[North-Holland Mathematics Studies] Orders: Description and Roles - In Set Theory, Lattices, Ordered Groups, Topology, Theory of Models and Relations, Combinatorics, Effectiveness,

Annals of Discrete Mathematics 23 (1984) 39-60 0 Elsevier Science Publishers B.V. (North-Holland) 39

ORDERED SETS AND DUALITY FOR DISTRIBUTIVE LATTICES

H . A . P r i e s t l e y

M a t h e m a t i c a l I n s t i t u t e U n i v e r s i t y of Oxford

England

An a c c o u n t i s g i v e n o f t h e c a t e g o r i c a l d u a l i t y which e x i s t s be tween bounded d i s t r i b u t i v e l a t t i c e s and compact t o t a l l y o r d e r d i s c o n n e c t e d s p a c e s . D u r i n g t h e p a s t d e c a d e , a w i d e r a n g e of s t r u c t u r a l p roblems c o n c e r n i n g d i s t r i b u t i v e l a t t i c e s h a v e b e e n s o l v e d by t h e t o p o l o g i c a l and o r d e r t h e o r e t i c t e c h n i q u e s p r o v i d e d by d u a l i t y , and a r e p r e s e n t a t i v e s e l e c t i o n o f t h e s e i s p r e s e n t e d . I n a d d i t i o n , c e r t a i n r e l a t e d d u a l i t i e s are b r i e f l y c o n s i d e r e d , as are compact t o t a l l y o r d e r d i s c o n n e c t e d s p a c e s i n t h e i r own r i g h t . The p a p e r e n d s w i t h a ' d i c t i o n a r y ' o f m u t u a l l y d u a l p r o p e r t i e s .

ENSEMBLES O R D O N N ~ S E T DUAL IT^ POUR LES TREILLIS DISTRIBUTIFS

Au c o u r s d e ces d e r n i P r e s a n n g e s , l es d u a l i t 6 s du t y p e P o n t r y a g i n o n t p r o l i f 6 r 6 . L 'une d ' e l l e s e s t c e l l e e x i s t a n t e n t r e l a c a t e ' g o r i e d e s t r e i l l i s d i s t r i b u t i f s e t b o r n 6 s e t la c a t 6 g o r i e d e s e s p a c e s compacts e t t o t a l e m e n t s g p a r g s p o u r l ' o r d r e . (Un e s p a c e t o p o l o g i q u e X a v e c une r e l a t i o n d ' o r d r e p a r t i e l g e s t t o t a l e m e n t s 6 p a r 6 pour l ' o r d r e s i , q u e l s q u e s o i e n t l e s p o i n t s x,y d e X a v e c LI: $ y , il e x i s t e un ensemble d g c r o i s s a n t ( c ' e s t - & d i r e , un i d g a l p o u r l ' o r d r e ) U q u i e s t o u v e r t e t ferm6 e t t e l s que z 4 U. y E U.) e n t r e 1) e t P , 2 ses r e p e r c u s s i o n s e t 2 s a p l a c e d a n s l e p a y s a g e math6mat ique .

Au n i v e a u d e s o b j e t s , l a d u a l i t 6 p e r m e t d ' i d e n t i f i e r un t r e i l l i s L 6 D a v e c l e t r e i l l i s d e s e n s e m b l e s d g c r o i s s a n t - o u v e r t s e t fe rm6s d ' u n e s p a c e XL 6-z; p o u r XL, o n p e u t p r e n d r e l ' e n s e m b l e d e s ide 'aux p r e m i e r s , o rdonn6 p a r i n c l u s i o n e t convenablement t o p o l o g i s 6 . On a donc l a g 6 n 6 r a l i s a t i o n n a t u r e l l e s i m u l t a n 6 e d e l a r e p r g s e n t a t i o n d e B i r k h o f f d e s t r e i l l i s d i s t r i b u t i f s f i n i s (oh l a t o p o l o g i e e s t d i s c r P t e e t n e j o u e aucun r 6 1 e ) e t l a r e p r 6 s e n t a t i o n d e S t o n e d e s a l g P b r e s d e Boole (oh l ' o r d r e e s t d i s c r e t ) .

Une d u a l i t 6 c a t 6 g o r i q u e p a r f a i t e u n i t l a t h 6 o r i e des t r e i l l i s d a n s l a l a n g u e d e s e s p a c e s t o p o l o g i q u e s o r d o n n g s . La r e p r s s e n t a t i o n imag6e q u e ces e s p a c e s f o u r n i s s e n t ( b i e n q u e c e l l e - c i ne s o i t p l u s guPre c o n s t i t u 6 e q u e d e diagrammes d e Venn s o p h i s t i q u s s ) e s t 6 t o n n a m e n t p u i s s a n t e . E l l e e s t i l l u s t r 4 e p a r un mblange d ' a p p l i c a t i o n s e t d ' e x e m p l e s q u i m e t t e n t B nu l a s t r u c t u r e d e s t r e i l l i s e t l a s t r u c t u r e d e s v a r i e t 6 s d ' a l g P b r e s dans u. L ' o r d r e j o u e un rsle c r u c i a l e n d u d l i t 6 c-f. c e t t e d u a l i t 6 d e l a d u a l i t 6 e q u i v a l e n t e p o u r m e t t a n t e n j e u l es e s p a c e s s p e c t r a u x . L ' a c t i o n r k c i p r o q u e e n t r e l a t o p o l o g i e e t l ' o r d r e s u r un o b j e t d e p e u t 6 t r e a s s e z s u b t i l e . c o n c e r n a n t l ' o r d r e ( i l y a , p a r exemple , d e s d e s c r i p t i o n s d i v e r s e s d e c e u x d e s ensembles o r d o n n 6 s q u i p e u v e n t d e v e n i r o b j e t s d e c ) , beaucoup d e p r o b l s m e s e n c o r e e n s u s p e n s s e m b l e n t a u s s i a t t i r a n t s q u ' 6 p i n e u x !

Le p r 6 s e n t d6veloppement s ' i n t 6 r e s s e 2 l a d u a l i t 6

- -

e t P e t p e r m e t d e t r a d u i r e l es c o n c e p t s d e

En e f f e t , c ' e s t l ' o i d r e q u i d i s t i n g u e

A l o r s qu'on a don& une r 6 p o n s e a c e r t a i n e s q u e s t i o n s

40 H. A . Priestle y

1. INTRODUC'I'LO?:

Dur ing t h e p a s t t e n y e a r s , d u a l i t i e s of P o n t r y a g i n t y p e have p r o l i c e r a t e d . One such d u a l i t y i s t h a t be tween t h e c a t e g o r y o f bounded d i s t r i b u t i v e l a t t i c e s and t h e c a t e g o r y 2 of compact t o t a l l y o r d e r d i s c o n n e c t e d s p a c e s , w h i c h , a t t h e o b j e c t l e v e l , p r o v i d e s t h e n a t u r a l s i m u l t a n e o u s g e n c r a l i s a t i o n of B i r k h o f f ' s r e p r e s e n t a t i o n o f f i n i t e d i s t r i b u t i v e l a t t i c e s and S t o n e ' s r r p r e s e n L a t i o n o f Boolean a l g e b r a s , The f u l l c a t e g o r i c a l d u a l i t y which l i n k s and a l l o w s l a t t i c e t h e o r e t i c c u n c e p t s and problems t o be t r a n s l a t e d i n t o t h e language of o r d e r e d t o p o l o g i c a l s p a c e s . The f i n a l s e c t i o n of t h i s s u r v e y a s s e m b l e s f o r r e f e r e n c e some of t h e most f r e q u e n t l y used items i n t h e LLE ' d i c t i o n a r y ' . s e c t i o n s we d e s c r i b e tiow t h e d u a l i t y works and i l l u s t r a t e some of t h e ways i n which i t has b e e n used t o r e v e a l t h e s t r u c t u r e of l a t t i c e s i n and of c e r t a i n s u b c l a s s e s of !. t o p o l o g y and o r d e r i n R :-object .3nd, i n b a r e s t o u t l i n e , we c o n s i d e r t h e g e n e r a l s e t t i n g i n which d i s t r i b u t i v e l a t t i c e d u a l i t y s h o u l d b e p l a c e d .

I t h a s n o t been p o s s i b l e i n t h e s p a c e a v a i l a b l e t o p r o v i d e :I f u l l y comprehens ive s u r v e y . I n s e l e c t i n g m a t e r i a l we have t r i e d t o complement t h e a d m i r a b l e a c c o u n t o f d u a l i t y p r e s e n t e d by B . A . Uavcy and D . Duffus i n t h e i r p a p e r ' E x p o n e n t i a t i o n and d u a l i t y ' p u b l i s h e d i n t h e P r o c e e d i n g s of t h e Banff Symposium on Ordered S e t s 1371; o u r approach i s somewhat more t o p o l o g i c a l and p e r h a p s l e s s i n f l u e n c e d by u n i v e r s a l a l g e b r a . We h a v e t r i e d t o make t h e b i b l i o g r a p h y c o m p l e t e i n r e s p e c t o f t h e E-E d u a l i t y and i t s a p p l i c a t i o n s to problems c o n c e r n i n g d i s t r i b u t i v e - l n t t i r r - o r d e r e d a l g e b r a s , b u t h a v e n o t s o u g h t t o g i v e full r e f e r e n c e s f o r t h e same problems t r e a t e d a l g e b r a i c a l l y .

