An Approach to Fuzzy Frames Via Fuzzy Posets Wei Yao∗

Fuzzy Sets and Systems ( ) – www.elsevier.com/locate/fss

ଁ An approach to fuzzy frames via fuzzy posets Wei Yao∗

Department of Mathematics, Hebei University of Science and Technology, Shijiazhuang 050018, China

Received 10 October 2009; received in revised form 18 November 2010; accepted 19 November 2010

Abstract In this paper, based on a complete Heyting algebra L, we define a fuzzy version of frames, called L- frames. Then we construct an adjunction between the category of stratified L- topological spaces and that of L- locales (the opposite category of L- frames), which is a fuzzy counterpart of the Isbell-adjunction between topological spaces and locales. We also study the corresponding sobriety of stratified L- topologies and spatiality of L- frames and show the dual equivalence between them. © 2010 Published by Elsevier B.V.

Keywords: Topology; Adjunction; L- ordered set; (Stratiﬁed); L- topology; L- frame; L- locale; Sober space; Spatial L- frame

1. Introduction

A topological duality is a correspondence between two mathematical structures involving points and predicates such that isomorphic structures can be identified. Stone [29] firstly finds such a duality between topology and logic. In [16] Isbell gives an adjunction between the category of topological spaces with continuous functions and the opposite category of frames with frame homomorphism (which yields a duality between sober spaces and spatial frames). Stone duality in mathematical context is studied in a book of Johnstone [17]. Since topological ideas combined with those of fuzzy sets in 1968 [6], fuzzy topology has made a great progress now. It is natural to ask that whether or not it is possible to establish a category to play the same role with respect to a given notion of fuzzy topology as that locales play for the classical topological spaces. Rodabaugh is the first person to embark this question. In [21,22,25], he essentially captures the lattice theoretic behavior of the category L-Top of L-topology (not necessary stratified) by locales. The main result, as is stated in [34], is the fuzzification of the adjunction −→ pt via the introduction of L-fuzzy points of a crisp locale A, which are defined to be the frame morphisms from A to L. That is to say that he obtains an adjunction (L, Lpt):L-Top −→ Loc, where L takes every L-topological space to the locale of its open sets, and Lpt takes every locale A to the L-topological space of L-fuzzy points of A. Consequently, Rodabaugh establishes the Stone representation theorem for distributive lattices by means of this adjunction.

ଁ This paper is supported by the NNSF of China (10926055), the Foundation of Hebei Province (A2010000826, 09276158) and the Foundation of HEBUST (XL200821, QD200957). ∗ Tel.: +86 311 81668514. E-mail address: [email protected]

Please cite this article as: W. Yao, An approach to fuzzy frames via fuzzy posets, Fuzzy Sets and Systems (2010), doi: 10.1016/j.fss.2010.11.010 2 W. Yao / Fuzzy Sets and Systems ( ) –

In [34], Zhang and Liu define a kind of an L-frame by a pair (A, i A), where A is a classical frame and i A : L −→ A is a frame morphism. For a stratified L-topological space (X, ), the pair (, iX ) is one of this kind of L-frames, where iX : L −→ is a map which sends a ∈ L to the constant map with the value a. Conversely, a point of an L-frame (A, i A) is frame morphism p :(A, i A) −→ (L, idL ) satisfying p ◦ i A = idL and Lpt(A) denotes the set of all points of (A, i A). Then {x : Lpt(A) −→ L|∀p ∈ Lpt(A), x (p) = p(x)} is a stratified L-topology on Lpt(A). By these two assignments, Zhang and Liu construct an adjunction between SL-Top and L-Loc and consequently establish the Stone representation theorem for distributive lattices by means of this adjunction. Zhang and Liu [34] point out that, from the viewpoint of lattice theory, Rodabaugh’s fuzzy version of Stone representation theory just is and has nothing differ from the classical one. While in our opinion, Zhang–Liu’s L-frames preserve many features of and also seem to have no strong difference from a crisp one. In [18,20] for L a complete chain, Pultr and Rodabaugh introduce a new approach to describing L-topological spaces using categorical constructs called lattice-valued frames or also L-frames. This approach not only gives new description of previously known type of sober spaces, but also leads naturally to a new type of sober spaces not previously documented in the literature. In [13], the kind of lattice-valued frames is extended from a complete chain to a completely distributive lattice. But for the case L = 2, their 2-Frm is categorical isomorphic to Frm2, not to Frm. In this paper, firstly we aim to define an L-frame by an L-ordered set equipped with some further conditions. For explicit, since a classical frame is a complete lattice satisfying the infinite distributive law of binary meets over arbitrary joins or the meet operation has a right adjoint, by means of fuzzy Galois connections in [32],anL-frame in this paper will be a complete L-ordered set with the meet operation having a right fuzzy adjoint. We then aim to establish an adjunction between the category of stratified L-topological spaces and the category of L-locales, the opposite category of this kind of L-frames. Corresponding sobriety of stratified L-topological spaces and spatiality of L-locales are defined and their duality is obtained.

2. Preliminaries

2.1. Category theory

For category theory, we refer to [1]. For two objects A, B in a category C,wewouldliketouse[A, B]C to denote the set of all C-morphism from A to B, and by |C| the class of C-objects and by Mor(C) the class of C-morphisms. Let F : A −→ B be a functor and B ∈|B|.Apair(u, A) with A ∈|A| and u : B −→ F(A) ∈ Mor(B) is called universal for B w.r.t. F provided that for each A ∈|A| and each B-morphism f : B −→ F(A) there exists a unique A-morphism f : A −→ A such that F( f )◦u = f . Dually, A pair (A, u) with A ∈|A| and u : F(A) −→ B ∈ Mor(B) is called co-universal for B w.r.t. F provided that for each A ∈|A| and each B-morphism f : F(A) −→ B there exists a unique A-morphism f : A −→ A such that u ◦ F( f ) = f . Let F : A −→ B and G : B −→ A be two functors. F is called a left adjoint of G (or G a right adjoint of F) or (F, G) is an adjunction between A, B, in symbols FٜG : AB, if for each A ∈|A|, there exists a universal pair (u A, F(A)) w.r.t. G (or equivalently, for each B ∈|B|, there exists a co-universal pair (G(B), u B) w.r.t. F).

