfies the following condition: (FQU4) FU(supX ) = 1, where supX : LX → LX is defined by supX (A)(x) = y∈X A(y) for A ∈ LX . (LX , FU) is called an (L, M)-fuzzy quasi-uniform space. A mapping F : (LX , FU) → (LY , FU 1 ) is called uniformly continuous if FU(F ⇐ (d))FU 1 (d) for all d ∈ H (LY ). The categories of (L, M)-fuzzy quasi-uniform spaces and stratified (L, M)-fuzzy quasi-uniform spaces are denoted by (L, M)-FHuQUnif and (L, M)-SFHuQUnif, respectively. When H (LX ) is replaced by H (2X ) in Definition 2.5, it will reduce to the definition of (2, M)-fuzzy quasi-uniform spaces. The category of (2, M)-fuzzy quasi-uniform spaces and uniformly continuous mappings is denoted by MFQUnif. A mapping FU : H (LX ) → M satisfying (FQU1) and (FQU2) is called an (L, M)-fuzzy semi-quasi-uniformity. Uniformly continuous mappings between (L, M)-fuzzy semi-quasi-uniform spaces are defined similarly. Let (L, M)FHsQUnif denote the category of (L, M)-fuzzy semi-quasi-uniform spaces. Similarly, when L = {0, 1}, the definition of (2, M)-fuzzy semi-quasi-uniformity is obtained and let M-FsQUnif denote the category of (2, M)-fuzzy semi-quasiuniform spaces. A functor T : A → B is called topological provided every T-source {fj : X → (Xj , j )}j ∈J has a unique Tinitial lift. For undefined notions about category, please refer to [1]. In [24], it was shown that (L, M)-FHuQUnif is a topological category over SET. In fact, the following theorem is also valid.


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Theorem 2.6. (L, M)-FHsQUnif is a topological category over SET. Proof. The key is to show that it is initially complete. Given a source {Ft : LX → (LXt , FU t )}t∈T , it is easy to verify that FU : H (LX ) → M defined by ∧i=n FU(d) = i=1 FU ti (eti ) ⇐ d ∧i=n i=1 Fti (eti ), n∈N

is just the unique initial lift on X with respect to the given source.

Definition 2.7. A mapping FB : H (LX ) → M is called fuzzy normal if FB(d)

e◦e d

FB(e) for all d ∈ H (LX ).

It is obvious that FU : H (LX ) → M is an (L, M)-fuzzy quasi-uniformity if and only if FU is both an (L, M)-fuzzy semi-quasi-uniformity and a fuzzy normal mapping. ¯ : H (LX ) → M defined by FB(d) ¯ Lemma 2.8. If FB : H (LX ) → M is fuzzy normal, then FB = i=n ¯ ∧ FB(di ) is also fuzzy normal and satisfies FB FB.

d=∧i=n i=1 di

i=1

¯ FB FB ¯ is obvious. It suffices to show that Proof. From the definition of FB, ¯ ¯ FB(d) FB(e) = ∧i=n i=1 FB(ei ) e◦e d

e◦e d e=∧i=n ei i=1

¯ Let tFB(d). Since FB is fuzzy normal, we have ¯ tFB(d) = ∧i=n ∧i=n FB(ei ). i=1 FB(di ) i=1

for all d ∈

H (LX ).

d=∧i=n i=1 di

d=∧i=n i=1 di

ei ◦ei di

X Then there exists a finite family {di }i=n i=1 of H (L ) such that

(i) d = ∧i=n i=1 di ; (ii) For each 1 i n, there is some ei ∈ H (LX ) such that ei ◦ ei di and t FB(ei ). Let e = ∧i=n i=1 ei . Then we have i=n i=n i=n e ◦ e = ∧i=n i=1 ei ◦ ∧i=1 ei ∧i=1 (ei ◦ ei ) ∧i=1 di d. ¯ ¯ ¯ Thus t ∧i=n e◦e d FB(e). From the arbitrariness of t, we have FB(d) e◦e d i=1 FB(ei ) FB(e) ¯ FB(e). This completes the proof.