We assume f a m i l i a r i t y w i t h t h e r u d i m e n t s of t h e t h e o r y of d i s t r i b u t i v e l a t t i c e s a s s e t o u t i n 1 1 4 7 , 1191 and C541. Our r e p r e s e n t a t i o n s p a c e s w i l l be o r d e r e d s e t s , a p p r o p r i a t e l y t o p o l o g i s e d . The n o t a t i o n ( X , C , d w i l l d e n o t e a s e t X e q u i p p e d w i t h a t o p o l o g y e and a p a r t i a l o r d e r <. The b a s i c p r o p e r t i e s of s u c h s p a c e s were d e v e l o p e d i n 1681; s e e a l s o C 2 4 1 , [ 4 9 1 and 1711. I t i s p e r h a p s w o r t h e m p h a s i s i n g now t h a t t h e p r o o f s u s i n g d u a l i t y of t h e v a s t m a j o r i t y of t h e a l g e b r a i c theorems s t a t e d i n t h i s p a p e r r e q u i r e t o p o l o g i c a l e x p e r t i s e no g r e a t e r t h a n t h a t i m p a r t e d by a f i r s t ( u n d e r g r a d u a t e l e v e l ) c o u r s e i n g e n e r a l t o p o l o g y .

I f iP,d i s an o r d e r e d s e t , we w r i t e , f o r :r E I>,

111 t h e e a r l i e r

In a d d i t i o n we b r i e f l y i n v e s t i g a t e t h e i n t e r a c t i o n be tween t h e

4r= i y E P I y < 5 } t Z = i y E z ' I ~ { > a l and , f o r 12 & P ,

4Q = U i C Z I a E Q), I-& = U[+x I 5 E Q I . A s u b s e t Q o f P i s s a i d t o be decreasing ( increasing) i f +Q = Q (+Q = Q). In t h e l i t e r a t u r e , d e c r e a s i n g s e t s are a l s o c a l l e d o r d e r i d e a l s , h e r e d i t a r y s e t s , i n i t i a l segments , down se t s , l o w e r s e t s and lower e n d s . I f t' i s a n o r d e r e d se t we d e n o t e by Pop t h e o r d e r e d s e t o b t a i n e d by r e v e r s i n g t h e o r d e r .

2 . BASIC DUALITY FOR DISTRIBUTIVE LATTICES

The c a t e g o r y h a s as o b j e c t s t h e bounded d i s t r i b u t i v e l a t t i c e s ; t h e u n i v e r s a l bounds o f any L, t i, a r e d e n o t e d by O,1. l a t t i c e homomorphisms. W i t h i n a r e two i m p o r t a n t f u l l s u b c a t e g o r i e s : c o n s i s t s of t h e f i n i t e d i s t r i b u t i v e l a t t i c e s and of t h e Boolean a l g e b r a s . A s l o n g ago a s t h e 1 9 3 0 ' s key r e p r e s e n t a t i o n theorems were proved f o r t h e s e s u b c a t e g o r i e s :

Morphisms i n 0, a r e t h e 0 , l - p r e s e r v i n g

THEOREM 2 . 1 ( G . B i r k h o f f 1181; or see L l 9 1 ) . to t h c latticc, of decrea::lng .:uhsct:: of t h e S L L J ( L ) o f Join- irreducible elements of L .

Let L € L&.. Then I , i s isomorpJLic

Ordered sets and duality for distributive lattices 41

13irI<ltoff s h o w c ~ d t h . t t ,itiy t 1 i s isontorpl t i ( . L O i~ r i n g of s e t s . T h i s w a s rn.ldc, i i i o r t ' prLst-isL, by S t o n i , , w l i o g ~ v c i t i I 8 5 I a p u r e l y t o p o l o g i c ; ~ l ~ - c ' p r e s c n t ; ~ ~ i o n f o r N I '-cjI).j t,c t s L 11.1 t 11~1cs nu t t ir;in s j i ; i i -c ' t t t 1 y r e d ucc t o 13 i r k h o r f ' s r e p resc7.n t ii t i o r OII c,), . I n r c t respect ri ntot-c. ti;ittii-Lil appi-oa<,lt set'tns o b v i o u s . For f, f i n i t c , t h e m:ip

f ( ++ 1 ' t , i s i ' t s u p ; in i somorphisni ht.twt.cn , 7 f / , j a n d t h e pr imc i d e a l s o f /, o r d c r e d by i n c l u s i o n . F o r i Bmi lean , thc. maxinr,iI idea ls c t l i n t , i d c ' w i t h t h e p r i t w id txa ls . 'Thus r t t i Lipprupt- iaLe r c - p r c s e n L n t i o n spac.e f ~ r 1, E w i ~ u l d seL>m tcl h e t h c se t o f pr ime i d e c i l s o l 1 , c , q u i p p d wit11 an ordctr ( r ' f . Thc.clrc,nt 2 . 1 ) - a n d 3 t o p o l o g y (i,J'. Theorem 2 . 2 ) .

A c c u r d i n g l y , t c i eaclt 1, t 1 ' w e a s s o c i a t e i t s Ti', (Xi,, k, G) , where

H i s t h c s e t or pr ime idt,:ils o f /,,

,i i s s e t i n c l u s i o n

k h a s : I S .I bas t , t h c s t ' t s t , r t .X I .I'

/

and

IJe c a n i d c n t i t y (k'L,P,2) w i t h thr' honi-set p,(l,,z) of morphisnts on to t h e 2-elemcnt c h a i n , p o i n t w i s e o r d e r e d and t o p o l o g i s c d a s ;I s u h s p a c t ~ o f ?-'I, w h e r e 2 d e n o t e s the, 2-elfmcxnt ch,iin w i t h tlic d i s c r c t c t o p o l o g y .

T h e t o p o l o g y P i s compact . A l s o , t h e t h e c r u c i , i l p r o p e r t y of Lctfil c ? r c / c ' 2 3 L

p r o p e r t i e s o f compact t o t a l l y o r d e r n e c t e d s p a c e s a r e c o l l e c t e d t o g e t h e r i n P r o pi's i t i o t i 2 . 6 .

We d e f i n e t o he t h e c a t e g o r y o r conipnct t o t a l l y o r d e r d i s c o n n e c t e d s p a c e s ( P r i e s t l e y s p a c e s ) and c o n t i n u o u s o r d e r - p r e s e r v i n g maps. For i' E we d e n o t e by U ( P ) t h e l a t t i c e o f P-clopen d e c r c a s i n g s u b s e t s of P . We d e f e r u n t i l 5 7 t h e f o r m a l c a t e g o r i c a l s t a t e m e n t of t h e Q-E d u a l i t y . F o r o u r immediate p u r p o s e s we need o n l y t h e r e p r e s e n t a t i o n f o r o b j e c t s (which giv6.s t h e common g e n e r a l i s a t i o n of Theorems 2 .1 and 2 . 2 ) and t h e basic f a c t s a b o u t d u a l morphisms.

THEOREM 2 . 3 ( 1 7 0 1 ) . ( i ) L e t L € c. Y ' / w ? ? ,Y E I' nizd I, 2 O(.Y ) 7iin tiit, niq,

( i i ) Lr:t T ' E c. Tt' /di O(/)) E

c z ! , is€ A'), I . I ' 3 ( T I , f o r ii t l , . /

l o g y of (Xl,:k,G) a r e l i n k e d by : g i v e n p o i n t s .I,,;/ i n XL w i t h

3: & y, t h e r e e x i s t s ;I &-clopen d c c r r , t h a t : I .$ v, !/ € 1'. Basic

I, T, a +-+ in E X L 11' + (21.

t r i i d F' 2 X i ) (,,) (2 1ic'r.c nimning h o n ; c w r i ( iryii L:ni trntl or&l~ i s o ~ I ~ L i Y ~ p l 1 ism) .

We s h a l l h e n c e f o r t h i d e n t i f y i, w i t h O ( X ) and I ' w i t h ( L E Q, 1' € L l . L*

The p r o o f of Theorem 2 . 3 i n v o l v e s a c o n s t r u c t i o n of pr ime i d e a l s which n e c e s s i t a t e s i n v o k i n g snme form of t h e Axiom of C h o i c e . I n f a c t t h e d u a l i t y theorem f o r is e q u i v a l e n t t o t h e Roolean Prime I d e a l Theorem, which i s s t r i c t l y weaker t h a n ( A C ) . However t h e s t a t e m e n t t h a t e v e r y d i s t r i b u t i v e l a t t i c e w i t h 1 h a s a maximal i d e a l i m p l i e s (AC) ( s e e 1 1 7 1 ) . A f u l l e r d i s c u s s i o n of t h e s e p o i n t s c a n be found i n I h O l .

The d u a l i t y t h e o r e m c a n e a s i l y be a d a p t e d t o h a n d l e l a t t i c e s which l a c k e i t h e r z e r o o r i d e n t i t y ( s e e 1 7 1 1 ) . However i n p r a c t i c e i t i s u s u a l l y e a s i e r t o a d j o i n b o u n d s , s o as t o be a b l e t o work i n E .

For morphisms we h a v e

PROPOSITION 2.4 ([70], 1711) . Thcrc exi::ts a ii- c Lion Oc h l L ' f 3 1 2 ~ - 1 m ~ p 1 1 i : ; i n s f:L + Id and ~-rr~orpl~i:;ins $ : X M + X givcn l ) ~ ! E j "(u) i f arid onZ!/ if $(!/) E n L j

42 H.A. Priestley

(a E L, y E XM). f is i n j e c t i v e if and only if $ is sur jec t ivc .