2.2. Lattices

In this paper, if there is no further statement, L always denotes a complete Heyting algebra with , ⊥ as the top and bottom elements respectively. Thus there is an implication operation →: L × L −→ L induced by the binary meets ∧ on L, which is given by → = { ∈ L| ∧ ≤ }(∀, ∈ L). A complete Heyting algebra is equivalent to a frame, i.e., a complete lattice satisfying the distributive law of binary meets over arbitrary joins: ∧ = ∧ ∀ ∈ , ∀ ⊆ (IDL) a S s∈S(a s)( a L S L). Properties of complete Heyting algebras can be found in lots of literatures, e.g., [17]. Clearly, a complete lattice is a frame iff ∧ (−) has a right adjoint → (−)forany ∈ L. A frame morphism is a map preserves ﬁnite meets and arbitrary joins. Note that for a frame morphism f : A −→ B,wehave f () =

Please cite this article as: W. Yao, An approach to fuzzy frames via fuzzy posets, Fuzzy Sets and Systems (2010), doi: 10.1016/j.fss.2010.11.010 W. Yao / Fuzzy Sets and Systems ( ) – 3

(note that is the meet of empty set). The category of complete lattices with frame morphisms is called the category of semi-frames [23], denoted by SFrm, which ﬁrstly appeared in [22] under the symbol CSLF.AndFrm is its full subcategory of frames. Clearly, for objects a semi-frame has no difference from a complete lattice.

2.3. Fuzzy sets

For a set X, L X denotes the set of all L-fuzzy subsets of X.For ∈ L, the constant map with the value is X denoted by X . All operations on L can be extended onto L . For example, if L is a frame, deﬁne (A ⊗ B)(x) = A(x) ⊗ B(x)(∀A, B ∈ L X , x ∈ X)for⊗∈{∧, ∨, →},thenL X is also a frame. For f : X −→ Y an ordinary map, the usual forward and backward power set operators are respectively deﬁned by f → :2X −→ 2Y , f →(A) ={f (x)|x ∈ A}(∀A ∈ 2X )and f ← :2Y −→ 2X , f ←(B) ={x ∈ X| f (x) ∈ B} (∀B ∈ Y → X −→ Y 2 ). Then their L-counterparts are fL : L L (called L-valued Zadeh function or L-forward powerset operator, ← Y −→ X → = cf. [24])and fL : L L (called L-backward powerset operator, cf. [24])by fL (A)(y) f (x)=y A(x)for ∈ X ∈ ← = ◦ ∈ Y A L and y Y ,and fL (B) B f for B L , respectively. −→ → X −→ Y ⊆ X → → ⊆ Foranordinarymap f : X Y , consider fL : L L as an ordinary map too, if L and ( fL ) ( ) ⊆ Y → −→ ← Y −→ X L , then we still use fL : to denote the restriction. Similarly, for the usual map fL : L L ,if ⊆ Y ← → ⊆ ⊆ X ← −→ L and ( fL ) ( ) L , then we still use fL : to denote the restriction.

2.4. (Stratiﬁed) L-topology

For materials related to fuzzy topology, we refer to [15]. X An L-topology on X is a family ⊆ L satisfying that (O1) X , ⊥X ∈ ; (O2) for any A, B ∈ , A ∧ B ∈ ; { | ∈ }⊆, ∈ ∈ (O3) for any Ai i I i Ai .AnL-topology on X is called stratified if it satisfies that (OS) X for any ∈ L. For an (resp., a stratified) L-topology on X, the pair (X, ) is called an (resp., a stratified) L-topological space. Amap f : X −→ Y is called continuous (resp., open) with respect to two given L-topological spaces (X, X )and , ← ∈ ∈ → ∈ ∈ (Y Y )iff fL (B) X for all B Y (resp., fL (A) Y for all A X ). The category of L-topological spaces with continuous maps is denoted by L-Top and by SL-Top its full subcategory of all stratified L-topological spaces. Amap f :(X, X ) −→ (Y, Y ) is called a homeomorphism between two L-topological spaces if f is a bijection −1 and both f :(X, X ) −→ (Y, Y ), f :(Y, Y ) −→ (X, X ) are continuous. Since for a bijection f : X −→ Y ,it → = −1 ← , −→ , is easy to show that fL ( f )L , we have that a bijection f :(X X ) (Y Y ) is a homeomorphism iff f is a categorical isomorphism in L-Top iff f is both open and continuous (cf. Theorem 5.1.2 in [23]).

2.5. L-ordered sets

Based on a (complete) Heyting algebra, Fan and Zhang [11,33] have studied quantitative domains under the frame- work of fuzzy set theory. Their approach firstly defines a fuzzy partial order on a non-empty set. Based on a (complete) residuated lattice L, in order to study fuzzy relational systems, Bˇelohlávek [2,3] has defined and studied an L-ordered set. In fact, a fuzzy partial order in the sense of Fan–Zhang (when being extended onto a residuated lattice) and an L-order in the sense of Bˇelohlávek are equivalent to each other (see Section 3 in [30]). Let P be a set and e : P × P −→ L be a map. The pair (P, e)iscalledanL-ordered set if for all x, y, z ∈ P, (E1) e(x, x) =; (E2) e(x, y) ∧ e(y, z) ≤ e(x, z); (E3) e(x, y) ∧ e(y, x) =implies x = y.

