Algebraic representation of L-valued continuous lattices via the open filter monad
aa r X i v : . [ m a t h . GN ] D ec Algebraic representation of L -valued continuous latticesvia the open filter monad Wei Yao
School of Mathematics and Statistics, Nanjing University of Information Science andTechnology, Nanjing, China
Yueli Yue
School of Mathematical Sciences, Ocean University of China, Qingdao, China
Bin Pang
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China
Abstract
With a complete Heyting algebra L as the truth value table, we prove thatthe collections of open filters of stratified L -valued topological spaces form amonad. By means of L -Scott topology and the specialization L -order, we getthat the algebras of open filter monad are precisely L -continuous lattices. Keywords:
Complete Heyting algebra; stratified L -topology; open filtermonad; Eilenberg-Moore algebra; L -Scott topology; L -continuous lattice
1. Introduction
Monad (also called triple) is an endofunctor over a base category, togetherwith two natural transformations satisfying two coherence conditions [10].The categories of algebras arising from a monad seem to be the most naturalgeneralization of Birkhoff’s equational classes [4]. Nowadays, the monadapproach becomes a very important way to study topological structures.In [19], Manes proved that compact Hausdorff spaces can be obtainedas the Eilenberg-Moore algebras of the ultrafilter monad. Later Barr [2]showed that by going from
Set (the category of sets) to
Rel (the categoryof binary relational sets) and relaxing the axioms on the monad, and then
Preprint submitted to Advances in Mathematics December 10, 2019 he related algebras derived, it was possible to obtain all topological spacesas lax algebras of a suitable lax extension of the ultrafilter monad to
Rel .Domain Theory initiated by Dana Scott in the late seventies, from aviewpoint of pure mathematics, can be considered as a combined branch oforder/lattice theory, topology, logic, category theory and so on. In [20], DanaScott characterized the continuous lattices endowed with the Scott topologyprecisely as the spaces that are injective over all subspace embedding. Thiswas the first important result of connections between algebraic structuresand topological structure in domain theory in the literature. For anotherperspective, Alan Day [5] and Oswald Wyler [25] independently character-ized the continuous lattices as the algebras of the open filter monad on thecategory of T topological spaces. One uses the fact that the filter monad isof Kock-Z¨oberlein type, and that in any poset-enriched category with sucha monad structure, the injective objects over a certain class of embeddingsdefined in terms of monad structures are precisely the algebras. The conclu-sion follows from the fact that the embeddings associated to the filter monadare exactly topological embeddings. Escardo and Flagg [6] studied variouskinds of injective spaces over different kinds of topological embeddings.Quantitative domains are extensions of classical domains by generaliz-ing ordered relations to other more general structures, such as enrichedcategories, generalized metrics and lattice-valued fuzzy ordered relations.Lattice-valued fuzzy order approach to domain theory is originated by Fanand Zhang [7, 29], and then mainly developed by Yao [26, 27], Zhang [12, 13,14, 15, 30], Li Qingguo [8, 16, 21, 22, 23], Zhao [17, 18, 24], etc.In [28], Yao showed that frame-valued continuous lattices are categoricalisomorphic to injective lattice-valued T spaces. The aim of this paper is togeneralize Day’s result to lattice-valued setting to show that frame-valuedcontinuous lattices are exactly the algebras of the open filter monad on thecategory of T frame-valued topological spaces.
2. Preliminaries
We refer to [9] for the contents of lattice theory.A poset L is called a complete lattice if the supermum W S exists for each S ⊆ L , or equivalently, the infimum V T exists for each T ⊆ L . For the caseof ∅ , W ∅ and V ∅ are the least and the greatest elements, denoted by 0 and1, respectively. 2 efinition 2.1. A complete lattice L is called a frame, or a complete Heyt-ing algebra, if it satisfies the first infinite distributive law, that is, a ∧ _ S = _ s ∈ S a ∧ s ( ∀ a ∈ L, ∀ S ⊆ L ) . For a frame L , the related implication operation → : L × L −→ L is givenby a → b = _ { c ∈ L | a ∧ c ≤ b } ( ∀ a, b ∈ L )Then we get an adjoint pair ( ∧ , → ) satisfying that c ≤ a → b ⇐⇒ a ∧ c ≤ b ( ∀ a, b, c ∈ L ) . In this paper, L always denotes a frame. Proposition 2.2.
For all a, b, c ∈ L, { a i | i ∈ I } , { c i | i ∈ I } , { b j | j ∈ J } ⊆ L , it holds that (1) a → b = 1 ⇐⇒ a ≤ b ; (2) 1 → a = a ; (3) a ∧ ( a → b ) = a ∧ b ; (4) b ≤ a → ( a ∧ b ) , a ≤ ( a → b ) → b ; (5) ( a → b ) ∧ ( b → c ) ≤ a → c ; (6) ( W i a i ) → b = V i ( a i → b ) ; (7) a → ( V j b j ) = V j ( a → b j ) ; (8) ( V i a i ) → ( V i c i ) ≥ V i ( a i → c i ) ; (9) ( W i a i ) → ( W i c i ) ≥ V i ( a i → c i ) ; (10) ( c → a ) → ( c → b ) ≥ a → b ; (11) ( a → c ) → ( b → c ) ≥ b → a ; (12) a → ( b → c ) = ( a ∧ b ) → c = b → ( a ∧ c ) ; (13) b → c ≤ ( a ∧ b ) → ( a → c ) . Example 2.3. (1)
Every finite distributive lattice is a frame. (2)
The unit interval [0 , is a frame. (3) For every ordinary topological space ( X, T ) , the pair ( T , ⊆ ) is a frame. .2. L -subsets and L -valued topological spaces We refer to [11] for the contents of L -subsets and L -valued topologicalspaces.Let X be a set. Every mapping A : X −→ L is called an L - subset of X ,denoted by A ∈ L X . For an element a ∈ L , the notation a X denotes theconstant L -subset of X with the value a , that is, a X ( x ) = a ( ∀ x ∈ X ).Let f : X −→ Y be a mapping between two sets. Define f → L : L X −→ L Y and f ← L : L Y −→ L X respectively by f ← L ( B ) = B ◦ f, f → L ( A )( y ) = _ f ( x )= y A ( x ) . Definition 2.4.