COROLLARY 2 .5 . Let L,M E c. ' I ' h m ( u p to isomorphism i n am! p ) ,

Further, f is surtji.c2f if trnd onl!j if $ 7s an cmb<dJirig, atid

( i ) M i:: a ~ 0 , l l - s u h l a l t i r r of L if and only if X is i i i c i m a ~ 7 ~ X undcrz a M I, continuous order-prescJroing map;

( i i ) M = L / B for some congruence 8 on 1, if and on7y if XM i:; a clo;ed sitbsrt of

*

We conclude t h i s s e c t i o n wi th a p ropos i t i on c o n t a i n i n g key elementary f a c t s about P-spaces. These r e s u l t s provide a l l t h e b a s i c t o o l s used i n d u a l i t y p roo t s o f a l g e b r a i c theorems.

PROPOSITION 2 .6 (See [681, 1711). l;rt (X,C,<) b t compact and t o i a l l g order disconnected. l'izen

( i ) t he graph of < is closed i n X x X;

( i i ) if Q is C-closed, +Q, +Q are r e s p c c t i w l y C-closed dccreasing, Z-c2oscd incwasing;

( i i i ) if Q i s C-clospd, Q contains a point idhich i s minimal (maximal) i n Q w i t / ) respect t o G;

( i v ) if Q,R are C-closed .substts of X such that x there ex i s t s a C-clopen decreasing subset V of X such that Q 5 X

LJ for all x E Q, y E R, then V , h' c V .

3. EQUATIONAL SUBCATEGORIES OF 0: THE INFLUENCE OF CATEGORY THEORY AND UNIVERSAL ALGEBRA

This s e c t i o n b u i l d s on t h e b a s i c d u a l i t y theory i n 52 and focuses on concepts and problems wi th t h e i r r o o t s i n ca t egory theory and un ive r sa l a lgeb ra . In te r a l i a , w e b r i e f l y cons ide r f r e e o b j e c t s and coproducts i n and i t s subca tegor i e s , and s u b d i r e c t l y i r r e d u c i b l e a lgeb ras i n va r ious e q u a t i o n a l subca tegor i e s of E. Suppose ,C i s a subca tegory o f k . Provided w e can i d e n t i f y t h e P-objec ts a r i s i n g a s dua l s of C-objects and can d e s c r i b e the dua l s of C-morphisms, we can r e s t r i c t t h e g-z d u a l i t y t o o b t a i n a d u a l i t y f o r ,C and thereby a technique f o r ana lys ing C. This approach has proved p r o f i t a b l e f o r , f o r example, d i s t r i b u t i v e p-a lgebras and double p-a lgebras , Stone a lgeb ras , de Morgan a l g e b r a s , Ockham a l g e b r a s and tukas iewicz a lgeb ras ; r e fe rences can be found i n the b ib l iog raphy .

We i l l u s t r a t e wi th a s imple example. The ca t egory R of ( d i s t r i b u t i v e ) p-algebras c o n s i s t s of those l a t t i c e s L E complementation s a t i s f y i n g c h a = 0 i f and only i f c Q u*. t he g-morphisms p rese rv ing *. THEOREM 3.1 (C731). is open whenever U E O(XL); for a E L,

Let f E D_(L,M) have duaZ $ E ~ ( X M , XI,). the minimal elements beZow y onto the minimal etements below $(y).

Theorem 3.1 provides a dua l d e s c r i p t i o n of &-congruences from which one s e e s t h a t L E i s s u b d i r e c t l y i r r e d u c i b l e i f and on ly i f L i s a Boolean a lgeb ra wi th a new i d e n t i t y ad jo ined ( s e e [351, C371 and C731, o r , f o r an a l g e b r a i c approach, C641, C651).

The s u b v a r i e t i e s of & were f i r s t desc r ibed by K. B . Lee i n C651. cha in

which posses s a unavy o p e r a t i o n , *, of pseudo- The B -morphisms a r e

"W

Let L E g, Then L E Bw if and only if XL is such that f U

Then f E & ( L , M ) if and only if $ maps

a* = XL +a.

I They form a


!-l C k0 ( = e ) c El (= Stone a l g e b r a s ) c R

f o r 1 n < w, B c o n s i s t s of those a l g e b r a s L i n H f o r which X i s such t h a t each of i t s elements ma jo r i se s a t most n minimal p o i n t s .

The c l a s s HW of d i s t r i b u t i v e double p-a lgebras ( t h a t i s , a l g e b r a s 1, such t h a t L and Lop a r e w i n h' ) i s much less t r a c t a b l e than . However major advances have been made i n r ec2n t yea r s by R . Beazer, B . A . Davey, T . Katriiidk and A . Urquhar t , us ing both a l g e b r a i c and topo log ica l methods. In p a r t i c u l a r , Urquhart has shown i n 1971tha t RW has an uncountable l a t t i c e of s u b v a r i e t i e s ; h i s proof uses d u a l i t y backed up by-gome graph t h e o r e t i c i d e a s .

Anyone of a c a t e g o r i c a l bent would not f e e l f u l l y acquain ted wi th a g iven ca tegory u n t i l he had desc r ibed the f r e e o b j e c t s ( i f a n y ) , and i n v e s t i g a t e d l i m i t s and c o l i m i t s . Under our d u a l i t y , l i m i t s t ransform i n t o c o l i m i t s , e t c . , so we can ana lyse v i a c. THEOREM 3.2 (See 1371, 1 4 1 1 ) . The free bounded d i s t r i b u t i v e l a t t i c e on K

generators, F ~ ( K ) , e s i s t s and +as duaZ space ZK. For any ordwcd s e t Q, t h e free bounded d i s t r i b u t i v e Zatticc generated by Q, FED(&), e x i s t s and has dual spacP (2')OP (where zQ i s thc s p t of order-preservfR funct ions from Q t o 2, ordered pointwise and topologisrd as u subspace o f 2 i.

Here Fg(Q) i s r equ i r ed t o possess the fo l lowing u n i v e r s a l p rope r ty : t h e r e e x i s t s an order-embedding e : Q -+ FD(&) and, f o r each 1, and each o rde r -p rese rv ing map a:& -+ L , t h e r e e x i s t s a D-morphism G:FD(Q) -+ L such t h a t U 0 e = a. embedding e of Q i n O ( ( 2 q P ) sends z E Q t o { h E 24 1 h(z) = 01.

Coproducts e x i s t i n 0.

c . . . c Rw; -2

7 2 -w L

The

E x p l i c i t l y , we have

THEOREM 3.3. ( i ) Let be a fafarm:ly o f g-objects. Then the coproduct

L A i s O( JTx, ) (where the product o f the dual spaces i s ordered point- At A X E A X wise i.

( i i ) Let L,M E 0. Then the coproduct L * M is O ( X L x X M ) , which i s isomorphic t o C G ( X L , M ) , the l a t t i c e of continuous order-preserving funct ions from XL t o M , uhere M i s equipped with the d iscre te topology.

For a d i s c u s s i o n of t hese and o t h e r r e s u l t s on coproducts i n g, s e e 1 2 7 1 , 1321, L791, 1801. We cons ide r f u r t h e r i n 54 the s p e c i a l ca se of Pos t a l g e b r a s (coproducts B * C, where B E and C i s a f i n i t e c h a i n ) . For c e r t a i n subca tegor i e s ,C of _D, t h e C-coproduct conven ien t ly c o i n c i d e s wi th t h e _D-coproduct (de Morgan a l g e b r a s provide one example 1281). i n s t a n c e , a s p e c i a l d e s c r i p t i o n of t h e coproduct is needed ( s e e 1393).

An element of ,D i s a (weak) p r o j e c t i v e i f and only i f i t i s a r e t r a c t of some F ~ ( K ) . However, R . Balbes has proved i n 1131 t h a t every g -p ro jec t ive l a t t i c e i s N D-ca ta ly t i c ( L i s g-catalyt ic i f _D(L,M) i s a l a t t i c e under t h e po in twi se o r d e r i n g f o r each M E E'; s e e L631), and t h a t t he converse a l s o ho lds f o r coun tab le l a t t i c e s . ; - ca t a ly t i c l a t t i c e s have been c h a r a c t e r i s e d a l g e b r a i c a l l y i n C131. We a l s o have

I n o t h e r s u b c a t e g o r i e s , & f o r

Such l a t t i c e s have proved d i f f i c u l t t o d e s c r i b e e x p l i c i t l y ( s e e 191, C151).

THEOREM 3.4 ( 1 7 4 1 ) . Let L E ;. Then the following are equivalent:

(i) L i s 2-catalyt ic;

( i i ) X L i s a compact zero-dimensional topologica2 l a t t i c e ;

( i i i ) L i s a r e t m c t of FD(Q), for some ordered s e t Q.

44 H.A. Priestley

This theorem impl i e s , a s i s well-known, t h a t t he dua l space of a i , -pro jec t ivc l a t t i c e i s a complete l a t t i c e , and t h a t L t g),, i s p r o j e c t i v e i f and only i f i t s s e t o f j o i n - i r r e d u c i b l e s J ( L ) i s a l a t t i c e . More s p e c i f i c a l l y , an ordered s e t i s isomorphic t o t he set of prime i d e a l s under i n c l u s i o n of a U_-projective l a t t i c e i f and only i f i t i s a r e t r a c t of some z", where both maps involved p rese rve up-d i rec ted sups and down-directed i d s ( C 7 4 1 ; see also 5 6 ) .

