An Invitation to Priestley Duality Outline Outline Concrete Examples of Bounded Distributive Lattices

Outline

Bounded distributive lattices

Priestley duality for ﬁnite distributive lattices Lecture 1: an invitation to Priestley duality Priestley duality via homsets

Brian A. Davey Priestley duality for inﬁnite distributive lattices

Examples of Priestley spaces TACL 2015 School Campus of Salerno (Fisciano) The translation industry: restricted Priestley duals 15–19 June 2015

Useful facts about Priestley spaces

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Outline Concrete examples of bounded distributive lattices 1 1. The two-element chain Bounded distributive lattices 2 = 0, 1 ; , , 0, 1 . 0 Priestley duality for ﬁnite distributive lattices h{ } _ ^ i

Priestley duality via homsets 2. All subsets of a set S: }(S); , , ?, S . h [ \ i Priestley duality for inﬁnite distributive lattices 3. Finite or coﬁnite subsets of N: Examples of Priestley spaces }FC(N); , , ?, N . h [ \ i The translation industry: restricted Priestley duals 4. Open subsets of a topological space X: Useful facts about Priestley spaces O(X); , , ?, X . h [ \ i

3 / 35 4 / 35 12 1

4 6 3

2 3 2

1 1

lcm max _ _ gcd min ^ ^ 6 division 6 usual

max _ min ^ 6 usual

More examples of bounded distributive lattices Drawing distributive lattices

Any distributive lattice L; , , 0, 1 has a natural order h _ ^ i corresponding to set inclusion: a b a b = b. 6 () _ 5. T , F ; or, and, F, T . 1, 2, 3 h{ } i { } f 6. N 0 ; lcm, gcd, 1, 0 . h [ { } i 1, 2 1, 3 2, 3 (Use the fact that, lcm(m, n) gcd(m, n)=mn, for all { }{}{} · m, n N 0 , and that a lattice is distributive iff it satisﬁes 2 [ { } x z = y z & x z = y z = x = y.) _ _ ^ ^ ) 1 2 3 { }{}{} 7. Subgroups of a cyclic group G, Sub(G); , , e , G , where H K := sg (H K ). ? h _ \ { } i _ G [ union _ intersection ^ 6 inclusion 5 / 35 6 / 35

Drawing distributive lattices Drawing distributive lattices

Any distributive lattice L; , , 0, 1 has a natural order Any distributive lattice L; , , 0, 1 has a natural order h _ ^ i h _ ^ i corresponding to set inclusion: a b a b = b. corresponding to set inclusion: a b a b = b. 6 () _ 6 () _ 1, 2, 3 1, 2, 3 { } { } 1 12 f 12 f

1, 2 1, 3 2, 3 1, 2 1, 3 2, 3 { }{}{} 4 6 { }{}{} 4 6 3

1 2 3 2 3 1 2 3 2 3 2 { }{}{} { }{}{}

1 1 1 ? ?

union lcm union lcm max _ _ _ _ _ intersection gcd intersection gcd min ^ ^ ^ ^ ^ 6 inclusion 6 division 6 inclusion 6 division 6 usual 6 / 35 6 / 35 Note: Every distributive lattice embeds into 2S, for some set S. Note: Every distributive lattice embeds into 2S, for some set S.

A sublattice L of 24 A sublattice L of 24

24 24 L

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A sublattice L of 24 Outline

Bounded distributive lattices

Priestley duality for ﬁnite distributive lattices

Priestley duality via homsets

Priestley duality for inﬁnite distributive lattices

Examples of Priestley spaces

4 2 The translation industry: restricted Priestley duals

Note: Every distributive lattice embeds into 2S, for some set S. Useful facts about Priestley spaces

7 / 35 8 / 35 In fact, we can choose P to be the ordered set (L); of hJ 6i join-irreducible elements of L.

Theorem [G. Birkhoff] Theorem [G. Birkhoff] Let L be a finite distributive lattice and let P be a finite ordered Let L be a finite distributive lattice and let P be a finite ordered set. Then set. Then

I L is isomorphic to ( (L)), and I L is isomorphic to ( (L)), and O J O J I P is isomorphic to ( (P)). I P is isomorphic to ( (P)). J O J O

Representing ﬁnite distributive lattices Representing ﬁnite distributive lattices

Birkhoff’s representation for a finite distributive lattice L Birkhoff’s representation for a finite distributive lattice L Let L be a finite distributive lattice. Let L be a finite distributive lattice. L is isomorphic to the collection (P) of all down-sets of an L is isomorphic to the collection (P) of all down-sets of an O O ordered set P = P; , under union, intersection, ? and P. ordered set P = P; , under union, intersection, ? and P. h 6i h 6i In fact, we can choose P to be the ordered set (L); of hJ 6i join-irreducible elements of L.

