On adjunctions between Lim, SL-Top, and SL-Lim

Fuzzy Sets and Systems 182 (2011) 66 – 78 www.elsevier.com/locate/fss

On adjunctions between Lim, SL-Top, and SL-Lim夡 Lingqiang Li∗ , Qiu Jin Department of Mathematics, Liaocheng University, Liaocheng 252059, PR China Available online 14 October 2010

Abstract Consider (L , ∗, 1) be a commutative, strictly two-sided quantale with the underlying lattice L being meet-continuous. Two adjunctions, one is between limit spaces and stratified L-limit spaces and the other is between stratified L-limit spaces and stratified L-topological spaces, are established. The first adjunction can be viewed as an extension of Lowen’s adjunction between the category of topological spaces and stratified [0,1]-topological spaces. The second is an extension of an adjunction between limit spaces and (stratified) L-topological spaces established in U. Höhle and T. Kubiak (Höhle–Kubiak, Semigroup Forum (2007)). © 2010 Elsevier B.V. All rights reserved. Keywords: Commutative; Strictly two-sided quantale; Adjunction; Many valued limit space; Meet-continuous lattice

1. Introduction Let (L , ∗, 1) be a commutative, strictly two-sided quantale such that the underlying complete lattice L is meetcontinuous. A notion of stratified L-limit spaces is introduced based on the fuzzy subsethood orders on the fuzzy ˜ L ⵫˜ L between limit spaces and powerset L X and the set FsL (X ) of all the stratified L-filters on X. An adjunction  stratified L-limit spaces and an adjunction R L ⵫E L between stratified L-limit spaces and stratified L-topological spaces are established. These two adjunctions imply that the category SL-Lim of stratified L-limit spaces includes the category Lim of limit spaces as coreflective concrete subcategory; and that the category SL-Lim includes the category SL-Top of stratified L-topological spaces as reflective concrete subcategory. The adjunction  ˜ L ⵬˜ L is an extension of Lowen’s adjunction [22] between the category of topological spaces and stratified [0,1]-topological spaces; and the adjunction R L ⵫E L is an extension of the adjunction  L ⵫ L between limit spaces and (stratified) L-topological spaces in Höhle and Kubiak [12]. The contents are arranged as follows. Section 2 recalls some basic notions as preliminary, including commutative, strictly two-sided quantales, L-preorders, and Galois correspondences. Section 3 recalls the adjunction  L ⵫ L between limits spaces and stratified L-topological spaces. Section 4 introduces the notion of stratified L-limit spaces. Sections 5 and 6 present the adjunctions  ˜ L ⵬˜ L : LimSL-Lim and R L ⵫E L : SL-LimSL-Top. Section 7 summarizes the ˜ L ⵬˜ L , and R L ⵫E L . relationship between the adjunctions  L ⵫ L ,  夡 This work is partially supported by the Tianyuan Found for Mathematic of NSFC (10926044) and the Doctor Foundation of Liaocheng University (31805). ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (L. Li), [email protected] (Q. Jin). 0165-0114/$ - see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2010.10.002

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It should be noted that the paper of Fang [3] has appeared during the time that this paper is under review. The paper [3] studies stratified L-ordered convergence structures when the underlying lattice is a frame (i.e., ∗ = ∧). The stratified L-ordered convergence structures in [3] are special cases of the stratified L-convergence structures considered in this paper. More importantly, there is little overlap in the main results even in the case that ∗ = ∧. While [3] focuses on the properties of the category of stratified L-ordered convergence spaces (for ∗ = ∧), this paper is concerned with adjunctions between the categories of limit spaces, stratified L-topological spaces, and stratified L-limit spaces. 2. Preliminaries A complete lattice L is said to be meet-continuous [5] if the binary meet operation ∧ distributes over directed joins. In this paper, if not otherwise specified, (L , ∗, 1) is always a commutative, strictly two-sided quantale with the underlying lattice being meet-continuous. This is, L is a complete lattice with a top element 1 and a bottom element 0; ∗ is a binary operation on L such that (i) (L , ∗, 1) is a commutative monoid; and (ii) ∗ distributes over arbitrary joins. Since the binary operation ∗ distributes over arbitrary joins, the mapping  ∗ (−) : L −→ L has a right adjoint  → (−) : L −→ L given by →=

 { ∈ L :  ∗  ≤ }.