The i n j e c t i v e s i n a r e j u s t the complete Boolean a lgeb ras (L9l; f o r a t opo log ica l p roo f , s e e C717). R e l a t i n g t o completeness i n we have

THEOREM 3.5. Let I, E g. 3'hen the following are eqiu7:vali?nt:

( i ) L is complete;

( i i ) X L is eziremZZy order disconnectad ( t i iut is, tho C-closwe: of U E 11 i : ; U-open, where C, U arc r c s p e c t i v d ! j Lhc C-opt?ii dccrca:; h g , i n c m m in9 stit::

in (XL, C,Q;

(with po in tu i se ordar) 7::: condi t ional ly cornplc (i i i) t h e l a t t i c e CG(XLJR) of continuous ordcr-prcscw)ing function:: from XL t o W

For a proof of ( i ) Q ( i i ) , s ee 1711; f o r ( i ) Q (i i i) see 1781, where f u n c t o r i a l p r o p e r t i e s of t he assignment L *+ C (X ,R) a r e a l s o cons idered . M . J . Canfe l l i n h i s t h e s i s 1231 proves a r e s u l t ver? l i k e ( i i ) * ( i i i ) . we remark t h a t i t was i n C231 t h a t compact t o t a l l y o rde r d i sconnec ted spaces were f i r s t in t roduced , i n connec t ion wi th a s tudy of semi-algebras of cont inuous func t ions .

< c On a h i s t o r i c a l n o t e ,

4 . DUALITY I N A C T l O N : POST ALGEBRAS AND BEYOND

Sec t ion 5 w i l l be a po t p o u r r i of a p p l i c a t i o n s of d u a l i t y , chosen f o r t h e i r d i v e r s i t y . In t h i s s e c t i o n , by c o n t r a s t , we cons ide r a r a t h e r r e s t r i c t e d , bu t neve r the l e s s impor tan t , c l a s s of l a t t i c e s : t he Pos t a lgeb ras and g e n e r a l i s a t i o n s of t hese . We begin wi th Pos t a l g e b r a s . These a r e a l g e b r a i c a l l y well-understood ( see , f o r example, 1141) and d u a l i t y can do no more than make s t anda rd r e s u l t s obvious. We then cons ide r chain-based l a t t i c e s , f o r which a dua l space approach provides new s t r u c t u r a l in format ion . A f t e r a cu r so ry d i s c u s s i o n of o t h e r Post a lgeb ra g e n e r a l i s a t i o n s we f i n a l l y u n v e i l a p rev ious ly e l u s i v e c l a s s : t he d i s t r i b u t i v e l a t t i c e s of o rde r < m. Our account is based on work of W . G . Bowen (C201, c211, C221).

We no te t h a t t he dua l space of a Boolean a lgeb ra i s , qua ordered s e t , an a n t i c h a i n , whi le t h e dua l space of a cha in i s a cha in . For convenience we denote the elements of t h e z e l e m e n t cha in n by O , l , ...$- 1 and those of i t s dua l space n;l ( t h e d i s c r e t e l y topologised (n-i)-element cha in ) by 1 , 2 , . . . , n-1.

Let L = B >k - n (where B E g) be a t y p i c a l Post a lgeb ra . Then ,YL = XB x nZ1, which can be v i s u a l i s e d as t h e d i s j o i n t

t n-l X n- 1

x2 1 ; x1

xB

Figure 1

union of n clopen ' l a y e r s ' X i , each homeomorphic t o t h e Boolean space xn, where

( z , i ) Q (y,j)

i f and only i f z = y and i so t h a t each p o i n t of XL i s major i sed p r e c i s e l y by the cor responding p o i n t s i n h ighe r l a y e r s ( see f i g u r e 1 ) . L a t t i c e elements a r e r ep resen ted by clopen dec reas ing s e t s . These a r e b u i l t

j,


up from ' c h a i n e l e m e n t s ' a n d ' c e n t r a l e l e m e n t s ' . L e t co = 0 , e . = u X . (i > 1).

0 < c'l < * * * < c' n- 1

Then j<i 3

= 1 O = t

forms d c h d i n i n L i somorphic t o ". The c e n t e r o f 1, ( i t s complemented e l e m e n t s ) i s {ij xn-1 I b E ii); i t forms A Boolean a l g e b r a i somorphic t o 8. t h e s e s p e c i a l e l e m e n t s .

F i g u r e 2 d e p i c t s

c h a i n e lement c e n t r a l e lement

F i g u r e 2

A g e n e r a l e l e m e n t a i n L may be r e p r e s e n t e d d i a g r a m m a t i c a l l y a s i n F i g u r e 3 and i t i s e a s i l y s e e n t h a t i t h a s a unique r e p r e s e n t a t i o n i n e a c h of t h e f o l l o w i n g forms.

D i s j o i n t r e p r e s e n t t i on : n- 1

Monotone r e p r e s e n t a t i o n : n- 1

a = v ( c j i A e.), c. E B, L' c f o r i j. i j i=o

I n F i g u r e 3 t h e d i s j o i n t r e p r e s e n t a t i o n i s i n d i c a t e d ; t h e monotone r e p r e s e n t a t i o n is o b t a i n e d i n a s i m i l a r f a s h i o n .

F i g u r e 3

More i m p o r t a n t l y , t h e d u a l s p a c e makes t r a n s p a r e n t t h e r e p r e s e n t a t i o n of e l e m e n t s of L a s c o n t i n u o u s f u n c t i o n s from XB t o hand d iagram i n F i g u r e 4 d e p i c t s a n e lement o f O(Xn x n--l); t h e r i g h t hand d iagram shows t h e graph o f t h e a s s o c i a t e d c o n t i n u o u s f u n c t i o n .

( s e e Theorem 3 . 3 and C801): t h e l e f t

46 H. A. Priestle y

Figure 4

The coproduct n k 2 i s genera ted by i t s c e n t e r and an n-element cha in i n a very s p e c i f i c way. We now cons ide r a more gene ra l s i t u a t i o n . Suppose t h a t C i s a f ixed cha in

0 = co < el < ... < c = 1 n- 1

i n a l a t t i c e L E 2. be chain-based i f L is genera ted by ii U C, where I3 i s t h e c e n t e r of I,; t h e d i s t i n g u i s h e d cha in C i s c a l l e d a c i a i n basc f o r I,. We s h a l l c a l l I, E U_ a P - l a t t i c e i f some f i n i t e subchain of I, i s a c h J i n base . Such l a t t i c e s were in t roduced by T . Traczyk 1891 and have been s t u d i e d by G . Eps t e in and A . Horn in C441. It is e a s i l y proved from the obse rva t ions conta ined i n

LEMMA 4 . 1 . of the natural l'nclusion maps from U,C in to L .

Then (L;C) (o r where no confus ion a r i s e s j u s t i.) i s s a i d t o

0

Theorem 4 . 2 g ives the dua l c h a r a c t e r i s a t i o n of P o - l a t t i c e s .

Lcl ( L ; C ) be chain-based and l e t v:XL + X b J a:XL + n-1 be thc duuls

-1 ( i ) ( a ) a E B Q a = II ( h ) for some clopen subsel 0 o$ Xu; -1 (b) each e

t h c Lopology on X L ;

-1

E C, ei f 0, i c c q r e s s i b ~ e as a (ri); i ( i i ) { i r - l (~ i ) I b clopen i n xB} u {a- '( i) I 7: = 1 , 2 , ...,n- 11 forms a s h h a s e f o r

-1 ( i i i ) ( a ) for i = 1,2, ..., n-1, a (i) is a n antichain;

(b) for z E XB, ~i (z) is a f i n i t r chain;

( i v ) 3: I-- (n(3:),a(x)) is a homeomorphic order embedding of XI, i n t o XB x n i l .

THEOREM 4 . 2 (W.G.Bowen [20], [221). only if for some Boolean space Z and some n > 1, X I , is a closed subspace of

Let L E g. Then L is a P0-lattice if and

2 x n-1.

The d u a l space of a P O - l o t t i c e may be dep ic t ed a s i n F igure 5:

N

prime i d e a l s l i e i n d i s j o i n t maximal cha ins and i n d i s j o i n t c lopen ' l a y e r s ' . The l a y e r i n g of t h e space depends on t h e choice of cha in base ( see Lemma 4 . 1 ( i i ) ) . The dua l space of L i s ( t o w i t h i n homeomorphism and o rde r isomorphism) una l t e red i f a c lopen s u b s e t of a l a y e r i s moved t o occupy a vacant space i n a n ad jacen t l a y e r , bu t t h e cha in base changes.

Let C denote the c o l l e c t i o n of cha in bases of a P o - l a t t i c e L of t h e maximal p o s s i b l e l eng th , o rdered elementwise.

n:l Y

-Ij Figure 5


We say 1, i s a E’l-l,atticc. i f C has a l a r g e s t e lement ,

a F -2utticc i f C has l a r g e s t and s m a l l e s t elements and a P:i-Lalticc i f /Ci = 1.