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Representing ﬁnite distributive lattices More examples

Birkhoff’s representation for a ﬁnite distributive lattice L Let L be a ﬁnite distributive lattice. Distributive L is isomorphic to the collection (P) of all down-sets of an O lattice ordered set P = P; 6 , under union, intersection, ? and P. h i L = ( (L)) ⇠ O J In fact, we can choose P to be the ordered set (L); of hJ 6i join-irreducible elements of L.

[G. Birkhoff] Theorem Ordered set Let L be a ﬁnite distributive lattice and let P be a ﬁnite ordered P = ( (P)) set. Then ⇠ J O I L is isomorphic to ( (L)), and O J I P is isomorphic to ( (P)). J O

9 / 35 10 / 35 surjections embeddings ! embeddings surjections ! products disjoint unions !

In fact, we have the following dual order-isomorphism: (L) =@ D(L, 2). J ⇠

Duality for ﬁnite distributive lattices Duality for ﬁnite distributive lattices

The classes of The classes of finite distributive lattices and finite ordered sets finite distributive lattices and finite ordered sets are dually equivalent. are dually equivalent.

surjections embeddings ! embeddings surjections ! products disjoint unions !

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Outline Duals of ﬁnite bounded distributive lattices

Bounded distributive lattices Let L = L; , , 0, 1 be a finite bounded distributive lattice. h _ ^ i Priestley duality for finite distributive lattices We can define its dual D(L) to be either

I (L) — the ordered set of join-irreducible elements of L Priestley duality via homsets J or I D(L, 2) — the ordered set of 0, 1 -homomorphisms from Priestley duality for inﬁnite distributive lattices { } L to the two-element bounded lattice 2 = 0, 1 ; , , 0, 1 . h{ } _ ^ i Examples of Priestley spaces

The translation industry: restricted Priestley duals Here D denotes the category of bounded distributive lattices.

Useful facts about Priestley spaces

12 / 35 13 / 35 In fact, we have the following dual lattice-isomorphism: @ (P) ⇠= P(P, 2). O ⇠

I There is a bijection between the order-preserving maps from P to Q and the 0, 1 -homomorphisms from E(Q) { } to E(P). Given ': P Q, we deﬁne ! 1 f : (Q) (P) by f (A):=' (A), O ! O f : P(Q, 2) P(P, 2) by f (↵):=↵ '. ⇠ ! ⇠ We denote the map f by E(').

Duals of ﬁnite bounded distributive lattices Duals of ﬁnite ordered sets

Let P = P; be a finite ordered set. h 6i Let L = L; , , 0, 1 be a finite bounded distributive lattice. h _ ^ i We can define its dual E(P) to be either We can define its dual D(L) to be either I (P) – the lattice of down-sets (= order ideals) of P O I (L) — the ordered set of join-irreducible elements of L J or or I P(P, 2) – the lattice of order-preserving maps from P to I D(L, 2) — the ordered set of 0, 1 -homomorphisms from the two-element⇠ ordered set 2 = 0, 1 ; 6 . { } h{ } i L to the two-element bounded lattice 2 = 0, 1 ; , , 0, 1 . ⇠ h{ } _ ^ i In fact, we have the following dual order-isomorphism: (L) =@ D(L, 2). J ⇠ Here P denotes the category of ordered sets. Here D denotes the category of bounded distributive lattices. (Warning! The definitions of 2 and P will change once we consider the infinite case.) ⇠

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Duals of ﬁnite ordered sets Duals of morphisms

Let L and K be a finite distributive lattices and let P and Q be a Let P = P; be a finite ordered set. finite ordered sets. h 6i I There is a bijection between the 0, 1 -homomorphisms We can define its dual E(P) to be either { } from L to K and the order-preserving maps from D(K) I (P) – the lattice of down-sets (= order ideals) of P to D(L). Given f : L K, we define O ! 1 or ': (K) (L) by '(x):=min(f ( x)), J ! J " I P(P, 2) – the lattice of order-preserving maps from P to ': D(K, 2) D(L, 2) by '(x):=x f . ! the two-element⇠ ordered set 2 = 0, 1 ; 6 . h{ } i We denote the map ' by D(f ). In fact, we have the following dual⇠ lattice-isomorphism: @ (P) ⇠= P(P, 2). O ⇠ Here P denotes the category of ordered sets. (Warning! The definitions of 2 and P will change once we consider the infinite case.) ⇠

14 / 35 15 / 35 Theorem [G. Birkhoff, H. A. Priestley] Every ﬁnite distributive lattice is encoded by an ordered set:

L ⇠= ED(L) and P ⇠= DE(P), for each finite distributive lattice L and finite ordered set P. Indeed, the categories of finite bounded distributive lattices and finite ordered sets are dually equivalent.