A subfamily O ( X ) ⊆ L X is called an L -valued topology if (O1) A, B ∈ O ( X ) implies A ∧ B ∈ O ( X ) ; (O2) { A j | j ∈ J } ⊆ O ( X ) implies W j A j ∈ O ( X ) ; (O3) a X ∈ O ( X ) for any a ∈ L .The pair ( X, O ( X )) is called an L -valued topological space. Let O ( X ) be an L -valued topology. A subfamily B ⊆ O ( X ) is called a base of X if for every A ∈ O ( X ), there exists { ( B j , a j ) | j ∈ J } ⊆ O ( X ) × L such that A = W j B j ∧ ( a j ) X . An L -valued topological space X is called T if A ( x ) = A ( y ) ( ∀ A ∈ O ( X )) implies x = y .A mapping f : ( X, O ( X )) −→ ( Y, O ( Y )) between two L -valued topolog-ical spaces is called continuous if f ← L ( B ) ∈ O ( X ) for each B ∈ O ( X ). If Y has a base B , then it is routine to show that f : ( X, O ( X )) −→ ( Y, O ( Y )) iscontinuous iff f ← L ( B ) ∈ O ( X ) for every B ∈ B .Let L - Top denote the category of all T L -valued topological spaces andcontinuous mappings as morphisms. We refer to [1, 10] for contents of category theory.A functor F : A −→ B between two categories is an assignment sendingevery A -object A to a B -object F ( B ) and every A -morphism f : A −→ A ′ to a B -morphism F ( f ) : F ( A ) −→ F ( A ′ ), which preserves composition andidentity morphisms. For any category A , there is an identity functor id A sending every f : A −→ A ′ to itself.Let F, G : A −→ B be two functors. A natural transformation τ from F to G (denoted by τ : F −→ G ) is a function that assigns to each A -object A a4 -morphism τ A : F ( A ) −→ G ( A ) such that the following diagram commutes,that is, for each A -morphism f : A −→ A ′ , G ( f ) · τ A = τ A ′ · F ( f ). G ( A ) G ( A ′ ) G ( f ) / / F ( A ) G ( A ) τ A (cid:15) (cid:15) F ( A ) F ( A ′ ) F ( f ) / / F ( A ′ ) G ( A ′ ) τ A ′ (cid:15) (cid:15) A monad over a category X is a triple ( T, η, µ ) consisting of a functor T : X −→ X and natural transformations η : id X −→ T and µ : T ◦ T −→ T such that the following diagrams commute, that is, µ · T µ = µ · µT and µ · ηT = id T = µ · T η . T T T µ / / T T TT T µT (cid:15) (cid:15) T T T T T
T µ / / T TT µ (cid:15) (cid:15) T T T ηT / / T T T o o T η
T TT µ (cid:15) (cid:15) T TT id T (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄ id T (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ Given a monad (
T, η, µ ) over X , a T -algebra (or an Eilenberg-Moorealgebra) is a pair ( X, r ), where X is an X -object and the structured morphism r : T ( X ) −→ X satisfies, r · T ( r ) = r · µ X and id X = r · η X . T ( X ) X r / / T T ( X ) T ( X ) µ X (cid:15) (cid:15) T T ( X ) T ( X ) T ( r ) / / T ( X ) X r (cid:15) (cid:15) X T ( X ) η X / / X X id X (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄ T ( X ) X r (cid:15) (cid:15) L -ordered sets We refer to [ ?
26, 27] for the contents of L -ordered sets. Definition 2.5.
A mapping e : X × X −→ L is called an L -order if (E1) e ( x, x ) = 1 ; (E2) e ( x, y ) ∧ e ( y, z ) ≤ e ( x, z ) ; (E3) if e ( x, y ) ∧ e ( y, x ) = 1 , then x = y .The pair ( X, e ) is called an L -ordered set. A l , A u ∈ L X respectively by A l ( x ) = ^ y ∈ X A ( y ) → e ( x, y ) and A u ( x ) = ^ y ∈ X A ( y ) → e ( y, x ) , and define ↑ x and ↓ x respectively by ↑ x ( y ) = e ( x, y ) and ↓ x ( y ) = e ( y, x ). An L -subset S ∈ L X is called a lower set (resp., upper set ) if S ( x ) ∧ e ( y, x ) ≤ S ( y )(resp., S ( x ) ∧ e ( x, y ) ≤ S ( y )) for all x, y ∈ X . Clearly, A l and ↓ x (resp., A u and ↑ x ) are lower (resp., upper) sets for all A ∈ L X and x ∈ X . Definition 2.6. (1)
An element x is called a supremum of A ∈ L X , insymbols, x = ⊔ A , if e ( x, y ) = A u ( y ) ( ∀ y ∈ X ) . (2) An element x is called an infimum of A ∈ L X , in symbols, x = ⊓ A ,if e ( y, x ) = A l ( y ) ( ∀ y ∈ X ) . Clearly, ⊔↓ x = x = ⊓↑ x for every x ∈ X . It is easy to show the followingproposition. Proposition 2.7. (Yao Specialization order) (1) If A ≤ B ≤ C and ⊔ A = ⊔ C = x , then ⊔ B = x . (2) V z ∈ X e ( x, z ) → e ( y, z ) = e ( y, x ) . (3) V z ∈ X e ( z, x ) → e ( z, y ) = e ( x, y ) . Definition 2.8. An L -ordered set ( X, e ) is called complete if every L -subsethas a supremum, or equivalently, every L -subset has an infimum. The equivalence in Definition 2.8 is because ⊔ A = ⊓ A u , ⊓ A = ⊔ A l holdfor A ∈ L X in an L -ordered set X . Example 2.9. (1)
Let ( X, O ( X )) be a T L -valued topological space. Define e O ( X ) : X × X −→ L by e O ( X ) ( x, y ) = ^ A ∈O ( X ) A ( x ) → A ( y ) ( ∀ x, y ∈ X ) . Then e O ( X ) is an L -order on X , called the specialization L -order of ( X, O ( X )) . (2) Define a mapping e L : L × L −→ L by e L ( x, y ) = x → y . Then ( L, e L ) is a complete L -ordered set, where for A ∈ L L , ⊔ A = _ a ∈ L a ∧ A ( a ) , ⊓ A = ^ a ∈ L A ( a ) → a. et S, A ∈ L X be two L -subsets. By considering S as a mapping from X to L , it is easy to verify that ⊔ S → L ( A ) = W x ∈ X S ( x ) ∧ A ( x ) . (3) Let X be a nonempty set. Define a mapping sub X : L X × L X −→ L by sub X ( A, B ) = ^ x ∈ X A ( x ) → B ( x ) . Then ( L X , sub X ) is a complete L -ordered set, where for A ∈ L ( L X ) , ⊔A = _ A ∈ L X A ∧ A ( A ) , ⊓A = ^ A ∈ L X A ( A ) → A. If the background set is clear, then we always drop the subscript in sub X tobe sub . In a complete L -ordered set ( X, e ), there are a top element ⊤ and abottom element ⊥ such that e ( x, ⊤ ) = e ( ⊥ , x ) = 1 for every x ∈ X [ ? ]. Remark 2.10. (1) x = ⊔ A iff e ( x, y ) = sub( ↓ y, A ) ( ∀ y ∈ X ) iff (J1) A ( y ) ≤ e ( y, x ) ( ∀ y ∈ X ) ; (J2) V z ∈ X A ( z ) → e ( z, y ) ≤ e ( x, y ) ( ∀ y ∈ X ) . (2) x = ⊓ A iff e ( y, x ) = sub( A, ↑ y ) ( ∀ y ∈ X ) iff (M1) A ( y ) ≤ e ( x, y ) ( ∀ y ∈ X ) ; (M2) V z ∈ X A ( z ) → e ( y, z ) ≤ e ( y, x ) ( ∀ y ∈ X ) . (3) A ( x ) ≤ e ( x, ⊔ A ) ∧ e ( ⊓ A, x ) ( ∀ x ∈ X ) , that is, A ≤ ↑ ( ⊓ A ) ∧ ↓ ( ⊔ A ) . (4) Let f : X −→ Y be a mapping between sets. Then sub( f → L ( A ) , B ) =sub( A, f ← L ( B )) for all A ∈ L X , B ∈ L Y .2.5. Open filters of L -valued topological spaces Definition 2.11.
Let ( X, O ( X )) be an L -valued topological space. An openfilter of X is a mapping u : O ( X ) −→ L satisfying that (F1) u ( A ∧ B ) = u ( A ) ∧ u ( B ) ( ∀ A, B ∈ O ( X )) ; (F2) u ( a X ) ≥ a ( ∀ a ∈ L ) .Let Φ L ( X ) denote the set of all open filters on X . Example 2.12.
