Lattice Topologies with Interval Bases

Transcription

1 Lattice Topologies with Interval Bases Marcel Erné Faculty for Mathematics and Physics Leibniz University Hannover D Hannover, Germany Abstract We study topologies τ on a lattice L with open subbases of intervals [y) and (z] (y Y, z Z). As a basic result, we show that τ coincides with the order topology (and (L, τ) is a totally order-disconnected topological lattice) iff Y is a join-dense set of compact elements and Z is a meet-dense set of cocompact elements; moreover, that (L, τ) is a compact totally disconnected topological lattice iff L is a complete lattice with Y as a join-dense set of hypercompact elements and Z as a meet-dense set of hypercocompact elements, iff L is complete and τ agrees with the interval topology. Our results lead to various large classes of order-topological lattices with interval bases, to topological characterizations of bi-algebraic lattices and, on the other hand, to algebraic characterizations of compact totally (order-)disconnected topological lattices. Mathematics Subject Classification. 06F30, 54A10, 54H12 Key Words. (Bi-)algebraic lattice, (hyper-)compact, interval, Lawson topology, order topology, Scott topology, topological lattice, totally (order-) disconnected. 0 Introduction In this note we investigate certain topologies on lattices that admit a topological treatment of various algebraic decomposition properties. It is well known that the existence of join-decompositions into compact elements is closely related to total disconnectedness of the involved topologies (see e.g. the Fundamental Theorem for Compact Totally Disconnected Semilattices [12, VI 3.13]). Here we generalize some classical results of continuous lattice theory to the setting of arbitrary order-consistent topologies. For example, if such a topology has an interval base then it agrees with the Scott topology, and the underlying lattice is compactly generated (and vice versa). Concerning the two-sided situation, one central observation will be that for a topology τ on a lattice L having a subbase consisting of principal ideals and principal dual ideals, τ-convergence agrees with order convergence iff (L, τ) is a totally (order-)disconnected topological lattice whose topology is coarser than the order topology, and that any such lattice is bi-compactly generated. 1

2 This leads to a topological description of bi-algebraic lattices as totally orderdisconnected topological lattices with an open base of intervals whose complements are compact in the interval topology. As another consequence, we shall obtain diverse characterizations of totally order-disconnected compact topological lattices. Some of these characterizations are new, others are known, but our approach essentially facilitates proofs. In this context, it turns out that in the presence of enough indecomposable elements (atoms, inaccessible elements, irreducible elements, etc.) seemingly weak separation axioms like T 1 already entail total (order-)disconnectedness. For example, a bi-atomistic semitopological lattice with closed universal bounds is already totally order-disconnected. Furthermore, we improve and generalize some results of Choe and Greechie [2] and of Riečanová [17] [19] from orthomodular to arbitrary lattices (see also [11]). The Axiom of Choice will often apply without particular emphasis. 1 Intrinsic topologies of lattices In what follows, L denotes a lattice and L op its dual or opposite, obtained by reversing the order; often, it would suffice that L is a semilattice or merely a poset. Principal ideals (z] = {x L : x z}, principal dual ideals [y) = {x L : y x}, and their intersections [y, z] = [y) (z] = {x L : y x z} are called intervals. By an interval base of a topology τ on L, we mean an open base for τ consisting of intervals. Notice that these intervals need not be closed with respect to the topology τ, though they are sometimes referred to as closed intervals. A subset U of L is an upper set (upper end, increasing set, upset) if y U implies [y) U; lower sets (lower ends, decreasing sets, downsets) are defined dually. If X is an arbitrary subset of L then is the upper set generated by X, and X = {[x) : x X} X = {(x] : x X} is the lower set generated by X. By J L we denote the set of all ideals (that is, of all directed lower sets) possessing a join. The upper (resp. lower) sets form a topology τ U = α(l) (resp. τ L = α(l op )), the so-called upper (resp. lower) Alexandroff topology. Besides these two topologies, we shall consider the following intrinsic topologies on L (cf. [4] [12], [14]): the upper topology τ u = υ(l) is generated by the complements of all principal ideals, and the lower topology τ l = υ(l op ) by the complements of all principal dual ideals; the Scott topology τ S = σ(l) consists of all upper sets U such that 2