The binary operation → is called the residuation with respect to ∗. We collect here some basic properties of the binary operations ∗ and →. Proposition 2.1 (Hájek [7] and Höhle [8]). Let (L , ∗, 1) be a commutative, strictly two-sided quantale. Then (I1) (I2) (I3) (I4) (I5) (I6) (I7) (I8)

0 ∗  = 0 and 1 →  = ;  →  = 1 ⇔  ≤ ;  ∗ ( → ) ≤  and ( → ) ∗ ( → r ) ≤  → r ;  → ( → r ) = ( ∗ ) → r =  → ( → r ); ≤ ( → ) → ;   j ) →  =  j∈J ( j → ); ( j∈J  → ( j∈J  j ) = j∈J ( →  j );  ≤  ⇒  →  ≥  →  and  →  ≤  → .

Because the operations ∗ and → are interlocked with each other by the adjoint property  ∗  ≤  ⇐⇒  ≤  → , (L , ∗, 1) can play the role of truth-value table in a “many valued” logic, where 1 ∈ L is interpreted as true, 0 ∈ L as absurd, ∗ as the logic connective conjunction, and → as implication [2,6–8,20]. This is why L-limit space, L-topological spaces, L-preordered sets are also called many valued limit spaces, many valued topological spaces, and many valued preordered sets, respectively [6,11,19,21,28,30]. For a set X, the set L X of functions X −→ L with the pointwise order becomes a complete lattice. Each element of X X X we denote by L is called   an L-subset (or a fuzzy subset) of X. For any  ∈ L , K ⊆ L and  ∈ L,  ∗ ,  → , K and K the L-subsets defined by ( ∗ )(x) = ∗(x), ( → )(x) =  → (x), ( K)(x) = ∈K (x) and   ( K)(x) = ∈K (x). Also, we make no difference between a constant function and its value since no confusion will arise. For a crisp subset A ⊆ X , let 1 A be the characteristic function, i.e. 1 A (x) = 1 if x ∈ A and 1 A (x) = 0 if x ∈ / A. If A = {x} is a singleton, 1 A is denoted by 1x for short. Clearly, the characteristic function 1 A of a subset A ⊆ X can be regarded as a map from X to L. Definition 2.2 (Bˇelohlávek [2], Gottwald [6], and Valverde [27]). Let X be a set. A fuzzy preorder (or, an L-preorder) on X is a function R : X × X −→ L such that (1) R(a, a) = 1 for every a ∈ X (reflexivity); (2) R(a, b) ∗ R(b, c) ≤ R(a, c) for all a, b, c ∈ X (transitivity). The pair (X, R) is called an L-preordered set. An L-order-preserving function f : (X, R) −→ (Y, S) between L-preordered sets is a function f : X −→ Y such that R(a, b) ≤ S( f (a), f (b)) for all a, b ∈ X .

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Let R : L X × L X −→ L be a mapping defined by  R(A, B) = (A(x) → B(x)), x∈X

then R is an L-preorder on L X . The value R(A, B) ∈ L is the degree that A is contained in B, so the fuzzy preorder R is called fuzzy inclusion order, e.g. [2]. The L-preordered set (L X , R) is called the fuzzy powerset of X, denoted by [L X ] for short. A concrete category is a pair (A, U), where A is a category and U : A −→ Set is a faithful functor. A concrete functor F : (A, U ) −→ (B, V ) between concrete categories is a functor F : A −→ B such that U = V ◦ F. That means, F only changes the structures on the underlying sets, leaving the underlying sets and morphisms untouched. All the categories and functors considered in this paper are concrete ones. Definition 2.3 (Adámek et al. [1]). Suppose that A and B are concrete categories; F : A −→ B and G : B −→ A are concrete functors. The pair (F,G) is called a Galois correspondence if either of the following equivalent conditions holds: (1) {idY : F ◦ G(Y ) −→ Y |Y ∈ B} is a natural transformation from the functor F ◦ G to the identity functor on B; and {id X : X −→ G ◦ F(X )| X ∈ A} is a natural transformation from the identity functor on A to G ◦ F. (2) F ◦ G ≤ id in the sense that for each Y ∈ B, idY : F ◦ G(Y ) −→ Y is a B-morphism; and id ≤ G ◦ F in the sense that for each X ∈ A, id X : X −→ G ◦ F(X ) is an A-morphism. If (F, G) is a Galois correspondence, then F is a left adjoint of G (equivalently, G is a right adjoint of F) [1], hence F and G form an adjunction F⵫G : AB. 3. Limit spaces and stratified L-topological spaces For each set X, let F(X ) denote the set of filters on X. For each x ∈ X , let x˙ = {A ⊆ X | x ∈ A} denote the principal filter generated by x. A convergence structure [4,13,14,23] on X is a function T : F(X ) −→ 2 X such that (C1) x ∈ T (x) ˙ for all x ∈ X ; (C2) F ⊆ G ⇒ T (F) ⊆ T (G). The pair (X,T) is called a convergence space. A convergence space (X,T) is called a limit space if it satisfies (C3) T (F)(x) ∩ T (G)(x) ⊆ T (F ∩ G)(x). T