Duals of morphisms The duality at the ﬁnite level

Let L and K be a finite distributive lattices and let P and Q be a I 2 = 0, 1 ; , , 0, 1 is the two-element lattice, finite ordered sets. h{ } _ ^ i I 2 = 0, 1 ; 6 is the two-element ordered set with 0 6 1. I There is a bijection between the 0, 1 -homomorphisms ⇠ h{ } i { } from L to K and the order-preserving maps from D(K) Define either to D(L). Given f : L K, we define ! D(L):= (L) and E(P):= (P) 1 J O ': (K) (L) by '(x):=min(f ( x)), or J ! J " ': D(K, 2) D(L, 2) by '(x):=x f . D L P P ! D(L):= (L, 2) 6 2 and E(P):= (P, 2) 6 2 . We denote the map ' by D(f ). ⇠ ⇠ I There is a bijection between the order-preserving maps from P to Q and the 0, 1 -homomorphisms from E(Q) { } to E(P). Given ': P Q, we define ! 1 f : (Q) (P) by f (A):=' (A), O ! O f : P(Q, 2) P(P, 2) by f (↵):=↵ '. ⇠ ! ⇠ We denote the map f by E(').

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The duality at the ﬁnite level Outline

I 2 = 0, 1 ; , , 0, 1 is the two-element lattice, h{ } _ ^ i I 2 = 0, 1 ; 6 is the two-element ordered set with 0 6 1. Bounded distributive lattices ⇠ h{ } i Define either Priestley duality for finite distributive lattices D(L):= (L) and E(P):= (P) J O or Priestley duality via homsets L P D(L):=D(L, 2) 6 2 and E(P):=P(P, 2) 6 2 . ⇠ ⇠ Priestley duality for infinite distributive lattices Theorem [G. Birkhoff, H. A. Priestley] Every finite distributive lattice is encoded by an ordered set: Examples of Priestley spaces

L ⇠= ED(L) and P ⇠= DE(P), The translation industry: restricted Priestley duals for each finite distributive lattice L and finite ordered set P. Indeed, the categories of finite bounded distributive lattices Useful facts about Priestley spaces and finite ordered sets are dually equivalent.

16 / 35 17 / 35 I Since L is inﬁnite, the ordered set would have to be inﬁnite. I So there would be at least 2N down-sets.

I But }FC(N) is countable.

Proof.

I Since L is complemented, the ordered set would have to be an anti-chain. I Since L is inﬁnite, the ordered set would have to be inﬁnite. I So there would be at least 2N down-sets.

I But }FC(N) is countable.

But it can be obtained as the clopen down-sets of a topological But it can be obtained as the clopen down-sets of a topological ordered set. ordered set.

123 4 5 123 4 5 1 1

I So there would be at least 2N down-sets.

I But }FC(N) is countable. I But }FC(N) is countable.

But it can be obtained as the clopen down-sets of a topological But it can be obtained as the clopen down-sets of a topological ordered set. ordered set.

123 4 5 123 4 5 1 1

Inﬁnite distributive lattices Inﬁnite distributive lattices

Example Example The finite-cofinite lattice L = }FC(N); , , ?, N cannot be The finite-cofinite lattice L = }FC(N); , , ?, N cannot be h [ \ i h [ \ i obtained as the down-sets of an ordered set. obtained as the down-sets of an ordered set. Proof.

I Since L is complemented, the ordered set would have to be an anti-chain.

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Inﬁnite distributive lattices Inﬁnite distributive lattices

I Since L is complemented, the ordered set would have to I Since L is complemented, the ordered set would have to be an anti-chain. be an anti-chain. I Since L is infinite, the ordered set would have to be infinite. I Since L is infinite, the ordered set would have to be infinite. I So there would be at least 2N down-sets.

18 / 35 18 / 35 But it can be obtained as the clopen down-sets of a topological ordered set.

123 4 5 1

Inﬁnite distributive lattices Inﬁnite distributive lattices

I But }FC(N) is countable. I But }FC(N) is countable.

But it can be obtained as the clopen down-sets of a topological ordered set.