Let (P, ≤) be a classical poset. Then (P, ≤)isanL-ordered set, where ≤ is the characteristic function of ≤.Foran L-ordered set (P, e), ≤e={(x, y)| e(x, y) =}is a crisp partial order on X.If(P, e)isanL-ordered set and Q ⊆ P, then the restriction of e on Q × Q, still denoted by e (since there is no confusion will arise), is an L-order on Q.

Example 2.1. (1) Deﬁne eL : L × L −→ L by eL (x, y) = x → y for all x, y ∈ L.Then(L, eL )isanL-ordered set [2,33].

Please cite this article as: W. Yao, An approach to fuzzy frames via fuzzy posets, Fuzzy Sets and Systems (2010), doi: 10.1016/j.fss.2010.11.010 4 W. Yao / Fuzzy Sets and Systems ( ) – , ∈ X , = → (2) Let X be a set. For any S T L , the subsethood degree [12] of S in T is deﬁned by subX (S T ) x∈X S(x) X , ≤ ≤ X T (x). Then (L subX )isanL-ordered set and subX is just the pointwise order on L [2].

P Deﬁnition 2.2. Let (P, e)beanL-ordered set and S ∈ L .Anelementx0 ∈ P is called a join (resp., meet) of S,in symbols x0 =S (resp., x0 =S), if for all x ∈ P, ≤ , ≤ , (J1) S(x) e(x x0) (resp., (M1) S(x) e(x0 x)); → , ≤ , → , ≤ , (J2) y∈X S(y) e(y x) e(x0 x) (resp., (M2) y∈X S(y) e(x y) e(x x0)).

Remark 2.3. (1) It is easy to verify by (E3) that if x1, x2 are two joins (or meets) of S,thenx1 = x2. That is, each S ∈ L P has at most one join (or meet). (2) In [4,5],forT a t-norm on the unit interval [0,1], Bodenhofer introduces and studies T .E-partial orders w.r.t. a T -equivalence E.In[8,9], Demirci extends Bodenhofer’s T–E-partial orders from [0,1] to an integral commutative complete quasi-monoidal-lattice [7] (L, ∗, ≤) and then studies vague lattices for L an integral, commutative cl-monoid. Proposition 3.2 in [31] shows that an L-ordered set is exactly an L–E-poset satisfying the condition that there exists at most one join (or meet) for any L-fuzzy subsets.

The following deﬁnitions and results can be found in [2–5,7–9,11,33]. Let (P, e)beanL-ordered set and S ∈ L P . Su ∈ L P is deﬁned by ∀x ∈ P, Su(x) = S(y) → e(y, x). y∈P

Sl ∈ L P is deﬁned by ∀x ∈ P, Sl (x) = S(y) → e(x, y). y∈P

For all S ∈ L P ,supS ∈ L P (resp., inf S ∈ L P )isdeﬁnedby

∀x ∈ P, sup S = Su ∧ Sul (resp., inf S = Sl ∧ Slu).

The value Su(x) (resp., Sl (x)) can be interpreted as the degree of x being an upper (resp., a lower) bound of S.More general deﬁnitions of sups and infs of an L-fuzzy subset in L–E-ordered sets can be found in [8,9].

Proposition 2.4. Let (P, e) be an L-ordered set and S ∈ L P . Then = = ∈ , , = → , (1) x0 Siffsup S(x0) 1 iff for all x P e(x0 x) y∈P S(y) e(y x). = = ∈ , , = → , (2) x0 Siffinf S(x0) 1 iff for all x P e(x x0) y∈P S(y) e(x y).

u Proof. Some materials of the proof can be found in [2,3,11,33]. We here give a detailed proof for (1). Clearly, S (x0) = 1 ul u is equivalent to the condition (J1). And S (x0) = 1iffforanyx ∈ P, S (x) ≤ e(x0, x) iff (J2) holds. Hence x0 =S iff sup S(x0) = 1. P In the following, we only need to show that (J1) and (J2) hold for S ∈ L and x0 ∈ P,orsupS(x0) = 1ifffor u u u any x ∈ P, e(x0, x) = S (x). For the right direction, by (J2) we have e(x0, x) ≥ S (x)(∀x ∈ P). And S (x0) ≥ sup S(x0) = 1 and then for any x ∈ P, u u u S (x) = S (x0) → S (x) ≥ (S(y) → e(y, x0)) → (S(y) → e(y, x)) y∈P ≥ e(y, x0) → e(y, x) y∈P

≥ e(x0, x).

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u For the left direction, (J2) is equivalent to that S (x) ≤ e(x0, x)(∀x ∈ P). And for any x ∈ P, S(x) → e(x, x0) ≥ u S (x0) = e(x0, x0) = 1andthenS(x) ≤ e(x, x0), (J1) also holds. ç

Deﬁnition 2.5. An L-ordered set (P, e)iscalledcompleteifforallS ∈ L P , S and S exist. , = ∧ = For example, (L eL ) is a complete L-ordered set (cf. Example 3.4(1)), where S ∈L S( ) and S → ∈ L , , ≤ ∈L S( ) for all S L .If(P e) is a complete L-ordered set, then (P e) is a complete lattice, where = = ⊆ S S and S S for any S P.

Proposition 2.6 (Yao and Lu [32, Theorem 2.10 and Corollary 2.11]). For an L -ordered set (P, e), the following state- ments are equivalent: (1) (P, e) is complete. (2) For any S ∈ L P , Sexists. (3) For any S ∈ L P , Sexists.