If x ∈ T (F), we say that F converges to x with respect to T, and write it as F → x (or, F → x if no confusion would arise). A continuous function f : (X, TX ) −→ (Y, TY ) between convergence spaces is a function f : X −→ Y such that F → x ⇒ f (F) → f (x), where f (F) is the filter on Y generated by the filterbase { f (A)|A ∈ F}. The category of convergence spaces and continuous functions is denoted by Con. The full subcategory of Con consisting of limit space is denoted by Lim. Let (X, ) be a topological space. Define e( ) : F(X ) −→ 2 X by x ∈ e( )(F) ⇐⇒ F converges to x in the topological space (X, ). Then (X, e( )) is a limit space. The correspondence (X, )(X, e( )) defines a full and faithful concrete functor e : Top −→ Lim. The functor e : Top −→ Lim has a concrete left adjoint r: Lim −→ Top given by r (X, T ) = (X, r (T )), where r (T ) = {U ⊆ X | ∀x ∈ U, F → x ⇒ U ∈ F}. Therefore, Top is a concretely reflective subcategory of Lim [23].

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An L-topology on a set X is a subset T of L X such that (LT1) 0, 1 ∈ T ;  (LT2) { j | j ∈ J } ⊆ T ⇒ j∈J  j ∈ T ; (LT3) ,  ∈ T ⇒  ∧  ∈ T . An L-topology is said to be stratified [9,11,15] if it also fulfills (LTs)  ∈ T ,  ∈ L ⇒  ∗  ∈ T. The pair (X, T) is called a (stratified) L-topological space. A continuous function f : (X, T X ) −→ (Y, TY ) between stratified L-topological spaces is a mapping f : X −→ Y such that  ◦ f ∈ T X for all  ∈ TY . The category of stratified L-topological spaces and continuous functions is denoted by SL-Top. Limit spaces and stratified L-topological spaces are extensions of topological spaces in quite different directions. However, they are closely related to each other via the Scott convergence on complete  lattices.  A filter F on a complete lattice L is Scott convergent to x ∈ L [5] if x ≤ A∈F y∈A y. Then the function L : F(L) −→ 2 L given by x ∈ L(F) ⇐⇒ F is Scott convergent to x is a convergence structure on L. Moreover, (L , L) is a limit space if and only if L is meet-continuous [25]. For every limit space (X, T), let  L (T ) = { : (X, T ) −→ (L , L) |  is continuous}.

  Then  L (T ) is a stratified L-topology on X. We observe that  ∈  L (T ) if and only if (x) ≤ A∈F y∈A (y) for all F ∈ F(X ) with x ∈ T (F). The correspondence (X, T )(X,  L (T )) defines a concrete functor  L : Lim −→ SL-Top. Conversely, for each stratified L-topological space (X, T), define  L (T) : F(X ) −→ 2 X by x ∈  L (T)(F) ⇐⇒ ∀  ∈ T, (x) ≤

 

(y).

A∈F y∈A

Then (X,  L (T)) is a limit space. The correspondence (X, T)(X,  L (T)) defines a concrete functor  L : SL-Top −→ Lim. Proposition 3.1 (Höhle [11], Höhle and Kubiak [12], and Zhang [29]). The pair ( L ,  L ) is a Galois correspondence. If L = {0, 1}, then SL-Top = Top and the adjunction  L ⵫ L reduces to the adjunction r ⵫e. 4. Stratified L-limit spaces Definition 4.1. A stratified L-filter on a set X is a mapping F : L X −→ L such that for each ,  ∈ L X and  ∈ L, (F0) (F1) (F2) (Fs)

F(0) = 0, F(1) = 1;  ≤  ⇒ F() ≤ F(); F() ∧ F() ≤ F( ∧ ); F( ∗ ) ≥  ∗ F().

The above definition is a special case of Definition 6.1.4 in Höhle [9] for ⊗ = ∧. In the presence of (F1), the condition (Fs) is equivalent to (Fs ) F( → ) ≤  → F().