123 4 5 1 18 / 35 18 / 35

More examples More examples

Distributive lattice: Distributive lattice: All finite subsets of N, as well as N itself, All finite subsets of N, as well as N itself, }fin(N) N ; , , ?, N . N 0 ; lcm, gcd, 1, 0 . h [ { } [ \ i h [ { } i

Topological ordered set: Topological ordered set:

1 1 23

2 5 22 3 4 5 3 1 2 3 2

19 / 35 20 / 35 We first endow 0, 1 ; with the discrete topology: h{ } 6i I 2 = 0, 1 ; , T is the two-element ordered set I 2 = 0, 1 ; , T is the two-element ordered set h{ } 6 i h{ } 6 i with⇠ 0 6 1 endowed with the discrete topology T. with⇠ 0 6 1 endowed with the discrete topology T. Then we define Then we define L L I D(L):=D(L, 2) 6 2 . I D(L):=D(L, 2) 6 2 . We put the pointwise order⇠ and the product topology on 2 L. We put the pointwise order⇠ and the product topology on 2 L. ⇠ ⇠ Then D(L):=D(L, 2) inherits its order and topology from 2 L. Then D(L):=D(L, 2) inherits its order and topology from 2 L. ⇠ ⇠ D(L, 2) is a topologically closed subset of 2 L (easy exercise). D(L, 2) is a topologically closed subset of 2 L (easy exercise). ⇠ ⇠ Hence D(L) is a compact ordered topological space. Hence D(L) is a compact ordered topological space.

Then we deﬁne L I D(L):=D(L, 2) 6 2 . We put the pointwise order⇠ and the product topology on 2 L. We put the pointwise order and the product topology on 2 L. ⇠ ⇠ Then D(L):=D(L, 2) inherits its order and topology from 2 L. Then D(L):=D(L, 2) inherits its order and topology from 2 L. ⇠ ⇠ D(L, 2) is a topologically closed subset of 2 L (easy exercise). D(L, 2) is a topologically closed subset of 2 L (easy exercise). ⇠ ⇠ Hence D(L) is a compact ordered topological space. Hence D(L) is a compact ordered topological space.

The duality in general The duality in general

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The duality in general The duality in general

21 / 35 21 / 35 Then D(L):=D(L, 2) inherits its order and topology from 2 L. ⇠ D(L, 2) is a topologically closed subset of 2 L (easy exercise). D(L, 2) is a topologically closed subset of 2 L (easy exercise). ⇠ ⇠ Hence D(L) is a compact ordered topological space. Hence D(L) is a compact ordered topological space.

Hence D(L) is a compact ordered topological space.

The duality in general The duality in general

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The duality in general The duality in general

I 2 = 0, 1 ; , , 0, 1 is the two-element bounded lattice. I 2 = 0, 1 ; , , 0, 1 is the two-element bounded lattice. h{ } _ ^ i h{ } _ ^ i In general, we need to endow the dual D(L) of a bounded In general, we need to endow the dual D(L) of a bounded distributive lattice L with a topology. This is easy if we define distributive lattice L with a topology. This is easy if we define the dual of L to be D(L):=D(L, 2). the dual of L to be D(L):=D(L, 2). We first endow 0, 1 ; with the discrete topology: We first endow 0, 1 ; with the discrete topology: h{ } 6i h{ } 6i I 2 = 0, 1 ; , T is the two-element ordered set I 2 = 0, 1 ; , T is the two-element ordered set h{ } 6 i h{ } 6 i with⇠ 0 6 1 endowed with the discrete topology T. with⇠ 0 6 1 endowed with the discrete topology T. Then we define Then we define L L I D(L):=D(L, 2) 6 2 . I D(L):=D(L, 2) 6 2 . We put the pointwise order⇠ and the product topology on 2 L. We put the pointwise order⇠ and the product topology on 2 L. ⇠ ⇠ Then D(L):=D(L, 2) inherits its order and topology from 2 L. Then D(L):=D(L, 2) inherits its order and topology from 2 L. ⇠ ⇠ D(L, 2) is a topologically closed subset of 2 L (easy exercise). D(L, 2) is a topologically closed subset of 2 L (easy exercise). ⇠ ⇠ Hence D(L) is a compact ordered topological space. 21 / 35 21 / 35 The following result is very easy to prove. Lemma L I D(L)=D(L, 2) 6 2 is a Priestley space, for every bounded distributive⇠ lattice. X I E(X)=P(X, 2) 6 2 is a bounded distributive lattice, for every Priestley⇠ space X.

D and E are deﬁned on morphisms via composition exactly as they were in the ﬁnite case:

I given f : L K, we deﬁne ! D(f ): D(K, 2) D(L, 2) by D(f )(x):=x f ; ! I given ': X Y, we deﬁne ! E('): P(Y, 2) P(X, 2) by E(')(↵):=↵ '. ⇠ ! ⇠ The functors are contravariant as they reverse the direction of The functors are contravariant as they reverse the direction of the morphisms: D(f ): D(K) D(L) and E('): E(Y) E(X). the morphisms: D(f ): D(K) D(L) and E('): E(Y) E(X). ! ! ! !

