Generalized continuous and hypercontinuous lattices

ROCKY MOUNTAIN JOURNA L OF MATHEMATICS Volume 11, Number 2, Spring 1981 GENERALIZED CONTINUOUS AND HYPERCONTINUOUS LATTICES G. GIERZ AND J. D . LAWSON* A class of complete lattices which have recently received a considerable deal of attention is the class of continuous lattices introduced by D. Scott [13] (see also [3]). One of the interesting features of this class of lattices is the fact that these lattices admit a unique compact Hausdorff topology for which the meet operation is continuous (i.e., they admit the structure of a compact topological semilattice). This topology turns out to be an "intrinsic" topology, i.e., one that can be defined directly from the lattice structure. We refer to this topology as the CL-topology. A major goal of this paper is to give a more detailed examination of this CL-topology. For any complete lattice this topology is always compact and 7\. We characterize those complete lattices for which it is Hausdorff; because these lattices have many characteristics reminiscent of continuous lattices, we call them generalized continuous lattices. They seem to be an interesting class of lattices in their own right; hence we develop some of their fundamental properties. One of the oldest of the intrinsic topologies is Frink's interval topology. We address ourselves to the question of for what continuous lattices do the CL-topology and the interval topology coincide. This turns out to be precisely the class of lattices which we call "hypercontinuous". We turn our attention to these and point out some surprising connections between these lattices and generalized continuous lattices. 0. Preliminaries. In this preliminary section we collect some well-known notations, definitions and results needed later on. DEFINITION 0.1. If L is any lattice and if A E L is a subset of L, then A is called on upper set provided that for a, b e L, a ^ b and a e A implies b e A. If A is any subset of L, then we denote by \A the smallest upper set of L which contains A9 i.e., ]A = {x: there is an a e A with a ^ x}. Lower sets and \A are defined dually. An upper set (lower set), which is at the same time a sublattice of L is called a, filter {ideal), A subset / E L is called down-directed {up-directed), if for each pair of elements a, b e I *The latter author acknowledges the support of the National Science Foundation through NSF No. MCS 76-06537 AOL Received by the editors on February 14,1979. Copyright © 1981 Rocky Mountain Mathematics Consortium 271 272 G. GIERZ AND J. D. LAWSON there is a c e I such that a A b ^ c (a V b ^ c). Note that / is downdirected (up-directed) if and only if \I is a filter ([I is an ideal) of L. DEFINITION 0.2. Let L be a complete lattice and let a, be Lbe two elements. Then we say that a is way below b (and write a < è), provided that every up-directed subset / of L with a supremum greater than or equal to b gets eventually above a, i.e., b ^ sup / implies the existence of a del such that a ^ d. An element which is way below itself is called compact. A complete lattice L is called a continuous lattice, if every element e e l is the supremum of all elements way below a. An algebraic lattice is a complete lattice in which every element is the supremum of compact elements. Note that every algebraic lattice is in fact continuous. The way below relation on every continuous lattice has the following interesting interpolation property. If a, be L are two elements of a continuous lattice L with a < b, then there is an element e e L such that a < c and c < b. DEFINITION 0.3. Among the several possibilities of defining a topology on a complete lattice L, we introduce in this preliminary section only the following one. Let U ü L be a subset. Then U is called Scott open if (i) U is an upper set and (ii) if D is an up-directed subset of L and if sup D e U, then there is de D with de U. It is easy to see that the Scott open subsets of L form a topology, which we shall call the Scott topoplogy. In terms of the Scott topology we can give the following characterization of continuous lattices. A complete lattice L is a continuous lattice if and only if for each a e l w e have a = sup {inf U: a e U, Uis Scott open}. In fact this characterization was the original definition given by D. Scott. EXAMPLE 0.4. An important example of continuous lattices is the following. Let I b e a locally compact topological space and let O(X) denote the (complete) lattice of open subsets of X. Then 0(X) is a continuous lattice and we have U < V if and only if U is compact and U E V. Moreover, for HausdorfT spaces X these are the only examples of this type. If we consider spaces X which are no longer HausdorfT, the question whether or not O(X) is a continuous lattice is more complicated. All we can say in full generality is the following. If X is any topological space, then 0(X) is a continuous lattice if and only if for every point xeX and every neighborhood U of x there is a neighborhood K o f i which is relatively compact in U. For a proof of this simple fact and more details concerning O(X) we refer to [7]. 1. Generalized continuous lattices. We give here a generalization of continuous lattices which we stumbled upon in another context (to which we turn later on). Besides having application to later situations, these GENERALIZED CONTINUOUS LATTICES 273 objects appear worthy of study in their own right. Their theory bears rather striking resemblances to that of continuous lattices and gives a new perspective to earlier work. DEFINITION 1.1. Let L be a complete lattice. If F E L is a finite subset and if x e L is an element, we write F < x if for every up-directed set Z), sup D ^ x implies y S d for some de D, y e F. In this case we say F guards x (from below). The idea here is that x cannot be penetrated without overrunning some member of F. The following is our original example. EXAMPLE 1.2. Consider the following subset of the unit square ordered by coordinatewise ordering, (xh yi) ^ (x2, y^ if *i ^ *2 a n d y\ S Ï2- (0, 1) (1, 1) (0, 0) (1, 0) Thus L is an upside down U\ L is a complete lattice with respect to the induced order although the meet operation is somewhat peculiar. Now let F = {(1, 1/2), (1/2, 1)}. Then F < 1 ( = (1, 1)) and as a matter of fact such sets (but not singletons) can be picked arbitrarily close to 1. Note that sets of the form (1/2 — e, 1), (1, 1 — e) guard the point (1/2, 1) and that the second points are necessary to prevent rear attacks. We turn now to the definition of a generalized continuous lattice. The idea is that each point can be guarded from finitely many locations "arbitrarily near" to x. In continuous lattices this is always possible from even one position. One has simply to pick an element way below x, and these elements can be picked arbitrarily close to x, i.e., x is the supremum of all those elements. DEFINITION 1.3. Let L be a complete lattice. Then L is a generalized continuous lattice (henceforth denoted GCL) if for all x, y e L such that x ^ y, there exists a finite set F such that F < x and l y fi F = 0 . This definition is a smoother version of our first one which we include in the following proposition (whose proof is quite straightforward and hence omitted). 