The following proposition will be used later. −→ ∈ X → = ∧ Proposition 2.7. Let X be a set and f : X L be a map. Then for any S L , fL (S) x∈X f (x) S(x). → = → ∧ = ∧ = ∧ ç Proof. fL (S) a∈L fL (S)(a) a a∈L f (x)=a S(x) a x∈X f (x) S(x).

3. L-semi-frames and L-frames

In this section, we will introduce L-frames and L-semi-frames by means of L-ordered sets. Amap f :(P, eP ) −→ (Q, eQ) between two L-ordered sets is called monotone if for all x, y ∈ P, eP (x, y) ≤ eQ( f (x), f (y)).

Deﬁnition 3.1 (Yao [30], Yao and Lu [32]). Let (P, eP ), (Q, eQ)betwoL-ordered sets and f : P −→ Q, g : Q −→ P two monotone maps. The pair ( f, g) is called a fuzzy Galois connection between P and Q if

eQ( f (x), y) = eP (x, g(y)) for all x ∈ P, y ∈ Q,where f is called the fuzzy left adjoint of g and dually g the fuzzy right adjoint of f .

Proposition 3.2 (Yao [30], Yao and Lu [32]). Let f :(P, eP ) −→ (Q, eQ) and g :(Q, eQ) −→ (P, eP ) be two monotone maps. Then = → ∈ P (1) If P is complete, then f is monotone and has a fuzzy right adjoint if and only if f ( S) fL (S) for all S L . = → ∈ Q (2) If Q is complete, then g is monotone and has a fuzzy left adjoint if and only if g( T ) gL (T ) for all T L .

Deﬁnition 3.3. Let (P, e) be a complete L-ordered set and ∧ be the meet operation on (P, ≤e) such that for any a ∈ P, the map ∧a() = a ∧ (), b ∧a (b) = a ∧ b is monotone. We call (P, e)anL-frame if for any a ∈ P, ∧a has a fuzzy right adjoint, or equivalently, the following identity holds: ∧ =∧ → ∀ ∈ , ∀ ∈ P (FIDL) a( S) ( a)L (S)( a P S L ).

The condition (FIDL) could be called the fuzzy inﬁnite distributive law of binary meets over arbitrary (fuzzy) joins, which is the fuzzy counterpart of (IDL). Clearly, for L = 2, (FIDL)=(IDL). For an arbitrary frame L,ifS is a crisp ∧ → { ∧ | ∈ } , subset of P,then( a)L (S) turns to the ordinary subset a s s S . It follows that if (P e)isanL-frame, then (P, ≤e) is a crisp one. But the converse need not be true as the following example shows.

Example 3.4. (1) The frame L itself is an L-frame with respect to eL . (2) Let be a stratiﬁed L-topology on X.Then(, subX )isanL-frame.

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(3) Suppose that L ={⊥, , , } is a diamond lattice, that is ⊥ < , < and , . Clearly, (L, ≤)isan ordinary frame, but (L, ≤) is neither an L-frame nor a complete L-ordered set. , ∈ L = ∧ ∈ Proof. (1) Step 1: (L eL ) is complete, where for any S L ,wehave S ∈L S( ) . In fact, for any L, ⎛ ⎞ ⎝ ⎠ eL S() ∧ , = S() → ( → ) = S() → eL (, ). ∈L ∈L ∈L

Step 2: For a ∈ L,deﬁne∧a : L −→ L by ∧a() = a ∧ . On one hand, for any , ∈ L,

eL (∧a(), ∧a()) = (a ∧ ) → (a ∧ ) ≥ → = eL (, ). ∧ , −→ , ∈ L ∧ = ∧ ∧ = Then a :(L eL ) (L eL ) is monotone. On the other hand, for any S L , a( S) a ∈L S( ) ∧ ∧ ∈ ∈L S( ) a .Andforany L, ∧ → → , = → → ( a)L (S)( ) eL ( ) S( ) ( ) ∈L ∈L a∧= = ((S() ∧ a ∧ ) → ) ∈L

=∧a(S) →

= eL (∧a(S), ).

Thus, the condition (FIDL) follows from Proposition 2.4(1). , A ∈ A = A ∧ A ∧ ∈ (2) Step 1: ( subX ) is complete, where for any L , U∈( (U))X U. Obviously, U∈( (U))X U since is stratiﬁed. And for any V ∈ , ⎛ ⎞ ⎛ ⎞ ⎝ ⎠ ⎝ ⎠ subX (A(U))X ∧ U, V = (A(U))X ∧ U (x) → V (x) U∈ x∈X U∈ = ((A(U) ∧ U(x)) → V (x)) x∈X U∈ = A(U) → U(x) → V (x) U∈ x∈X = A(U) → subX (U, V ). U∈ A = A ∧ Thus U∈( (U))X U. Step 2: For any U ∈ ,let∧U : −→ be deﬁned by ∧U (V ) = U ∧ V (∀V ∈ ). On one hand, for any V1, V2 ∈ , subX (∧U (V1), ∧U (V2)) = (U(x) ∧ V1(x)) → (U(x) ∧ V2(x)) ≥ V1(x) → V2(x) = subX (V1, V2). x∈X x∈X

∧ , −→ , A ∈ ∧ A =∧ → A Then U :( subX ) ( subX ) is monotone. On the other hand, for any L ,wehave U ( ) ( U )L ( ), ∧ → −→ ∧ where ( U )L : L L is the L-forward operator of U . In fact, ⎛ ⎞ ⎝ ⎠ ∧U (A) = U ∧ (A(V ))X ∧ V = U ∧ (A(V ))X ∧ V V ∈ V ∈

Please cite this article as: W. Yao, An approach to fuzzy frames via fuzzy posets, Fuzzy Sets and Systems (2010), doi: 10.1016/j.fss.2010.11.010 W. Yao / Fuzzy Sets and Systems ( ) – 7 and for any W ∈ , ∧ → A → , = A → → ( U )L ( )(W1) subX (W1 W) (V ) (W1(x) W(x)) ∧ = ∈ W1∈ W1∈ U V W1 x X = ((A(V ) ∧ U(x) ∧ V (x)) → W(x)) x∈X V ∈ ⎛ ⎞ = ⎝ A(V ) ∧ U(x) ∧ V (x)⎠ → W(x) x∈X V ∈ = ∧U (A)(x) → W(x) x∈X

= subX (∧U (A), W).