Let ( X, O ( X ) be an L -valued topological space. (1) For A ∈ L X , define [ A ] : O ( X ) −→ L by [ A ]( B ) = sub( A, B ) ( ∀ B ∈O ( X )) . Then [ A ] ∈ Φ L ( X ) . For x ∈ X , Define [ x ] : O ( X ) −→ L by [ x ]( B ) = B ( x ) ( ∀ B ∈ O ( X )) .Then [ x ] ∈ Φ L ( X ) , called the pointed filter of x , which is an L -valued case ofthe open neighborhood system of x . Proposition 2.13.
Let X be an L -valued topological space. For every u ∈ Φ L ( X ) , it holds that: for every B ∈ O ( X ) , (F3) u ( B ) = W A ∈O ( X ) u ( A ) ∧ sub( A, B ) ; (F3 ′ ) u = W A ∈O ( X ) u ( A ) ∧ [ A ] . (F4) u ( B ) = V A ∈O ( X ) sub( B, A ) → u ( A ) ; (F4 ′ ) u ( B ) = sub([ B ] , u ) . Proof. (F3) ⇐⇒ (F3 ′ ) is clear. Firstly, _ A ∈O ( X ) u ( A ) ∧ sub( A, B ) ≥ u ( B ) ∧ sub( B, B ) = u ( B ) . Secondly, for every A ∈ O ( X ), it is easily seen that A ∧ (sub( A, B )) X ≤ B and then u ( A ) ∧ sub( A, B ) ≤ u ( A ) ∧ u ((sub( A, B )) X ) = u ( A ∧ (sub( A, B )) X ) ≤ u ( B ) . Hence, u ( B ) = W A ∈O ( X ) u ( A ) ∧ sub( A, B ).(F4) ⇐⇒ (F4 ′ ) is clear. Firstly, ^ A ∈O ( X ) sub( B, A ) → u ( A ) ≤ sub( B, B ) → u ( B ) = u ( B ) . Secondly, for every A ∈ O ( X ), by the proof of (F3), u ( B ) ∧ sub( B, A ) ≤ u ( A )and then u ( B ) ≤ sub( B, A ) → u ( A ). Hence, u ( B ) = V A ∈O ( X ) sub( B, A ) → u ( A ). (cid:3) Proposition 2.14.
Let
A, B ∈ L X . If A ≤ B , then sub( B, C ) ≤ sub( A, C ) for each C ∈ L X and so [ B ] ≤ [ A ] . Proof.
This is trivial since sub(
A, B ) = 1. (cid:3)
For an L -valued topological space ( X, O ( X )), we can define the conver-gence of open filters as a mapping lim X : Φ L ( X ) × X −→ L by lim X u ( x ) =sub X ([ x ] , u ), that is, lim X u ( x ) = V A ∈O ( X ) A ( x ) → u ( A ).8 . The open filter monad of L -valued topological spaces In this section will establish the open filter monad for L -valued topologicalspaces.Let X be an L -valued topological space. For every A ∈ O ( X ), define φ ( A ) : Φ L ( X ) −→ L by φ ( A )( u ) = u ( A ). Equipping Φ L ( X ) with an L -valued topology generated by { φ ( A ) | A ∈ O ( X ) } as a base, denoted by O (Φ L ( X )). Clearly, g : Φ L ( X ) −→ Φ L ( Y ) is continuous iff g ← L ( φ ( B )) ∈O (Φ L ( X )) ( ∀ B ∈ O ( Y )).For Φ L ( X ), α ∈ Φ L ( X ) means an open filter α : O (Φ L ( X )) −→ L . For A ∈ O ( X ), the mapping φ ( φ ( A )) : Φ L ( X ) −→ L is given by φ ( φ ( A ))( α ) = α ( φ ( A )). Proposition 3.1.
Let B be a base of an L -topological space ( X, O ( X )) .Then For every A ∈ O ( X )) and every x ∈ X , it holds that A ( x ) = _ B ∈B B ( x ) ∧ sub( B, A ) . Proof.
Firstly, for every B ∈ B and every x ∈ X , we have B ( x ) ∧ sub( B, A ) ≤ A ( x ) and then A ( x ) ≥ W B ∈B B ( x ) ∧ sub( B, A ) . Secondly, there exists { ( B j , a j ) } j ∈ J ⊆ B × L such that A = W j B j ∧ ( a j ) X .For each j ∈ J , a j ≤ sub( B j , A ) and then ( B j ∧ ( a j ) X )( x ) = B j ( x ) ∧ a j ≤ B j ( x ) ∧ sub( B j , A ). Thus A ( x ) ≤ W B ∈B B ( x ) ∧ sub( B, A ) . Hence, A ( x ) = W B ∈B B ( x ) ∧ sub( B, A ) . (cid:3) Let f : X −→ Y be a mapping. Define Φ L f : Φ L ( X ) −→ Φ L ( Y ) by(Φ L f )( u )( B ) = u ( f ← L ( B )) . It is easily shown that Φ L f is a well-defined mapping. Lemma 3.2. (1)
The space Φ L ( X ) is T . (2) If f : X −→ Y is a continuous mapping, then so is Φ L f : Φ L ( X ) −→ Φ L Y .Consequently, Φ L can be considered as an endofunctor on L - Top . roof. (1) Let u, v ∈ Φ L ( X ) and W ( u ) = W ( v ) for all W ∈ O (Φ L ( X )).Then φ ( A )( u ) = φ ( A )( v ) for all A ∈ O ( X ). That is, u ( A ) = v ( A ) for all A ∈ O ( X ), and so u = v . Therefore, Φ L ( X ) is T .(2) For every B ∈ O ( Y ) and every u ∈ Φ L ( X ). Then(Φ L f ) ← L ( φ ( B ))( u ) = ( φ ( B ))((Φ L f )( u ))= (Φ L f )( u )( B )= u ( f ← L ( B ))= φ ( f ← L ( B ))( u ) . Therefore, (Φ L f ) ← L ( φ ( B )) = φ ( f ← L ( B )) ∈ O (Φ L ( X )). Hence, Φ L f : Φ L ( X ) −→ Φ L ( Y ) is continuous. (cid:3) Define η X : X −→ Φ L ( X ) by η X ( x ) = [ x ] ( ∀ x ∈ X ). Lemma 3.3.
Let X be an L -valued topological space. Then for each U ∈O ( X ) , it holds that ( η X ) ← L ( φ ( U )) = U . Proof.
For each x ∈ X ,( η X ) ← L ( φ ( U ))( x ) = φ ( U )( η X ( x )) = φ ( U )([ x ]) = [ x ]( U ) = U ( x ) , as desired. (cid:3) Theorem 3.4. η : id L - Top −→ Φ L is a natural transformation. Proof. (1) For every U ∈ O ( X ), by Lemma 3.3, we have ( η X ) ← L ( φ ( U )) = U ∈ O ( X ) and then η X : X −→ Φ L ( X ) is continuous.(2) Let f : X −→ Y be a continuous mapping. We need to show that η Y · f = (Φ L f ) · η X . In fact, for every x ∈ X and every B ∈ O ( Y ), we have(Φ L f ) · η X ( x )( B ) = (Φ L f )([ x ])( B ) = [ x ]( f ← L ( B )) = f ← L ( B )( x ) = B ( f ( x )) =[ f ( x )]( B ) = η Y · f ( x )( B ) . (cid:3) Let α ∈ Φ L ( X ), define µ X ( α )( A ) = α ( φ ( A )) ( ∀ A ∈ O ( X )). Theorem 3.5. µ X ( α ) ∈ Φ L ( X ) for every α ∈ Φ L ( X ) . Proof.