Lemma 2.9. If {FB t }t∈T is a family of fuzzy normal mappings, then ( t∈T FBt )(d) = t∈T (FBt (d)) is also fuzzy normal.

t∈T

FBt : H (LX ) → M defined by

Lemma 2.10. For every mapping FB : H (LX ) → M, there is a biggest fuzzy normal mapping FB ∗ : H (LX ) → M smaller than FB. Furthermore, if FB : H (LX ) → M is an (L, M)-fuzzy semi-quasi-uniformity, then FB∗ is an (L, M)-fuzzy quasi-uniformity. Proof. The first conclusion is trivial from Lemma 2.9. We show the second result and need to check that FB ∗ satisfies (FQU1)–(FQU3) if FB : H (LX ) → M is an (L, M)-fuzzy semi-quasi-uniformity. ˆ : H (LX ) → M given by (FQU1) It is easy to verify that FB 1, d = d1 , ˆ FB(d) = 0 others ˆ FB. Hence, FB ˆ FB∗ . Therefore, FB∗ (d1 ) = 1. is fuzzy normal and satisfies FB

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¯ ∗ FB, we know that FB∗ = FB ¯ ∗ . Hence, FB∗ (d ∧e) = FB ¯ ∗ (d ∧e)FB∗ (d)∧ (FQU2) From Lemma 2.8 and FB ∗ ∗ ∗ ˆ : H (LX ) → M FB (e). Conversely, we show that FB (e1 ) FB (e2 ) when e1 e2 . In fact, let e1 e2 and define FB as follows: FB∗ (e1 ) ∨ FB ∗ (e2 ), d = e2 , X ˆ ∀d ∈ H (L ), FB(d) = FB∗ (d) others. ˆ is fuzzy normal and FB ˆ FB. Hence FB ˆ FB∗ . Thus Then FB ˆ 2 ) = FB∗ (e1 ) ∨ FB ∗ (e2 ) FB∗ (e1 ). FB∗ (e2 ) FB(e Therefore, FB∗ (d ∧ e)FB ∗ (d) ∧ FB ∗ (e). (FQU3) From (FQU2), it is trivial. The readers can easily show that * defined in Lemma 2.10 is a functor from (L, M)-FHsQUnif to (L, M)-FHuQUnif. Furthermore, we have the following lemma. Lemma 2.11. ∗ is a left adjoint functor of the embedding functor i : (L, M)-FHuQUnif FHsQUnif. Hence, when L = {0, 1}, M-FQUnif is a reflective full subcategory of M-FsQUnif.

→ (L, M)-

Proof. (L, M)-FHuQUnif is a full subcategory of (L, M)-FHsQUnif is trivial. Next, let (LX , FU) be an (L, M)fuzzy semi-quasi-uniform space. Since (L, M)-FHuQUnif is topological over SET, from Lemma 2.10, we claim that the (L, M)-FHuQUnif reflection is given by idX : (LX , FU) → (LX , FU ∗ ), where FU ∗ is the biggest fuzzy normal mapping smaller than FU. Lemma 2.12 (Yue and Fang [24]). Let (LX , FU) be an (L, M)-fuzzy quasi-uniform space and define F U : LX → M as follows: ∀A ∈ LX , F U (A) = FU(d). x ∈c(LX ), x A x d(A )

Then F U is an (L, M)-fuzzy topology on X. Similarly, when U is a (2, M)-fuzzy on X, then U is an M-fuzzifying topology on X, where quasi-uniformity U : 2X → M is defined by U (U ) = x∈U x ∈f U(f ). / (X−U ) 3. The relationship between (2, M)-fuzzy quasi-uniformities and (L, M)-fuzzy quasi-uniformities The aim of this section is to study the relationship between (2, M)-fuzzy quasi-uniform spaces and (L, M)-fuzzy quasi-uniform spaces. We will show that M-FQUnif can be embedded in (L, M)-SFHuQUnif as a both reflective and coreflective full subcategory. Lemma 3.1 (Zhang [25]). The mapping L : H (2X ) → H (LX ) defined by L (f )(A) = f (supp A) satisfies: (1) f1 f2 implies L (f1 ) L (f2 ); (2) L (f1 ◦ f2 ) = L (f1 ) ◦ L (f2 ). Lemma 3.2. Let (X, U) be a (2, M)-fuzzy quasi-uniform space and L (U) : H (LX ) → M be defined by L (U)(d) = X L (f ) d U(f ) for d ∈ H (L ). Then L (U) is an (L, M)-fuzzy quasi-uniformity. Lemma 3.3. If F : (X, U1 ) → (Y, U2 ) is uniformly continuous, then F : (LX , L (U1 )) → (LY , L (U2 )) is uniformly continuous. Hence, L is a functor from M-FQUnif to (L, M)-FHuQUnif.