2

The P . - l a t t i c e s a r e i n v e s t i g a t e d i n C 4 4 1 , 1901. Dual space diagrams make i t easy t o e shnb l i sh many p r o p e r t i e s of P . - l a t t i c e s , i nc lud ing those s e t o u t i n t h e t a b l e below; thc new r e s u l t s a r e provea i n [ 2 0 1 . a l g e b r a s : those L E f o r which

We denote by

0 + b = maxic E L 1 c A a G 0 1

t h e c l a s s of Heyting

e x i s t s f o r a l l (l,b E /,. Dual ly ,

a + h = m ‘ l n t c E i, 1 c v a > b } ,

In t h e dua l space a + 11 is r ep resen ted by X L where t h i s e x i s t s . a + b by + ( b \ t i ) .

I n t hc t a b l e (1,;L‘) i s a chain-based l a t t i c e , wi th C E C the cha in

+ ( a -. b ) and

0 =

For any a t L ,

< d1 < ... < en-, = I,

and n i s the c e n t e r of L . Note t h a t Theorem 4 . 2 and Coro l l a ry 2 . 5 ( i i ) imply t h a t ( L ; C ) is (isomorphic t o ) t he q u o t i e n t o f ii x - n by a congruence.

! iz = max{b t H I b G u } , i f t h i s e x i s t s .

A 1 geb r a i c e q u i v a l e n t

P - l a t t i c e

e i + e = e . i-1 7,

( L ; C ) a

P - l a t t i c e V i 1

e + c = e

and ! ei e x i s t s

V i

i i-1 I: ( L ; C ) a

P - l a t t i c e 2

/I and ! en-2 = 0

Dual e q u i v a l e n t

X c losed i n

X, x n,--1 L,

X L c losed

dec reas ing i n

XB x n z l

X c lopen

dec reas ing i n

XB x n-1

L

Rela t ion of L

a lgebra B * ~f

Pos t

Quot ien t by a

congruence

Quot ien t by a

f i l t e r

Quot ien t by a

p r i n c i p a l

f i l t e r

Heyting a l g e b r a

equiva len t

L,LOPE

The dua l space c h a r a c t e r i s a t i o n s make i t easy t o c o n s t r u c t examples of P . - l a t t i c e s . The fo l lowing ones a l l have t h e same c e n t e r , viz. t he Boolean a lgeb ra o f ’ f i n i t e and c o f i n i t e s u b s e t s of IN, whose dua l space 2 i s the one-point cornpac t i f ica t ion of t h e d i s c r e t e spacelN, wi th the d i s c r e t e o r d e r . g ives a l a t t i c e which is i n t h e i n d i c a t e d c l a s s b u t n o t i n the c l a s s above. A l l a r e s u b s e t s of X = 2 x 2, which i s the dua l of a P - l a t t i c e (Pas t a l g e b r a ) .

Each of the dua l spaces below

I I I 111:::1 X 3

48 H A . Priestley

(: 1 d S s

1'. - l a t t i c e s L

P - l a t t i c e s 1

I'"-lat t i c e s

F i g u r e h

'I'hc f o l l o w i n g space p r o v i d e s a c o n t r a s t i n g example . Wc t a k e X < / j h l orrlercci s e t t o be A x - 2 , where A i s t h e Cai i tur s e t , d i s c r e t e l y u r d e r c d . i n t o a compact t o t a l l y o r d e r d i s c o n n e c t e d s p a c e by t d k i n g as ;i c L i ~ p e n s \ h b a s c > f u r ;I t o p o l o g y t h e s e t s I ( . c , 2 ) I .I: E A] U l i : x - 2 1 (c is c l o p e n i n A w i t h i t s u s u a l t o p o l o g y } . 'The r e s u l t i n g s p a c e i s made u p o f two copi t l s o f A forming two l a y e r s , w i t h Lhe d i s c r e t e t o p o l o g y induced on t h e upper l a y e r and the u s u a l t o p o l o g y on t h e lower one . The l a y e r s a r e , o f c o u r s c , n o t t o p o l o g i c a l l y d i s j o i n t a n d i t is p o s s i b l e t o show t h a t t l ie d i s t r i b u t i v e l a t t i c e X r e p r e s e n t s h a s - no ch.iin hdsc; in f a c t i t i s n o t even a s u b l a t t i c e o f I' - l a t t i c e . 0

The dual space o f a Post a l g e b r a L hi i s t h e f o l l o w i n g p r o p e r t i e s :

'This s e t can bc made

(7:) XI, i s o r d e r e d a s a d i s j o i n t un ion o f maximal c h a i n s ;

(-I-) XI, i s t o p o l o g i s e d iis ;1 f i n i t e d i s j o i n t un ion of Boolean s p a c e s .

B e w i l d e r i n g l y many s u b c l a s s e s o f h a v e been c o n s i d e r e d which s h a r e t o a g r e a t c ' r d e g r e e t h e a l g e b r a i c p r o p e r t i e s of P o s t a lgcsbras (n-va lued C u k a s i e w i c z a l g e b r a s , S t o n e l a t t i c e s L I E o r d e r n , ... ) . A L L , d u a l l y , r e t a i n e i t h e r p r o p e r t y (") o r p r o p e r t y ( - 1 ) . A s y s t e m a t i c s t u d y O F t h e i n t e r a c t i o n be tween t h e s e c l a s s e s has been u n d e r t a k e n by W . G . Bowen in 1 2 0 1 .

From among l a t t i c e s s a t i s f y i n g (") we s i n g l e o u t f o r b r i e f ment ion t h e n-valued f .ukas iewicz a l g e b r a s . The dual s p a c e X I of such a n a l g e b r a i. looks l i k e a Post a l g e b r a d u a l s p a c e 7 x n.=.,l i n which som; o f t h e l a y e r s have b e e n p i n c h e d Logethcr by a map &:(:r ,L) ++ i\.(x:), where h . : Z -+ X1 s u r j e c t i v e , o r d e r - p r e k e r v i n g and s & c h t l i a i r,!] a r e i n c o m p a r a b l e i t and on1.y i f rT . (s ) , ii (!y) are incomparab I e .

( i = l , 2 , ..., Y L - ~ ) i s c o n t i n u o u s ,

The r i g l i t , ipproach Lo c 1 , i s s i F y i n g l , i t L i c e s w i t h p r o p e r t y ( I ) seems t o be L o mike


Tire du , i l sp : i c~> oI 'I 1 : i t t ic .c ui th i s t y p c i s t h e d i s j o i n t un ion of t h e I!oolenn sp.Ic<'s 'Y ( [ ' = 1,2, . . . , j / - l ) . 'I'lit, d i s t r i b u t i v e l a t t i c e s o f o r d e r ni form

"~;-l,* I ' I s u b c a t e g o r y o f c. I t s n l g e b r ' i s Iicive b i n a r y o p e r a t i o n s V , A , u n a r y , ( I ' = 1 , 2 ,..., : r t ) .ind n u l l a r y o p e r a t i o n s i'. ( i = 0 , 1 , ..., m-1); i f

' i i - l a r e as i n D e t i n i t i o k 4 . 3 , i . = (2 1 , i s n f o r d e r i i < m, t h e n , ( ] ' . . . 9 '

v rr) A f y ; i n r t ~ j - i , ~ ~ , l l l ~

f o r i n-1

;t . 2 ' > '1 ,ind ( r i s t h e comptement U T (.

Frequent l y thr. cei1tc.r d i s t r i b u t i v e l a t t i c e L w i l l b e t o o s m a l l Tor 1, t o be

s u b l a t t i c c bl s u c h t h a t e n c r a t e d b y bl u i?. I J ~ assume !, rind L"p a r e t i e y t i n g n l g e b r n s .ind d c f i u e t h e i i i , I' 01 1, t o b e t h e s u b l a t t i c e g e n e r a t e d by a11 C' leme1t t s

F o r any i i E !i, (a + ~ 2 ' ) + ((2' + n) = ii, so t h e c e n t e r i s t .un ta ined i n the c x o c e n t e r . F u r t h e r n i o t i v a t i o n f o r the p e r h a p s c u r i o u s l o o k i n g d e f i n i t i o n i s p r o v i d e d by F i g u r e 8 , which i l l u s t r a t e s t h e l o o s e s e n s e i n which t h e e x o c e n t r a l g i 'ncr r i to r (1' + c) ( ( 2 '- 1 1 ) c n n be t h o u g h t of as h a v i n g (c + b ) + ( f J + c) as an ' approxi t i la te coiiipteriren~' .

i- I

gcllel-.ltcd by ii ; I d '1 g n i t e rlic3in i:. We may t h e n seek some o t h e r ' m i n i m a l '

(11 --f < 2 ) 4 (c + / I ) (b,c t I , ) .

__ . . . .

5 . A M I S C E L I A N Y OF IIUALLTY APPLICATIONS

An i m p o r t a n t use t c ) which d u a l i t y tias been p u t tins been t h e p r o d u c t i o n o f

The e x o c e n t e r w a s i n t r u d u c e d by C . Epstein 1 4 3 I ; h i s d e f i . n i t i o n is d i f f e r e n t irtm,

b u t e q u i v a l e n t t o , t h e onc g i v e n here . ProperLics o f t h e e x o c e n t e r h a v e been i n v e s t i g a t e d i n d e p e n d e n t l y by W . G . Bowen i n 1 2 0 1 , I 2 1 I a n d by G . 1 7 p s t r i n a n d A . Horn ( s e e I 4 5 I ) .