3 any Y J L with Y U intersects U. Thus, τ S is finer than (and sometimes, but not always, identical with) τ u but usually coarser than τ U. The order topology τ o = Ω(L) consists of all sets U such that (o) for any up-directed set Y and any down-directed set Z with Y = Z U, there exist elements y Y in z Z with [y, z] U. Hence, the τ S -open sets are precisely the τ o -open upper sets. We call a topology o-coarse if it is contained in the order topology, i.e., each open set U enjoys (o). A filter F order converges (o-converges) to a point x (cf. [4, 8]) iff x is both the limit inferior and the limit superior of F, that is, x = {y L : [y) F} = {z L : (z] F}. In the present note, order convergence always refers to filters. (For the definition of net-theoretical order convergence, see e.g. [1, X 9]). The order topology is the finest topology τ such that order convergence implies τ-convergence (see [8, Thm. 4.6]). Order convergence is said to be topological if it agrees with convergence in some topology, which is then necessarily the order topology. This coincidence often fails; for example, a Boolean lattice has topological order convergence if and only if it is atomic (see [4, Thm. 4]). A necessary and sufficient criterion for order convergence to be topological was given in [8, Cor. 4.13]: Proposition 1.1 The order convergence on a lattice L is topological iff each neighborhood filter with respect to the order topology has a base of intervals. In particular, this happens whenever the order topology has an interval base. From the previous topologies one may build new ones by forming joins in the lattice T (L) of all topologies on L. The following notation will be convenient: given any topology τ on L, we denote by τ the topology generated by τ and the lower topology τ l ; in other words, τ τ l is an open subbase for τ. For example, the join τu = τ u τ l of the upper and the lower topology is the interval topology τ i = ι(l), the coarsest topology in which every interval is closed. Of particular importance for continuous lattice theory is the Lawson topology τs = λ(l) (see [12, III]). Sometimes the half-open interval topologies τu = τ U τ l and τ u τ L are also of interest (cf. [20, Ex. 51 and 84]). Notice that each of the topologies τ u, τ l, τ i, τ S and τs is o-coarse, on account of the inclusions τ u τ l = τ i τ S τ o. Moreover, order convergence, order topology and interval topology are related as follows (see [8, ]): Proposition 1.2 If the interval topology τ i is T 2 then it agrees with the order topology τ o. Furthermore, τ i is compact and T 2 iff every ultrafilter o-converges. In the present context (and also in the subsequent considerations), the meaning of the word compact does not include the Hausdorff separation axiom T 2. The following generalization of Frink s theorem on the compactness of the interval topology (cf. [1, X Thm. 20]) is often useful: Proposition 1.3 A lattice L with a topology between the interval topology and the Lawson topology is compact iff L is complete. 3

4 By Alexander s subbase theorem, the complement of a set U is τ i -compact iff for each filterbase β of intervals with β U, there is a B β with B U. On the other hand, a set U is open in the order topology iff for any filterbase β of intervals with β = {x} U, there is a B β with B U. This proves Proposition 1.4 Any τ i -compact set is τ o -closed. In particular, any compact topology finer than the interval topology is o-coarse, and so is any topology τ such that τ is compact. In the subsequent diagram, we display seven of the previously introduced topologies together with the indiscrete topology. They form a join-subsemilattice of the lattice T (L). Diagram 1.5 Intrinsic topologies on a lattice L. Alexandroff topology τu half-open interval =τ U τ l topology τ U =α(l) τs =λ(l) Lawson topology Scott τ S =σ(l) τ i =ι(l) interval topology topology upper τ u =υ(l) τl =υ(l op ) topology τ in ={, L} indiscrete topology lower topology For complete lattices L, this is even a sublattice of the lattice T (L), on account of Proposition 1.6 In a complete lattice, the Lawson-open upper sets are the Sott-open upper sets, the Lawson-open lower sets are the τ l -open sets, and the τ i -open sets are the τ u -open sets. For the first and the second statement, see [12, III 1.6 and 3.16]. The third one is obtained by dualizing the second and observing that τ i is always coarser than the Lawson topology. Without the assumption of completeness, the second and the third assertion in 1.5 fail to be true in general: Example 1.7 The pairs ( 1 n, 1 n ), ( 1 n, 0) and ( 1 n, 0), where n runs through the positive integers, form a sublattice of the real plane (partially ordered componentwise by the usual ), and the interval topology of this sublattice is discrete. Hence the element x = ( 1, 0) generates a τ i -open principal dual ideal which is neither τ u -open nor τ S -open. 4

