Algebraic representation of continuous lattices via the open filter monad, revisited
aa r X i v : . [ m a t h . GN ] D ec Algebraic representation of continuous lattices via theopen filter monad, revisited
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
In [A. Day, Filter monads, continuous lattices and closure systems, Can.J. Math. 27 (1975) 50–59], Day showed that continuous lattices are preciselythe algebras of the open filter monad over the category of T spaces.The aim of this paper is to give a clean and clear version of the wholeprocess of Day’s approach.
1. Preliminaries
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 a B -morphism τ A : F ( A ) −→ G ( A ) such that G ( f ) · τ A = τ A ′ · F ( f ) holdsfor each A -morphism f : A −→ A ′ .A monad over a category X is a triple ( T, η, µ ) consisting of a functor T : X −→ X and two natural transformations η : id X −→ T and µ : T ◦ T −→ T such that µ · T µ = µ · µT and µ · ηT = id T = µ · T η .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 r : T ( X ) −→ X is astructured morphism satisfying that r · T ( r ) = r · µ X and id X = r · η X . T topological spaces Definition 1.1.
Let ( X, O ( X )) be a topological space. A subset v ⊆ O ( X ) is called an open filter of X if v is a lattice-theoretic filter of ( O ( X ) , ⊆ ) , thatis, Preprint submitted to Topology and Its Applications December 30, 2019 F) A ∩ B ∈ u iff A, B ⊆ u ( ∀ A, B ∈ O ( X )) . Let Φ( X ) denote the set of all open filters of X . Then the pair (Φ( X ) , ⊆ )is a dcpo, where for each directed subset A of Φ( X ), W A = S A . Example 1.2.
Let ( X, O ( X )) be a topological space. For A ∈ X , define [ A ] ⊆ O ( X ) by [ A ] = { B ∈ O ( X ) | A ⊆ B } . Then [ A ] ∈ Φ( X ) . For every x ∈ X , denote [ { x } ] by [ x ] . It is easy to show that, (1) for every v ∈ Φ( X ) , v = S A ∈ v [ A ] ; (2) for each B ∈ O ( X ) , B ∈ v iff [ B ] ⊆ v . Let X be a topological space. For every A ∈ O ( X ), define φ ( A ) ⊆ Φ( X )by φ ( A ) = { v ∈ Φ( X ) | A ∈ v } . Then { φ ( A ) | A ∈ O ( X ) } becomes atopological base, which generates a T topology, denoted by O (Φ( X )). Itis easy to see that every member of O (Φ( X )) is an upper set in the dcpo(Φ( X ) , ⊆ ).Let f : X −→ Y be a mapping. Define Φ( f ) : Φ( X ) −→ Φ( Y ) byΦ( f )( u ) = { B ∈ O ( Y ) | f ← ( B ) ∈ u } . It is easily shown that Φ( f ) is a well-defined mapping. Then Φ is an endo-functor on Top .Define η X : X −→ Φ( X ) and µ X : Φ ( X ) −→ Φ( X ) respectively by η X ( x ) = [ x ] ( ∀ x ∈ X ) and µ X ( α )( A ) = α ( φ ( A )) ( ∀ α ∈ Φ ( X ) , ∀ A ∈ O ( X ))). Proposition 1.3.
Both η : id Top −→ Φ and µ : Φ −→ Φ are naturaltransformations, and further (Φ , η, µ ) is a monad over Top .1.3. Continuous lattices and the Scott topology Definition 1.4. (1)
A nonempty subset I of a poset X is called an ideal ifit is a lower directed subset. Denote by I ( X ) the set of all ideals of X . Aposet X is called a dcpo if W I exists for every I ∈ I ( X ) . (2) Let X be a dcpo and x, y ∈ X . If for every I ∈ I ( X ) , y ≤ W I alwaysimplies x ∈ I , then x is called way-below y , in symbols, x ≪ y . For each x ∈ X , denote ⇑ x = { y ∈ X | x ≪ y } and ⇓ x = { y ∈ X | y ≪ x } . (3) A complete lattice X is called a continuous lattice if x = W ⇓ x holdsfor every x ∈ X . efinition 1.5. On a dcpo X , a subset V ⊆ X is called Scott open if V isan upper set and W I ∈ V implies I ∩ V = ∅ for every I ∈ I ( X ) . The set ofall Scott open subsets of X forms a topology, called the Scott topology on X ,denoted by σ ( X ) . Proposition 1.6.