Priestley spaces Priestley spaces

L L The ordered space D(L):=D(L, 2) 6 2 is more than a The ordered space D(L):=D(L, 2) 6 2 is more than a compact ordered space. It is a Priestley⇠ space. compact ordered space. It is a Priestley⇠ space. A topological structure X = X; , T is a Priestley space if A topological structure X = X; , T is a Priestley space if h 6 i h 6 i I X; , is an ordered set, I X; , is an ordered set, h 6 i h 6 i I T is a compact topology on X, and I T is a compact topology on X, and I for all x, y X with x ⌦ y, there is a clopen down-set A I for all x, y X with x ⌦ y, there is a clopen down-set A 2 2 of X such that x / A and y A. of X such that x / A and y A. 2 2 2 2 The category of Priestley spaces is denoted by P. The category of Priestley spaces is denoted by P. The following result is very easy to prove. Lemma L I D(L)=D(L, 2) 6 2 is a Priestley space, for every bounded distributive⇠ lattice. X I E(X)=P(X, 2) 6 2 is a bounded distributive lattice, for every Priestley⇠ space X.

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The functors The functors

We now have functors D : D P and E : P D given by We now have functors D : D P and E : P D given by ! ! ! ! L X L X D(L)=D(L, 2) 6 2 and E(X)=P(X, 2) 6 2 . D(L)=D(L, 2) 6 2 and E(X)=P(X, 2) 6 2 . ⇠ ⇠ ⇠ ⇠ D and E are deﬁned on morphisms via composition exactly as they were in the ﬁnite case:

I given f : L K, we deﬁne ! D(f ): D(K, 2) D(L, 2) by D(f )(x):=x f ; ! I given ': X Y, we deﬁne ! E('): P(Y, 2) P(X, 2) by E(')(↵):=↵ '. ⇠ ! ⇠

23 / 35 23 / 35 namely,

I eL : L P(D(L, 2), 2) given by a eL(a), ! ⇠ 7! where eL(a): D(L, 2) 2 : x x(a), ! ⇠ 7! I "X : X D(P(X, 2), 2) given by x "X(x), ! ⇠ 7! where "X(x): P(X, 2) 2 : ↵ ↵(x). ⇠ ! 7! Priestley duality tells us that these maps are isomorphisms (in D or P, as appropriate).

The functors The functors

I given f : L K, we define I given f : L K, we define ! ! D(f ): D(K, 2) D(L, 2) by D(f )(x):=x f ; D(f ): D(K, 2) D(L, 2) by D(f )(x):=x f ; ! ! I given ': X Y, we define I given ': X Y, we define ! ! E('): P(Y, 2) P(X, 2) by E(')(↵):=↵ '. E('): P(Y, 2) P(X, 2) by E(')(↵):=↵ '. ⇠ ! ⇠ ⇠ ! ⇠ The functors are contravariant as they reverse the direction of The functors are contravariant as they reverse the direction of the morphisms: D(f ): D(K) D(L) and E('): E(Y) E(X). the morphisms: D(f ): D(K) D(L) and E('): E(Y) E(X). ! ! ! !

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The functors The natural transformations

We now have functors D : D P and E : P D given by ! ! Let L D and let X P. There are natural maps L X 2 2 D(L)=D(L, 2) 6 2 and E(X)=P(X, 2) 6 2 . e : L ED(L) and " : X DE(X) ⇠ ⇠ L ! X ! D and E are deﬁned on morphisms via composition exactly as they were in the ﬁnite case: to the double duals;

23 / 35 24 / 35 namely,

I eL : L P(D(L, 2), 2) given by a eL(a), ! ⇠ 7! where eL(a): D(L, 2) 2 : x x(a), ! ⇠ 7! I "X : X D(P(X, 2), 2) given by x "X(x), I "X : X D(P(X, 2), 2) given by x "X(x), ! ⇠ 7! ! ⇠ 7! where "X(x): P(X, 2) 2 : ↵ ↵(x). where "X(x): P(X, 2) 2 : ↵ ↵(x). ⇠ ! 7! ⇠ ! 7! Priestley duality tells us that these maps are isomorphisms Priestley duality tells us that these maps are isomorphisms (in D or P, as appropriate). (in D or P, as appropriate).

Priestley duality tells us that these maps are isomorphisms (in D or P, as appropriate).

The natural transformations The natural transformations

Let L D and let X P. There are natural maps Let L D and let X P. There are natural maps 2 2 2 2 e : L ED(L) and " : X DE(X) e : L ED(L) and " : X DE(X) L ! X ! L ! X ! to the double duals; to the double duals; namely,

I eL : L P(D(L, 2), 2) given by a eL(a), ! ⇠ 7! where eL(a): D(L, 2) 2 : x x(a), ! ⇠ 7!