(3) Put S = L , that is the constant map with the value .ThenS has no join in the L-ordered set (L, ≤). In fact, if = ∈ , = → , = → , = a S,thenwehaveforany b L, ≤(a b) c∈L S(c) ≤(c b) ( c∈L ≤(c b)). Put b a,wehave = , = → , ≤ , = = 1 ≤(a a) ( c∈L ≤(c a)) and c∈L ≤(c a), which implies that a ,thatis S. Now put b = ,wehave ⊥= ≤(, ) = → ≤(c, ) = →⊥=. c∈L It is impossible. ç

We call a map f :(A, eA) −→ (B, eB) between two complete L-ordered sets an L-frame homomorphism if f : , ≤ −→ , ≤ A ∈ (A eA ) (B eB ) is a frame morphism and f preserves joins of arbitrary L-fuzzy subset of A (i.e., for any A, A = → A L f ( ) fL ( )). The category of complete L-ordered sets (resp., L-frames) and L-frame homomorphisms is called the category of L-semi-frames (resp., L-frames), denoted by L-SFrm (resp., L-Frm). The opposite category is denoted by L-SLoc (resp., L-Loc). Obviously, L-Frm (resp., L-Loc) is a full subcategory of L-SFrm (resp., L-SLoc).

, −→ , ← , −→ , Proposition 3.5. Let f :(X ) (Y ) be a morphism in SL-Top. Then fL :( subY ) ( subX ) is a morphism in L-SFrm.

Proof. Obviously, f ← :(, ≤) −→ (, ≤) is a morphism in SFrm. We only need to show that f ← preserves arbitrary L L joins of L-fuzzy subsets of (, subY ). In fact, for A ∈ L ,foranyU ∈ ,wehave ← → A = A . ( fL )L ( )(U) (V ) ← = fL (V ) U Then ← → A = ← → A ∧ ( fL )L ( ) (( fL )L ( )(U))X U U∈ ⎛ ⎞ = ⎝ A(V )⎠ ∧ U ∈ f ←(V )=U U L X = (A(V ))X ∧ U ∈ ← = U fL (V ) U = A ∧ ← ( (V ))X fL (V ) V ∈

Please cite this article as: W. Yao, An approach to fuzzy frames via fuzzy posets, Fuzzy Sets and Systems (2010), doi: 10.1016/j.fss.2010.11.010 8 W. Yao / Fuzzy Sets and Systems ( ) – and ⎛ ⎞ ← A = ← ⎝ A ∧ ⎠ = ← A ∧ ← = A ∧ ← . fL ( ) fL ( (V ))Y V fL (( (V ))Y ) fL (V ) ( (V ))X fL (V ) V ∈ V ∈ V ∈ ← = ∈ ← A = ← → A ç Notice that fL ( Y ) X for all L. Hence fL ( ) ( fL )L ( ).

Let M be a distributive lattice. A map F : M −→ L is called an L-fuzzy ﬁlter of M if F(a ∧ b) = F(a) ∧ F(b)for a, b ∈ M. Denote the set of all L-fuzzy ﬁlters on M by FL (M). Obviously, for any ∈ L, M ∈ FL (M).

Lemma 3.6 (See in Zhang and Liu [34] for L-fuzzy ideals). Let :(M, ≤) −→ (L, ≤) be a monotone map. Then ∗ ∈ L M defined by ∗ (m) = (a)(∀m ∈ M) A∈[m] a∈A is the minimal L-fuzzy filter of M which is larger than or equal to (under pointwise order), where [m] ={A ⊆ M|A is finite and m ≥ A}.

∗ ∗ Proof. (1) is an L-fuzzy ﬁlter. Since it is easily shown that :(M, ≤) −→ (L, ≤) is monotone, we only need to ∗ ∗ ∗ show that for any m , m ∈ M, (m ) ∧ (m ) ≤ (m ∧ m ). In fact, 1 2 1 2 1 2 ∗ ∗ (m1) ∧ (m2) = (a) ∧ (b) A∈[m1],B∈[m2] a∈A,b∈B ≤ (c)

A∪B∈[m1∧m2] c∈A∪B ≤ (c)

C∈[m1∧m2] c∈C ∗ = (m1 ∧ m2). ∗ ∗ (2) Minimality of . Clearly, ≥ since {m}∈[m]foranym ∈ M. Suppose that F is an L-fuzzy ﬁlter of M such ∗ that F ≥ . Then for any m ∈ M, (m) = ∈ ∈ (a) ≤ ∈ ∈ F(a) = ∈ F( A) ≤ F(m) ∗ A [m] a A A [m] a A A [m] and hence ≤ F. ç

F F , A ∈ L (M) A = Proposition 3.7. ( L (M) subM ) is an L-frame, where for any L , we have ( F∈F (M) ∗ L (A(F))M ∧ F) .

∗ Proof. Step 1: Put A = ∈F (A(F))M ∧ F,wehaveA = A. Obviously A : M −→ L is monotone and by ∗ F L (M) Lemma 3.6, A ∈ FL (M)andforanym ∈ M, ∗ A(m) = A(a) = A(F) ∧ F(a).