In a continuous lattice X , it holds that (1) ∀ x, y ∈ X , if x ≪ y , then there exists z ∈ X such that x ≪ z ≪ y ; (2) the family {⇑ x | x ∈ X } forms a base of σ ( X ) . Proposition 1.7.
Let X be a dcpo, I ∈ I ( X ) and x ∈ X . Then (1) If x ≤ W I , then ⇓ x ⊆ I ; (2) If x ≪ W I , then x ∈ I . Let X be a complete lattice equipped with the Scott topology σ ( X ). Forevery v ∈ Φ( X ), define v ↓ = S A ∈ v A ↓ , where A ↓ = { y ∈ X | A ⊆ ↑ y } . It is aroutine to show that v ↓ ∈ I ( X ) for every v ∈ Φ( X ). Proposition 1.8.
Let X be a complete lattice. Then X is a continuouslattice iff x = W [ x ] ↓ holds for every x ∈ X . Proof. = ⇒ : We only need to show that ⇓ x ⊆ [ x ] ↓ ⊆ ↓ x . Firstly,the inclusion [ x ] ↓ ⊆ ↓ x is obvious. Secondly, for every y ≪ x , we have x ∈ ⇑ y ∈ σ ( X ) and then ⇑ y ∈ [ x ]. Thus y ∈ ( ↑ y ) ↓ ⊆ ( ⇑ y ) ↓ ⊆ [ x ] ↓ . By thearbitrariness y , ⇓ x ⊆ [ x ] ↓ , as desired. ⇐ =: We only need to show that [ x ] ↓ ⊆ ⇓ x . If y ∈ [ x ] ↓ , then there exists A ∈ σ ( X ) such that y ∈ A ↓ , and then y ≤ x . Let I ∈ I ( X ) and x ≤ W I .Then W I ∈ A and so there exist z ∈ A ∩ I . Thus y ≤ z and then y ∈ I .Hence, y ≪ x , and consequently, [ x ] ↓ ⊆ ⇓ x . (cid:3)
2. The representation of continuous lattices as monad algebrasTheorem 2.1.
Let X be a continuous lattice equipped with the Scott topologyand define r : Φ( X ) −→ X by r ( v ) = W v ↓ . Then the pair ( X, r ) is a Φ -algebra over the monad (Φ , µ, η ) . Lemma 2.2.
For every A ∈ σ ( X ) , it holds that r ← ( A ) ⊆ φ ( A ) . Proof. If v ∈ r ← ( A ), then W v ↓ = r ( v ) ∈ A and v ↓ ∩ A = ∅ since v ↓ isan ideal. Thus there exists x ∈ A such that x ∈ v ↓ . For x ∈ v ↓ , there exists B ∈ v such that x ∈ B ↓ . Then B ⊆ ↑ x ⊆ A . Therefore, A ∈ v and v ∈ φ ( A ).Hence, r ← ( A ) ⊆ φ ( A ). (cid:3) emma 2.3. For every A ∈ σ ( X ) and every x ∈ X , if x ≪ V A , then φ ( A ) ⊆ r ← ( ⇑ x ) . Proof. If v ∈ φ ( A ), then A ∈ v . Then V A ∈ v ↓ and V A ≤ W v ↓ = r ( v ).If x ≪ V A , then x ≪ r ( v ) and v ∈ r ← ( ⇑ x ). By the arbitrariness of v , wehave φ ( A ) ⊆ r ← ( ⇑ x ). (cid:3) Proposition 2.4. r · Φ( r ) = r · µ X . Proof.