1A. Let L be a complete lattice. For each xeL, let <F'x = { F i L: \F\ < oo and F < x). Then the following conditions are equivalent: (Ì) Lis a GCL. PROPOSITION 274 G. GIERZ AND J. D. LAWSON (2) For each x e L and for each choice function a e x ^ sup{a(F):Fe^x}. UFŒ^XF, we have One of the key properties of continuous lattices which makes everything work nicely is the interpolation property for < . We need also an interpolation property for GCL's, but it is more elusive in this setting. Hence we need to introduce and develop additional machinery. Let X be a TQ topological space. Then X has a partial ordering induced by the topology by defining x ^ y if and only if y e c\{x}. Conversely, let (X, S) be a poset. Then this ordering induces a topology on X by defining all sets of the form jx to be a subbase for the closed sets. The ordering induced by this topology is precisely the original ordering. This topology is not new, but it seems to have no standard name. In [7] it is called the INF-topology while in [11] it is called the closure of points (COP) topology. Since the open sets are lower sets, we refer to this topology as the lower topology. The upper topology is defined dually. Let X be a topological space, A E B E X. We say that A is precompact in B if every open cover of B contains a finite subcover of A. The following is a mild generalization of Alexander's Lemma. PROPOSITION 1.5. Let y be a subbasis for the topology on X. If A is precompact in Bfor the subbasis SF, then A is precompact in B. PROOF. Suppose A is not precompact in B. Then there exists an ultrafilter SF with A e BF such that & does not cluster (equivalently, converge) to any point in B. Hence for each point x e B, there exists a basic open set Ux sich that x e Ux $ BF. Since Ux is basic, there exist subbasic open sets S l5 . . . , Sn e £f such that Ux = Sx f] • • • f) Sn. If each S,- e BF, then Ux e BF. Hence there is an Sx e £f such that x e Sx, Sx $ BF. Since A is precompact in B with respect to «9*, there exist xi, . . . , xk e B such that ASXX1\J • • • U SXh, but Sx. £ BF for all 1 ^ / ^ k. Thus X\SX. e & for all /, and hence 0 = A f] (X\SX1) fl • • • fi (x\SXk) e &, a contradiction. We now present a mild generalization of definition (1.1). DEFINITION 1.6. Let L be a complete lattice, let F, G E L. Then we write F < G is sup D ^ g for some directe set D and some element gsG, then y ^ d for some y e F, d e D. In this case F is said to guard G. PROPOSITION 1.7. Let L be a complete lattice, F, G E L. Then F < G if and only if L\] F is precompact in L\]Gfor the lower topology on L. PROOF. Suppose L\]Fis precompact in L\]G. Let D be a directed set such that sup D ^ g for some g eG. Then L\]G g [j{L\}d: deD}9 since ]G ü }g =i O l î ^ ' de D). Since each L\\d is open in the lower GENERALIZED CONTINUOUS LATTICES 275 topology, there exist dl9 ..., d„eD such that L\]F S L\]di U * • * U L\\dn. Let (ie Z) such that du ...,dn£ d. Then L\]F g L\\dy i.e., \d g \F and hence d ^ / for some / e F . Conversely suppose F < G. Let {L\î*} xe 4 be a collection of subbasic open sets which covers L\]G. Let D be the up-directed set of all suprema of finite subsets of A. If B E A is a finite subset, then L\|(sup B) ü (J oeEß L\t6. Hence {L\|rf: deZ>} is a cover of L\fG. Thus sup D e ]G. Since F < G, there exist j> G F and a finite set B ë ^4 such that y S sup ,5. Now we can conclude that L\]F E Î J E L\î(sup 5) = {J^ßL^b. Hence finitely many of the collection {L\]x: xeA} cover L\F. Thus by Proposition (1.5) L\]F is precompact in L\ÎC7. 1.8. Let Lbe a generalized continuous lattice. (i) If F < A and G « ^, fAe/i F v G « ^ (wAere F V G = {x V y: xeF,yeG}). (ii) 7/*v4 w closed in the lower topology and if B < A, then there exists a finite set F such that A g \F g fi? a«tìf furthermore B < F < A. In particular, ifxeL, and if G is a finite set such that G < x, then there exists a finite set F such that G < F < x (the interpolation property). THEOREM PROOF. Let F, G < A and let D be a directed set such that a ^ sup D for some a e A. Then there exist x G F and J G G and rfl5 d2eD such that x g rfi and j> ^ d2. Pick Side D such that ^ and d2 are less than or equal to d. Then we have x V y ^ d. This proves (i). (ii) Suppose that A is closed in the lower topology, and that B < A. Let & = {]F: Fis finite and there is a finite set G such that F < G <. A}. We first show A — Ç\SF. Let y e L\^. Since A is closed in the lower topology, A is the intersection of sets of the form \G where G is finite. Hence there exists some set G which is finite such that y <£ G and A g \G. Since L is a GCL, for each g e G there is a finite set Fg such that Fg <. g and j> £ |F^. Let F = [j{Fg: geG}. Then F < G and >> £ | F . Repeat this process with F to obtain an Fx < F such that y $ Fx. Then Fx < F < A, and hence Fx e $F. Thus j^ £ n ^ * Since y was arbitrary, we have A — Ç\3F. Since each member of J* is closed in the lower topology and A = f]^9 {L\]F: ]Fe^} is an open cover of L\A. By proposition (1.7) L\\B is precompact in L\A. Hence we can find finite sets F l5 ..., Fn such that each }F{ is a member of & and such that L \ î # g L\ÎF X U • U F\ÎF M . For each /. there exists G{ finite such that Ft- < Gt < ^ . Let F = Fx v • • • V FM and G = Gx V • • • V Gn. By part (i) we have G < A and F < G. Since | F = î^! fi • • • fl î ^ we can conclude that L\\B g L\]F, i.e., Î F g ]B. For the final part let ^ = Î*. COROLLARY equivalent: 1.9. Let L be a complete lattice. The following statements are 276 G. GIERZ AND J. D. LAWSON (1) L is a GCL. (2) The lattice Oi(L) of sets open in the lower topology is a continuous lattice (with respect to the operation of intersection). PROOF. (1) => (2). Let U be open in the lower topology, and let xe U. Then L\U is closed and x £ L\U. Hence as in the proof of theorem (1.8), we can find a finite set F such that L\U E ]F and F < L\U (equivalently }F < L\U) and x $ ]F. By proposition (1.7) we have x e V = L\]Fç U and V is precompact in U. Since x was arbitrary, U = (J{K: Fis precompact in U). This is precisely the condition needed for Ox(L) to be a continuous lattice. (2) => (1). Let x, y e L, x $ y. Then y e L\}x ••= U, which is open in the lower topology. By hypothesis there exists an open set V such that y e V and V is precompact in U. Since O^L) is a continuous lattice and hence has the interpolation property, there is an open set W such that V is precompact in Pfand Wis precompact in U. Since K = L\Wis closed, as in the proof of the preceding theorem K is the intersection of a descending family of upper sets of finite sets. Since V is precompact in W, there exists a finite set F such that K g ]F and V g L\\F. Thus y$ \F and by Proposition (1.7) we have K < x and hence F < x since AT E î^7. Thus L is a GCL. 2. Topologies on generalized continuous lattices. DEFINITION 2.1. Let (X, ^ ) be a partially ordered set. A topology 0 on X is said to be order consistent if (i) for all x e X, c\({x}) = [x, and (ii) if D is an ascending subset of X and z = sup D, then considering D as a net, D converges to z in (9. If X is a T0 space, then a partial order may be defined on X by y ^ x if y e cl({x}). If a topology on a partially ordered set is order consistent, then this induced order is precisely the original order. PROPOSITION 2.2. On a partially ordered set (X, ^ ) the upper topology is the coarsest of the order consistent topologies and the Scott topology is the finest. PROOF. Since for every closed set A in the upper or Scott topology [A = A and since [x is closed for all x, we have cl({x}) = [x for both of these topologies. Let D be an ascending subset of X. If D does not converge to z = sup D in the upper topology, then there exists a subbasic open set L\\x such that z G L\[X but some cofinal subset of D misses L\[x. But since D is ascending, we have that there exists d e D such that d' ^ x for d ^ d'. Hence we have x ^ sup D, i.e., x ^ z. This contradicts z e L\[x. Therefore the upper topology is order consistent. GENERALIZED CONTINUOUS LATTICES 277 If D does not converge to z in the Scott topology, then there is an open set U such that z e U, but some cofinal subset of D misses U. Since U = Î U, D itself misses U. Since U is open, z = sup D $ U, a contradiction. Thus the Scott topology is order consistent. Now let (9 be any order consistent topology. Since for all x e X, [x = cl({*}), which is closed, the topology (9 is finer than the upper topology. Let C be a closed set in Xfor the ^-topology. Then [C = C (from condition (i) for order consistency). Let D be an up-directed set, D E C. Then D converges to sup D since 0 is order consistent and hence sup D e C since C is closed. Thus C is closed in the Scott topology. We return to these "one-sided" topologies shortly, but first we have need to introduce some "two-sided" ones. DEFINITION 2.3. Let L be a complete lattice. We define the CL (or Lawson) topology on L to be the topology for which all Scott open and all lower open sets form a subbase. We define the Limlnf topology (LI) by declaring a set A to be closed if every ultrafilter which has A as a member has the liminf of the ultrafilter in A (if J* is an ultrafilter, liminf SF = sup {inf F: Fe &?