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The natural transformations The natural transformations

Let L D and let X P. There are natural maps Let L D and let X P. There are natural maps 2 2 2 2 e : L ED(L) and " : X DE(X) e : L ED(L) and " : X DE(X) L ! X ! L ! X ! to the double duals; namely, to the double duals; namely,

I eL : L P(D(L, 2), 2) given by a eL(a), I eL : L P(D(L, 2), 2) given by a eL(a), ! ⇠ 7! ! ⇠ 7! where eL(a): D(L, 2) 2 : x x(a), where eL(a): D(L, 2) 2 : x x(a), ! ⇠ 7! ! ⇠ 7! I "X : X D(P(X, 2), 2) given by x "X(x), I "X : X D(P(X, 2), 2) given by x "X(x), ! ⇠ 7! ! ⇠ 7! where "X(x): P(X, 2) 2 : ↵ ↵(x). where "X(x): P(X, 2) 2 : ↵ ↵(x). ⇠ ! 7! ⇠ ! 7! Priestley duality tells us that these maps are isomorphisms (in D or P, as appropriate).

24 / 35 24 / 35 I C; T is the Cantor space created by successively deleting h i middle thirds of the unit interval, and I x < y if and only if x and y are the endpoints of a deleted middle third.

I [A. Stralka] Let C = C; , T , where h 6 i

Then is closed in C C, but C = C; , T is not a 6 ⇥ h 6 i Priestley space. I You can’t draw the Stralka space. I [Bezhanishvili, Mines, Morandi] Any ordered compact space X that you can draw in which 6 is topologically closed in X X is a Priestley space. ⇥

Priestley duality Outline

Bounded distributive lattices

Priestley duality for ﬁnite distributive lattices Theorem (Priestley duality)

I The functors D : D P and E : P D give a dual Priestley duality via homsets ! ! category equivalence between D and P. Priestley duality for inﬁnite distributive lattices I In particular, e : L ED(L) and " : X DE(X) are L ! X ! isomorphisms for all L D and X P. 2 2 Examples of Priestley spaces

The translation industry: restricted Priestley duals

Useful facts about Priestley spaces

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Some ordered spaces A subtlety The following ﬁgure comes from Chapter 11 of Davey268 and Priestley:Representation: Introduction the general to Lattices case and Order. I If X = X; , T is a Priestley space, then is a 6 6 b1 b2 b3 b 1 h i topologically closed subset of X X. X X 1 2 ⇥ 3 4 a1 a2 a3 a 1 2

b1 b2 b3 b X3 a a3 a2 a1

b1 b2 b3 b X4 c a1 a2 a3 a

b1 b3 b5 b b6 b4 b2 X5 a1 a3 a5 a a6 a4 a2 Figure 11.5 27 / 35 28 / 35

11.10 Consider the ordered spaces shown in Figure 11.5. In each case the order and the topology should be apparent. For example, in X1 the only comparabilities are a1 < b1 and an < bn 1,an < bn for n 2; note, in particular, that a b .Asatopological space, X1 is the disjoint union of two copies on N ,namely a1,a2,... a and b1,b2,... b .Theotherexamples { } [ { } { } [ { } are built similarly from one or two copies of N . (i) Show that X / P. 1 2 (ii) Show that X P and describe all the elements of (X ). 2 2 O 2 (iii) Consider the Priestley space Y given in Figure 11.2. Show that (Y )isasublatticeofFC(N)anddescribetheelements of (OY )(intermsofoddandevennumbers). O (iv) Show that X P.Showthat (X )isisomorphictoa 3 2 O 3 sublattice of FC(N) FC(N). [Hint. Find a continuous order- ⇥ . preserving map from N N onto X3 then use the duality.] Give an explicit description [ of the elements of this sublattice of FC(N) FC(N). ⇥ (v) Show that X P and X P. 4 2 5 2 I C; T is the Cantor space created by successively deleting h i middle thirds of the unit interval, and I x < y if and only if x and y are the endpoints of a deleted I x < y if and only if x and y are the endpoints of a deleted middle third. middle third. Then is closed in C C, but C = C; , T is not a Then is closed in C C, but C = C; , T is not a 6 ⇥ h 6 i 6 ⇥ h 6 i Priestley space. Priestley space. I You can’t draw the Stralka space. I You can’t draw the Stralka space. I [Bezhanishvili, Mines, Morandi] Any ordered compact I [Bezhanishvili, Mines, Morandi] Any ordered compact space X that you can draw in which 6 is topologically space X that you can draw in which 6 is topologically closed in X X is a Priestley space. closed in X X is a Priestley space. ⇥ ⇥