For all
A, B ∈ O ( X ), µ X ( α )( A ) ∧ µ X ( α )( B ) = α ( φ ( A )) ∧ α ( φ ( B ))= α ( φ ( A ) ∧ φ ( B ))= α ( φ ( A ∧ B ))= µ X ( α )( A ∧ B ) . For all a ∈ L , µ X ( α )( a X ) = α ( φ ( a X )) ≥ α ( a Φ L ( X ) ) ≥ a . Notice here φ ( a X )( u ) = u ( a X ) ≥ a ( ∀ u ∈ Φ L ( X )) and then φ ( a X ) ≥ a Φ L ( X ) . (cid:3) heorem 3.6. µ : Φ L −→ Φ L is a natural transformation. Proof.
Step 1. For every L -valued topological space X and every A ∈O ( X ), we have ( µ X ) ← L ( φ ( A ))( α ) = φ ( A )( µ X ( α ))= µ X ( α )( A )= α ( φ ( A ))= φφ ( A )( α ) ∈ O (Φ L ( X )) . Hence, µ X : Φ L ( X ) −→ Φ L ( X ) is a continuous mapping.Step 2. Let f : X −→ Y be a mapping. We need to prove that µ Y · (Φ L f ) = Φ L f · µ X . For every α ∈ Φ L ( X ), for every B ∈ O ( Y ), for every u ∈ Φ L ( Y ), by the proof of Lemma 3.2(2), we have (Φ L f ) ← L ( φ ( B )) = φ ( f ← L ( B )),and then ( µ Y ) · (Φ L f )( α )( B ) = (Φ L (Φ L f ))( α )( φ ( B ))= α ((Φ L f ) ← L ( φ ( B )))= α ( φ ( f ← L ( B )))= µ X ( α )( f ← L ( B ))= Φ L f · µ X ( α )( B ) . Hence, µ Y · (Φ L f ) = Φ L f · µ X . (cid:3) Theorem 3.7. (Φ L , η, µ ) is a monad over L - Top . Proof. (1) µ ◦ (Φ L µ ) = µ ◦ ( µ Φ L ), that is, µ · (Φ L µ X ) = µ · ( µ Φ L ( X ) ) for every T L -valued topological space, where Φ L µ X : Φ L (Φ L ( X )) −→ Φ L (Φ L ( X ))and µ Φ L ( X ) : Φ L (Φ L ( X )) −→ Φ L (Φ L ( X )). Notice that for every α ∈ Φ L ( X ),( µ X ) ← L ( φ ( U ))( α ) = φ ( U )( µ X ( α )) = µ X ( α )( U ) = φφ ( U )( α ) . Then ( µ X ) ← L ( φ ( U )) = φφ ( U ) and for every Ξ ∈ Φ L (Φ L ( X )), we have µ [(Φ L µ X )(Ξ)]( U ) = µ [( µ X ) → L (Ξ)]( U )= ( µ X ) → L (Ξ)( φ ( U ))= Ξ(( µ X ) ← L ( φ ( U )))= Ξ( φφ ( U ))= ( µ Φ X )(Ξ)( φ ( U ))= µ [( µ Φ X )(Ξ)]( U ) . µ · (Φ L µ X ) = µ · ( µ Φ L ( X ) ).(2) µ ◦ η Φ L = id Φ L = µ ◦ Φ L η , that is, µ ( η Φ L ( X ) )( u ) = u = µ (Φ L η )( u )for each L -valued topological space X and every u ∈ Φ L ( X ). In fact, for all u ∈ Φ L ( X ) and U ∈ O ( X ), we have µ ( η Φ L ( X ) ( u ))( U ) = µ ([ u ])( U ) = [ u ]( φ ( U )) = φ ( U )( u ) = u ( U )and by Lemma 3.3, µ [Φ L η ( u )]( U ) = µ [( η X ) → L ( u )]( U ) = ( η X ) → L ( u )( φ ( U )) = u [( η X ) ← L ( φ ( U ))] = u ( U ) . Hence, µ ◦ η Φ L = id Φ L = µ ◦ Φ L η . (cid:3) L -valued continuous lattices and L -valued Scott topology In order to characterize L -valued domain structures via the monad (Φ L , η, µ ),we will recall some basic concepts and results about L -valued domains and L -valued Scott topology [26, 27]. Definition 4.1.
Let X be an L -ordered set. An L -subset D ∈ L X is calleddirected if (D1) W x ∈ X D ( x ) = 1 ; (D2) D ( x ) ∧ D ( y ) ≤ W z ∈ X D ( z ) ∧ e ( x, z ) ∧ e ( y, z ) .An L -subset I ∈ L X is called an ideal of X if it is a directed lowerset. Denote by D L ( X ) (resp., mathcalI L ( X ) ) the set of all directed L -subsets (resp., ideals) of X . An L -ordered set X is called an L -valued dcpoif every directed L -subset has a supremum, or equivalently, every ideal has asupremum. Definition 4.2.
For an L -valued dcpo X , A ∈ L X is called Scott open if itsatisfies one of the following equivalent conditions: (1) A ( ⊔ D ) = ⊔ A → L ( D ) for every D ∈ D L ( X ) ; (2) A ( ⊔ I ) = ⊔ A → L ( I ) for every I ∈ I L ( X ) ; (3) A is an upper set and A ( ⊔ D ) ≤ ⊔ A → L ( D ) for every D ∈ D L ( X ) ; (4) A is an upper set and A ( ⊔ I ) ≤ ⊔ A → L ( I ) for every I ∈ I L ( X ) .The family σ L ( X ) of all Scott open L -subsets of X forms an L -valuedtopology, called the L -valued Scott topology on X . efinition 4.3. Let ( X, e ) be a complete L -ordered set. For every x ∈ X ,define ⇓ x ∈ L X by ⇓ x ( y ) = ^ I ∈I L ( X ) e ( x, ⊔ I ) → I ( y ) . A complete L -ordered set ( X, e ) is called an L -valued continuous lattice if ⊔⇓ x = x for each x ∈ X . The L -relation ⇓ is an L -valued version of the classical way-below relation. Proposition 4.4. (1) ⇓ x ( y ) ≤ e ( y, x ) ( ∀ x, y ∈ X ) ; (2) e ( y , y ) ∧ ⇓ x ( y ) ∧ e ( x, x ) ≤ ⇓ w ( z ) ( ∀ x, x , y, y ∈ X ) . Proposition 4.5. If ( X, e ) is an L -valued continuous lattice, then (1) The mapping ⇓ is interpolative, that is, ⇓ x ( y ) = W z ∈ X ⇓ z ( y ) ∧ ⇓ x ( z ) ; (2) {⇑ x | x ∈ X } is a base of σ L ( X ) , where ⇑ x ( y ) = ⇓ y ( x ) ( ∀ x, y ∈ X ) ; (3) σ L ( X ) is a T L -valued topological space. Lemma 4.6.