}). Equivalently A is closed if for every universal net in A, the liminf of the net is again in A. PROPOSITION 2.4. Let L be a complete lattice. Then the CL and the LI topologies are compact and 7\. The identity mapping is continuous from (L, LI) to (L, CL). PROOF. We first show the latter statement. Let K be a closed set in the lower topology. Let ^ be an ultrafilter such that K e<F.Let M be a finite set such that K E \M. It suffices to show liminf 3F e \M since by definition of the lower topology K is the intersection of such sets. Since <F is an ultrafilter and since ]M = (J{î* : * e M } and since this latter collection is finite, we can find x e M such that ]x e gF. Thus x ^ liminf 2F and hence liminf J* e \M. Now let A be a Scott closed set and let again & be an ultrafilter such that / l € , f . For any F e « f , F Ç] A ^ 0 and hence inf F S a for some a e A. Since A = [A9 inf Fe A. Since A contains sups of up-directed sets, liminf ZF e A. Thus both lower closed sets and Scott closed sets are closed in the LI topology and therefore the identity function is continuous. Furthermore, let L be equipped with the LI topology. Then in this topology ultrafilters still converge to their liminf<F(a general characteristic of defining a topology in terms of filters or nets), although additional limit points may also exist in the topology. In particular since every ultrafilter has a point of convergence, (L, LI) is compact. By continuity (L, CL) is compact, too. Finally, (L, CL) is Tx since {x} = [x f] }x and ]x is closed in the 278 G. GIERZ AND J. D. LAWSON lower topology and }x is closed in the Scott topology. Hence by continuity of the identity mapping (L, LI) is also Tv We come at this point to a major theorem. THEOREM 2.5. Let L be a complete lattice. Then the following statements are equivalent: (1) Lis a GCL. (2) (L, CL) is Hausdorff. (3) (L, LI) is Hausdorff. Furthermore if any of these equivalent conditions are satisfied, then the LI and CL topology agree and the partial order ^ has closed graph for this topology. Furthermore, this topology has a subbase of open sets all sets of the form {s: F < s) where Fis some finite set and L\]x where x e L. PROOF. (1) => (2). Let L be a GCL. Let x, yeL and suppose that x S y. Then there is a finite set F such that F < x and y $ ]F. Let U = {s: F < s} and let V = L\\F. Then U f| V = 0 and V is open in the lower, and hence CL, topology. To finish the proof we show that U is Scott open. Let D be an up-directed set such that p = sup D e U. By the interpolation property (1.8 (ii)), we can find a finite set G such that F < G < p. Thus there exists de D such that b ^ d for some b eG. But F < G implies F < b and thus F < d. Hence sup D e U implies de U for some J G D. Thus £/ is Scott open. Note that by the preceding paragraph sets of the form {s: F < s} and L\]x, x e L generate a Hausdorff topology if L is a GCL, and that these sets are open in the CL topology. Since by proposition (2.4) the CL topology is compact, these sets must generate precisely the CL topology. Hence the last statement of the theorem holds. (2) => (3). This is immediate since by (2.4) the identity function from (L. LI) to (L, CL) is continuous. (3) => (\). Note first that if SF is an ultrafilter on L, then $F converges to liminf 3F and to that point alone in the LI topology (since the convergence by definition implies convergence in the topology and since by Hausdorffness there is at most one point of convergence). We show now that the relation ^ is closed in L x L (with each factor equipped with the LI topology). Let ^ be an ultrafilter contained in {(x, y): x ^ y}9 i.e., the set ^ is a member of ^ . Let &x = {^(G): Ge&} and ^2 = W ß ) : Ge&}. For each G e <g such that G g {(x, y):x ^ y}, we have for the first and second projections, 7C\{G) and ft2(G), inf(^x(G)) S infO^G)). Thus liminf{ffi((j): Ge&} ^ liminf{TC2(G) : Ge&}. Since ^ is an ultrafilter, %<S&) and TZ2{&) are also ultrafilters. Hence if GENERALIZED CONTINUOUS LATTICES 279 a = liminf{ici(G): Ge&} and b = liminf{tf2(G): G e ^ } , then n^g) and TZïdg) converge to a and b respectively, and therefore ^ converges to {a, b). Hence the relation ^ is closed. We now have that under the hypothesis (3) L is a compact Hausdorff space with a closed partial order with respect to the LI topology. Hence we may invoke the known properties of such structures. Let x, y e L with x $ y. We wish to find a finite set F such that F < x and y $ \F. It suffices to find a finite F such that j> <£ j F a n d there exists an open set U such that U = ÎUand xe U ^ |F(for if D is an up-directed set with sup D ^ x, then since D converges to its supremum, de U E ]F for some deD). Thus suppose for every open set U ü ]U with xe Uand for every finite F such that J ; ^ }F, U $ | F , i.e., C/\|F =£ 0 . Then these sets form a filter base; extend this base to an ultrafilter <F. Since<Fcontains all open neighborhoods U = | U of x, the limit of BF is a member of |x. Since {inf L: Le <F} converges to the liminf of J^ which must be the limit of 8F) there exists L e J* such that inf L % y. We thus have \z e 3F where z = inf L and L\|z e J^ from the original definition. However this is impossible; so the argument is complete. We have shown in this argument that ^ is closed. Since (L, LI) is compact, if (L, CL) is Hausdorff, then the two topologies agree by (2.4). The next corollary is due to K.H. Hofmann. COROLLARY 2.6. Let L be a meet continuous complete lattice. Then if L is a GCL, it is a continuous lattice. PROOF. By [5] a meet continuous complete lattice for which the CL topology is Hausdorff is a continuous lattice. Hence the corollary follows from theorem (2.5). We return now to a more detailed consideration of the lower topology. 