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Indeed, if (Fs) holds then  ∗ F( → ) ≤ F( ∗ ( → )) ≤ F(), whence F( → ) ≤  → F(). Conversely, if (Fs ) holds then  → F( ∗ ) ≥ F( →  ∗ ) ≥ F(), whence  ∗ F() ≤ F( ∗ ). Remark 4.2. Let F : L X → L be a mapping satisfying the condition F(1) = 1 and the axiom (F2). When ∗ = ∧, Fang [3] shows that F satisfies the axioms (F1) and (Fs) if and only if it satisfies the axiom (OF1) [L X ](, ) ≤ F() → F(), ∀,  ∈ L X . Indeed, this equivalence also holds even if ∗  ∧. The proof is similar to Lemma 3.1 in [3]. This shows that the stratification condition of L-filter is compatible with the fuzzy inclusion order of L-subsets. Remark 4.3. If we take ⊗ to be ∗ in Definition 6.1.4 in [9] then we obtain the notion of stratified L-filters studied in Yao [28]. Precisely, a stratified L-filter on X in the sense of Yao is a function F : L X −→ L satisfying (F0), (F1), (Fs) and (F2 ) F() ∗ F() ≤ F( ∗ ). The set of stratified L-filters (resp., stratified L-filters in sense of Yao) on a set X is denoted by FsL (X ) (resp., Fs∗ (X )). Example 4.4. (1) Let  be an L-subset of a set X. It is easily seen if  is a characteristic function or 0 ∈ L is a prime element, then the mapping [] : L X −→ L , []() = [L X ](, ) is a stratified L-filter on X. 1 In particular, letting  be the characteristic function of the singleton {x} we get a stratified L-filter [x] : L X −→ L , [x]() = [L X ](1x , ) = (x), which is called the principal stratified L-filter generated by x. (2) Let F be a filter on a set X. The mapping E(F) : L X −→ L given by   [L X ](1 A , ) = (a),  ∈ L X E(F)() = A∈F

A∈F a∈A

is a stratified L-filter (cf. Section 4.4 in [11]). (3) Let f : X −→ Y be a function and F ∈ FsL (X ). Then the function f (F) : L Y −→ L defined by F( ◦ f ) is a stratified L-filter on Y, called the image of F under f. X

Since FsL (X ) is a subset of the fuzzy powerset L (L ) , hence, there is a natural fuzzy preorder on FsL (X ) inherited X from L (L ) . Precisely, if we let  X [FsL (X )](F, G) = [L (L ) ](F, G) = (F() → G()) ∈L X

for all F, G ∈ FsL (X ), then [FsL (X )] is an L-preordered set. Replacing the inclusion order of the subsets in the condition (C2) by the fuzzy inclusion order, we obtain the notion of L-convergence structures: Definition 4.5. A stratified L-convergence structure on X is a function lim X : FsL (X ) → L X satisfying (LC1) lim X [x](x) = 1 for every x ∈ X ; and 1 This is pointed out by an anonymous referee.

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(LC2) lim X : FsL (X ) −→ L X is L-order-preserving, that is, [FsL (X )](F, G) ∗ lim X F(x) ≤ lim X G(x) for all x ∈ X and F, G ∈ FsL (X ). The pair (X, lim X ) is called a stratified L-convergence space. If no confusion would arise, we write lim instead of lim X . For each F ∈ FsL (X ) and each x ∈ X , the value lim F(x) is the extent to which F converges to x. Remark 4.6. The axiom (LC2) can be rewritten as (OL2) : [FsL (X )](F, G) ≤ lim X F(x) → lim X G(x). Fang [3] calls a mapping lim X : FsL (X ) → L X with (LC1) and (OL2) as a stratified L-ordered convergence structure on X. Remark 4.7. For a frame (L , ∗, 1), replacing the inclusion order of the subsets in (C2) by the pointwise order in L X , Jäger [15] defined a stratified L-convergence structure on a set X as a function lim X : FsL (X ) → L X satisfying (LC1) and (LC2 ) : F ≤ G ⇒ lim F(x) ≤ lim G(x). In [28], Yao defined a stratified L-convergence structures on a set X as a function lim : Fs∗ (X ) → L X satisfying (LC1) and (LC2 ). Clearly, when ∗ = ∧, Yao’s definition of stratified L-convergence structure coincides with that of Jäger. It is easily seen that (LC2) implies (LC2 ) since F ≤ G ⇒ [FsL (X )](F, G) = 1. This shows that a stratified L-convergence structure is also a stratified L-convergence structure in Jäger’s sense. But the following example (which appeared in Example 3.7 of [17] for another purpose) shows that the converse is not true. Example 4.8 (Fang [3] and Jäger [17]). Let L be the linearly ordered frame ({0, , 1}, ∧, 1) with 0 <  < 1. Assume X = {x, y}. For each F ∈ FsL (X ) and z ∈ X , let  lim F(z) =

1, F ≥ [z], 0, others.

Then (X, lim) is a stratified L-convergence space in the sense of Jäger. However, (X, lim) is not a stratified L-convergence space in our sense. To see this, let ⎧ 1,  = 1 X , ⎪ ⎪ ⎪ ⎪ ⎨ , (x) = 1, (y)  1, F0 () = ⎪ , (x) = , ⎪ ⎪ ⎪ ⎩ 0, (x) = 0, for all  ∈ L X . Then F0 is a stratified L-filter on X. Let 0 ∈ L X be given by 0 (x) = 1 and 0 (y) = 0. Since [x] ⱕ F0 , we have that lim F0 (x) = 0. However,   [FsL (X )]([x], F0 ) = [x]() → F0 () = (x) → F0 () = , ∈L X

∈L X

whence lim[x](x) ∧ [FsL (X )]([x], F0 ) = 1 ∧  =  ⱕ 0 = lim F0 (x). Thus, (X, lim) does not satisfy (LC2).