The . ippropri , i tcness of the d c f i i i i t i o n is c o n f i r m e d by

50 H.A. Priestley

counterexamples and of l a t t i c e s s a t i s f y i n g s p e c i f i e d cond i t ions . We g ive some i l l u s t r a t i o n s , r e f e r r i n g the r eade r t o the c i t e d papers f o r t he (sometimes i n t r i c a t e ) d e t a i l s of t h e c o n s t r u c t i o n s .

A s a f i r s t example cons ide r the problem of f ind ing which 1, E have the p rope r ty (*): every i d e a l ( f i l t e r ) i n L i s t h e i n t e r s e c t i o n o i maximal i d e a l s ( f i l t e r s ) . I t was poin ted ou t by A . Monteiro t h a t (*) holds i f L is Boolean. Using d u a l i t y , M . E . Adams proved i n C21 t h a t i t holds i f e i t h e r L E R, o r L is s c a t t e r e d ( t h a t i s , t he r a t i o n a l cha in does not embed i n L ) , He a l s o cons t ruc t ed an example of t he f a i l u r e of (*); t he dua l space of a s u i t a b l e l a t t i c e was obta ined by t ak ing the Cantor s e t i n i t s usua l topology and superimposing an appropr i a t e o r d e r . A d i f f e r e n t example was found by R . Balbes 1111 us ing a l g e b r a i c techniques . This is no t t he only case of a problem's be ing so lved independently and almost s imul taneous ly by a l g e b r a i c and topo log ica l methods which a r e n o t mut a l l y d u a l . Another i s provided by q u a s i v a r i e t i e s of p-a lgebras : t h a t t h e r e a r e 2'0 of these was shown a l g e b r a i c a l l y by A. Wroriski I1001 and t o p o l o g i c a l l y by M. E . Adanis c31. This r e s u l t was r e f i n e d by G . GrBtzer, H . Lakser and R. W . Quackenbush C551 who shnwed t h a t the q u a s i v a r i e t i e s w i t h i n form a non-modular l a t t i c e , of c a r d i n a l i t y 2% ; t h e i r t echniques inc lude d u a l i t y , ?haugh app l i ed t o f i n i t e a lgeb ras on ly .

We now look a t s i i b l a t t i c e s . M a r i s e s from (XL,E,4) as fo l lows .

Fur M a { 0 , I ) - s u b l a t t i c e of 1, E U,, t he dua l space of L e t

J" = I ( z , y ) I z + y , (VU E M)(y E a =) x E a)}

be the separating s e t f o r M. Then S U 4 i s a quas io rde r r e l a t i o n on X m d P given by x p y i f and only i f (.z,y), (y,z) E S U 4 is an equiva lence r e t a t i o n . equipping thep-equiva lence c l a s s e s wi th the obvious q u o t i e n t topology and o r d e r , we o b t a i n M ' s dual space . Sepa ra t ing s e t s ( i n c l u d i n g t h e i r ex tens ion t o not n e c e s s a r i l y bounded l a t t i c e s ) were f i r s t e x p l i c i t l y employed by M . E . Adams c11. The technique i s perhaps no t a s widely known a s i t deserves t o be ( i t r e c e n t l y proved u s e f u l f o r s tudy ing the exocenter (Theorem 4 . 4 ) ) . As a simple a p p l i c a t i o n we mention Adams' dua l cons t ruc t ion of a d i s t r i b u t i v e l a t t i c e wi th no maximal proper s u b l a t t i c e s : t he dua l space c o n s i s t s of t he n a t u r a l numbers s u i t a b l y ordered . The main r e s u l t s i n Cll concern the Fratt in? sublat t icc @ ( L ) of a d i s t r i b u t i v e l a t t i c e L ; by d e f i n i t i o n , @ ( L ) is t h e i n t e r s e c t i o n of t h e maximal proper s u b l a t t i c e s of L . I t is shown, in ter a l i a , t h a t any d i s t r i b u t i v e l a t t i c e L is Q ( L ' ) f o r some d i s t r i b u t i v e l a t t i c e I,'.

Many yea r s ago, A . Tarsk i posed h i s 'cube problem': does t h e r e e x i s t B E B, such t h a t B B x B x B bu t B $ B X B ? I n a remarkable paper C621, J . Ketonen r e c e n t l y c h a r a c t e r i s e d the countable Boolean a l g e b r a s and provided a ( coun tab le ) counterexample t o t h e cube problem. For coproducts of countable Boolean a l g e b r a s , V. Trnkov5 C921 proved t h a t B p B * B * B does imply B B * B . By - o n t r a s t , M. E . Adams and V. Trnkovd have shown i n C81 t h a t i t is p o s s i b l e t o f i n d 2'0 non- isomorphic countable d i s t r i b u t i v e l a t t i c e s L E 0, such t h a t L 2 L * L * L bu t L $ L * L ; f u r t h e r , given p resc r ibed countable bounded d i s t r i b u t i v e l a t t i c e s B,C, each of t h e l a t t i c e s L above can be chosen t o have B a s a s u b l a t t i c e and C a s a homomorphic image. The c o n s t r u c t i o n uses d u a l i t y i n a n o n - t r i v i a l way.

Our l a s t example i s somewhat d i f f e r e n t and involves t h e i n t e r v a l topology. On any ordered s e t P, t h e interval topology has a c losed subbase of t h e s e t s +x, +x f o r z E P. I t w i l l be of some importance i n 56; he re we a r e concerned wi th i t s appearance i n a r e c e n t paper by G . Gierz and J . D . Lawson C5ll on so-ca l led gene ra l i zed continuous l a t t i c e s . They prove t h a t a d i s t r i b u t i v e l a t t i c e L is the q u o t i e n t of a completely d i s t r i b u t i v e l a t t i c e L' under a map p rese rv ing up-d i rec ted supsanddown-directed i n f s p r e c i s e l y when the i n t e r v a l topology on L is Hausdorf f . The proof of s u f f i c i e n c y i s obta ined by t ak ing L' t o be t h e l a t t i c e of order - p re se rv ing maps from (X r equ i r ed q u o t i e n t map; i t is unknown whether t h e theorem remains v a l i d wi thou t t h i s assumption.

By

)OP t o 2. D i s t r i b u t i v i t y of L is used t o produce t h e F$?'e(4,

Ordered sets and duality for distributive lattices 5 1

6. TOPOLOGY AND ORDER IN PRIESTLEY SPACES

A P r i e s t l e y space i s s imul taneous ly an ordered s e t and a (Boolean) t o p o l o g i c a l space . It must a l s o be t o t a l l y o rde r d i sconnec ted . To s e e the s t r e n g t h of t h i s l a t t e r requi rement we d e a l b r i e f l y wi th t h e problem of topo log i s ing an ordered s e t t o c r e a t e Li g-ob jec t . s e t ( I 3 , < ) i s wprc.;cvitclDle, t h a t i s , i s isomorphic t o (X C) f o r some L E t . A deep a n a l y s i s of r i n g s p e c t r a by M . t lochster [561 r evea led t h a t (F ,G) i s r e p r e s e n t a b l e p r e c i s e l y when i t i s isomorphic t o t h e s e t of prime i d e a l s , o rdered by i n c l u s i o n , o f a commutative r i n g wi th i d e n t i t y . Thus r e p r e s e n t a b i l i t y i s of i n t e r e s t bo th t o l a t t i c e t h e o r i s t s and to r i n g t h e o r i s t s ; f o r a sample of t h e l i t e r a t u r e s e e [ l o ] , 1121, C311, C481, C561, 1661, C871, where f u r t h e r r e f e r e n c e s can a l s o be found.

Equ iva len t ly , we a sk under what c o n d i t i o n s a g iven ordered

L’-

Any r e p r e s e n t a b l e s e t (P,<) is such t h a t

( i ) every up-d i rec ted subse t of P has a sup , every down-directed s u b s e t has an i n f ;

rzd,ul = {u,u1 ( t h a t i s , L J . , ~ ] c o n t a i n s a cove r ing p a i r ) ; ( i i ) I’ is weakly a tomic : i f .r < y , t h e r e e x i s t u,v such t h a t x < u < u y and

( i i i ) t he i n t e r v a l topology on P is compact;

( i v ) Fop i s a l s o r e p r e s e n t a b l e .

Condi t ions ( i ) - ( i v ) e s s e n t i a l l y appear i n C561. I t i s easy t o prove t h a t they hold i n the under ly ing ordered s e t of any _P-object ( s e e 1701 and C491, 5VI.1). By ( i ) , any r e p r e s e n t a b l e s e m i l a t t i c e i s n e c e s s a r i l y a complete l a t t i c e . From ( i ) , ( i i ) we s e e immediately t h a t , f o r example, n e i t h e r o nor w+w* i s r e p r e s e n t a b l e . I n f a c t , a cha in is r e p r e s e n t a b l e i f and on ly i f i t s a t i s f i e s ( i ) , ( i i ) . I n g e n e r a l , even ( i ) , ( i i ) and ( i i i ) a r e i n s u f f i c i e n t t o gua ran tee r e p r e s e n t a b i l i t y ( s e e 1661).