2.7. Let Lbea complete lattice. (i) If 3F is an ultrafilter on L, the set of cluster ( = convergence) points is \(liminf <F) in the lower topology. (ii) A subset M of L is closed in the lower topology if and only if \M = M and if for every ultrafilter gF with M e 3F, liminf BF e M. (iii) A subset M of L is closed in the lower topology if and only if ]M = M and if M is closed in the CL topology. PROPOSITION PROOF, (i) The cluster points of an ultrafilter consists of all points in the intersection of the closure of all sets in the ultrafilter. If A is the set of cluster points, we have A = C\{F: FeBF) g f | { î ( i n f F ) : F e BF} = î(liminf BF). Conversely, let y ^ liminf &. If y $ F, then by definition of the lower 280 G. GIERZ AND J. D. LAWSON topology, there is a finite set K such that F E ]K and y i \K. Since F E F E ]K, we have ]Ke£F. Since ^ is an ultrafilter, ]xe^ for some x e K, because K is finite. Hence x ^ liminf 3F, a contradiction to x $ y. Thus y e F for all Fe ^F and therefore jus a cluster point. (ii) Suppose M is closed in the lower topology. It follows immediately that M = ]M. Since M is closed, every ultrafilter containing M converges to points which are contained in M. Hence by part (i), we have liminf <F e M. Conversely suppose \M = M and M e , f implies liminf«^ G M for every ultrafilter <F. Let yeM. Then we can find an ultrafilter 3F with M e J*7 converging to y. By part (i), y j> liminf ^ \ Since liminf 3F e M and M = \M, we can conclude that J G ¥ . Thus M is closed. (iii) If M is closed in the lower topology, then it is closed in the CL topology by definition. Conversely suppose M = \M\s closed in the CL topology. Then by proposition (2.4) it is closed in the LI topology and hence by part (ii) in the lower topology, too. COROLLARY 2.8. Let L be a complete lattice and let 0 be an order consistent topology on L. The set of cluster points for an ultrafilter gF is contained in Î(liminf J^). PROOF. This corollary follows easily from (2.7. (i)) and (2.2.) PROPOSITION 2.9. Let L be a complete lattice. A subset U of L is open in the Scott topology if and only if U — ]U and U is open in the CL topology. Furthermore the following conditions are equivalent: (l)LisaGCL. (2) For every ultrafilter SF the set of cluster points of gF for the Scott topology is |(liminf £F). PROOF. By definition of the CL topology, if U is Scott open, then U is CL open. Conversely suppose U = ]U is CL open. Let D be an updirected set (with x = sup D) in L\U. It is easily verified that as a net D converges to every point of [x in the lower topology, to every point of [x in the Scott topology and hence precisely to x in the CL topology. Since L\U is CL closed, xeL\U. Hence L\U is Scott closed, i.e., U is Scott open. (1) => (2). Let SF be an ultrafilter, y = liminf <F. If x $ y, then we may find a finite set K such that K < x and y $ ]K. Set W = {w: K<t:w}. As in the proof of theorem (2.5), Wis Scott open. Hence x is not a cluster point of 8F for the Scott topology. Let F e $F and let z ^ y with U a Scott open set such that z e U. Since U is Scott open, yeU and hence inf F' e U for some F' e &. Since F' E î(inf F% F' E U. Also F Ç] Ff ^ 0 implies F f] U ^ 0 . Since U was an arbitrary Scott open neighborhood of z, the Scott closure F of F contains z. Since F was arbitrary, z is a cluster point of !F (this inclusion holds even if L is not a GCL). 281 GENERALIZED CONTINUOUS LATTICES (2) => (1). By (2.7. (i)), the hypothesis and the definition of the CL topology, the only possible cluster point for an ultrafilter !F is liminf J*. Hence the CL topology is Hausdorff. The conclusion follows now from theorem (2.5). PROPOSITION 2.10. Let L be a complete lattice equipped with a compact topology for which the closed upper sets are precisely the sets closed in the lower topology andfor which the partial order is closed. Then L is a GCL. PROOF. It is well known that the closed upper sets of a compact space with closed order form a continuous lattice (see [4]). The proposition follows now from proposition (1.9). 3. Morphisms between generalized continuous lattices. In this section we want to describe under which conditions a mapping between generalized continuous lattices is continuous for the various topologies. In this context it seems to be reasonable to restrict ourselves to monotone mappings. Let us start with a well-known result (see [13]): PROPOSITION 3.1. Let £: L -> L' be a monotone mapping. Then £ is continuous for the Scott topology on L and L' resp. if and only if'£ preserves suprema of up-directed families. PROPOSITION 3.2. Let £: L -» L' be a mapping preserving up-directed suprema. Then £ is continuous for the lower topologies on L and L' resp. if and only if £ preserves liminf s of ultrafilters. Moreover, any monotone mapping preserving liminf s of ultrafilters is continuous. PROOF. Let us first assume that £ is continuous. Pick an ultrafilter SF on L. Then £(liminf SF) is a clusterpoint of the ultrafilter £ ( ^ ) by the continuity of £. This implies liminf Ç(^) S £(liminf g>) by corollary (2.8). and the latter supreConversely, £(liminf &?) = £(sup{inf F: Fe^}) mum is up-directed. Hence £(liminf &) = sup £{(inf F): Fee?} S sup{inf Ç,(F): Fe^} = liminf C(J^) and this inequality holds because £ is monotone. Now let £ by any monotone mapping preserving liminf's of ultrafilters. Then £ is continuous. Let A' E L' be closed in the lower topology and let A — Zrl(A'). Then A is an upper set because £ is monotone and A is closed under liminf's of ultrafilters containing A because A1 is. Hence A is closed in the lower topology by (2.7). PROPOSITION 3.3. Let £: L -> L' be a monotone mapping between two complete lattices L and L'.IfC, preserves up-directed suprema and liminf s 282 G. GIERZ AND J. D. LAWSON of ultrafilters, then Ç is continuous for the CL topologies on L and L resp. Moreover, ifL is a GCL, then the converse also holds. PROOF. By (3.1), (3.2) and the definition of the CL topology we need only give a proof of the second statement. Let us assume that L is a GCL and that Ç: L -> L' is CL continuous If U g L is Scott open, then Zr\U) is an open upper set and hence open in the Scott topology by (2.9). Hence £ is Scott continuous and therefore preserves suprema of up-directed sets by (3.1). Next, let a' e L' and let A = Zr\\a'). Then A is a CL closed upper set of L and therefore closed in the lower topology on L by (2.7). Now we can conclude that £ is continuous in the lower topology and hence preserves liminf's of ultrafilters by (3.2). In the remainder of this section we will show that generalized continuous lattices are preserved under a weak kind of quotients, subojects, and products. PROPOSITION 3.4. Let L be a GCL, L a complete lattice, and £: L -» L be a surjective mapping which preserves upwards directed suprema and which is continuous for the lower topology (i.e., £ is monotone and continuous for the CL topologies). Then L is a GCL. PROOF. Endow L with the liminf topology. Then (L, LI) is compact and Hausdorff by (2.5). Moreover, ker Ç = {(a, ò):£(fl) = £(ò)} is closed in L x L. Indeed, let <g be an ultrafilter o n l x l containing ker £. We have to show that lim <& = (lim %fé, lim %2^) = (liminf %x%, liminf %2<g) is contained in ker £, where %x. L x L -> L and %<i'. L x L -• L denote the first and second projection resp. First, note that M ü ker £ implies £ o 7Ci(M) = C ° %2ÌM) a n d that £ preserves liminf's of ultrafilters by the assumptions (3.4) and proposition (3.2). Hence £(liminf %i&) = liminf £ o 7 ^ = liminf £ ° 7 ^ = Ç(liminf 7r2^)Now we can conclude that the quotient topology of £ on L is Hausdorff. A similar argument shows that the order ^ is closed in the quotient topology. If A' ü L' is closed in the lower topology, then Zrl(A') is closed in the lower topology of L by the assumptions of (3.4). Hence Z~l(A') is closed in the CL topology of L and therefore Ä is closed in the quotient topology. Conversely, let A' = \Ä Ü L' be an upper set which is closed in the quotient topology. Then A' is closed under liminf's of ultrafilters. Indeed, let SF' be an ultrafilter on L containing Ä. Pick an ultrafilter <F containing Xr\&'\ Then Zr\A') e & and ÇGF) = 3F\ As Zrl(A') is closed in the CL topology of L, liminf«^" e Ç - 1 ^')- Hence GENERALIZED CONTINUOUS LATTICES 283 Ç(liminfJÊ') = liminf C(^) = liminf J^' e A', because £ preserves liminf's of ultrafilters by (3.3). Now (2.7) yields that A' is closed in the lower topology. Therefore the lower closed sets of L are