Let I ∈ I L ( X ) and x ∈ X . Then (1) If x = ⊔ I , then ⇓ x ≤ I ; (2) ⇓ ( ⊔ I )( x ) ≤ I ( x ) . Proof. (1) For each y ∈ X , ⇓ x ( y ) = V J ∈I L ( X ) e ( x, ⊔ J ) → J ( y ) ≤ e ( x, ⊔ I ) → I ( y ) = 1 → I ( y ) = I ( y ) . (2) ⇓ ( ⊔ I )( x ) = V J ∈I L ( X ) e ( ⊔ I, ⊔ J ) → J ( x ) ≤ e ( ⊔ I, ⊔ I ) → I ( x ) = I ( x ) . (cid:3) Let X be a complete L -ordered set and equipped with the L -valued Scotttopology σ L ( X ). For every u ∈ Φ L ( X ), define u l ∈ L X by u l ( x ) = _ A ∈O ( X ) u ( A ) ∧ A l ( x ) , and further define lim S : Φ L ( X ) × X −→ L bylim S u ( x ) = _ I ∈I L ( X ) sub X ( I, u l ) ∧ e ( x, ⊔ I ) . This is the fuzzy Scott convergence of open filters, which is a modification ofScott convergence of stratified L -filters in [Yao].13 emma 4.7. Let X be a complete L -ordered set and u be an open filter of σ L ( X ) . Then u l ∈ I L ( X ) . Proof.
Firstly, it is routine to show that u l is a lower set.Secondly, W x ∈ X u l ( x ) ≥ u l ( ⊥ ) = W A ∈O ( X ) u ( A ) ∧ sub( A, ↑⊥ ) ≥ u (1 X ) = 1.Thirdly, for all x, y ∈ X , u l ( x ) ∧ u l ( y ) = W A ∈ σ L ( X ) u ( A ) ∧ sub( A, ↑ x ) ∧ W B ∈ σ L ( X ) u ( B ) ∧ sub( B, ↑ y )= W A,B ∈ σ L ( X ) u ( A ) ∧ u ( B ) ∧ sub( A, ↑ x ) ∧ sub( B, ↑ y ) ≤ W A,B ∈ σ L ( X ) u ( A ∧ B ) ∧ sub( A ∧ B, ↑ x ) ∧ sub( A ∧ B, ↑ y ) ≤ W C ∈ σ L ( X ) u ( C ) ∧ sub( C, ↑ x ) ∧ sub( C, ↑ y ) ≤ W z ∈ X W C ∈ σ L ( X ) u ( C ) ∧ sub( C, ↑ z ) ∧ e ( x, z ) ∧ e ( y, z ) . Hence, u l ∈ I L ( X ) (cid:3) For every open filter u of X with respect to the L -valued Scott topology.Define r : Φ L ( X ) −→ X by r ( u ) = ⊔ u l . Theorem 4.8. lim S u ( x ) = e ( x, r ( u )) . Proof.
On one hand,lim S u ( x ) = _ I ∈I L ( X ) sub X ( I, u l ) ∧ e ( x, ⊔ I ) ≥ sub X ( u l , u l ) ∧ e ( x, ⊔ u l ) = e ( x, ⊔ u l ) . On the other handlim S u ( x ) = _ I ∈I L ( X ) sub X ( I, u l ) ∧ e ( x, ⊔ I ) ≤ _ I ∈I L ( X ) e ( ⊔ I, ⊔ u l ) ∧ e ( x, ⊔ I ) ≤ e ( x, ⊔ u l ) . Hence, lim S u ( x ) = e ( x, r ( u )). (cid:3) Remark 4.9.
The meaning of Theorem 4.8 is that, the set of limit point ofan open filter u is a lower set with r ( u ) as the largest element. Proposition 4.10.
Let X be a complete L -ordered set. Then X is an L -valued continuous lattice iff x = ⊔ [ x ] l holds for every x ∈ X . roof. = ⇒ : On one hand, for every y ∈ X ,[ x ] l ( y ) = _ A ∈O ( X ) [ x ]( A ) ∧ sub( A, ↑ y ) ≤ A ( x ) ∧ ( A ( x ) → e ( y, x )) ≤ e ( y, x ) . On the other hand, V z ∈ X [ x ] l ( z ) → e ( x, y )= V z ∈ X V A ∈ σ L ( X ) ([ x ]( A ) ∧ sub( A, ↑ z )) → e ( z, y ) (Prop . . ≤ V z ∈ X V x ∈ X ( ⇑ x ( x ) ∧ sub( ⇑ x , ↑ z )) → e ( z, y ) (Prop . . V x ∈ X ⇓ x ( x ) → [ V z ∈ X sub( ⇑ x , ↑ z ) → e ( z, y )] (Prop . . , ≤ V x ∈ X ⇓ x ( x ) → [ V z ∈ X sub( ↑ x , ↑ z ) → e ( z, y )] (Prop . . ≤ V x ∈ X ⇓ x ( x ) → [ V z ∈ X e ( z, x ) → e ( z, y )] ≤ V x ∈ X ⇓ x ( x ) → e ( x , y ) (Prop . . e ( ⊔⇓ x, y ) = e ( x, y ) . Hence, x = ⊔ [ x ] l . ⇐ =: By Proposition 2.7(1), we only need to show that [ x ] l ≤ ⇓ x ≤ ↓ x .Firstly, ⇓ x ≤ ↓ x is obvious by Proposition 4.4(1). Secondly, since ⇓ x ( y ) = V I ∈I L ( X ) e ( x, ⊔ I ) → I ( y ) and [ x ] l ( y ) = W A ∈ σ L ( X ) A ( x ) ∧ sub( A, ↑ y ), we have A ( x ) ∧ sub( A, ↑ y ) ∧ e ( x, ⊔ I ) ≤ A ( ⊔ I ) ∧ sub( A, ↑ y ) (Def . . W z ∈ X A ( z ) ∧ I ( z ) ∧ sub( A, ↑ y ) (Def . . , Rem . . ≤ W z ∈ X I ( z ) ∧ ↑ y ( z )= W z ∈ X I ( z ) ∧ e ( y, z ) ≤ I ( y ) . (Def . . A ( x ) ∧ sub( A, ↑ y ) ≤ e ( x, ⊔ I ) → I ( y ) and consequently, [ x ] l ≤ ⇓ x bythe arbitrariness of y ∈ X . (cid:3)
5. The first main theoremThe First Main Theorem If X is an L -valued continuous lattice andequipping X with the L -valued Scott topology, then the pair ( X, r ) is a Φ L -15lgebra over the monad (Φ L , µ, η ), that is, r · Φ L r = r · µ X and r · η X = id X ,where r : Φ L ( X ) −→ L is given by r ( u ) = ⊔ u l .First of all, by Proposition 4.5(3), σ L ( X ) is a T L -valued topologicalspace. Lemma 5.1.
Let ( X, e ) be a complete L -ordered set. For every A ∈ σ L ( X ) ,it holds that r ← L ( A ) ≤ φ ( A ) . Proof.
For all u ∈ Φ L ( X ), we have r ← L ( A )( u ) = A ( r ( u )) = A ( ⊔ u l ) = ⊔ A → L ( u l ) (Def . . W x ∈ X A ( x ) ∧ u l ( x ) (Exam . . W x ∈ X A ( x ) ∧ W B ∈ σ L ( X ) u ( B ) ∧ sub( B, ↑ x )= W x ∈ X W B ∈ σ L ( X ) sub( ↑ x, A ) ∧ u ( B ) ∧ sub( B, ↑ x ) ≤ W B ∈ σ L ( X ) u ( B ) ∧ sub( B, A ) (Exam . . u ( A ) = φ ( A )( u ) . (F3)Hence, r ← L ( A ) ≤ φ ( A ). (cid:3) Lemma 5.2.
Let ( X, e ) be an L -valued continuous lattice. For every A ∈ σ L ( X ) and every x ∈ X , it holds that ⇓ ( ⊓ A )( x ) ≤ sub( φ ( A ) , r ← L ( ⇑ x )) . Proof.