Among o rde red s e t s s a t i s f y i n g ( i ) and ( i i ) a r e c l e a r l y t h e a l g e b r a i c l a t t i c e s ( s e e 1491 f o r the d e f i n i t i o n and b a s i c p r o p e r t i e s ) . An a l g e b r a i c l a t t i c e P becomes a P r i e s t l e y space i f i t i s given t h e (Lawson) topology X ( P ) wi th open subbase c o n s i s t i n g of t h e s e t s + k ( k E K(P), t h e compact e lements i n P) and P 93: (J. i’) ( see [491 , C741, 1751) . Among the dua l spaces of t h i s type a r e t h e zero-dimensional t opo log ica l l a t t i c e s a r i s i n g i n Theorem 3.4.

In gene ra l a r e p r e s e n t a b l e ordered s e t can c a r r y many d i f f e r e n t t opo log ie s which make i t a P r i e s t l e y space , and so r e p r e s e n t many non-isomorphic l a t t i c e s . A c l a s s i c example i s provided by M a d j o i n e d ) ; A(M ) and A(MwOP) yieYd non-isomorphic then LI con ta in2 the i n t e r v a l topology IV(P) on P (by P r o p o s i t i o n 2 .6 ( i i ) ) ; t h i s i s how r e p r e s e n t a b i l i t y cond i t ion ( i i i ) comes about . Thus C i s uniquely determined (as IV(P)) i f IV(P) i s Hausdorf f . When t h i s happens, we have ( P , Q) ( X L , ~ ) f o r (up t o isomorphism) only one L E 0. Among l a t t i c e s uniquely determined i n t h i s way by t h e i r o rdered s e t s of prime i d e a l s a r e t h e ; - ca t a ly t i c l a t t i c e s and i n p a r t i c u l a r any ;-object f r e e l y genera ted by an ordered s e t and any g -p ro jec t ive l a t t i c e . The i n t e r v a l topology on P i s Hausdorff whenever P has no i n f i n i t e a n t i c h a i n ; necessary and s u f f i c i e n t cond i t ions f o r IV(P) t o be Hausdorff a r e given i n C461, C51l.

We have been cons ide r ing t o p o l o g i s i n g a g iven ordered s e t t o produce a P r i e s t l e y space . produce a _P-object may be t r i v i a l o r hard depending on t h e type of o r d e r i n g demanded. making i t a g -ob jec t . [671; a l i n e a r o r d e r can be imposed on a Boolean space (X,,t) i f and on ly i f t h e Boolean a l g e b r a B has an ordered base .

We conclude t h i s s e c t i o n wi th a cau t iona ry example, due t o A. R . S t r a l k a [ 8 6 1 .

(a countable a n t i c h a i n wi th u n i v e r s a l bounds - o b j e c t s . I f (P,C,s) E p ,

We remark i n p a s s i n g t h a t t h e problem of o r d e r i n g a Boolean space t o

For example, no l i n e a r o r d e r e x i s t s on t h e Tychonov p lank (w+l) X (w,+l) This is e s s e n t i a l l y due t o R . D . Mayer and R . S . P i e r c e

52 H.A. Priestley

C o n s i d e r t h e C a n t o r s e t A w i t h i t s u s u a l t o p o l o g y C and o r d e r Q d e r i n e d by 2: L/ i f and o n l y i f 5: = y o r J = 1, !/ = 0 o r i i !i c o v e r s .c i n t h e n a t u r a l o r d e r on A . 'The s p a c e (A,C,G) i s c a l l e d t h e Lr:;aL?/wc/~s; i t i s made up or a c o u n t a b l e c o l l e c t i o n of 2-element c h a i n s and an u n c o u n t a b l e a n t i c h a i n . Lt t u r n s o u t t h a t , g i v e n 5: $ y i n A ,

( i ) 3 d i s j o i n t C-clopen s e t s i l l , Vl w i t h .c t U1, !/ 6 V1,

V 2 , r e s p e c t i v e l y i n c r e a s i n g , d e c r e a s i n g , w i t h , 2'

( i i ) 3 d i s j o i n t C-open s e t s U x E u2, y E v 2 ,

b u t

( i i i ) 3 d i s j o i n t C-clopen se t s 0, V , r e s p e c t i v e l y i n c r e a s i n g , d e c r e a s i n g , w i t h

I n f a c t o (a ) = { $ , a ) !

5: 6 u, E V .

7 . DUALITY IN A WIDER PERSPECTIVE

?he c r e d i t f o r i n i t i a t i n g t h e s t u d y o f ~ o p o l o g i c a l r e p r e s e n t a t i o n s of a l g e b r a i c o b j e c t s goes t o M . H . S t o n e , H i s c o n t r i b u t i o n i s p u t in h i s t o r i c a l p e r s p e c t i v e i n t h e i n t r o d u c t i o n t o 1'. T . J o h n s t o n e ' s book S t o n e Spaces [ 601. We recommend t h i s s t i m u l a t i n g and w i d e - r a n g i n g a s s e s s m e n t as complementary re , id ing t o t h e f o l l o w i n g s h o r t d i s c u s s i o n of d u a l i t i e s r e l a t e d t o t h e !)-z d u a l i t y .

We b e g i n w i t h some h i s t o r i c a l remarks on d u a l i t i e s f o r 0. S t o n e ' s o r i g i n a l t o p o l o g i c a l r e p r e s e n t a t i o n e x t e n d s t o a d u a l i t y be tween f! and t h e c a t e g o r y 2 o t s p e c t r a l s p a c e s ( s e e [ 1 4 1 , [541 , [ 8 5 1 ) . W . H . C o r n i s h proved i n [261 t h a t m d S a r e i s o m o r p h i c ( n o t j u s t e q u i v a l e n t ) c a t e g o r i e s . I f t h e c - d u a l of L i s TXL,C,Q) t h e n i t s 2-dual i s ( X L , C ) where C i s t h e t o p o l o g y o f C-open d e c r e a s i n g sets ; c o n v e r s e l y , C i s c o n s t r u c t e d f r o m a s p e c t r a l t o p o l o g y by t a k i n g d S a s u b b a s e t h e C-compact-open s e t s and t h e i r complements , and Q is g i v e n by s y Q

y E c l ~ { 3 c 1 . M . H o c n s t e r [561 . For p r a c t i c a l p u r p o s e s seems t o h a v e d i s t i n c t a d v a n t a g e s o v e r S . However t h e r e a r e s i t u a t i o n s where a ' h u l l - k e r n e l ' a p p r o a c h (as w i t h 2) i s a p p r o p r i a t e ; one s u c h is t h e s p e c t r a l t h e o r y f o r d i s t r i b u t i v e c o n t i n u o u s l a t t i c e s r e c e n t l y d e v e l o p e d by K . H . Hofmann and J . D . Lawson ( [ 5 8 ] ; o r s e e [ 4 9 1 ) . One can r e c a p t u r e d u a l i t y f o r f rom t h i s by r e g a r d i n g t h e pr ime i d e a l s of 1, E as t h e pr ime e l e m e n t s of T d ( L ) , t h e i d e a l l a t t i c e of L , which i s a d i s t r i b u t i v e c o n t i n u o u s l a t t i c e ( s e e [ 4 9 1 , p . 325) . An e x t e n s i o n of d u a l i t y i n a q u i t e d i f f e r e n t d i r e c t i o n is s u g g e s t e d by G.-C. R o t a i n 1761.

We now s e e k a more s y s t e m a t i c u n d e r s t a n d i n g of how t h e d u a l i t y be tween i n some s e n s e p r o t o t y p i c a l . We b e g i n w i t h a f o r m a l s t a t e m e n t o f t h e d u a l i t y be tween o u r c a t e g o r i e s

The t r a n s i t i o n f rom 2 t o was p a r t i a l l y made by A . Nerode [691 and

and P i s

i) : d i s t r i b u t i v e l a t t i c e s w i t h 0,1, - 0 , l - p r e s e r v i n g l a t t i c e homomorphisms

c o n t i n u o u s o r d e r - p r e s e r v i n g maps. and ,F : compact t o t a l l y o r d e r d i s c o n n e c t e d s p a c e s

We n o t e t h a t t h e o b j e c t "2" a p p e a r s i n two g u i s e s : as 2 i t l i e s i n L l , as 2 i t l i e s i n P. u n d g r s t o o d t o be t h e 2-e lement c h a i n w i t h e l e m e n t s 0 , l . I t w i l l now b e more c o n v e n i e n t t o work w i t h p o i n t w i s e - o r d e r e d hom-sets r a t h e r t h a n pr ime i d e a l s .

DUALITY THEOREM. T?w categ0ric.s varianl f inetor;

(where Liic image c ( L , z ) of L has the pointwise o r d m and topology inherli ted from

The n o t a t i o n h e r e is d i f f e r e n t f rom t h a t a d o p t e d i n § 5 ; 2 i s t o b e

and ,P are d u a l l y equivalent u n d m t h e contra-

- D(-,L) : + p


2')

aiui - P(-,?) : + i'

:'YlC, J O i t i : 1

r t ~ ' ~ ' g{Ii.h ri ( J h r l I, t 2, i' E [) L J ; ~ f L J I,,:

-+ <( f, 1 1 : 1 ~ + I i ( p ( - , ? ) ,2) '3s t-hc nl j i i~nci ior~ - - - I J 4

( E 1 , ( / I ) ) (f) = f ( a )

a d (rtl,(x)) (9) = o ( . I - ) (.r 11, $ E C ( P , ? ) ) .