exactly the upper sets which are closed in the quotient topology. This completes the proof by (2.10). DEFINITION 3.5. A CL-morphism between complete lattices L and V is a mapping preserving arbitrary infima and up-directed suprema. Clearly, every CL-morphism preserves liminf's of ultrafilters. This yields the following proposition. PROPOSITION 3.6. Every CL-mrophism is continuous for the lower topologies, the Scott topologies, the CL topologies, and the LI topologies on L and L' resp. PROPOSITION 3.7. Let Ç: L -> L' be a CL-morphism between complete lattices L and L. (i) IfL, is surjective and ifL is a GCL, then so is L'. (ii) ffZ is injective and ifL is a GCL, then so is L. PROOF, (i) follows from (3.4) and (3.6) and (ii) follows from the observation that Ç(L) is a subsemilattice of L, which is closed in the LI topology. Now let L be a complete lattice and let P(L) its ideal lattice. From [8] we know that L is a continuous lattice if and only if the mapping / H-> sup /: P(L) -> L is a CL-morphism and that L is meet continuous if and only if / •-> sup / : P(L) -> L preserves finite infima. We add one more condition. THEOREM 3.8. Let Lbe a complete lattice. Then L is a GCL if and only if the mapping 11-> sup /: P(L) -• L is continuous for the lower topology (or equivalently for the CL topology) on P(L) and L resp. PROOF. AS the mapping / •-» sup I always preserves up-directed suprema, one direction is clear by (3.4). Conversely, let L be a GCL and let a e L be a point. We have to prove that {/ e P(L) : a "$ sup /} is open in the lower topology of P(L). Let Je {le P(L) : a $ sup / } . Then sup J ^ a. Hence we can find a finite set F ü L\|sup / which guards a from below. Now let O = {/e P(L): î / g / for a l l / e F } . Then / e O and O is open in the lower topology of P(L). Moreover, I e O implies sup I <£ a, because otherwise we would have \f f| / ^ 0 for s o m e / e F, and t h u s / e /, i.e., [f E /. Hence Je O c {Ie P(L): a $ sup / } . COROLLARY 3.9. A complete lattice L is a GCL // and only if it is the quotient of a continuous lattice L under a mapping which is continuous for 284 G. GIERZ AND J. D. LAWSON the lower topologies on L and U resp. and which preserves up-directed suprema. We conclude this section with two results, which may perhaps illustrate that many properties holding for continuous lattices are also true for generalized continuous lattices. PROPOSITION 3.10. (See [8]). If L and L are GCL and ify:L-±L is a mapping preserving arbitrary infima, then y preserves up-directed suprema if and only ij F < G in L' implies öF < öG in L, where d: L' -> L is the right adjoint of y (i.e., d(a) = inf y~l(]a)). PROOF. Assume that y preserves up-directed suprema and that sup D ^ ö(g) for some geG and an up-directed family D g L. Then ^(sup D) = sup y(D) ^ g, hence y(d) ^ / f o r some de D and s o m e / e F This implies d ^ 5(f) for some de D and s o m e / e F, i.e., dF < öG. Conversely, assume that F < G implies ô(F) < ö(G). Let D g L be an up-directed set. We have to show that y (sup D) = sup y(D). Let F ü L be a finite subset which guards ^(sup D) from below. By hypothesis we have OF < <?7-(sup D) S sup D, and hence we can find a n / e F and a d e D such that ôf ^ d, i.e.,/ ^ yd. Therefore we can conclude that sup y(D) e f]{]F: F < y sup D) = ]y sup D; the last equality follows from (1.4). But this yields y sup D ^ sup y(D). The other inequality holds for every monotone mapping. Our second example is the lemma on primes. LEMMA 3.11. (The Lemma on Primes. See [4]). Let Lbea complete lattice and let L' be a GCL. Furthermore, let £ : L' -> L be a mapping preserving up-directed suprema and let p e L be an element ofL satisfying the following primality condition. (P) For every finite subset A g L\ inf Z(A) ^ p implies the existence of an a e A such that Z(a) ^ p. Then for every CL compact subset K g Z/, inf L,(K) ^ p implies the existence of a k e Ksuch that^(k) ^ p. PROOF. Let K g L' be a CL compact subset of L such that inf C,(K) ^ p and assume that K f] Ç,~l(ÏP) = 0 - Then K is contained in the Scott open set L \ C - 1 ( 1 / J ) . Hence for every k e K we can find a finite subset Fk g L\Zrl([p) with Fk < k (if every finite subset F < k would intersect £>~l(lp), then f]{]F f| C~1(l/?): F < /:} would be non-empty, because Z/ is CL compact. But n{î^nc-lap):^«^} = c-11(iJp)nn{î^^«^} = c- u/>)nî* = 0Now we have K g [Jk(=K{x: Fk < x} and hence GENERALIZED CONTINUOUS LATTICES 285 K g Û I*- Fk< « A for certain ku ...., kn e K by the compactness of K (Recall from the proof of (2.5) that the set {x: Fk < x] is open in the Scott topology). Now, clearly, />^infÇ(/OàinfC(Û^,) 1=1 and \Jf=1Fk. is finite. Hence the primality condition (p) yields an element / e U ? = i ^ / w i t h Z(f)èp, a contradiction to \J»=1Fkirg ZAC^Ü/O. Therefore there is an element k e K with £(/:) ti p. 4. Complete lattices in the interval topology. In this section we discuss the symmetric interval topology and its relations to generalized continuous lattices. DEFINITION 4.1. Let L be a complete lattice. We define the interval topology ( = IV topology) on L to be the topology which has as a subbase for the closed sets the collection of all closed intervals [a, b] = {x: a ^ x g b, a, be L}. It is immediate that the IV topology is the supremum of the lower and the upper topology on L. This observation proves a large part of the following proposition. PROPOSITION 4.2. Let Lbe a complete lattice and let gF be an ultrafilter on L. Then the set of all cluster points of ^ in the IV topology is exactly the set [liminf J% limsup <F\ and this set is not empty. Hence L is a compact Tx space in the IV topology and the IV topology is coarser than the LI topology and the CL topology on L {or L°P). PROOF. Clearly, for every filter ZF we have supjinf M:Me^} 5g inf{sup M: M e <F) and hence [liminf SF^ limsup SF\ ^ 0 . By (2.7) and the above remark, [liminf &?, limsup 3F\ is exactly the set of all cluster points of an ultrafilter $F. PROPOSITION 4.3. Let Lbe a complete lattice. Then the following statements are equivalent. (i) The IV topology on L is Hausdorff. (ii) For every ultrafilter<Fon Lwe have liminf !F = limsup <F. (iii) L is a generalizedbicontinuous lattice {i.e., L and L°P are both GCL and the CL topologies on L and L°P agree). In each of these cases, the IV topology and the CL topology agree. PROOF, (i) o (ii) holds by (4.2). (i) => (iii). By (4.2), the IV topology is coarser than both the CL topology 286 G. GIERZ AND J. D. LAWSON on L and the CL topology on L0*. If the IV topology is Hausdorff, then all the three topologies agree. Hence (iii) follows by (2.5). (iii) => (ii) is an easy consequence on (2.5). COROLLARY 4.4. Let Lbe a meet andjoin continuous lattice. Then the following conditions are equivalent. (i) L is bicontinuous. (ii) The interval topology on L is T2. (iii) The CL topology on L and the interval topology on L agree. Here again, a bicontinuous lattice L is a complete lattice with the property that L and L°P are continuous lattices and the CL topologies on L and L°* resp. agree. We will give another description of bicontinuous lattices later on. For the moment let us return to lattices which are Hausdorff in their interval topology. This type of lattice is preserved under a certain kind of quotients. PROPOSITION 4.5. Let L andL' be two complete lattices and let £ : L -> L be a sürjective mapping preserving up-directed supprema and down-directed infima. Assume moreover that the IV topology on L is Hausdorff. Then the IV topology on L is also Hausdorff. PROOF. Let 3F be an ultrafilter on L and let 3F' be an ultrafilter on L which is mapped onto ^ under Ç. As limsup &?' = liminf <F' in