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A stratified L-convergence space (X, lim X ) is called a stratified L-limit space if it satisfies   (LC3) ∀F, G ∈ FsL (X ), ∀x ∈ X, lim X F(x) lim X G(x) ≤ lim X (F G)(x). A function f : X −→ Y between stratified L-convergence spaces (X, lim) and (Y, lim ) is continuous if lim F(x) ≤ lim f (F)( f (x)) for each F ∈ FsL (X ) and each x ∈ X . The category of stratified L-convergence spaces and continuous functions is denoted by SL-Con. The full subcategory of SL-Con consisting of stratified L-limit spaces is denoted by SL-Lim. x : Let (X, lim) be a stratified L-convergence space in the sense of Jäger. For each x ∈ X , define a mapping Ulim X L −→ L as  x Ulim () = (lim F(x) → F()). F∈FsL (X ) x is a stratified L-filter on X. A space (X, lim) in the sense of Jäger is called Then it is easy to check that Ulim L-pretopological [15,16] if it satisfies the axiom x (Lp) lim F(x) = [FsL (X )](Ulim , F).

Proposition 4.9. A stratified L-pretopological convergence space is a stratified L-limit space. Proof. It suffices to prove that (Lp) ⇒ (LC2) and (Lp) ⇒ (LC3). (Lp) ⇒ (LC2). ∀F, G ∈ FsL (X ), ∀x ∈ X (Lp)

x [FsL (X )](F, G) ∗ lim F(x) = [FsL (X )](F, G) ∗ [FsL (X )](Ulim , F) (Lp)

x ≤ [FsL (X )](Ulim , G) = lim G(x).

(Lp) ⇒ (LC3). Similar to the case that ∗ = ∧ in Jäger [16, Section 5].  5. The adjunction x ˜ L ⵫˜i L : Lim SL-Lim In this section, we embed the category Lim in the category SL-Lim as a coreflective concrete subcategory. Example 4.4(2) shows that the correspondence FE(F) embeds the set of ordinary filters on a set X into that of stratified L-filters on X. The following proposition collects some basic properties of the embedding E which shall be needed in this paper. Proposition 5.1. Let F, G ∈ F(X ). Then (1) (2) (3) (4) (5) (6)

E(x) ˙ = [x]. For each A ⊆ X , E(F)(1 A ) = 1 ⇔ A ∈ F. For all  ∈ L X , the set { a∈A (a)|A ∈ F} ⊆ L is directed. E(F) ≤ E(G) if and only if F ⊆ G.

(E(F)) = E( (F)) for each function : X −→ Y . E(F ∩ G) = E(F) ∧ E(G).

Proof. (1) For each  ∈ L X , we have that   E(x)() ˙ = (a) = (a) = (x) = [x](). A∈x˙ a∈A

a∈{x}

(2) If E(F)(1 A ) = 1 then there exists B ∈ F such that B ⊆ A, thus A ∈ F since F is a filter on X. The converse implication is obvious. (3) Trivial. (4) The implication F ⊆ G ⇒ E(F) ≤ E(G) is obvious. Conversely, assume that E(F) ≤ E(G). For each A ∈ F, we have E(F)(1 A ) = 1, hence E(G)(1 A ) = 1 and then A ∈ G by (2). Thus, F ⊆ G by the arbitrariness of A.

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(5) This is easy since for each  ∈ L Y , E( (F))() =

 

(b) =



( f (a)) = (E(F))(),

A∈F a∈A

B∈ (F) b∈B

where the second equality holds because B ∈ (F) if and only if (A) ⊆ B for some A ∈ F. (6) The inequality E(F ∩ G) ≤ E(F) ∧ E(G) follows (4). Conversely, for each  ∈ L X , we have         (a) ∧ (b) = (x) (E(F) ∧ E(G))() = A∈F a∈A

≤





B∈G b∈B

A∈F,B∈G x∈A∪B

(x) = E(F ∩ G)(),

D∈F∩G x∈D

where the second equality follows from the meet-continuity of L.  Proposition 5.2. Let (X,T) be a limit space. Then the mapping 

lim T : FsL (X ) −→ L X , lim T F(x) =

[FsL (X )](E(F), F)

{F|x∈T (F)}

is a stratified L-limit structure on X. The above equality can be interpreted as “F converges to x iff there exists some filter F converging to x such that E(F) is contained in F”. Proof. (LC1): For each x ∈ X , since x ∈ T (x) ˙ and E(x) ˙ = [x], then we get lim T [x](x) =



[FsL (X )](E(F), [x]) ≥ [FsL (X )](E(x), ˙ [x]) = 1.