A∈[m] a∈A A∈[m] a∈A F∈FL (M) ∗ In order to show A = A, we need to show (J1) and (J2) hold for any F0 ∈ FL (M). ∗ ∗ (J1) For all m ∈ M, A(F0) ∧ F0(m) ≤ A(m) ≤ A(m), then A(F0) ≤ F0(m) → A(m) and by the arbitrariness of ∗ m,wehaveA(F0) ≤ subM (F0, A). ∈ ∈ (J2)⎛ For any m M and any A [m],⎞ ⎛ ⎞ ⎝ ⎠ ⎝ ⎠ A(G) → subM (G, F0) ∧ A(F) ∧ F(a)

G∈FL (M) a∈A F∈FL (M) ≤ ((A(F) → subM (F, F0)) ∧ (A(F) ∧ F(a)))

a∈A F∈FL (M)

a∈A F∈FL (M) ≤ F0(a) = F0( A) ≤ F0(m). ∈ a A Then A(G) → sub (G, F ) ≤ ( A(F) ∧ F(a)) → F (m). By the arbitrariness of m and G∈FL (M) M 0 a∈A F∈FL (M) 0 A,wehave ⎛ ⎞ ⎝ ⎠ A(G) → subM (G, F0) ≤ A(F) ∧ F(a) → F0(m)

G∈FL (M) m∈M A∈[m] a∈A F∈FL (M) ⎛ ⎞ ⎝ ⎠ = A(F) ∧ F(a) → F0(m)

m∈M A∈[m] a∈A F∈FL (M) ∗ = subM (A, F0).

F ∈ F ∧ A ∈ L (M) ∈ Step 2: Let F0 L (M). Clearly,⎛ F0 is monotone. For any ⎞ L and any m M, ∧ A = ∧ ⎝ A ∧ ⎠ F0 ( )(m) F0(m) (F) F(a) A∈[m] a∈A F∈FL (M) = A(F) ∧ F(a) ∧ F0(m)

A∈[m] a∈A F∈FL (M) and ∧ → A = ∧ → A ∧ ( (( F0 )L ( )))(m) ( F0 )L ( )(G) G(a) A∈[m] a∈A G∈FL (M) = A(F) ∧ G(a)

A∈[m] a∈A F0∧F=G = A(F) ∧ F0(a) ∧ F(a).

A∈[m] a∈A F∈FL (M) Indeed, on one hand, for any A ∈ [m]wehave ⎛ ⎞ ⎝ ⎠ A(F) ∧ F0(a) ∧ F(a) ≤ F0(b) ∧ A(F) ∧ F(a)

a∈A F∈FL (M) b∈A a∈A F∈FL (M) ⎛ ⎞ ⎝ ⎠ = F0( A) ∧ A(F) ∧ F(a)

a∈A F∈FL (M) ≤ A(F) ∧ F0(m) ∧ F(a).

a∈A F∈FL (M) ∧ A ≥∧ → A ∈ By the arbitrariness of A and m,wehave F0 ( ) ( F0 )L ( ). On the other hand, for any A [m], put ={ ∨ | ∈ } = ∨ = ∈ BA m aa A ,then BA m A mand thus BA [m]. Then A(F) ∧ F(b) ∧ F0(b) = A(F) ∧ F(m ∨ a) ∧ F0(m ∨ a)

b∈BA F∈FL (M) a∈A F∈FL (M) ≥ A(F) ∧ F(a) ∧ F0(m).

a∈A F∈FL (M) ∧ A ≤ ∧ → A ∧ A ≤ ∧ → A ç Hence, ( F0 ( ))(m) ( ( F0 )L ( ))(m) and by the arbitrariness of m,wehave F0 ( ) ( F0 )L ( ).

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4. An adjunction between SL-top and L-SLoc (resp., L-loc)

For (X, ) a stratiﬁed L-topological space, since (, subX )isanL-frame, we obtain a map L : |SL-Top|−→ |L-SLoc|, (X, )(, subX ). By Proposition 3.5, such a map is a functor. That is

Proposition 4.1. The assignment

L : SL-Top −→ L-SLoc, , −→ , ← op , −→ , ( f :(X ) (Y )) (( fL ) :( subX ) ( subY )) is a functor.

Conversely, for (A, e)anL-semi-frame, put ptL (A) = [(A, e), (L, eL )]L-SFrm. We call each member of ptL (A)an L-fuzzy points of (A, e). For all a ∈ A,deﬁneL (a):ptL (A) −→ L by pp(a). Then

Proposition 4.2. L (A) =: {L (a)| a ∈ A} is a stratiﬁed L-topology on ptL (A), the corresponding topological space is denoted by PtL (A).

Proof. (O2) and (O3) are trivial since ptL (A) ⊆ [(A, ≤e), (L, ≤)]SFrm and L (a) ∧ L (b) = L (a ∧ b)and = ∧, , ≤ i L (ai ) L ( i ai ), where are taken in (A e). ∈ A = ∈ A = A ∈ = ∈ (OS) For any L, put A L and a0 : A.Then L (a0) ptL (A). In fact, for any p ptL (A), = = A = → A = ∧ A = ∧ = . L (a0)(p) p(a0) p( ) pL ( ) p(a) (a) p(a) a∈A a∈A , ≤ −→ , ≤ = = , Notice that p :(A e) (L )isanFrm-morphism and then a∈A p(a) p( A) ,where A are the top elements of (A, ≤A)and(L, ≤) respectively. ç

op Proposition 4.3. Let f :(A, eA) −→ (B, eB) be a morphism in L-SLoc, i.e., f :(B, eB) −→ (A, eA) is a morphism op in L-SFrm. Then PtL ( f ):PtL (A) −→ PtL (B), pp ◦ f is a continuous map.