Let α ∈ Φ ( X ). Then r (Φ( r )( α )) = _ (Φ( r )( α )) ↓ , r ( µ X ( α )) = _ ( µ X ( α )) ↓ . Firstly,(Φ( r )( α )) ↓ = { x ∈ X | ∃ A ∈ σ ( X ) s.t. A ∈ Φ( r )( α ) and A ⊆ ↑ x } = { x ∈ X | ∃ A ∈ σ ( X ) s.t. r ← ( A ) ∈ α and A ⊆ ↑ x } , and ( µ X ( α )) ↓ = { x ∈ X | ∃ A ∈ σ ( X ) s.t. A ∈ µ X ( α ) and A ⊆ ↑ x } = { x ∈ X | ∃ A ∈ σ ( X ) s.t. φ ( A ) ∈ α and A ⊆ ↑ x } . Firstly, by Lemma 2.2, (Φ( r )( α )) ↓ ⊆ ( µ X ( α )) ↓ and then r (Φ( r )( α )) ≤ r ( µ X ( α )) . Secondly, let x = r [ µ X ( α )]. Then x = W ⇓ x and by Proposition 1.7(1), ⇓ x ⊆ ( µ X ( α )) ↓ . We will show that ⇓ x ≤ (Φ( r )( α )) ↓ , so that r ( µ X ( α )]) x ≤ W (Φ( r )( α )) ↓ = r (Φ( r )( α )).If x ≪ x , then there exists y ∈ X such that x ≪ y ≪ x . Since( µ X ( α )) ↓ is an ideal, by Proposition 1.7(2), we have y ∈ ( µ X ( α )) ↓ and thusthere exists A ∈ σ ( X ) such that φ ( A ) ∈ α and A ⊆ ↑ y . Then y ≤ V A and x ≪ V A . By Lemma 2.3, r ← ( ⇑ x ) ∈ α . Since ⇑ x ∈ σ ( X ) and ⇑ x ⊆ ↑ x , wehave x ∈ Φ( r )( α ). Hence, ⇓ x ≤ (Φ( r )( α )) ↓ . This completes the proof ofthis step.Therefore, r · Φ r = r · µ X . (cid:3) Proposition 2.5. r · η X = id X . Proof.
Let x ∈ X . Then by Proposition 1.8, it holds that r · η X ( x ) = r ([ x ]) = W [ x ] ↓ = x . (cid:3) heorem 2.6. If ( X, r ) is a Φ -algebra over Top , then by considering X with the specialization order, X is a continuous lattice and r ( v ) = W v ↓ . Lemma 2.7.
For every A ∈ O ( X ) , it holds that A ⊆ r → ( φ ( A )) . Proof.
For every x ∈ A , we have [ x ] ∈ φ ( A ) and then x = r ( η X ( x )) = r ([ x ]) ∈ r → ( φ ( A )). Hence, A ⊆ r → ( φ ( A )). (cid:3) Lemma 2.8.
For all v, w ∈ Φ( X ) , if v ⊆ w , then r ( v ) ≤ O ( X ) r ( w ) . Proof.
Let A ∈ O ( X ) and r ( v ) ∈ A . Then v ∈ r ← ( A ) ∈ O (Φ( X )).Since r ← ( A ) is an upper set, we have w ∈ r ← ( A ) and then r ( w ) ∈ A . By thearbitrariness of A , r ( v ) ≤ O ( X ) r ( w ). (cid:3) Proposition 2.9.
The pair ( X, ≤ O ( X ) ) is a complete lattice, where V A = r ([ A ]) . Proof.
Let A ⊆ X . Firstly, if x ∈ A , then [ A ] ⊆ [ x ] and then by Lemma2.8, r ([ A ]) ≤ r ([ x ]) = x . That is to say, r ([ A ]) is a lower bound of A .Suppose that y is another lower bound of A , i.e., A ⊆ ↑ y . If B ∈ [ y ], then ↑ y ⊆ B and then B ∈ [ A ]. Thus [ y ] ⊆ [ A ] and y = r ([ y ]) ≤ r ([ A ]).Hence, V A = r ([ A ]). (cid:3) Lemma 2.10.