The P o n t r y a g i n d u a l i t y f o r comp;i~-t g r o u p s e x h i b i t s e s s e n t i a l l y t h e same f e a t u r e s a s t h e c-[ d u a l i t y . i s o m o r p h i c t o t h e group or c o n t i n u o u s homomorphisms f rom i t s c h a r a c t e r g r o u p - A ( / l , 3 ' ) t o x; h e r e t h e c i r c l e g r o u p 2' qim d i s c r e t e group T l i e s i n 4, quo. compact g r o u p 2' i t l i e s i n t h e d u a l c a t e g o r y . o b j e c t ; in e a c h of t h e m u t u a l l y d u a l c a t e g o r i e s c a n b e i d e n t i f i e d w i t h s t r u c t u r e d hom-sets (morphisms i n t o Y,X, morphisms i n t o 2,2); A(A,r) i n h e r i t s a t o p o l o g i c a l and o p e r a t i o n a l s t r u c t u r e - f r o m TI, w h i l e i ) ( L , T ) i n h e r i t s a t o p o l o g i c a l and r e l a t i o n a l s t r u c t u r e f rum 2". been s e t up by B . A . Ilavey and 11. Werner i n [ 4 2 1 . The t h e o r y h a s i t s r o o t s i n a v a r i a n t of F r e y d ' s r e p r e s e n t a b i l i t y theorem and f o l l o w s e a r l i e r work by , among o t h e r s , B . A . Davey [ 3 4 1 and I<. H. Hofmann and K . Keimel [ 5 7 1 . The t h e o r y d e v e l o p e d by Davey and Werner e n c o m p a s s e s , b e s i d e s t h e P o n t r y a g i n and c-' d u a l i t i e s , M o r i t a d u a l i t y f o r modules , t h e Hofmann-Mislove-Stralka d u a l i t y f o r compact z e r o - d i m e n s i o n a l s e m i l a t t i c e s [591 and d u a l i t i e s (some o l d , some new) f o r c e r t a i n o t h e r v a r i e t i e s which a r e g e n e r a t e d by a s i n g l e f i n i t e a l g e b r a . A s u b c l a s s of U may h a v e two d u a l i t i e s , one by r e s t r i c t i o n of t h e {-f d u a l i t y , t h e o t h e r p r o v i d e d d i r e c t l y by t h e Davey/Werner a p p r o a c h . Sometimes t h e s e d u a l i t i e s are e s s e n t i a l l y t h e same ( f o r S t o n e a l g e b r a s , f o r e x a m p l e ) , somet imes t h e y a re d i f f e r e n t (as i s t h e c a s e f o r Kleene a l g e b r a s ) .

B r a n c h i n g o u t i n a d i f f e r e n t d i r e c t i o n , o n e n a t u r a l l y s e e k s an e x t e n s i o n of ,U-p d u a l i t y t o l a t t i c e s i n g e n e r a l . V a r i o u s r e p r e s e n t a t i o n s have b e e n d e v i s e d , e a c h w i t h i n t e r e s t i n g f e a t u r e s ( [ 3 0 1 , [ 5 0 1 , [ 9 4 1 ) . G . G i e r z and K . Keimel i n [ 5 0 ] e x t e n d t o t h e i n f i n i t e c a s e t h e n o t i o n o f scaffoZding i n t r o d u c e d f o r f i n i t e l a t t i c e s by R . Wille [ 9 8 1 ; t h e i r r e p r e s e n t a t i o n , which i s b a s e d on compact H a u s d o r f f t o p o l o g i c a l p a r t i a l l a t t i c e s , h a s proved u s e f u l i n p r o b l e m s on e x p o n e n t i a t i o n ( s e e [161 , [ 9 9 1 ) . U r q u h a r t ' s d u a l i t y [ 9 4 1 ; t h i s u s e s as r e p r e s e n t a t i o n s p a c e s c e r t a i n t o p o l o g i c a l s p a c e s e q u i p p e d w i t h a p a i r o f q u a s i o r d e r s l i n k e d w i t h t h e t o p o l o g y and w i t h e a c h o t h e r . Any l a t t i c e c a n be i d e n t i f i e d w i t h a c o l l e c t i o n of s u b s e t s o f s u c h a s p a c e u n d e r o p e r a t i o n s A , v, where A i s i n t e r s e c t i o n , b u t , a s i s i n e v i t a b l e f o r n o n - d i s t r i b u t i v e l a t t i c e s , v i s n o t g e n e r a l l y u n i o n . F o r n o n - s u r j e c t i v e morphisms t h e n o t i o n of a d u a l morphism d o e s n o t e x i s t , b u t t h e d u a l i t y d o e s e x t e n d f a r enough t o e n a b l e U r q u h a r t t o p r o v e t h e i n d e p e n d e n c e o f t h e c o n g r u e n c e l a t t i c e and t h e au tomorphism g r o u p o f a f i n i t e l a t t i c e .

The U-P d u a l i t y seems t o s t a n d i n a s t r a t e g i c p o s i t i o n . e x h i b i t many o f t h e s t r u c t u r a l f e a t u r e s t h a t p e r v a d e d u a l i t y t h e o r y , y e t i s s p e c i a l enough t o p r o v i d e a p r a c t i c a l method of s o l v i n g p r o b l e m s , and t h i s method h a s t h e mer i t o f b e i n g p i c t o r i a l .

I t a s s e r t s t h a t any A E /1 ( t h e v a r i e t y o f a b e l i a n g r o u p s ) i s

I n b o t h P o n t r y a g i n and F - t d u a l i t i e s t h e

A g e n e r a l f ramework f o r d u a l i t i e s o f t h i s s o r t h a s

C l o s e s t i n s p i r i t t o t h e C-z d u a l i t y i s A .

I t i s g e n e r a l enough t o

8 . A D-p DICTIONARY

The t a b l e be low summar ises some of t h e most commonly u s e d d u a l e q u i v a l e n t s . Many o f t h e s e now come i n t o t h e realm o f f o l k l o r e and w e h a v e n o t a t t e m p t e d t o

54 H.A. Priestley

a t t r i b u t e them. Most a r e easy consequences of r e s u l t s i n 92 o r appear a s theorems elsewhere i n the t e x t . Throughout, L E and has dua l space (XL,C,G); we i d e n t i f y L with O(XL). The topo log ie s of &-open dec reas ing ( i n c r e a s i n g ) s e t s a r e denoted by i: (U). In the t a b l e each s ta tement i n the f i r s t cslumn i s equ iva len t t o the cor responding s ta tement i n the second column.

D - L a t t i c e types

L f i n i t e

Boolean

a cha in

countable

comp 1 e t e

f r e e on K gene ra to r s

Spec ia l elements i n L

Jo in - i r r educ ib le (# 0)

Meet - i r reducib le (# 1)

Morphisms f E g ( L , M )

f i n j e c t i v e

f s u r j e c t i v e

I d e a l s , f i l t e r s , congruences

I d e a l i n L

p r i n c i p a l i d e a l

prime i d e a l

maximal i d e a l

F i l t e r

Congruence

L a t t i c e s r e l a t e d t o L

I d e a l l a t t i c e

F i l t e r l a t t i c e

Congruence l a t t i c e

LOP

Minimal Boolean ex tens ion

F r a t t i n i s u b l a t t i c e

P -

C d i s c r e t e

( X u , < ) an a n t i c h a i n

(XL,G) a cha in

C m e t r i s a b l e

C l U V 6 c v v E 1:

x = ZK L -

&-clopen s e t +;c

t -c lopen s e t X L \ tr

@ E _ P ' X M J L '

$ s u r j e c t i v e

$ a n embedding

C-open dec reas ing s e t

&-clopen dec reas ing s e t

s e t X L fz (3: E XL)

se t X L \ {XI (X maximal)

C-open i n c r e a s i n g s e t

t-open s e t

No tes / r e fe rence

Theorem 3 .5

Theorem 3.2

see [I]


n i' Notes l r e fe rence

Ca tegor i ca l no t ions

S u b l a t t i c e of L

Quot ien t l a t t i c e of L

F i n i t e d i r e c t p roduct

Coproduct

Tensor product

Equat iona l Subca tegor i e s

B (p-a lgebras) "W

L Hu Pseudocomplement a*

R -morphism -W

R," (dua l p-a lgebras)

L E n," Dual pseudocomplement a+

2 (Heyting a l g e b r a s )

L E E

R e l a t i v e pseudocomplement a + b

- H-morphism

Hop (dua l Heyting a l g e b r a s )

L E HOP

R e l a t i v e dua l pseudocomplement a + b

HoP-morphism

de Morgan a l g e b r a s

L de Morgan

-a

L Separa t ing s e t f o r X

Closed subspace of XL

D i s j o i n t union

Di rec t product

XI, ' +(a ' b )

Dual $ such t h a t $ o + = + . $

V E & * + V E C

+ ( b ' a )

$ 0 4 = + 0 $ Dual $ such t h a t

3 o rde r - r eve r s ing homeomorphism g on X L wi th g2 = 1

Coro l l a ry 2.5 & 111 Coro l l a ry 2.5

Theorem 3 . 3

see C771

Theorem 3.1

I n gene ra l , a unary homomorphic (dua l homomorphic) o p e r a t i o n on L corresponds t o a cont inuous o rde r -p rese rv ing (o rde r - r eve r s ing ) s e l f map on X L'

56 H.A . Priestley

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