Proof. For any b ∈ B and any p ∈ ptL (A), ← = = ◦ op = ◦ op = op (PtL ( f ))L ( L (b))(p) L (b)(ptL ( f )(p)) L (b)(p f ) p f (b) L ( f (b))(p) and then ← = op ∈ . ç (PtL ( f ))L ( L (b)) L ( f (b)) L (A)

By Propositions 4.2 and 4.3, we get a functor PtL : L-SLoc −→ SL-Top sending f :(A, eA) −→ (B, eB)to op PtL ( f ):PtL (A) −→ PtL (B)(pp ◦ f ). As expected, we have:

.Theorem 4.4. L ٜPtL : SL-TopL-SLoc

Proof. For any (B, eB) ∈|L-SLoc|, we know that PtL (B) ∈|SL-Top|. We will show that there exists an L-SLoc- morphism u B : L ◦ PtL (B) −→ B which is co-universal w.r.t. PtL . = op ◦ −→ ∈ −→ ◦ ∈ Step 1: Put u B L : L PtL (B) B,wehaveu B Mor(L-SLoc), i.e., L : B L PtL (B) Mor(L-SFrm). In fact, it is easily seen that L ◦ PtL (B) = (L (B), subB)andL :(B, eB) −→ (L (B), subB) ∈ Mor(SFrm). A ∈ B ∈ For any L and any p ptL (B), on one hand, A = A = → A = A ∧ L ( )(p) p( ) pL ( ) (b) p(b) ∈ b B = A(b) ∧ L (b)(p) = A(b) ∧ L (b) (p) b∈B b∈B

On the other hand, by Proposition 2.7 we have → A = A ∧ . ( L )L ( ) (b) L (b) b∈B A = → A Hence L ( ) ( L )L ( ). op Step 2: For any (X, ) ∈ SL-Top and any g : L (X, ) −→ (B, eB) ∈ Mor(L-SLoc), i.e., g :(B, eB) −→ (, subX ) ∈ Mor(L-SFrm), there is a unique f :(X, ) −→ PtL (B)suchthatu B ◦ L ( f ) = g. op (2.1) Deﬁne f : X −→ ptL (B)by f (x)(b) = g (b)(x)(∀x ∈ X, b ∈ B). Then f is a continuous map. Indeed, ∈ −→ ∈ op , ≤ −→ , ≤ ∈ (a) f is a map, i.e., for any x X, f (x):B L Mor(L-SFrm). Since g :(B eB ) ( ) Mor(SFrm), , ≤ −→ , ≤ ∈ A ∈ B we have f (x):(B eB ) (L ) Mor(SFrm). And for any L , f (x)(A) = gop(A)(x) → = ((gop) (A))(x) ⎛ L ⎞ = ⎝ op → A ∧ ⎠ ((g )L ( )(U))X U (x) ⎛U∈ ⎛ ⎞ ⎞ = ⎝ ⎝ A(b)⎠ ∧ U⎠ (x) U∈ gop(b)=U X op = (A(b))X ∧ g (b) (x) b∈B = A(b) ∧ gop(b)(x) b∈B = A(b) ∧ f (x)(b) b∈B and by Proposition 2.7, → A = A ∧ . ( f (x))L ( ) (b) f (x)(b) b∈B

Thus f (x) ∈ ptL (B). (b) f ∈ Mor(SL-Top). Indeed, for any L (b) ∈ L (B)andanyx ∈ X, ← = = = op , fL ( L (b))(x) L (b)( f (x)) f (x)(b) g (b)(x) ← = op ∈ thus fL ( L (b)) g (b) . = ← op −→ = op ◦ = (2.2) Since L ( f ) ( fL ) : L (B)andu B ( L ) , the equation u B L ( f ) g is equivalent to that ← ◦ = op ç fL L g , and this is guaranteed by (2.1)(b) and the uniqueness of f is clear.

Since for any stratiﬁed L-topological space (X, ), the pair (, subX )isanL-frame, restricted the two functors L , PtL on L-Loc,wehave

.Corollary 4.5. L ٜPtL : SL-TopL-Loc

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5. Corresponding sobriety and spatiality

The relations among different sobrieties in L-topology are discussed in [19]. Let A be a classical locale. Deﬁne

Lpt(A) ={p : A −→ L| p ∈ Frm}.

Let (X, )beanL-topological space. Deﬁne

Lptmod() ={p ∈ Lpt()|∀X ∈ , p(X ) = }.

X X For x ∈ X,deﬁne L (x):L −→ L by L (x)(U) = U(x)(∀U ∈ L ). The L-topological space (X, ) is called modiﬁed L-sober (resp., PR-L-sober) [18,19] if L : X −→ Lptmod() (resp., L : X −→ Lpt()) is a bijection. For any semi-locale A, Lpt(A)isaPR-L-sober space [18].

Deﬁnition 5.1. We call a stratiﬁed L-topological space (X, ) a sober space if L : X −→ ptL (, subX ) is a bijection.

Proposition 5.2. For any stratified L-topological space (X, ), ptL (, subX ) = Lptmod(). Thus the sobriety in Defi- nition 5.1 is equivalent to the modified L-sobriety.

Proof. If f ∈ ptL (, subX )then f :(, ≤) −→ (L, ≤)isanFrm-morphism. For ∈ L, put A = ,thatisA is the A = A ∧ = ∧ = ∧ = constant L-fuzzy subset on with the value .Then U∈( (U))X U U∈ X U X X X . And then = A = → A = A ∧ f ( X ) f ( ) fL ( ) (U) f (U) U∈ ⎛ ⎞ ⎝ ⎠ = ∧ f U = ∧ f (X ) = ∧=. U∈

Conversely, suppose that f : −→ L is an Frm-morphism and f (X ) = for all ∈ L. We only need to show that f preserves joins of arbitrary L-fuzzy subsets of (, subX ). For A ∈ L , ⎛ ⎞ ⎝ ⎠ f (A) = f (A(U))X ∧ U U∈ = f ((A(U))X ) ∧ f (U) U∈ = A(U) ∧ f (U) U∈ = → A . ç fL ( )

Proposition 5.3. Let (X, ) be a stratiﬁed L-topological space. The followings are equivalent: (1) (X, ) is a sober space; (2) L :(X, ) −→ PtL (, subX ) is a homeomorphism.