For every u ∈ Φ L ( X ), we have e ( ⊓ A, r ( u )) = e ( ⊓ A, ⊔ u l ) ≥ u l ( ⊓ A ) (M1)= W B ∈ σ L ( X ) u ( B ) ∧ sub( B, ↑ ( ⊓ A )) ≥ u ( A ) ∧ sub( A, ↑ ( ⊓ A ))= u ( A ) = φ ( A )( u ) . (Rem . . x ∈ X , by Proposition 4.4(2), we have φ ( A )( u ) ∧ ⇓ ( ⊓ A )( x ) ≤ e ( ⊓ A, r ( u )) ∧ ⇓ ( ⊓ A )( x ) ≤ ⇓ ( r ( u ))( x ) = ⇑ x ( r ( u )) = r ← L ( ⇑ x )( u ) . Hence, ⇓ ( ⊓ A )( x ) ≤ V u ∈ Φ L ( X ) φ ( A )( u ) → r ← L ( ⇑ x )( u ) = sub( φ ( A ) , r ← L ( ⇑ x )). (cid:3) roposition 5.3. r · Φ L r = r · µ X . Proof.
Let α ∈ Φ L ( X ). Then r [Φ L r ( α )] = ⊔ (Φ L r ( α )) l , r [ µ X ( α )] = ⊔ ( µ X ( α )) l . For every x ∈ X ,(Φ L r ( α )) l ( x ) = W A ∈ σ L ( X ) Φ L r ( α )( A ) ∧ sub( A, ↑ x )= W A ∈ σ L ( X ) r → L ( α )( A ) ∧ sub( A, ↑ x )= W A ∈ σ L ( X ) α ( r ← L ( A )) ∧ sub( A, ↑ x )and ( µ X ( α )) l ( x ) = W A ∈ σ L ( X ) µ X ( α )( A ) ∧ sub( A, ↑ x )= W A ∈ σ L ( X ) α ( φ ( A )) ∧ sub( A, ↑ x ) . Firstly, by Lemma 5.1, (Φ L r ( α )) l ≤ ( µ X ( α )) l . Hence, r [ µ X ( α )] = ⊔ (Φ L r ( α )) l ≤ ⊔ ( µ X ( α )) l = r [Φ r ( α )] . Secondly, let x = r [ µ X ( α )]. Then x = ⊔⇓ x and by Lemma 4.6(1), ⇓ x ≤ ( µ X ( α )) l . We will show that ⇓ x ≤ (Φ L r ( α )) l , so that r [ µ X ( α )] = x ≤ ⊔ (Φ L r ( α )) l = r [Φ r ( α )].Then ⇓ x ( x ) = W y ∈ X ⇓ x ( y ) ∧ ⇓ y ( x ) (Prop . . ≤ W y ∈ X ( µ X ( α )) l ( y ) ∧ ⇓ y ( x )= W y ∈ X W A ∈ σ L ( X ) α ( φ ( A )) ∧ sub( A, ↑ y ) ∧ ⇓ y ( x )= W y ∈ X W A ∈ σ L ( X ) α ( φ ( A )) ∧ e ( y, ⊓ A ) ∧ ⇓ y ( x ) (Rem . . ≤ W A ∈ σ L ( X ) α ( φ ( A )) ∧ ⇓ ( ⊓ A )( x ) (Prop . . ≤ W A ∈ σ L ( X ) α ( φ ( A )) ∧ sub( φ ( A ) , r ← L ( ⇑ x )) (Lem . . ≤ α ( r ← L ( ⇑ x )) (F4) ≤ W A ∈ σ L ( X ) α ( r ← L ( A )) ∧ sub( A, ↑ x ) (Prop . . L r ( α )) l ( x ) . ⇓ x ≤ (Φ L r ( α )) l and r [Φ L r ( α )] = x = ⊔⇓ x ≤ ⊔ (Φ L r ( α )) l = r [ µ X ( α )] . Therefore, r · Φ L r = r · µ X . (cid:3) Proposition 5.4. r · η X = id X . Proof.
For each x ∈ X , we have[ x ] l ( y ) = _ A ∈ σ L ( X ) [ x ]( A ) ∧ sub( A, ↑ y ) = _ A ∈ σ L ( X ) A ( x ) ∧ sub( A, ↑ y ) . Firstly, [ x ] l ( y ) = _ A ∈ σ L ( X ) A ( x ) ∧ sub( A, ↑ y ) ≤ ↑ y ( x ) = ↓ x ( y ) . Secondly, by Proposition 4.5(2),[ x ] l ( y ) = _ A ∈ σ L ( X ) A ( x ) ∧ sub( A, ↑ y ) ≥ ⇑ y ( x ) ∧ sub( ⇑ y, ↑ y ) = ⇓ x ( y ) . Thus, ⇓ x ≤ [ x ] l ≤ ↓ x . Since ⊔⇓ x = x = ⊔↓ x , by Proposition 2.7(1), we have r · η X ( x ) = r ([ x ]) = ⊔ [ x ] l = x . Hence, r · η X = id X . (cid:3)
6. The second main theoremThe Second Main Theorem
If (
X, r ) is a Φ L -algebra over L - Top ,then by considering X with the specialization L -order, X is an L -valuedcontinuous lattice and r ( u ) = ⊔ u l . Lemma 6.1.
For every A ∈ O ( X ) , it holds that A ≤ r → L ( φ ( A )) . Proof.
For every x ∈ X , we have r → L ( φ ( A ))( x ) = _ r ( u )= x φ ( A )( u ) = _ r ( u )= x u ( A ) ≥ [ x ]( A ) = A ( x ) . Hence, A ≤ r → L ( φ ( A )). (cid:3) In the following, we will study the property of the specialization L -order e O ( X ) of the L -valued topological space X . For simplicity, we write e insteadof e O ( X ) . 18 emma 6.2. Let u, v ∈ Φ L ( X ) . Then (1) V W ∈O (Φ L ( X )) W ( u ) → W ( v ) = V A ∈O ( X ) u ( A ) → v ( A ) ; (2) sub( u, v ) ≤ e ( r ( u ) , r ( v )) . Proof. (1) Firstly, ^ W ∈O (Φ L ( X )) W ( u ) → W ( v ) ≤ ^ A ∈O ( X ) φ ( A )( u ) → φ ( A )( v ) = ^ A ∈O ( X ) u ( A ) → v ( A ) . Secondly, V W ∈O (Φ L ( X )) W ( u ) → W ( v )= V { ( A j ,a j ) } j ∈ J ⊆O ( X ) × L ( W j φ ( A j ) ∧ ( a j ) X )( u ) → ( W j φ ( A j ) ∧ ( a j ) X ) ≥ V { ( A j ,a j ) } j ∈ J ⊆O ( X ) × L V j ( φ ( A j ) ∧ ( a j ) X )( u ) → ( φ ( A j ) ∧ ( a j ) X )( v ) (Prop . . V { ( A j ,a j ) } j ∈ J ⊆O ( X ) × L V j ( u ( A j ) ∧ a j ) → ( v ( A j ) ∧ a j ) ≥ V { ( A j ,a j ) } j ∈ J ⊆O ( X ) × L V j u ( A j ) → v ( A j ) (Prop . . V A ∈O ( X ) u ( A ) → v ( A ) . Hence, V W ∈O (Φ L ( X )) W ( u ) → W ( v ) = V A ∈O ( X ) u ( A ) → v ( A ).(2) For all u, v ∈ Φ L ( X ), e ( r ( u ) , r ( v )) = V A ∈O ( X ) A ( r ( u )) → A ( r ( v ))= V A ∈O ( X ) r ← L ( A )( u ) → r ← L ( A )( v ) ≥ V W ∈O (Φ L ( X )) W ( u ) → W ( v )= V A ∈O ( X ) u ( A ) → v ( A )= sub( u, v ) . The proof is completed. (cid:3)
Proposition 6.3. lim X u ( x ) = e ( x, r ( u )) . roof. Firstly, by Lemma 6.2(2),lim X u ( x ) = sub([ x ] , u ) ≤ e ( r ([ x ]) , r ( u )) = e ( x, r ( u )) . Secondly, e ( x, r ( u )) = ^ A ∈O ( X ) A ( x ) → A ( r ( u )) = ^ A ∈O ( X ) A ( x ) → r ← L ( A )( u ) . Since r ← L ( A ) ∈ O (Φ L ( X )) and u ∈ Φ L ( X ), by applying Proposition 3.1 tothe space (Φ L ( X ) , O (Φ L ( X ))) r ← L ( A )( u ) = W B ∈O ( X ) φ ( B )( u ) ∧ sub( φ ( B ) , r ← L ( A )) (Lem . . W B ∈O ( X ) u ( B ) ∧ sub( r → L ( φ ( B )) , A ) (Rem . . ≤ W B ∈O ( X ) u ( B ) ∧ sub( B, A ) (Lem . . u ( A ) (F3) . Hence, e ( x, r ( u )) ≤ V A ∈O ( X ) A ( x ) → u ( A ) = lim X u ( x ). (cid:3) Proposition 6.4.