{F|x∈T (F)}

(LC2): For all F, G ∈ FsL (X ) and x ∈ X [FsL (X )](F, G) ∗ lim T F(x) = [FsL (X )](F, G) ∗



[FsL (X )](E(F), F)

{F|x∈T (F)}



=

[FsL (X )](F, G) ∗ [FsL (X )](E(F), F)

{F|x∈T (F)}



≤

[FsL (X )](E(F), G) = lim T G(x).

{F|x∈T (F)}

(LC3): For all F, G ∈

FsL (X )

lim T F(x) ∧ lim T G(x) =

and x ∈ X , we have  [FsL (X )](E(F), F) ∧

{F|x∈T (F)}

=



[FsL (X )](E(G), G)

{G|x∈T (G)}



[FsL (X )](E(F), F) ∧ [FsL (X )](E(G), G)

{F,G|x∈T (F)∩T (G)}

≤



[FsL (X )](E(F ∩ G), F ∧ G) = lim T (F ∧ G)(x),

{F∩G|x∈T (F∩G)}

where the first equality holds because of the meet-continuity of L and the inequality holds because of (C3). 

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Proposition 5.3. Suppose (X,T) and (Y, T  ) are limit spaces. If : (X, T ) −→ (Y, T  ) is continuous, then so is

: (X, lim T ) −→ (Y, lim T  ). Hence, the correspondence (X, T )(X, lim T ) defines a concrete functor  ˜ L : Lim −→ SL-Lim. Proof. For all x ∈ T (F), we have (x) ∈ T  ( (F)) by continuity of . Thus, for each x ∈ X and each F ∈ FsL (X )   lim T  (F)( (x)) = [FsL (Y )](E(G), (F)) ≥ [FsL (Y )](E( (F)), (F)) {G| (x)∈T  (G)}



=









[FsL (Y )]( (E(F)), (F)) =

{F|x∈T (F)}

≥

{F|x∈T (F)}

( (E(F))() → (F)())

{F|x∈T (F)} ∈L Y



(E(F)() → F()) =

{F|x∈T (F)} ∈L X

[FsL (X )](E(F), F) = lim T F(x),

{F|x∈T (F)}

where the first inequality holds because of the continuity of and the second equality follows from Proposition 5.1(5). Therefore lim T F(x) ≤ lim T  (F)( (x)), and : (X, lim T ) −→ (Y, lim T  ) is thus continuous.  Proposition 5.4. Let (X, lim) be a stratified L-limit space. Then T lim : F(X ) −→ 2 X , T lim (F) = {x ∈ X | lim E(F)(x) = 1} is a limit structure on X. Proof. We check that (X, T lim ) satisfies (C3) for example. Suppose that x ∈ T lim (F) ∩ T lim (G). Then lim E(F ∩ G)(x) = lim(E(F) ∧ E(G))(x) ≥ lim E(F)(x) ∧ lim E(G)(x) = 1 by Proposition 5.1(6) and (LC3), hence x ∈ T lim (F ∩ G).  Proposition 5.5. Suppose (X, lim) and (Y, lim ) are stratified L-limit spaces. If : (X, lim) −→ (Y, lim ) is continuous,  then so is : (X, T lim ) −→ (Y, T lim ). Hence, the correspondence (X, lim)(X, T lim ) defines a concrete functor ˜ L : SL-Lim −→ Lim. Proof. If x ∈ T lim (F), then lim E(F)(x) = 1 by the definition of T lim . Hence, lim E( (F))( (x)) = lim (E(F))( (x)) ≥ lim E(F)(x) = 1 



by Proposition 5.1(5) and the continuity of . Therefore, (x) ∈ T lim ( (F)), and : (X, T lim ) −→ (Y, T lim ) is thus continuous.  Lemma 5.6. Let (X,T) be a limit space and F ∈ F(X ). Then lim T E(F)(x) = 1 ⇐⇒ x ∈ T (F). Proof. By definition we have  lim T E(F)(x) = [FsL (X )](E(G), E(F)) ≤ {G|x∈T (G)}

=





{G|x∈T (G)} B∈G

If lim T E(F)(x) = 1 then   E(F)(1 B ) = 1. {G|x∈T (G)} B∈G





{G|x∈T (G)} B∈G

E(F)(1 B ).