Let A be a directed subset of Φ( X ) . Define e A ⊆ O (Φ( X )) by e A = { W ∈ O (Φ( X )) | A ∩ W = ∅} . Then e A ∈ Φ ( X ) and µ X ( e A ) = W A . Proof.
For all W , W ∈ O (Φ( X )), it is clear that if W ∩ W ∈ e A then W , W ∈ e A . Conversely, suppose that W , W ∈ e A . Then A ∩ W = ∅ and A ∩ W = ∅ , and then there exist v , v ∈ A such that v ∈ W , v ∈ W .Since A is directed, there exists w ∈ A such that v , v ⊆ w . We knowthat W , W are upper sets of (Φ( X ) , ⊆ ), we have w ∈ W ∩ W . Thus, W ∩ W ∈ e A . Hence, e A ∈ Φ ( X ).For each A ∈ O ( X ), A ∈ µ X ( e A ) iff φ ( A ) ∈ e A iff φ ( A ) ∩ A 6 = ∅ iff thereexists v ∈ A such that A ∈ u , iff A ∈ S A . Hence, µ X ( e A ) = W A . (cid:3) Proposition 2.11. r : Φ( X ) −→ X preserves suprema of directed subsets. roof. Let A be a directed subset of Φ( X ). Then by Lemma 2.10, r ( _ A ) = r · µ X ( e A ) = r · Φ( r )( e A ) . We will prove that r · Φ( r )( e A ) ≤ W r → L ( A ) (the inverse inequality is routine).Since W r → ( A ) = V ( r → ( A )) ↑ = r ([( r → ( A )) ↑ ]), we only need to show thatΦ( r )( e A ) ⊆ [( r → ( A )) ↑ ].In fact, suppose that A ∈ Φ( r )( e A ). Then r ← ( A ) ∈ e A and then A ∩ r ← ( A ) = ∅ . Thus there exists v ∈ A and r ( v ) ∈ A . Suppose that x ∈ ( r → ( A )) ↑ . By v ∈ A , we know that r ( v ) ∈ r → ( A ) and then r ( v ) ≤ x , whichimplies that x ∈ A . Hence, by the arbitrariness of x , we have ( r → ( A )) ↑ ⊆ A ;and by the arbitrariness of A , we have Φ( r )( e A ) ⊆ [( r → ( A )) ↑ ]. This completesthe proof. (cid:3) Lemma 2.12.
For every v ∈ Φ( X ) , define A v ⊆ Φ( X ) by A v = { [ A ] | A ∈ v } . Then A v is directed in Φ( X ) and W A v = v . Proof.
Firstly, it is obvious that A v = ∅ . For all [ A ] , [ B ] ∈ A v with A, B ∈ v , we have A ∩ B ∈ v and then [ A ∩ B ] ∈ A v . Clearly, [ A ] , [ B ] ⊆ [ A ∩ B ]Hence, A v is directed. Secondly, v = S A ∈ v [ A ] = W A v . (cid:3) Proposition 2.13.
For v ∈ Φ( X ) , r ( v ) = W v ↓ . Proof.
We need to show that r ( v ) is the least upper bound of v ↓ .Firstly, for each x ∈ v ↓ , there exists A ∈ v such that x ∈ A ↓ . Then[ A ] ⊆ v and x ≤ V A = r ([ A ]) ≤ r ( v ). That is to say, r ( v ) is an upper boundof v ↓ .Secondly, suppose that y is another upper bound of v ↓ . Since r ( v ) = r ( W A v ) = W r → ( A v ), we only need to show that z ≤ y for each z ∈ r → ( A v ).In fact, for z ∈ r → ( A v ), there exists A ∈ v such that z = r ([ A ]) = V A .Then z ∈ A ↓ ⊆ v ↓ and then z ≤ y .Hence, r ( v ) is the least upper bound of v ↓ , i.e., r ( v ) = W v ↓ . (cid:3) Proposition 2.14. ( X, e O ( X ) ) is a continuous lattice. Proof.
By Proposition 2.13, x = r ([ x ]) = W [ x ] ↓ holds for every x ∈ X .By Propositions 1.8 and 2.9, X is a continuous lattice. (cid:3)(cid:3)