The pair ( X, e ) is a complete L -ordered set, where ⊓ A = r ([ A ]) . Proof.
Let A ∈ L X . Firstly, by Lemma 6.2(2), A ( x ) ≤ sub([ A ] , [ x ]) ≤ e ( r ([ A ]) , r ([ x ])) = e ( r ([ A ]) , x ) . Secondly, for every y ∈ X , V x ∈ X A ( x ) → e ( y, x )= V x ∈ X A ( x ) → ( V B ∈O ( X ) B ( y ) → B ( x ))= V B ∈O ( X ) B ( y ) → ( V x ∈ X A ( x ) → B ( x )) (Prop . . , V B ∈O ( X ) B ( y ) → sub( A, B )= V B ∈O ( X ) [ y ]( B ) → [ A ]( B )= sub([ y ] , [ A ]) ≤ e ( r ([ y ]) , r ([ A ])) (Lem . . e ( y, r ([ A ])) . Thus, ⊓ A = r ([ A ]). Hence, X is complete L -ordered set. (cid:3) emma 6.5. The pair (Φ L ( X ) , sub) is an L -valued dcpo, where for eachdirected L -subset A of Φ L ( X ) , ⊔A = W u ∈ Φ L ( X ) A ( u ) ∧ u . Proof.
Let A be a directed L -subset of (Φ L ( X ) , sub). Put w = _ u ∈ Φ L ( X ) A ( u ) ∧ u. We need to show that for every v ∈ Φ L ( X ), it holds that w ∈ Φ L ( X ) andsub( w, v ) = ^ u ∈ Φ L ( X ) A ( u ) → sub( u, v ) . Firstly, for each a ∈ L , by (F2) and (D1), w ( a X ) = _ u ∈ Φ L ( X ) A ( u ) ∧ u ( a X ) ≥ _ u ∈ Φ L ( X ) A ( u ) ∧ a = 1 ∧ a = a. For all
A, B ∈ O ( X ), clearly we have w ( A ∧ B ) ≤ w ( A ) ∧ w ( B ) and w ( A ) ∧ w ( B )= W u ,u ∈ Φ L ( X ) A ( u ) ∧ u ( A ) ∧ A ( u ) ∧ u ( B ) ≤ W u ,u ,v ∈ Φ L ( X ) A ( v ) ∧ sub( u , v ) ∧ sub( u , v ) ∧ u ( A ) ∧ u ( B ) (D2) ≤ W v ∈ Φ L ( X ) A ( v ) ∧ v ( A ) ∧ v ( B )= W v ∈ Φ L ( X ) A ( v ) ∧ v ( A ∧ B ) (F1)= w ( A ∧ B ) . Hence, w ( A ∧ B ) = w ( A ) ∧ w ( B ). Therefore, w ∈ Φ L ( X ).Secondly, for every v ∈ Φ L ( X ),sub( w, v ) = V A ∈O ( X ) w ( A ) → v ( A )= V A ∈O ( X ) V u ∈ Φ L ( X ) ( A ( u ) ∧ u ( A )) → v ( A ) (Prop . . V u ∈ Φ L ( X ) A ( u ) → V A ∈O ( X ) u ( A ) → v ( A ) (Prop . . , V u ∈ Φ L ( X ) A ( u ) → sub( u, v ) . In a summary, ⊔A = W u ∈ Φ L ( X ) A ( u ) ∧ u . (cid:3) emma 6.6. Let A be a directed L -subset of (Φ L ( X ) , sub) . Define e A : O (Φ L ( X )) −→ L by e A ( W ) = _ u ∈ Φ L ( X ) A ( u ) ∧ W ( u ) . Then (1) e A ∈ Φ L ( X ) ; (2) µ X ( e A ) = ⊔A . Proof. (1) Firstly, for each a ∈ L , by (D1), e A ( a Φ L ( X ) ) = _ u ∈ Φ L ( X ) A ( u ) ∧ a Φ L ( X ) ( u ) = a ∧ _ u ∈ Φ L ( X ) A ( u ) = a ∧ a. Secondly, for all W , W ∈ O (Φ L ( X )), it is clear that e A ( W ∧ W ) ≤ e A ( W ) ∧ e A ( W ) and e A ( W ) ∧ e A ( W )= W u,v ∈ Φ L ( X ) A ( u ) ∧ W ( u ) ∧ A ( v ) ∧ W ( v ) ≤ W u,v,w ∈ Φ L ( X ) A ( w ) ∧ sub( u, w ) ∧ sub( v, w ) ∧ W ( u ) ∧ W ( v ) (D2) ≤ W w ∈ Φ L ( X ) A ( w ) ∧ W ( w ) ∧ W ( w )= W w ∈ Φ L ( X ) A ( w ) ∧ ( W ∧ W )( w ) (F1)= e A ( W ∧ W ) . Then, e A ( W ) ∧ e A ( W ) = e A ( W ∧ W ). Hence, e A ∈ Φ L ( X ).(2) For each A ∈ O ( X ), we have µ X ( e A )( A ) = e A ( φ ( A )) = _ u ∈ Φ L ( X ) A ( u ) ∧ φ ( A )( u ) = _ u ∈ Φ L ( X ) A ( u ) ∧ u ( A ) = ( ⊔A )( A ) . Hence, µ X ( e A ) = ⊔A . (cid:3) Proposition 6.7. r : Φ L ( X ) −→ X preserves suprema of directed L -subsets. roof. Let A be a directed L -subset of Φ L ( X ). Then by Lemma 6.6(2), r ( ⊔A ) = r · µ X ( e A ) = r · Φ L r ( e A ) . We will prove that r · Φ L r ( e A ) ≤ ⊔ r → L ( A ) (the inverse inequality is routine).Since ⊔ r → L ( A ) = ⊓ ( r → L ( A )) u = r ([( r → L ( A )) u ]), we only need to show thatΦ L r ( e A ) ≤ [( r → L ( A )) u ].For every A ∈ O ( X ),Φ L r ( e A )( A ) = e A ( r ← L ( A )) = _ u ∈ Φ L ( X ) A ( u ) ∧ r ← L ( A )( u ) = _ u ∈ Φ L ( X ) A ( u ) ∧ A ( r ( u )) . For each x ∈ X ,( r → L ( A )) u ( x ) = V y ∈ X r → L ( A )( y ) → e ( y, x )= V u ∈ Φ L ( X ) A ( u ) → e ( r ( u ) , x )= V u ∈ Φ L ( X ) A ( u ) → V B ∈O ( X ) B ( r ( u )) → B ( x )= V u ∈ Φ L ( X ) V B ∈O ( X ) ( A ( u ) ∧ B ( r ( u ))) → B ( x ) (Prop . . V B ∈O ( X ) ( W u ∈ Φ L ( X ) A ( u ) ∧ B ( r ( u ))) → B ( x ) . (Prop . . r → L ( A )) u ]( A ) = sub(( r → L ( A )) u , A )= V x ∈ X ( r → L ( A )) u ( x ) → A ( x )= V x ∈ X { V B ∈O ( X ) ( W u ∈ Φ L ( X ) A ( u ) ∧ B ( r ( u ))) → B ( x ) } → A ( x ) ≥ V x ∈ X { ( W u ∈ Φ L ( X ) A ( u ) ∧ A ( r ( u ))) → A ( x ) } → A ( x ) (Prop . . ≥ W u ∈ Φ L ( X ) A ( u ) ∧ A ( r ( u )) (Prop . . L r ( e A )( A ) . This completes the proof. (cid:3)