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Theorem 3.4. Let (X, U) be a (2, M)-fuzzy quasi-uniform space. Then L (U ) = (U ), i.e., the following diagram is commutative. (−) - M-FYS M-FQUnif L

? (L, M)-FHuQUnif

(−)

? -(L, M)-FTOP

Proof. We need to show that L (U ) (A) = (U )(A) for all A ∈ LX , i.e., U(f ) L (U ) (A) = x ∈c(LX ),x A x d(A ) d L (f )

=

x∈U

x ∈f / (X−U ) U(f ),

0 = (U )(A).

A = U , others

Firstly, we show that L (U ) (A) (U )(A) for all A ∈ LX . We can assert that L (U ) (A) = 0 if A is not a characteristic function of some U ∈ 2X . In fact, if A = U , then there exists x ∈ U such that 0 < A (x) < 1. Hence there exists ∈ c(L) such that x A . For each d ∈ H (LX ) and f ∈ H (2X ) such that x d(A ) and d L (f ), we have x L (f )(A ) = f (supp A ) . Thus x ∈ / f (supp A ). Therefore, A (x) = 0. This is contradictory to 0 < A (x) < 1. So there are no d ∈ H (LX ) and f ∈ H (2X ) such that x d(A ) and d L (f ). Hence L (U ) (A) = 0. Assume A = U and let x ∈ U and tL (U ) (A). Take ∈ c(L). Then x A . Hence, there exist d ∈ H (LX ) and f ∈ H (2X ) such / f (X − U ). Therefore, that x d(A ), d L (f ) and t U(f ). From x L (f )(A ) = f (X−U ) , we know that x ∈ t (U )(A). From the arbitrariness of t, we have L (U ) (A) (U )(A), as desired. Conversely, it suffices to show that L (U ) (U ) L (U )(U ) for all U ∈ 2X . Let tL (U )(U ) and x (U ) = / f (X − U ) and t U(f ). Now let X−U . Then x ∈ U . From tL (U )(U ), there exists f ∈ H (2X ) such that x ∈ d = L (f ), then we have x d((U ) ). Thus, t L (U ) (U ). Therefore, L (U ) (U ) (U )(U ). This completes the proof. Lemma 3.5. Let(LX , FU) be an (L, M)-fuzzy semi-quasi-uniform space and L (FU) : H (2X ) → M be defined by L (FU)(f ) = L (d) f FU(d) for f ∈ H (2X ), where L (d) : 2X → 2X is given by L (d)(U ) = supp(d(U )). Then L (FU) is a (2, M)-fuzzy semi-quasi-uniformity. Lemma 3.6. If F : (LX , FU 1 ) → (LY , FU 2 ) is uniformly continuous between (L, M)-fuzzy semi-quasi-uniform spaces, then F : (X, L (FU 1 )) → (LY , L (FU 2 )) is uniformly continuous. Lemma 3.7. L : (L, M)-FHsQUnif → M-FsQUnif is a left adjoint functor of L : M-FsQUnif → (L, M)FHsQUnif. Proof. For each (L, M)-fuzzy semi-quasi-uniform space (LX , FU), we claim that the M-FsQUnif reflection is given by idX : (LX , FU) → (LX , L ( L (FU))). In fact: (1) idX : (LX , FU) → (LX , L ( L (FU))) is uniformly continuous. It suffices to show that L ( L (FU))FU, i.e., L ( L (FU))(d)FU(d) for all d ∈ H (LX ). From the definitions of L and L , we have L (FU)(f ) = FU(e). L ( L (FU))(d) = d L (f )

d L (f ) f L (e)

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Let d L (f ) and f L (e). Then d(A)L (f )(A)L ( L (e))(A) = L (e)(supp A) = supp(e(supp A )) e(A). Hence d L (f ) f L (e) FU(e)FU(d), as desired. (2) F : (LX , FU) → (LY , L (U)) is uniformly continuous implies F : (X, L (FU)) → (Y, U) is uniformly continuous for any (2, M)-fuzzy semi-quasi-uniform space (Y, U). We need to show that U(g) L (FU)(F (g)) for all g ∈ H (2Y ). From L (F ⇐ (L (g))) = F (g), we have L (FU)(F (g))FU(F ⇐ (L (g)))L (U)(L (g)) U(g). Thus the conclusion holds.