Proof. Obviously, (2) always implies (1). (1)⇒(2). Since L is a bijection, we only need to show that L :(X, ) −→ PtL (, subX ) is continuous and open. Firstly, for any U ∈ and any p = L (x) ∈ ptL ()(x ∈ X), we have → = → = = = . ( L )L (U)(p) ( L )L (U)( L (x)) U(x) p(U) L (U)(p)

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→ = ∈ ∈ ∈ ∈ Then ( L )L (U) L (U) L ( )and L is open. Secondly, for any L (U) L ( )(U )andanyx X,we have ← = ◦ = = . ( L )L ( L (U))(x) L (U) L (x) L (x)(U) U(x) ← = ∈ ç Then ( L )L ( L (U)) U and L is continuous.

Proposition 5.4. For any L-semi-locale (A, e), PtL (A) is sober.

Proof. We need to show that L : ptL (A) −→ ptL (L (A)) is an bijection. Firstly, for any p, q ∈ ptL (A), if p q, then there exists a ∈ A such that p(a) q(a). Then there exists L (a) ∈ L (A)suchthat

L (p)(L (a)) = L (a)(p) = p(a) q(a) = L (a)(q) = L (q)(L (a)).

Thus L is injective. Secondly, for any q ∈ ptL (L (A)), put p = q ◦ L , by the proof of Step 1 in Theorem 4.4, L :(A, e) −→ (L (A), subA)isanL-SFrm-morphism and then we have p ∈ ptL (A). In the following we will show that L (p) = q. In fact, for any U = L (a) with a ∈ A,wehavethat

L (p)(U) = L (a)(p) = p(a) = q(L (a)) = q(U).

Hence L is surjective. ç

Proposition 5.5. Let (A, e) be an L-locale. Then the followings are equivalent.

(1) L : A −→ L (A) is injective. (2) L :(A, e) −→ (L (A), subA) is an isomorphism in L-Frm.

Proof. Trivial since L : A −→ L (A) is surjective and L :(A, e) −→ (L (A), subA)isanL-Frm-morphism. ç

An L-locale is called spatial if it satisﬁes the conditions in Proposition 5.5.

Proposition 5.6. For any stratiﬁed L-topological space (X, ), (, subX ) is a spatial L-locale.

Proof. For U V in , there exists x ∈ X such that U(x) V (x). Put p = L (x) ∈ ptL (), we have

L (U)(p) = p(U) = L (x)(U) = U(x) V (x) = L (x)(V ) = p(V ) = L (V ).

Hence L : −→ ptL () is injective. ç

By Sob-SL-Top, we denote the full subcategory of SL-Top consisted of all sober stratiﬁed L-topological spaces. And by Spat-L-Loc we denote the full subcategory of L-SLoc consisted of all spatial L-locales. By Propositions 5.3–5.6,wehave:

Theorem 5.7. Sob-SL-Top is equivalent to Spat-L-Loc.

6. Conclusions and further work

We introduce an L-fuzzy version of frames and semi-frames, also called L-frames and L-semi-frames. Such an L-frame is a complete L-ordered sets satisfying a condition related to a fuzzy Galois connection. Then we successfully establish an adjunction between SL-Top and L-SLoc (resp., L-Loc), which is a fuzzy counterpart of the Isbell- adjunction between topological spaces and locales. We also study the corresponding sobriety of stratiﬁed L-topologies and spatiality of L-frames and show the dual equivalence between them. For generalizations of the classical Isbell-adjunction, besides Rodabaugh’s, Pultr–Rodabaugh’s, Liu–Zhang’s ap- proaches and the fuzzy-order-approach in this paper, there is another type of generalization studied by Demirci. 1 In

1 This is told by one anonymous reviewer, thanks!

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Demirci introduced an adjunction PqٜSq∗ between PCop and CGTop∗ [10, Theorem 6.6],whereC stands for the ,[10] category SQuant (resp., SSQuant, USQuant) of semi-quantales (resp., strong semi-quantales, unital semi-quantales) (cf. [26]). The objects in PC are small sources in the category C (cf. [10, Deﬁnition 5.1]), and the objects in CGTop∗ are generalized quasi-topological spaces (resp., strong generalized quasi-topological spaces, generalized topological spaces) which having the separation property T0 (cf. [10, Deﬁnitions 4.3, 4.4 and 5.6]). In [10, Corollary 6.7], Demirci ∗ ٜ showed that Pq Sq restricts to a dual equivalence between PCs and CGTop0,wherePCs is the full subcategory of ∗ PC consisted of spatial objects and CGTop0 is the full subcategory of CGTop consisted of all T0 objects. There are some further work using the approach in this paper, for example, (1) Tostudy the relations between an L-frame and that in the sense of Zhang–Liu or that in the sense of Pultr–Rodabaugh. (2) To establish an L-fuzzy version of Stone Representation and some other kinds of Stone dualities. (3) To study the difference between a crisp topology and an L-topology from the aspect of the subsethood degree operator sub. In detail, every crisp topology can be considered as an L-topology in which not only every open sets are crisp but also the subsethood degree operator sub is crisp (this means that the codomain of sub is {0, 1}). An L-topology with a crisp subset degree operator may has some similar (categorical) properties as a crisp one.

Acknowledgements

The author is grateful to the anonymous reviewers and Prof. S.E. Rodabaugh, the Area Editor, for the careful reading and constructive comments.

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