For every u ∈ Φ L ( X ), define A u ∈ L Φ L ( X ) by A u ( v ) = _ A ∈O ( X ) u ( A ) ∧ sub( v, [ A ]) ( ∀ v ∈ Φ L ( X )) . emma 6.8. A u is directed in (Φ L ( X ) , sub) and ⊔A u = u . Proof. (1) Firstly, by (F2), _ v ∈ Φ L ( X ) A u ( v ) = _ v ∈ Φ L ( X ) _ A ∈O ( X ) u ( A ) ∧ sub( v, [ A ]) ≥ u (1 X ) ∧ sub([1 X ] , [1 X ])) = 1 . Secondly, for all v , v ∈ Φ L ( X ), A u ( v ) ∧ A u ( v )= W A ,A ∈O ( X ) u ( A ) ∧ sub( v , [ A ]) ∧ u ( A ) ∧ sub( v , [ A ])= W A ,A ∈O ( X ) u ( A ∧ A ) ∧ sub( v , [ A ]) ∧ sub( v , [ A ]) (F1) ≤ W A ,A ∈O ( X ) u ( A ∧ A ) ∧ sub( v , [ A ∧ A ]) ∧ sub( v , [ A ∧ A ]) (Prop . . ≤ W A ∈O ( X ) u ( A ) ∧ sub( v , [ A ]) ∧ sub( v , [ A ])= W A ∈O ( X ) u ( A ) ∧ sub([ A ] , [ A ]) ∧ sub( v , [ A ]) ∧ sub( v , [ A ]) ≤ W v ∈ Φ L ( X ) W A ∈O ( X ) u ( A ) ∧ sub( v, [ A ]) ∧ sub( v , v ) ∧ sub( v , v )= W v ∈ Φ L ( X ) A u ( v ) ∧ sub( v , v ) ∧ sub( v , v ) . Hence A u is directed.Thirdly, by Lemma 6.5, for every B ∈ O ( X ), we have ⊔A u ( B ) = _ v ∈ Φ L ( X ) A u ( v ) ∧ v ( B ) = _ v ∈ Φ L ( X ) _ A ∈O ( X ) u ( A ) ∧ sub( v, [ A ]) ∧ v ( B ) . By (F3 ′ ), we only need to show that W v ∈ Φ L ( X ) sub( v, [ A ]) ∧ v ( B ) = [ A ]( B ).Firstly, by Proposition 2.2(3), _ v ∈ Φ L ( X ) sub( v, [ A ]) ∧ v ( B ) ≤ _ v ∈ Φ L ( X ) ( v ( B ) → [ A ]( B )) ∧ v ( B ) ≤ [ A ]( B ) . Secondly, _ v ∈ Φ L ( X ) sub( v, [ A ]) ∧ v ( B ) ≥ sub([ A ] , [ A ]) ∧ [ A ]( B ) = [ A ]( B )Thus, W v ∈ Φ L ( X ) sub( v, [ A ]) ∧ v ( B ) = [ A ]( B ).Hence, ⊔A u = u . (cid:3) roposition 6.9. For u ∈ Φ L ( X ) , r ( u ) = ⊔ u l . Proof.
We need to show that e ( r ( u ) , x ) = sub( u l , ↓ x ) for all x ∈ X .First of all,sub( u l , ↓ x )= V y ∈ X u l ( y ) → e ( y, x )= V y ∈ X V A ∈O ( X ) ( u ( A ) ∧ sub( A, ↑ y )) → e ( y, x ) (Prop . . V y ∈ X V A ∈O ( X ) ( u ( A ) ∧ sub( y, ⊓ A )) → e ( y, x ) (Rem . . V A ∈O ( X ) u ( A ) → ( V y ∈ X e ( y, ⊓ A ) → e ( y, x )) (Prop . . , V A ∈O ( X ) u ( A ) → ( e ( y, ⊓ A ) → e ( y, x )) (Prop . . , V A ∈O ( X ) u ( A ) → e ( ⊓ A, x ) . (Prop . . e ( r ( u ) , x ) = V A ∈O ( X ) u ( A ) → e ( ⊓ A, x ) for all x ∈ X .Firstly, for all A ∈ O ( X ), e ( r ( u ) , x ) ∧ u ( A ) = e ( r ( u ) , x ) ∧ sub([ A ] , u ) (F4 ′ ) ≤ e ( r ( u ) , x ) ∧ e ( r ([ A ]) , r ( u )) (Lam . . e ( r ( u ) , x ) ∧ e ( ⊓ A, r ( u )) (Prop . . ≤ e ( ⊓ A, x ) . (E3)By the arbitrariness of A ∈ O ( X ), we have e ( r ( u ) , x ) ≤ ^ A ∈O ( X ) u ( A ) → e ( ⊓ A, x ) . Secondly, for every x ∈ X , e ( r ( u ) , x )= e ( r ( ⊔A u ) , x ) = e ( ⊔ r → L ( A u ) , x ) (Prop . . , Lem . . V y ∈ X r → L ( A u )( y ) → e ( y, x ) (Rem . . V y ∈ X V r ( v )= y A u ( v ) → e ( y, x ) (Prop . . , Rem . . V v ∈ Φ L ( X ) V A ∈ O ( X ) ( u ( A ) ∧ sub( v, [ A ])) → e ( r ( v ) , x ) (Prop . . V A ∈ O ( X ) u ( A ) → ( V v ∈ Φ L ( X ) sub( v, [ A ]) → e ( r ( v ) , x )) (Prop . . , .
25e only need to show that e ( ⊓ A, x ) ∧ sub( v, [ A ]) ≤ e ( r ( v ) , x ) for all v ∈ Φ L ( X ). In fact, by Lemma 6.2(2) and (E3), e ( ⊓ A, x ) ∧ sub( v, [ A ]) ≤ e ( ⊓ A, x ) ∧ e ( r ( v ) , r ([ A ]))= e ( ⊓ A, x ) ∧ e ( r ( v ) , ⊓ A ) ≤ e ( r ( v ) , x ) . This completes the proof. (cid:3)
Proposition 6.10. ( X, e O ( X ) ) is an L -valued continuous lattice. Proof.
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