Theorem 3.8. L : M-FQUnif → (L, M)-FHuQUnif has a left adjoint functor. Proof. From Lemmas 2.11 and 3.7, we know that the composition of the functors L : (L, M)FHuQUnif → M-FsQUnif and the reflection functor ∗ : M-FsQUnif → M-FQUnif gives the desired reflection functor. Now we use Katsaras’s approach to study the relationship between (2, M)-fuzzy quasi-uniform space and (L, M)fuzzy quasi-uniform spaces. Lemma 3.9 (Katsaras [10]). The mapping : H (2X ) → H (LX ) defined by (f )(A) = satisfies:

x∈X

A(x) ∧ f ({x})

(1) f1 f2 implies (f1 ) (f2 ); (2) (f1 ◦ f2 ) = (f1 ) ◦ (f2 ). Remark 3.10. The readers can check that (f ) defined above can be also written by (f )(A) = where A = {x| A(x) }.

∈L

∧ f (A ) ,

Lemma 3.11. Let (X, U) be a (2, M)-fuzzy quasi-uniform space and (U) : H (LX ) → M be defined by (U)(d) = X d (f ) U(f ) for d ∈ H (L ). Then (U) is an (L, M)-fuzzy quasi-uniformity. Proof. We need to check (FQU1)–(FQU3). (FQU1) is obvious. (FQU2) From the definition of (U), we know that (U)(d1 ∧d2 ) (U)(d1 )∧ (U)(d2 ). Conversely, let t (U)(d1 ) ∧ (U)(d2 ). Obviously, t (U)(d1 ) and t (U)(d2 ). Hence, there exists f1 ∈ H (2X ) such that d1 (f1 ) and t U(f1 ). Similarly, there exists f2 ∈ H (2X ) such that d2 (f 2 ) and t U(f2 ). Thus, d1 ∧d2 (f1 )∧(f2 ) (f1 ∧ f2 ) and t U(f1 ) ∧ U(f2 ) = U(f1 ∧ f2 ). Therefore, t d1 ∧d2 (f ) U(f ). From the arbitrariness of t, we have ).

(U)(d1 ∧ d2 ) (U)(d1 ) ∧ (U)(d2 (FQU3) From (FQU2), (U)(d) e◦e d (U)(e) is obvious. Let t (U)(d) =

d (f )

U(f ) =

U(g).

d (f ) g◦g f

Then there exist f, g ∈ H (2X ) such that d (f ), g ◦ g f and t U(g). Hence we have d (f )(g ◦ g) = (g)◦ (g). Thus t (U)((g)). Therefore, t e◦e d (U)(e). From the arbitrariness of t, we have

(U)(d) e◦e d (U)(e). Lemma 3.12. If F : (X, U1 ) → (Y, U2 ) is uniformly continuous, then F : (LX , (U1 )) → (LY , (U2 )) is uniformly continuous.


Proof. This can be obtained by

(U2 )(d) = U2 (g) d (g)

U1 (F (g))

U1 (F (g))

F ⇐ (d) (F (g))

d (g)

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U1 (f ) = (U)(F ⇐ (d)).

F ⇐ (d) (f )

From Lemmas 3.11 and 3.12, we know that is a functor from M-FQUnif to (L, M)-FHuQUnif. Theorem 3.13. Let (X, U) be a (2, M)-fuzzy quasi-uniform space. Then (U ) = (U ), i.e., the following diagram is commutative. (−) - M-FYS M-FQUnif

? (L, M)-FHuQUnif

(−)

? -(L, M)-FTOP

Proof. We need to show that (U ) (A) = (U )(A) for all A ∈ LX , i.e., (U ) (A) = U(f ) x ∈c(LX ),x A x d(A ) d (f )

=

U(f )

r∈p(L) x∈r (A) x ∈f / (X−r (A))

