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 1 / 35 2 / 35 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 1 3 2 1 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 7 / 35 7 / 35 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. 9 / 35 9 / 35 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 ! 11 / 35 11 / 35 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.) ⇠ 13 / 35 14 / 35 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).