L and as Ç preserves up-directed suprema and down-directed infima, we can compute limsup <F = limsup C(^0 = inf{sup Ç(M): M e / } ^ inf{£(sup M):Me&'} = £(inf{sup M: Me &"}) = ^(limsup &>') = £(liminf #") = C(sup{inf M: M e &'}) = sup{£(inf M): M e &'} S sup inf{C(M): M e ^ ' } = liminf C(^") = liminf &?. Hence liminf !F = limsup 3F. COROLLARY 4.6. Let L' be a completely distributive lattice and let Lbe a complete lattice, Further, let £ : L' -> L be a sürjective mapping preserving up-directed suprema and down-directed infima. Then the interval topology on L is Hausdorff. PROOF. It is well known that the IV topology on a completely distributive lattice is Hausdorff (see [10]). Now apply (4.5). PROPOSITION 4.7. Let L andL' be two complete lattices, andletC,: L -• L be a monotone mapping. If the IV topologies on L andL resp. are Hausdorff, GENERALIZED CONTINUOUS LATTICES 287 then Ç is continuous if and only ij'£ preserves up-directed suprema and downdirected infima. PROOF. Every supremum of an up-directed family is the limit of the family, indexed by itself. Hence every continuous map preserves updirected suprema and down directed infima. Conversely, let 3F be an ultrafilter on L and let &' = CC^")- Then the proof of (4.5) shows that lim C(^") = limsup Ç(^0 g Ç(limsup SF) = Ç(lim &) S liminf C(^) = Hm ; ( F ) , i.e., lim C(^) = C(lini &). This is equivalent to the continuity of £. Corollary (4.6) gives rise to the question whether every complete lattice with the property that the IV topology is Hausdorffis a quotient of a completely distributive lattice under a mapping preserving upwards directed suprema and down-directed infima, i.e., a continuous and monotone mapping. We shall prove later on that this is at least true for meet (or join) continuous lattices, and we will see in a moment that this is true for distributive lattices. But we do not know the answer in general. Let us first recall some facts from the Priestley duality for distributive lattices. THEOREM 4.8. (Priestley). Let VDbe the category of all distributive lattices with 0 and 1 together with all lattice homomorphisms preserving 0 and 1 as morphisms, and let PZ be the category of all zero-dimensional compact (T2) ordered spaces with the property that two different points can be separated by a clopen upper set, together with all order preserving continuous mappings as morphisms. Then LD and PZ are dually equivalent under the contravariant functors LD( —, 2) : LD -> PZ, where for a lattice L, LD(L, 2) denotes the set of all LD-morphisms from L into 2, equipped with the pointwise ordering and the topology induced from the product topology of the discrete space 2, and PZ ( —, 2) : PZ -> LD, where for a VZ-object X, PZ(X, 2) is the lattice of all continuous and order preserving mappings into the discrete two element chain ordered by the pointwise ordering. The natural transformations ?7:1LD -> P Z ( L D ( - , 2), 2) and e: 1PZ -> L D ( P Z ( - , 2), 2) are given by 77L:L-PZ(LD(L,2),2) -> â, â(0 = «A) and ep:P-+LD(?Z(P,2),2) p^p, P(0 = COO- 288 G. GIERZ AND J. D. LAWSON As a consequence we have the following proposition. PROPOSITION 4.9. (see [1]). Let X be a partially ordered set. Then the free distributive lattice generated by X is isomorphic to PZ(X, 2), where X denotes the set of all order preserving maps £: X -> 2, equipped with the topology of pointwise convergence and thepointwise ordering. In the sequel, for a partially ordered set X the algebraic lattice X is always endowed with the CL topology. PROPOSITION 4.10. Let P be a VZ-object and let Ç : P -+ 2 be a monotone mapping. Then C = sup{inf{^: <f> e PZ(P, 2), 0 è * } : X ^ C, ^ d ) * <*w«/} = inf{sup{^: 0 G PZ(P, 2), 0 ^ %} : % à C, C - 1 0) is open}. Furthermore, all occurring suprema and infima are directed. First, let yr. P -> 2 be a monotone mapping such that x _ 1 0 ) *s closed. Then %~l(l) = ^ { ^ - % _1 0) = ^ ^ a clopen upper set} by the definition of PZ-objects and an easy compactness argument. Hence X — 'mî{iu'. x~l0) i t / , ( / a clopen upper set} = inf{^: cp e PZ(P, 2), (]j ^ ^ } , where ^ denotes the characteristic function of £/, and this inflmum is up-directed. Next, let £: P -» 2 be monotone. Then £ - 1 0 ) = LT {^4: Ç _1 0) ^ A, A & closed upper set}. Hence £ = sup{^ A : C _1 (0 = ^ = |y4, y4 is closed} = sup{^: x = ^ Z - 1 ( 0 i s closed}, where again XA denotes the characteristic function of A, and this supremum is also directed. This proves the first equality. The proof of the second equality is similar. PROOF. Now let us return to generalized continuous lattices. PROPOSITION 4.11. Let L be a GCL and let (/, ^ ) be a down-directed index set. For each i e I let D{ Ü L be an up-directed subset of L such that i ^ j implies D{ Ü Dj. Then the equation holds in L. AVA-= V A«(0 PROOF. The inequality " ^ " holds in every complete lattice. Conversely, let F e L be a finite subset that guards the left hand side from below. Then for every / G / there is a âf G D{ such that de }F. Hence we can find an fe F such that {/ G /: there is a de Dt with / ^ d) is cofinal in /. This certainly implies {/ G / : there is a de D{ w i t h / ^ d) = /. Hence there is an a G J]*e/ A such that A»e/ a(i) ^ fe F. But then the right hand side is contained in ]F. As Fis an arbitrary finite subset which guards Aie/ V A from below, the proof is complete by (1.4). GENERALIZED CONTINUOUS LATTICES 289 PROPOSITION 4.12. Let L be a GCL and let L denote the PZ-object of all monotone mappings £: L -> 2. Further, let X: PZ(L, 2) -> L be an Ahomomorphism and let L be the set of all monotone mappings £: L -> 2. 77^« f/ze mapping X*: L -+ L defined by h(Q = s u p ( i n f W ) : 4> e PZ(L, 2), 0 ^ z } : z g Ç, r K D * <*«"*} preserves down-directed infima. Dually, ifL°P is a GCL and if X: PZ(L, 2) -*• L is a V -homomorphism, then X*\L -* L defined by X*(Q = inf{sup{W): </, e PZ(L, 2), 0 g z ) : z ^ Ç, Ç-i(l) ft */**} preserves up directed suprema. PROOF. Let I E £ be down-directed and for every Ç e 7 let Z>c = { i n « ) : 0 e PZ(L, 2), 0 ^ z } : z g Ç, z - i ( l ) closed}. Then sup D^ = /l*(0 and Ç ^ Ç' implies Z>c E ^V- Hence we have inf{;i*(0: C e /} = AV^c = A V «(0 by (4.11). We want to show that V Aa(0^*(inf/). a^riDç C e / Let a e flc^/ ^c- Then we have to show that A a(Q ^ ^(inf / ) = {sup{inf X{cjj): 0 e PZ(L, 2), 0 ^ z } : c^/ _1 0 ) is closed}. z ^ inf 7, Z Now a(Q = i n « ) : </, e PZ(L, 2), ^ ^ Zc } for some closed. Choose Z o = inf{ Zc : £ e 7}. Then Z o = inf/and Zc ^ Ç, Zc'HD c^/ is closed. Hence it is enough to show that Ace/ cc(Q ^ A(^) for every 0 G PZ(L, 2), ç£ ^ Zo . But for every c[) e PZ(L, 2), 0-1(1) is clopen and ^ 0 ) 3 jßfW = H tfKl). c^/ Therefore we can find finitely many d , . . . , ÇM e 7 such that z& ! (i) n • • • n Z5KD Ü ^ K D , using the compactness of L. Again the compactness of L yields elements ^ l 5 . . . , </>„ e PZ(L, 2) such that Zc . ^ 0, and ^ A • • • A <jjn ^ <j). But now the fact that X is an A -homomorphism allows us to conclude A a(Q è K<h) A . . . A X(</>„) = Xifa A - - • A fa ^ Xty). c^/ 290 G. GIERZ AND J. D. LAWSON This proves inf A*(/) ^ /l*(inf /). The other inequality is always true. The second claim can be proven dually. PROPOSITION 4.13. If L and L°J are GCL and if I: PZ(L, 2) - * L is a lattice homomorphism, then /I* : L -> L and 1*: L -+ L agree. PROOF. We first show that for every C G £ the inequality 1*{Q ^ A*(Q holds. Indeed, this inequality is true for every complete lattice L and every monotone map h PZ(L, 2) -> L. Let ^ ^ £ ^ 7- such that % _1 0) is closed and f~l(\) is open. Then we have to show that inf W ) : <1> G PZ(£, 2), z ^ 0} ^ sup{A(0): 0 G PZ(L, 2), 0 ^ y). But x = 7 i m P n e s % -1 (l) = r _ 1 ( 0 - Now the compactness of L and the fact that L has enough clopen upper sets yield an open and closed upper set U such that x~l0) = U = r _ 1 ( 0 - Then the characteristic function xu of U satisfies x = Xu = 7 anc * hence i n f W ) : 0 e PZ(L, 2), z ^ 0} ^ X(Xu) ^ sup{A(0): 0 e PZ(L, 2), 0 ^ r } . To show the converse inequality, let / = {^ e L: ^ ^ £, % _1 0) is closed}, / = ( T - G L ; C â 7% r _ 1 ( 0 is open}. For every # G / let Dx = (Jl(0): 0 G PZ(L, 2), ^ g <[)} and for every 7- G / let Dr = {A(0): 0 G PZ(L, 2), 0 ^ 7*}. Then / i s up-directed, / i s down-directed and for ^ , ^ ' e / , ^ ^' implies Z)x E Ac a s w e ^ a s f ° r 7% 7*' G A 7* = 7*' implies Z)r g Z>r/. Moreover, by (4.11) we have UQ = WÄDX = A V a(z) and A*(0 = A \JDr = V A cc(7\ Fix a G n£>x and /3 G UDr. Then we have to show that Aß(r)£ V # But for every y e J we have /3(j) = A(0r) for some y ^ 0 r G PZ(L, 2) and for every # G / we have <%(%) = X((J>X) for some x = $xe PZ(L, 2). Moreover, 0(^(1): r e / }E n t r K D : r e / } = Ç-i(l) = I K r K O : Z ^ / } iUfc'd):^/}. Because both (j)~\\) and ^ O ) a r e clopen for all y G / , x e h and because L is compact, we can find yx, . . . , 7%, G / a n d ^ l5 . . . , %m e /such that 0-Ki) n • • • n 0^(1) s 0^(1) u • • • u 0^(1), 291 GENERALIZED CONTINUOUS LATTICES i.e., <pn A • • • A </>Tn ^ (pXl V • • • V (JjXm. Now the fact that X is a lattice homomorphism yields Aj3(r) ^ j8(n) A ••• A/3(r„) = ^ n ) A • • • A ktyj = A(0n) A • • • A 0 ( J g A(0Zl v • • • v $ J = X(<f>Xl) v • • • v AfeJ = a(Zi) V ••• V a(xm) ^ V <*(%)• Now we can prove a partial converse of corollary (4.6). THEOREM 4.14. Let Lbea distributive lattice such that L andL°P are GCL. Then (i) There exists a completely distributive lattice U and a surjective mapping £ : L' -> L preserving up-directed suprema and down-directed infima. Moreover, L can be chosen to be L and Ç: L -> L can be chosen to be an extension of the canonical map X: PZ (L, 2) -* L, which lifts the identity id : L -» L (Recall that PZ(L, 2) is a free lattice generated by the partially ordered set L). In this case, £ is a lattice homomorphism on a dense sublattice of L'. (ii) The CL topology on L and L°£ and the interval topology on L agree. In particular the interval topology on L is Hausdorjf. PROOF. Let L = L and let À: PZ(L, 2) -> L be the canonical map which lifts id : L -+ L. Then A is a lattice homomorphism. Let Ç = A* = A*. Then for 0O G PZ(L, 2) we have C(0o) = { s u p ( i n W ) : x £ </>ePZ(L, 2)}: x £ fa r \ \ ) is closed} = i n f W ) : 0o ^ 0 e PZ(L, 2)} = X{fa\ hence £ extends A. Moreover, £ preserves up-directed suprema and downdirected infima by (4.12). Further, PZ(L, 2) is dense in L by (4.10), because directed suprema and infima are limits in the CL topology. This proves (i). As L is a completely distributive lattice, (ii) is an immediate consequence of (4.6) and (4.2). COROLLARY 4.15. Let L be a distributive lattice. Then the following conditions are equivalent. (i) The IV topology on L is Hausdorjf. (ii) L is a quotient of a completely distributive lattice under a mapping preserving directed suprema and infima. 5. Characterization of generalized continuous lattices by the lattice of scott open subsets. It occurs regularly that every property of a complete lattice L can be characterized by a stronger property of the lattice 0(L) 292 G. GIERZ AND J. D. LAWSON of Scott open subset of L (see [2]). In this section we focus our attention on this phenomenon. Let L be a complete lattice, 0(L) (resp. C(L)) the lattice of all Scott open (resp. Scott closed) subsets of L and U(L) (resp. D(L)) the (completely distributive) lattice of all upper sets (resp. lower sets) of L. Then we have a mapping °d\ U(L) -+ 0(L) A i-> Aod where Aod denotes the largest Scott open subset contained in A. The mapping od is a kernel operator which preserves arbitrary infima. Dually, we have a mapping -«: D(L) - C(L) A^ Äd where Äd is the smallest Scott closed subset containing A. Clearly, ~d is a hull operator preserving arbitrary suprema. Note that od: U(L) -> O(L) (resp. ~d\ D(L) -» C(L)) is the left adjoint (resp. right adjoint) to the inclusion map. THEOREM 5.1. Let L be a complete lattice. Then the following conditions are equivalent. (i) L is a GCL. (ii) od: U(L) -+ O(L) preserves up-directed suprema. (ii') ~d: D(L) -+ C(L) preserves down-directed infima. (iii) (O(L), f| ) w # continuous lattice and the CL-topology agrees with the IV topology, i.e., the IV topology is Hausdorff. (iii') (C(L), \J)is a continuous lattice and the CL-topology agrees with the IV topology. PROOF. Clearly (ii) and (ii') as well as (iii) and (iii') are equivalent. (i) => (ii'). For A ü L let Ä be the closure of A in the CL-topology. Then for A e D(L) we have Äd = [Ä by (2.9). Let {^ : i e /} be a down-directed net of lower sets and let a e f)Äj. Then for every Scott open neighborhood U of a we have U fl A{ # 0 . As {^-: / e / } is down-directed, the set {U f] A{: i e I and a e U, U Scott open} forms a filter base. Let <F be an ultrafilter containing this base. Then Ai e BF for every i e / and hence for every M e £F we have A{ Ç\ M ^ 0. This implies inf M ^ inf(^4, f| M) e Ah i.e., inf M e ^ for every l e f and every ? e /. But this means inf M G Ç\A{ for every M e ^ . Therefore we can conclude lim J^ = liminf #" e f]Ad. On the other hand, 3F contains all Scott open neighborhoods of«. If lim SF would not be greater or equal to a, then we could find GENERALIZED CONTINUOUS LATTICES 293 a finite subset F<a with jlim & f| F= 0 . Because | F = ( J { î / : / 6 ^1 e «F, we can find a n / e F such that | / e J*7". But then lim & = liminf <F ^ / , a contradiction. Hence <z ^ lim &ef)Ai. This proves Ç\Àj ü P)y4f. The other inclusion is obvious. (ii') => (Hi') follows from (5.6) and (5.3), if we let U = D(L) and Ç = ~d. (iii') => (i). The mapping a 1-» | a : L -* C(L) preserves arbitrary infima and up-directed suprema and (C(L), f] ) is a GCL by (4.3). Now the result follows from (3.7). 6. CL-Quotients of completely distributive lattices, hypercontinuous lattices. Recall from (0.3) that a complete lattice L is a continuous lattice if and only if for each point * e L we have x = liminf 2f*, where %x denotes the filter of Scott open neighborhoods of x. If one replaces Scott open neighborhoods by neighborhoods which are open in the upper topology, one gets the following definition. DEFINITION 6.1. A complete lattice L is called hypercontinuous provided that for every point x e L we have x = liminf J?x, where S£x denotes the neighborhood filter in the upper topology of L. Clearly, every hypercontinuous lattice is a continuous lattice (Note that S£x ü %x implies x ^ liminf S£x ^ liminf %x ^ x). Before we state our main theorem on hypercontinuous lattices, let us give two definitions. DEFINITION 6.2. Let L be a complete lattice. Then the bi-Scott topology on L is the topology generated by the Scott open sets of L and L°P. DEFINITION 6.3. Let L be a complete lattice and U E L be a Scott open filter. Then we define Spec((7) = {y: y is maximal in L\U}. If k e L is a compact element, we let Spec(&) = Spec(|/:). Note that for every Scott open filter U we have Spec(C/) E IRR(L), where IRR(L) denotes the set of meet irreducible elements of L, i.e., the set of those elements of L which cannot be written down as an infimum of two strictly larger elements, (see [6]) Theorem 6.4. Let L be a complete lattice. Then the following conditions are equivalent (o) L is a hypercontinuous lattice. (i) L is a CL-quotient of a completely distributive lattice. (ii) L is a continuous lattice and the IV topology and the CL topology agree. (iii) L is meet continuous and the IV topology is Hausdorjf. (iv) L is a continuous lattice, L°P is a GCL, and the bi-Scott topology agrees with the CL topology. 