= (U )(A). Let t (U ) (A), r ∈ p(L) and x ∈ r (A). Then r A (x). Since r ∈ p(L), we have r ∈ c(L). Then there exist d ∈ H (LX ) and f ∈ H (2X ) such that xr d(A ), d (f ) and t U(f ). Now it suffices / f (X − r (A)). to show that x ∈ Since xr d(A ) and d (f ), we have xr (f )(A ), i.e., r (f )(A )(x) = x∈f ({y}) A (y). Hence, y ∈ r (A) for all y ∈ X with x ∈ f ({y}). This is to say x ∈ / f ({y}) if y ∈ / r (A). Therefore, x ∈ / f (X − r (A)). Hence t (U )(A). Thus (U ) (A) (U )(A) from the arbitrariness of t. Conversely, let t(U )(A) and xr ∈ c(LX ) such that xr A . Then there exist s ∈ c(L) such that sr and xs A . f ∈ H (2X ) such that x ∈ / f (X Thus, x ∈ s (A). Hence there exists − s (A)) and t U(f ). Now we show that xr (f )(A ), i.e., r(f )(A )(x) = x∈f ({y}) A (y). In fact, if r x∈f ({y}) A (y), then there is some y ∈ X such / f (X − s (A)), we know that y ∈ s (A), that x ∈ f ({y}) and s A (y) since sr. But, from x ∈ f ({y}) and x ∈ i.e., sA (y). This is a contradiction to s A (y). Therefore, t (U ) (A). Thus the conclusion holds. Lemma 3.14 (Katsaras [10]). Define : H (LX ) → H (2X ) as follows: ∀U ∈ 2X ,

(d)(U ) = {y ∈ X| ∃x ∈ U, A(x) d(A)(y) for all A ∈ LX }.

Then satisfies the following properties: (1) d1 d2 implies (d1 ) (d2 ); (2) (d1 ◦ d2 )(d1 ) ◦ (d2 ). Lemma 3.15. Let (LX FU) be an (L, M)-fuzzy quasi-uniform space and (FU) : H (2X ) → M be defined by (FU)(f ) = (d) f FU(d) for f ∈ H (2X ). Then (FU) is a (2, M)-fuzzy quasi-uniformity. Proof. The proof is similar to that of Lemma 3.11.

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Lemma 3.16. If F : (LX , FU 1 ) → (LY , FU 2 ) is uniformly continuous, then F : (X, (FU 1 )) → (LY , (FU 2 )) is uniformly continuous. Proof. This can be obtained by (FU 2 )(f ) = FU 2 (d)

FU 1 (F ⇐ (d))

FU 1 (F ⇐ (d))

F (f ) (F ⇐ (d))

f (d)

f (d)

FU 1 (e) = (FU 1 )(F (f )).

F (f ) (e)

From Lemmas 3.15 and 3.16, we know that is a functor from (L, M)-FHuQUnif to M-FQUnif. For and , we have one main theorem in this paper. Theorem 3.17. (1) ◦ = idM -FQUnif ; (2) ◦ id(L,M)-FHuQUnif ; (3) is a right adjoint functor of . Proof. (1) Let U be a (2, M)-fuzzy quasi-uniformity on X. We need to prove ∀f ∈ H (2X ), ◦ (U)(f ) =

U(g) = U(f ).

f (d) d (g)

On one hand, if d ∈ H (LX ) and g ∈ H (2X ) satisfy f (d) and d (g), then f (d) ◦ (g) = g. Hence U(g)U(f ). Thus ◦ (U)(f )U(f ), i.e., ◦ idM -FQUnif . On the other hand, let d = (f ), then f = ◦ (f ) = (d). Hence ◦ (U)(f )U(f ). This is to say ◦ idM -FQUnif . (2) Straightforward. (3) It suffices to prove that, given a (2, M)-fuzzy quasi-uniform space (X, U), for any (L, M)-fuzzy topological quasi-uniform space (LY , FU), if F : (X, U) → (Y, (FU)) is uniformly continuous, then there exists a unique uniformly continuous mapping F¯ : (LX , (U)) → (LY , FU) such that F = (F¯ ) ◦ idX . In fact, it is easy to verify that F : (X, U) → (Y, (FU)) is uniformly continuous if and only if F : (LX , (U)) → (LY , FU) is uniformly continuous. Hence if we put F¯ = F and (F¯ ) = F , then F = (F¯ ) ◦ idX . Therefore, is a right adjoint functor of . At the end of this section, we will show that M-FQUnif can be embedded in (L, M)-SFHuQUnif as a both reflective and coreflective full subcategory. Let FU be an (L, M)-fuzzy quasi-uniformity on X and define S(FU) : H (LX ) → M as follows: ∀d ∈ H (LX ),

S(FU)(d) =

∧i=n i=1 BF U (di ),

d ∧i=n i=1 di

where BF U (di ) : H (LX ) → M is defined by BF U (d) =

1, d = supX , FU(d) others.