Then is closed in C C, but C = C; , T is not a 6 ⇥ h 6 i Priestley space. I You can’t draw the Stralka space. I You can’t draw the Stralka space. I [Bezhanishvili, Mines, Morandi] Any ordered compact I [Bezhanishvili, Mines, Morandi] Any ordered compact space X that you can draw in which 6 is topologically space X that you can draw in which 6 is topologically closed in X X is a Priestley space. closed in X X is a Priestley space. ⇥ ⇥

A subtlety A subtlety

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A subtlety A subtlety

I If X = X; , T is a Priestley space, then is a I If X = X; , T is a Priestley space, then is a h 6 i 6 h 6 i 6 topologically closed subset of X X. topologically closed subset of X X. ⇥ ⇥ I [A. Stralka] Let C = C; , T , where I [A. Stralka] Let C = C; , T , where h 6 i h 6 i I C; T is the Cantor space created by successively deleting I C; T is the Cantor space created by successively deleting h i h i middle thirds of the unit interval, and middle thirds of the unit interval, and I x < y if and only if x and y are the endpoints of a deleted I x < y if and only if x and y are the endpoints of a deleted middle third. middle third. Then is closed in C C, but C = C; , T is not a 6 ⇥ h 6 i Priestley space.

28 / 35 28 / 35 I [Bezhanishvili, Mines, Morandi] Any ordered compact space X that you can draw in which 6 is topologically closed in X X is a Priestley space. ⇥

A subtlety A subtlety

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A subtlety Outline

I If X = X; , T is a Priestley space, then is a Bounded distributive lattices h 6 i 6 topologically closed subset of X X. ⇥ I [A. Stralka] Let C = C; , T , where Priestley duality for ﬁnite distributive lattices h 6 i I C; T is the Cantor space created by successively deleting h i middle thirds of the unit interval, and Priestley duality via homsets I x < y if and only if x and y are the endpoints of a deleted middle third. Priestley duality for inﬁnite distributive lattices Then is closed in C C, but C = C; , T is not a 6 ⇥ h 6 i Priestley space. Examples of Priestley spaces I You can’t draw the Stralka space. I [Bezhanishvili, Mines, Morandi] Any ordered compact The translation industry: restricted Priestley duals space X that you can draw in which 6 is topologically closed in X X is a Priestley space. ⇥ Useful facts about Priestley spaces

28 / 35 29 / 35 I Hence when translating properties of distributive lattices into properties of Priestley spaces, it is common to use clopen up-sets (or their complements, i.e., clopen down-sets).

I X is the dual of a distributive p-algebra, and called a p-space, iff

I U is clopen, for every clopen up-set U; # then U = X U in T(X). ⇤ \# U I If X and Y are p-spaces, then ': X Y is the dual of a I If X and Y are p-spaces, then ': X Y is the dual of a ! ! p-algebra homomorphism iff p-algebra homomorphism iff

I '(max(x)) = max('(x)), for all x X. I '(max(x)) = max('(x)), for all x X. 2 2 (Here max(z) denotes the set of maximal elements in z.) (Here max(z) denotes the set of maximal elements in z.) " "

The translation industry: restricted Priestley duals The translation industry: restricted Priestley duals

T T I Since P(X, 2) is isomorphic to the lattice (X) of clopen I Since P(X, 2) is isomorphic to the lattice (X) of clopen U U up-sets of X⇠, it is common to deﬁne the dual E(X) of a up-sets of X⇠, it is common to deﬁne the dual E(X) of a Priestley space X to be T(X). Priestley space X to be T(X). U U

I Hence when translating properties of distributive lattices into properties of Priestley spaces, it is common to use clopen up-sets (or their complements, i.e., clopen down-sets).

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The translation industry: restricted Priestley duals The translation industry: restricted Priestley duals

Examples: p-algebras Examples: p-algebras Let X and Y be a Priestley spaces and let ': X Y be Let X and Y be a Priestley spaces and let ': X Y be ! ! continuous and order-preserving. continuous and order-preserving.

I X is the dual of a distributive p-algebra, and called a p-space, iff

I U is clopen, for every clopen up-set U; # then U = X U in T(X). ⇤ \# U

31 / 35 31 / 35 I If X and Y are Heyting-spaces, then ': X Y is the dual ! of a Heyting algebra homomorphism iff

I '( x)= '(x), for all x X. " " 2

I Thus X = X; g, , T will be the restricted Priestley dual of h 6 i an Ockham algebra, known as an Ockham space, if X[ := X; , Tbis a Priestley space and g : X X is an h 6 i ! order-dual endomorphism of X[. I If X and Y are Ockham spaces, then ': bX Y is the dual ! of an Ockham algebra homomorphism if it is continuous, order-preserving and preserves the unary operation g.