294 G. GIERZ AND J. D. LAWSON (v) L is a continuous lattice and for open lower sets U, V g L we have U g V if there are finitely many au .. ., an e V such that £/ E |#i U • • • UK- (vi) L is a continuous lattice and for Scott open filters F\, F2 g L with Fi g F2 there exists a finite subset A g Spec(i<\) such that Spec(F2) g [A. (vii) L is a meet continuous and generalized bicontinuous lattice. PROOF, (i) -* (ii) is clear by (4.6) and (4.3). (ii) => (iii) is easy. (iii) => (iv). If the IV topology is Hausdorff, then L and L°P are GCL by (4.3). Moreover, every meet continuous GCL is a continuous lattice by (2.6), hence L is a continuous lattice. Furthermore, it follows from (5.3) that the bi-Scott topology is coarser than the CL-topology. A standard argument using the way below relation shows that the bi-Scott topology is Hausdorff. Hence the bi-Scott topology is a compact Hausdorff topology coarser than the (compact Hausdorff) CL-topology. Therefore both topologies agree. (iv) => (v). If the bi-Scott topology agrees with the CL topology, then the CL-topology on L°P is coarser than the CL topology on L. If L°P is a GCL, then the CL topology on L°P is Hausdorff, hence the compactness of both topologies yields that both topologies agree. This implies that the open down-sets of L in the CL-topology are exactly the Scott open sets of L°P. Furthermore U g ^is equivalent to U < Fin 0(L°P). Because od \ U(L°P) -> 0{L°P) is a CL-morphism by (5.1), its right adjoint/: 0(L°P) -> U(L°P), which is given by the inclusion map, preserves the way below relation. Hence U g V implies U < V in U(L°P). But this is the case if and only if U g I F for some finite subset F g V. (v) => (vi). Let Fi and F2 be Scott open filters with Fj g F2 and set U = L\F2, V = L\_Fi. Then we have F2 g F2, hence Ü = (L\F2)~ = L\F2 g L\F2 g L\Fi = V. Therefore we can find finitely many points tfl9 .. ., ane L\Fi such that U ^ [ax [] • • • U \an- For each a{ e L\Fi g L\Fi use Zorn's lemma to pick a bt e S p e c ^ ) with a{ ^ b£. Then L\F2 g l*i U • • • U [bH9 i.e., Spec(F2) g j ^ U • • • U [bn. (vi) => (vii). By the assumption (vi), L is clearly meet continuous and a GCL. Moreover, if b ^ Ö, pick open filters Fl9 F2 such that Fx g F2, be Fi, a$ F2. (We can do this by [9]). Then we can find a finite subset A g Spec Fi such that Spec F2 g [A. Clearly \b p A = 0 and L\F2 g L\F2 g I Spec F 2 g j/L But now A < a in L°P, because for a down-directed set D, inf D ^ a implies inf D = lim D e L\F2. Therefore deL\F2 g [A for some deD. Hence L** is a GCL. Finally, the CLtopologies on L and L°P agree. Let SF be an ultrafilter on L. Then liminf J^ = limsup <^\ because otherwise we would have liminf SF è limsup 3F. Now the above proof yields open filters Fu F2 such that / \ g F 2 , limsup ^ GENERALIZED CONTINUOUS LATTICES 295 e Fl9 liminf 3F # F2 and a finite subset A e Spec Fx with L\F 2 ü 1^As liminf i^ = lim <F in the CL-topology on L and as lim & e L\F2, we have L\F2 e J% hence [A e $F. Because A is finite, we can find an aeA such that [a e &. Therefore limsup & e [a, a contradiction to FX n i* = 0. (vii) => (i). Let L' = D(L) and let Ç: U -> L be defined by L,{A) = inf(L\^). Then £ is surjective and preserves arbitrary infima. Moreover, inf(L\^4) = infj(L\/4), where _ is the closure in the CL-topology. As L is a continuous lattice by (2.6), the mapping U *-* inf(L\(/) from the lattice of open lower sets into L preserves up-directed suprema by [4]. Further, by (5.1) the mapping U »-> Uod = L\Î(L\U)~ from the lattice of all lower sets into the lattice of open lower sets preserves up-directed suprema. Because £ is the composition of these two mappings, £ will preserve up-directed suprema. (o) => (ii). We show that every Scott open neighborhood of a point xe L contains a neighborhood in the upper topology. So let x e U be Scott open. As x = liminf £fx, we can find a Ve<gx such that inf V e U. But this implies V Ü (7, as desired. We now know that the upper topology and the Scott topology agree. As the IV topology is generated by the upper topology and the lower topology and the CL topology is generated by the Scott topology and the lower topology, the latter topologies agree, too. (v) => (o). It is enough to prove that for every point x e L the filter ( HX of Scott open neighborhoods is the same as the filter S£x of neighborhoods of x in the upper topology. So let W be a Scott open neighborhood of x. Choose Scott open neighborhoods U and V of x such that F i t / , U E W. Then U' = L\U and V = L/V are open lower sets satisfying U' E V'. By hypothesis (v) we can find a finite subset F i F' such that that U' g IF- Now U' = L\Ü E IF implies L\IF g U ^ W, i.e., L\[F is open in the upper topology and contained in W. Moreover, L\[F contains x, because otherwise we would have xe [F ^ L\V, contradicting xe V. This proves ty.x ü j£%. The other inclusion holds in every complete lattice. This completes the proof. COROLLARY 6.5. Let L be an algebraic lattice. Then the following statements are equivalent. (a) Conditions (o) => (vii) of'(6.4) hold. (b) For every compact element Ke L the set Spec(AT) is finite. PROOF. If (a) holds, (b) follows easily from condition (vi) in (6.3). Conversely, under the assumption (b), the proof of condition (vii) of (6.3) is an easy modification of the proof (vi) => (vii) in (6.3). Recall that a complete lattice L is called bicontinuous, if L and L°P 296 G. GIERZ AND J. D. LAWSON are both continuous lattices and if the CL-topologies on L and L0* coincide. The following corollary is now an easy consequence of (2.6) and (6.3). COROLLARY 6.6. Let L be a complete meet and join continuous lattice. Then the following conditions are equivalent. (i) L is bicontinuous. (ii) The interval topology on L is Hausdorjf. (iii) L is a CL-quotient of a completely distributive lattice. REFERENCES 1. G. Gierz, Construction of free lattices over partial lattices, Preprint No. 435, Technische Hochschule Darmstadt, 1978. 2. and K. H. Hofmann A lattice theoretical characterization of compact semilattices, Preprint, 1977. 3. G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott A Compendium of Continuous Lattices, Springer-Verlag, Berlin, Heidelberg, New York 1980. 4. G. Gierz and K. Keimel, A lemma on primes appearing in algebra and analysis, Houston Journal of Math. 3 (1977), 207-224. 5. K. H. Hofmann, Continuous lattices, topology and topological algebra, Topology Proceedings 2 (1977), 179-212. 6. and J. D. Lawson, Irreducibility and generation in continuous lattices, Semigroup Forum 13 (1977), 307-353. 7. , Spectral theory of distributive continuous lattices, Trans. Amer. Math. Soc. 246 (1978), 285-310. 8. K. H. Hofmann and A. Stralka, The algebraic theory of compact Lawson semilattices: Applications of Galois connections to compact semilattices, Diss. Math. 87 (1976), 1-54. 9. J. D. Lawson, Topological semilattices with small semilattices, J. London Math. Soc. II, Ser. I, (1969), 719-724. 10. , Intrinsic topologies in topological lattices and semilattices, Pac. J. of Math. 44(1973), 593-602. 11. W. J. Lewis and J. Ohm, The ordering o/Spec R, Can. J. of Math. 28 (1976), 820-835. 12. H. A. Priestley, Representation of distributive lattices by means of ordered Stone spaces, Bull. Lond. Math. Soc. Ill, Ser. 24, (1972), 507-530. 13. D. Scott, Continuous lattices, in: Toposes, algebraic geometry and logic, Springer Lecture Notes in Mathematics 274, Springer-Verlag, 1974, 97-136. G. GiERA, TECHNISCHE HOCHSCHULE, DARMSTADT, WEST GERMANY. J. D. LAWSON LOUISIANA STATE UNIVERSITY, BATON ROUGE, LA 70803.