Then it is easy to verify that S(FU) is the smallest stratified (L, M)-fuzzy quasi-uniformity bigger than FU and S is a functor from (L, M)-FHuQUnif to (L, M)-SFHuQUnif. Lemma 3.18. is the composition of L and S, i.e., = S ◦ L . Proof. Straightforward.


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Theorem 3.19. M-FQUnif can be embedded in (L, M)-SFHuQUnif as a both reflective and coreflective full subcategory. Proof. (1) : (L, M)-SFHuQUnif → M-FQUnif is a right adjoint functor of : M-FQUnif → (L, M)SFHuQUnif. See the proof of (3) in Theorem 3.17. (2) The composition functor ∗ ◦ L : (L, M)-SFHuQUnif → M-FQUnif is a left adjoint functor of : M-FQUnif → (L, M)-SFHuQUnif. For each stratified (L, M)-fuzzy quasi-uniform space (LX , FU ), we claim that the M-FQUnif reflection is given by idX : (LX , FU) → (LX , ◦ ∗ ◦ (FU )). In fact: Step 1: idX : (LX , FU) → (LX , ◦ ∗ ◦ (FU)) is uniformly continuous. Trivial and omitted. Step 2: F : (LX , FU) → (LY , (U)) is uniformly continuous implies F : (X, ∗ ◦ L (FU)) → (Y, U) is uniformly continuous for any (2, M)-fuzzy quasi-uniform space (Y, U). The readers can easily show that F (U) : H (2X ) → M defined by F (U)(f ) = f =F (g) U(g) for f ∈ H (2X ) is a normal mapping and satisfies F (U) L (FU). Hence from the definition of ∗, we have ∀g ∈ H (2Y ), U(g)F (U)(F (g)) ∗ ◦ L (FU)(F (g)). Therefore, F : (X, ∗ ◦ L (FU)) → (Y, U) is uniformly continuous. Thus the conclusion holds.

4. The relationship between (2, M)-fuzzy quasi-uniformities and pointwise (L, M)-fuzzy quasi-uniformities Shi [19,20] studied pointwise L-(quasi-)uniformities on fuzzy sets. The extension of Shi’s pointwise L-quasiuniformities in a Kubiak–Šostak sense was studied in [24] and was called pointwise (L, M)-fuzzy quasi-uniformity. The purpose of this section is to study the relationship between (2, M)-fuzzy quasi-uniform spaces and pointwise (L, M)fuzzy quasi-uniformities. We will show that M-FQUnif can be embedded in (L, M)-FShQUnif in two different ways. First, we recall some notions and results in [19,20,24]. Let D(LX ) denote the set of all mappings d : c(LX ) → LX such that x d(x ) for all x ∈ c(LX ). d0 is the smallest element of D(LX ), i.e., d0 (x ) = 0 for all x ∈ c(LX ). For d, e ∈ D(LX ), we define (1) d e if and only if d(x ) e(x ) for all x ∈ c(LX ), (2) (d ∨ e)(x ) = d(x ) ∨ e(x ) for all x ∈ c(LX ), (3) (d e)(x ) = {d(y )| y ∈ c(LX ), y e(x )} for all x ∈ c(LX ). Then d ∨ e ∈ D(LX ), d e ∈ D(LX ), d e d, d e e and the operations ∨ and satisfy associative law. Definition 4.1 (Yue and Fang [24]). A pointwise (L, M)-fuzzy quasi-uniformity is a mapping FU : D(LX ) → M such that (FQU1) FU(d0 ) = 1; (FQU2) FU(d ∨ e)= FU(d) ∧ FU(e) for all d, e ∈ D(LX ); (FQU3) FU(d) = ee d FU(e) for all d ∈ D(LX ). If FU is a pointwise (L, M)-fuzzy quasi-uniformity on X, the pair (LX , FU) is called a pointwise (L, M)-fuzzy quasiuniform space. A mapping F : (LX , FU) → (LY , FU 1 ) is called uniformly continuous if FU (F ⇐ (d)) FU 1 (d) for all d ∈ D(LY ), where F ⇐ (d) : c(LX ) → M is defined by F ⇐ (d)(x ) = FL← ◦ d ◦ FL→ (x ) . The category of pointwise (L, M)-fuzzy quasi-uniform spaces is denoted by (L, M)-FShQUnif. Lemma 4.2 (Yue and Fang [24]). Let (LX , FU) be a pointwise (L, M)-fuzzy quasi-uniform space and define F U : LX → M as follows: ∀A ∈ LX , F U (A) = FU(d).

x ∈c(LX ),x A d(x ) A

Then F U is an (L, M)-fuzzy topology on X.