E(G)(1 B ) → E(F)(1 B )

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Since E(F)(1 B ) is either 0 or 1, there exists some filter G such that x ∈ T (G) and E(F)(1 B ) = 1 for each B ∈ G. Hence, G ⊆ F by Proposition 5.1(2). Therefore, x ∈ T (F) by (C2). The converse implication is obvious.  ˜ L. Theorem 5.7. The pair ( ˜ L , ˜ L ) is a Galois correspondence and ˜ L is a left inverse of  Proof. It is sufficient to show that ˜ L ◦  ˜ L = id and  ˜ L ◦ ˜ L ≤ id. (1) Let (X,T) be limit a space. Then x ∈ T lim T (F) ⇔ lim T E(F)(x) = 1 ⇔ x ∈ T (F) ˜ L = id. by the above lemma. Thus, ˜ L ◦  (2) Suppose (X, lim) is a stratified L-limit space. Then for each F ∈ FsL (X ) and x ∈ X ,  [FsL (X )](E(F), F) lim T lim F(x) = {F|x∈T lim (F)}

=



lim E(F)(x) ∗ [FsL (X )](E(F), F) ≤ lim F(x),

{F|x∈T lim (F)}

where, the second equality holds because x ∈ T lim (F) implies lim E(F)(x) = 1; and the inequality follows from (LC2). Therefore,  ˜ L ◦ ˜ L ≤ id.  Furthermore, it is easily seen that  ˜ L : Lim −→ SL-Lim is an embedding and we obtain the following: Corollary 5.8. The category Lim is a concretely coreflective subcategory of SL-Lim. Remark 5.9. It is easily seen that if (L , ∗, 1) = ({0, 1}, ∧, 1), then  ˜ L = ˜ L =id. 6. The adjunction R L ⵫E L : SL-LimSL-Top In this section, we embed the category SL-Top in the category SL-Lim as a reflective concrete subcategory. Proposition 6.1. Let (X, T) be a stratified L-topological space. Then the function  limT : FsL (X ) −→ L X , limT F(x) = ((x) → F()) ∈T

is a stratified L-limit structure on X. Proof. It is routine to verify that limT satisfies (LC1)–(LC3) and the details are left to the reader.  Proposition 6.2. Let (X, lim) be a stratified L-limit space. Then the set Tlim ⊆ L X given by  ∈ Tlim

⇐⇒ (x) ∗ lim F(x) ≤ F(), ∀F ∈ FsL (X ), ∀x ∈ X

is a stratified L-topology on X. Proof. The conditions (LT1) and (LT2) are clear. To check (LT3), taking any ,  ∈ Tlim , then for all F ∈ FsL (X ) and x ∈ X , we have that ( ∧ )(x) ∗ lim F(x) = ((x) ∧ (x)) ∗ lim F(x) ≤ ((x) ∗ lim F(x)) ∧ ((x) ∗ lim F(x)) ≤ F() ∧ F() = F( ∧ ). Therefore,  ∧  ∈ Tlim .

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To check (LTs), let  ∈ L ,  ∈ Tlim . Then for all F ∈ FsL (X ) and x ∈ X ,  ∗ (x) ∗ lim F(x) ≤  ∗ F() ≤ F( ∗ ). Thus,  ∗  ∈ Tlim .  It is easily seen that the correspondence (X, T)(X, limT ) generates a concrete functor E L : SL-Top −→ SL-Lim; the correspondence (X, lim) (X, Tlim ) gives rise to a concrete functor R L : SL-Lim −→ SL-Top. Remark 6.3. (1) In the case that (L , ∗, 1) be a frame, the function limT has already appeared under the name convergence map in Stout [26]. (2) The condition  ∈ Tlim

⇐⇒ (x) ∗ lim F(x) ≤ F(), ∀F ∈ FsL (X ), ∀x ∈ X

can be interpreted as  is open if and only if for all x ∈ X and all stratified L-filters F, if x is in  and F converges to x then  must be in F. (3) If the operation ∗ is not idempotent, it is hard to check that the family Tlim is an L-topology in the sense of [12]. That is to say, we do not know whether  ∗  ∈ Tlim whenever ,  ∈ Tlim . Furthermore, even if we define an L-convergence structure as a function lim : Fs∗ (X ) −→ L X satisfying (LC1) and (LC2), we still do not know whether Tlim is an L-topology in the sense of [12]. We tend to believe the answer is negative. This is one of the reasons that we do not define stratified L-topology as in [12]. (4) The functors R L and E L are the equivalent formulations to the corresponding functors studied in [15, Sections 5 and 6]. Theorem 6.4. The pair (R L , E L ) is a Galois correspondence and R L is the left inverse of E L . Proof. It is sufficient to show that id ≤ E L ◦ R L and R L ◦ E L = id. (1) Let (X, lim) be a stratified L-limit space. Then for all x ∈ X and F ∈ FsL (X ),   limTlim F(x) = (x) → F() ≥ lim F(x) = lim F(x). ∈Tlim

∈Tlim

Thus, id ≤ E L ◦ R L . (2) Given a stratified L-topological space (X, T), let  int : L X −→ L X , int() = { ∈ T| ≤ } be the interior operator of T. For any x ∈ X , the function Fx : L X −→ L , Fx () = int()(x) is clearly a stratified L-filter on X with  limT Fx (x) = (x) → Fx () = 1. ∈T

Then for all  ∈ TlimT and x ∈ X , we have (x) = (x) ∗ limT Fx (x) ≤ Fx () = int()(x).