The translation industry: restricted Priestley duals The translation industry: restricted Priestley duals

Examples: p-algebras Examples: Heyting algebras Let X and Y be a Priestley spaces and let ': X Y be ! Let X and Y be a Priestley spaces and let ': X Y be continuous and order-preserving. ! continuous and order-preserving. I X is the dual of a distributive p-algebra, and called a I X is the dual of a Heyting algebra, and called a p-space, iff Heyting-space (or Esakia space), iff I U is clopen, for every clopen up-set U; # I U is open, for every open up-set U; T # then U⇤ = X U in (X). \# U then U V = X (U V ) in T(X). ! \ # \ U I If X and Y are p-spaces, then ': X Y is the dual of a ! p-algebra homomorphism iff

I '(max(x)) = max('(x)), for all x X. 2 (Here max(z) denotes the set of maximal elements in z.) "

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The translation industry: restricted Priestley duals The translation industry: restricted Priestley duals

Examples: Ockham algebras

Examples: Heyting algebras I A = A; , , g, 0, 1 is an Ockham algebra if [ h _ ^ i Let X and Y be a Priestley spaces and let ': X Y be A := A; , , 0, 1 is a bounded distributive lattice and ! h _ ^ i continuous and order-preserving. g satisﬁes De Morgan’s laws and is Boolean complement on 0, 1 ; in symbols, I X is the dual of a Heyting algebra, and called a { } Heyting-space (or Esakia space), iff g(a b)=g(a) g(b), g(a b)=g(a) g(b), g(0)=1, g(1)=0, _ ^ ^ _ I U is open, for every open up-set U; # i.e., g is a lattice-dual endomorphism of A[. then U V = X (U V ) in T(X). ! \ # \ U I If X and Y are Heyting-spaces, then ': X Y is the dual ! of a Heyting algebra homomorphism iff

I '( x)= '(x), for all x X. " " 2

32 / 35 33 / 35 I If X and Y are Ockham spaces, then ': X Y is the dual ! of an Ockham algebra homomorphism if it is continuous, order-preserving and preserves the unary operation g.

The translation industry: restricted Priestley duals The translation industry: restricted Priestley duals

Examples: Ockham algebras Examples: Ockham algebras

I A = A; , , g, 0, 1 is an Ockham algebra if I A = A; , , g, 0, 1 is an Ockham algebra if h _ ^ i h _ ^ i A[ := A; , , 0, 1 is a bounded distributive lattice and A[ := A; , , 0, 1 is a bounded distributive lattice and h _ ^ i h _ ^ i g satisﬁes De Morgan’s laws and is Boolean complement g satisﬁes De Morgan’s laws and is Boolean complement on 0, 1 ; in symbols, on 0, 1 ; in symbols, { } { } g(a b)=g(a) g(b), g(a b)=g(a) g(b), g(0)=1, g(1)=0, g(a b)=g(a) g(b), g(a b)=g(a) g(b), g(0)=1, g(1)=0, _ ^ ^ _ _ ^ ^ _ i.e., g is a lattice-dual endomorphism of A[. i.e., g is a lattice-dual endomorphism of A[. I Thus X = X; g, , T will be the restricted Priestley dual of I Thus X = X; g, , T will be the restricted Priestley dual of h 6 i h 6 i an Ockham algebra, known as an Ockham space, if an Ockham algebra, known as an Ockham space, if X[ := X; , Tbis a Priestley space and g : X X is an X[ := X; , Tbis a Priestley space and g : X X is an h 6 i ! h 6 i ! order-dual endomorphism of X[. order-dual endomorphism of X[. b I If X and Y are Ockham spaces, then ': bX Y is the dual ! of an Ockham algebra homomorphism if it is continuous, order-preserving and preserves the unary operation g. 33 / 35 33 / 35 b Outline Useful facts about Priestley spaces

Prove each of the following claims. The order-theoretic dual of Bounded distributive lattices each statement is also true. Let X = X; , T be a Priestley space. Priestley duality for ﬁnite distributive lattices h 6 i (1) The set Y := x X ( y Y ) x y is closed in X # { 2 | 9 2 6 } provided Y is closed in X. In particular, y is closed in X, Priestley duality via homsets # for all y X. 2 Priestley duality for inﬁnite distributive lattices (2) Every up-directed subset of X has a least upper bound in X. (3) The set Min(X) of minimal elements of X is non-empty. Examples of Priestley spaces (4) Let Y and Z be disjoint closed subsets of X such that Y is a down-set and Z is an up-set. Then there is a clopen down-set U with Y U and U Z = ?. The translation industry: restricted Priestley duals ✓ \ (5) Y is a closed down-set in X if and only if Y is an Useful facts about Priestley spaces intersection of clopen down-sets.

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