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Lemma 4.3. The mapping C : H (2X ) → D(LX ) defined by C(f )(x ) = X−f ({x}) satisfies: (1) f1 f2 implies C(f1 ) C(f2 ); (2) C(f1 ) ∨ C(f2 ) = C(f1 ∧ f2 ); (3) C(f1 ◦ f2 ) = C(f1 ) C(f2 ); Lemma 4.4. Let (X, U) be a (2, M)-fuzzy quasi-uniform space and (U) : D(LX ) → M be defined by (U)(d) = X d C (f ) U(f ) for d ∈ D(L ). Then (U) is a pointwise (L, M)-fuzzy quasi-uniformity. Lemma 4.5. If F : (X, U1 ) → (Y, U2 ) is uniformly continuous, then F : (LX , (U1 )) → (LY , (U2 )) is uniformly continuous. Proof. This can be obtained by U2 (g) (U2 )(d) = d C (g)

U1 (F ⇐ (g))

U1 (F (g))

F ⇐ (d) C (F (g))

d C (g) ⇐

U1 (f ) = (U)(F (d)).

F ⇐ (d) C (f )

Hence is a functor from M-FQUnif to (L, M)-FShQUnif.

Lemma 4.6. Define D : D(LX ) → H (2X ) as follows: ∀U ∈ 2X ,

D(d)(U ) = {y ∈ X|∃x ∈ U, s.t., y d(x ) for all ∈ c(L)}.

Then D satisfies the following statements: (1) d1 d2 implies D(d1 ) D(d2 ); (2) D(d1 ∨ d2 ) = D(d1 ) ∧ D(d2 ); (3) D(d1 d2 )D(d1 ) ◦ D(d2 ). Lemma 4.7. Let (LX , FU) be a pointwise (L, M)-fuzzy quasi-uniform space and (FU) : H (2X ) → M be defined by (FU)(f ) = f D(d) FU(d) for f ∈ H (2X ). Then D(FU) is a (2, M)-fuzzy quasi-uniformity. Lemma 4.8. If F : (LX , FU 1 ) → (LY , FU 2 ) is uniformly continuous, then F : (X, (FU 1 )) → (LY , (FU 2 )) is uniformly continuous. Proof. This can be obtained by FU 2 (e) (FU 2 )(f ) = f D (e)

FU 1 (F ⇐ (e))

f D (e)

FU 1 (d) = (U1 )(F (f )).

FU 1 (F ⇐ (e))

F (f ) D (F ⇐ (e))

F (f ) D (d)

Hence is a functor from (L, M)-FShQUnif to M-FQUnif. For the functors and , we have the following theorem. Theorem 4.9. ◦ = idM -FQUnif . Proof. Let U be a (2, M)-fuzzy quasi-uniformity on X. We need to prove U(g) = U(f ). ∀f ∈ H (2X ), ◦ (U)(f ) = f D (d) d C (g)


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We can assert that D ◦ C(f ) = f . In fact, D ◦ C(f )({x}) = {y| y C(f )(x ) for all ∈ L} = {y| y ∈ f ({x})} = f ({x}). On one hand, if d ∈ H (LX ) and g ∈ H (2X ) satisfy f D(d) and d C(g), then f D(d) D ◦ C(g) = g. Hence U(g)U(f ). Thus ◦ (U)(f )U(f ), i.e., ◦ idM -FQUnif . On the other hand, let d = C(f ), then f = D ◦ C(f ) = D(d). Hence ◦ (U)(f )U(f ). This is to say ◦ idM -FQUnif . Corollary 4.10. M-FQUnif can be embedded in (L, M)-FShQUnif. In the following discussion of this section, we give another approach to embed M-FQUnif in (L, M)-FShQUnif. Lemma 4.11. (1) Let A : H (2X ) → D(LX ) be defined by A(f )(x ) = {y|x ∈f / ({y})} . Then A satisfies the following properties: (i) f1 f2 implies A(f1 ) A(f2 ); (ii) A(f1 ) ∨ A(f2 ) = A(f1 ∧ f2 ); (iii) A(f ◦ f ) A(f ) A(f ). (2) Let B : D(LX ) → H (2X ) be defined by B(d)(U ) = {y ∈ X|∃x ∈ U, s.t., x d(y ) for all ∈ c(L)}. Then B satisfies: (i) d1 d2 implies B(d1 ) B(d2 ); (ii) B(d1 ∨ d2 ) = B(d1 ) ∧ B(d2 ); (iii) B(d d)B(d) ◦ B(d). Proof. We only prove (iii) of (1). We need to show that A(f ◦ f )(x ) A(f ) A(f )(x ) for all x ∈ c(LX ). From A(f ◦ f )(x ) = {w| x ∈f / ◦f ({w})} and A(f ) A(f )(x ) = {z|y ∈ / f ({z})}, y∈X,x∈f ({y})