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Hence,  = int(), then  ∈ T. This shows that TlimT ⊆ T. Conversely, if  ∈ T, then for all x ∈ X and F ∈ FsL (X ), ⎛ ⎞  (x) → F()⎠ ≤ (x) ∗ ((x) → F()) ≤ F(). (x) ∗ limT F(x) = (x) ∗ ⎝ ∈T

Thus,  ∈ TlimT . This shows that T ⊆ TlimT . Therefore, R L ◦ E L = id.  Corollary 6.5. The category SL-Top is a reflective concrete subcategory of SL-Lim. When L = {0, 1}, the pair (R L , E L ) coincides with (r,e). Thus, the Galois correspondence (R L , E L ) is an extension of the Galois correspondence (r,e). 7. The interrelationships between the adjunctions x ˜ L ⵫˜i L , R L ⵫E L , and x L ⵫i L The adjunctions discussed in this paper are collected in the following diagram. _ _ _ L_◦e_ _ _/ Top O o_ _ _r ◦_ L _ _ _tSL-Top t: O ttttt t t t  L tttttt t tt e r EL RL t t t ttttt  L t t tt t  ttz tt ˜ L  / Lim o SL-Lim ˜ L

Since both the functor E L and the functor  ˜ L are embedding, the following theorem shows that both the adjunction  ˜ L ⵫˜ L and the adjunction R L ⵫E L are extensions of the adjunction  L ⵫ L (cf. [10–12,21,29]). ˜ L =  L and ˜ L ◦ E L =  L . Theorem 7.1. R L ◦  Proof. To show the first equality, assume that (X,T) is a limit space. Noticing that   (a) = E(F)()  ∈  L (T ) ⇐⇒ (x) ≤ A∈F a∈A

for all (F, x) with x ∈ T (F), we obtain that if  ∈  L (T ), then ⎛ ⎞   E(F)()⎠ ∗ [FsL (X )](E(F), F) ≤ F(), (x) ∗ lim T F(x) ≤ ⎝ {F|x∈T (F)}

{F|x∈T (F)}

˜ L (T ). This shows that  L (T ) ⊆ R L ◦  ˜ L (T ). hence  ∈ R L ◦  ˜ L (T ). Then for any filter F on X with x ∈ T (F) we have Conversely, suppose  ∈ R L ◦    (a), (x) = (x) ∗ lim T (E(F))(x) ≤ E(F)() = A∈F a∈A

hence,  ∈  L (T ). This shows that R L ◦  ˜ L (T ) ⊆  L (T ). ˜ L =  L . The second equality follows from that both ˜ L ◦ E L and  L are right adjoint of  L .  Therefore, R L ◦  Corollary 7.2.  ˜ L ≤ E L ◦  L and ˜ L ≤  L ◦ R L . The inequalities in the above corollary can be strict, e.g. Example 3.2 in [21].

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Corollary 7.3.  L ◦ ˜ L ≤ R L and  ˜ L ◦ L ≤ EL . Remark 7.4. Let  =  L ◦ e. Then  assigns every ordinary topological space X to the L-topological space with the same underlying set and the L-topology consisting of all the lower semicontinuous L-valued maps from X to L [12]. The functor  L ◦ e does have a left adjoint [12, Theorem 5.1]. If L is a continuous lattice, then the composition  = r ◦  L : SL-Top −→ Top is a right adjoint of . Interestingly, it is demonstrated in [12, Theorem 5.2], that if  has a right adjoint, then L must be continuous. When (L , ∗, 1) = ([0, 1], ∧, 1), the functor  (resp., ) coincides with Lowen’s omega-functor (resp., iota-functor) [22], and the adjunction  ˜ L ⵫˜ L can be understood as an extension of Lowen’s adjunction between Top and S[0, 1]-Top. Acknowledgements The authors thank the reviewers for their valuable comments and suggestions. The authors also thank Dr. Hongliang Lai and the Guest editor Prof. Dexue Zhang for suggestions on the revision of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [19] [20] [21] [22] [23] [25]

[26] [27] [28] [29] [30]

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