it suffices to show that {w| x ∈ / f ◦ f ({w})} ⊆ {z| y ∈ / f ({z})} for all y ∈ X with x ∈ f ({y}). Let y0 ∈ X with x ∈ f ({y0 }). For each w ∈ X such that x ∈ / f ◦ f ({w}), we have x ∈ / f ({s}) for all s ∈ f ({w}). From x ∈ f ({y0 }), we know y0 ∈ / f ({w}). Hence, w ∈ {z| y0 ∈ / f ({z})}. This completes the proof. Example 4.12. Let X ⎧ ∅, ⎪ ⎪ ⎨ X, f (U ) = {y}, ⎪ ⎪ ⎩ X,

= {x, y} and f : 2X → 2X be defined by U U U U

= ∅, = {x}, = {y}, = X.

Then f ∈ H (2X ). The readers can easily verify that A(f )(x ) = {y} and C(f )(x ) = 0X . Hence A and C are different mappings. X Lemma 4.13. (1) Let (X, U) be an (2, XM)-fuzzy quasi-uniform space and (U) : D(L ) → M be defined by (U)(d) = d A(f ) U(f ) for d ∈ D(L ). Then (U) is a pointwise (L, M)-fuzzy quasi-uniformity. (2) Let (LX , FU) be a pointwise (L, M)-fuzzy quasi-uniform space and Υ (FU) : H (2X ) → M be defined by Υ (FU)(f ) = f B(d) FU(d) for f ∈ H (2X ). Then Υ (FU) is a (2, M)-fuzzy quasi-uniformity.

Lemma 4.14. (1) If F : (X, U1 )→(Y, U2 ) is uniformly continuous, then F : (LX , (U1 )) → (LY , (U2 )) is uniformly continuous.

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(2) If F : (LX , FU 1 )→(LY , FU 2 ) is uniformly continuous, then F : (X, Υ (FU 1 )) → (LY , Υ (FU 2 )) is uniformly continuous. Hence is a functor from M-FQUnif to (L, M)-FShQUnif and Υ is a functor from (L, M)-FShQUnif to M-FQUnif. Theorem 4.15. Let (X, U) be a (2, M)-fuzzy quasi-uniform space. Then (U ) = (U ), i.e., the following diagram is commutative. (−) - M-FYS M-FQUnif

? (L, M)-FShQUnif

(−)

Proof. Similar to the proof of Theorem 3.4.

? -(L, M)-FTOP

Theorem 4.16. Υ ◦ = idM -FQUnif . Corollary 4.17. M-FQUnif can be embedded in (L, M)-FShQUnif. Question 4.18. From Theorems 3.8 and 3.17, we know that L embeds M-FQUnif in (L, M)-FHuQUnif as a reflective full subcategory and embeds M-FQUnif in (L, M)-FHuQUnif as a coreflective full subcategory. But in Corollaries 4.10 and 4.17, it is only shown that M-FQUnif can be embedded in (L, M)-FShQUnif. Then, whether M-FQUnif can be embedded in (L, M)-FShQUnif as a reflective or coreflective full subcategory? Question 4.19. From [20], we know that a functor from (L, M)-FShQUnif to (L, M)-FHuQUnif can be defined in the following way: SU(e), (SU)(d) = d (e)

(L, M)-fuzzy quasi-uniformity on X and is a mapping from D(LX ) to H (LX ) where d ∈ H (LX ), SU is a pointwise given by (e)(A) = x A e(x ) for A ∈ LX . Then, whether there is some functor from (L, M)-FHuQUnif to

(L, M)-FShQUnif which can be employed to investigate the relationship between (L, M)-FShQUnif and (L, M)FHuQUnif? References [1] [2] [3] [4] [5] [6]

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