5 ( 1, 0) (1, 0) ( 1, 1) By a topological (respectively, semitopological) lattice, we mean a lattice endowed with a topology making the binary lattice operations (respectively, the unary operations y : y x y and y : x x y) continuous. Again, the Hausdorff property is not assumed a priori in these definitions. A lattice L satisfying the identity x Y = {x y : y Y } for all x L and Y J L (hence also for all directed sets Y possessing a join) is called meet-continuous or upper continuous. Notice that often these names are attributed to complete lattices only (cf. Birkhoff [1] or Crawley and Dilworth [3]). The dual notion is join-continuity or lower continuity. A topological justification of the notation meet-continuous has been given in [4, Prop. 5] and [9, Thm. 2.8]: Proposition 1.8 The following statements on a lattice L are equivalent: (a) L ist meet-continuous. (b) The binary meet operation is continuous with respect to o-convergence. (c) The unary meet operations are continuous with respect to o-convergence. (d) The unary meet operations are continuous with respect to τ o. (e) The unary meet operations are continuous with respect to τ S. In accordance with other authors we call a join- and meet-continuous lattice o-continuous, since by 1.7 and the dual statements, a lattice L is o-continuous iff its operations are continuous with respect to order convergence, or equivalently, if (L, τ o ) is a semitopological lattice. However, an o-continuous lattices need not be a topological lattice in its order topology (see [9, Cor. 4.11]). By an order-topological lattice we mean an o-continuous lattice with topological order convergence (of filters). Thus L is order-topological iff it is a topological lattice with a (unique) topology whose convergence agrees with order convergence. For a purely algebraic characterization of order-topological lattices as so-called bi-continuous lattices, see [4, Thm. 2]. Examples of order-topological lattices are bi-compactly generated lattices (in which every element is a join of compact elements and a meet of cocompact elements; (see [1, Thm. 3] and Section 24) and, in particular, chain-finite lattices (the latter having discrete order topologies); further, completely distrubitive complete lattices (hence arbitrary products of complete chains), and finite products of arbitrary chains. However, there are infinite products of chains which fail to be order-topological, for example, infinite powers of the chain ω of natural numbers (cf. [5, Section 4]). Notice also that a lattice with topological order convergence need not be order-topological (see Example 3.1). 5

6 2 One-sided topologies with interval bases Let us first recall some basic notions from topology. A topological space is said to be zero-dimensional if it has a base of clopen sets, totally disconnected if its connected components are singletons, and totally separated if for any two distinct points there exists a clopen set containing one of these points but not the other. Similarly, a partially ordered topological space is totally order-separated or -disconnected if for x y there is a clopen upper set containing x but not y. The compact totally order-disconnected spaces (Priestley spaces) are the duals of bounded distributive lattices in Priestley s duality (see, for example [16]). The following implications are obvious: zero-dimensional +T o totally separated totally disconnected +T 2 and it is well known that for compact spaces, these three properties coincide (see [14, II 4.2], though there exist examples of compact totally disconnected spaces which fail to be Hausdorff (see [20, Ex. 99]). Moreover, for compact topological meet-semilattices, the above three properties are equivalent to total order-disconnectedness. Indeed, if x z in a compact totally separated topological meet-semilattice then (z] is a closed, hence compact set disjoint from x, and we find a clopen set U containing x and disjoint from (z]. Thus U is a clopen upper set containing x but not z. Of course, every totally order-disconnectedness space is T 1 -ordered, that is, for x z there exists an open upper set containing x but not z, and dually. Apparently, the latter separation axiom simply means that the given topology is finer than the interval topology. Note also that a semitopological lattice is T 1 -ordered iff it is T 1 in the usual topological sense (singletons are closed). Now to the lattice-theoretical notions needed in the sequel. A subset Y of a lattice L is said to be join-dense or a join-base if each element of L is a join of elements from Y, or equivalently, if for any two elements x z, there is some y Y with y x yet y z; meet-dense subsets (meet-bases) are defined dually. The similarity with the definition of T 1 -ordered spaces and of totally order-disconnected spaces is evident and not casual, as we shall see in due course. The following types of elements are of particular importance for the decomposition theory of lattices (cf. [1, 3, 4, 6, 7, 10, 12, 13]). An element y of a lattice L is completely join-irreducible ( -irreducible) iff it belongs to each subset with join y; similarly, y is (join-)inaccessible if this condition holds at least for each ideal with join y, and y is compact (also finite or join-intranscessible) iff it belongs to each ideal having a join above y, or equivalently, iff for each subset X of L possessing a join with y X, there is some finite subset F of X with y F. The dual notions are completely meet-irreducible ( -irreducible), meet-inaccessible, and cocompact (cofinite, meet-intranscessible). Clearly, every atom (that is, every minimal non-zero element) is -irreducible, and every - irreducible element join-inaccessible. However, atoms need not be compact. Let us remark in passing that not only topological compactness may be regarded as a special instance of the corresponding lattice-theoretical notion (viewing topologies as complete lattices), but also vice versa: by Alexander s subbase theorem, an element y of a complete lattice is compact iff the com- 6

7 plement of [y) is compact in the lower topology. This equivalence extends to non-complete lattices if compactness is replaced with strong compactness: an element y is strongly compact if y X implies y F for some finite F X, where X = {(z] : X (z]} is the cut generated by X (cf. [11]). We shall also need two other strengthenings of lattice-theoretical compactness. Call an element y of L supercompact if the complement of [y) is a principal ideal, and call y hypercompact if the complement of [y) is a finite union of principal ideals. Supercocompact and hypercocompact elements are defined dually. Then we have the following implications: supercompact hypercompact strongly compact compact inaccessible In meet-continuous lattices, the inaccessible elements are exactly the compact elements. Furthermore, in arbitrary lattices, every supercompact element is completely join-irreducible, and the converse holds in frames, that is, in meetcontinuous distributive complete lattices. In fact, an element y of a complete lattice is supercompact iff it is completely join-prime (that is, y X implies y X for all X L). A less trivial observation is that in join-continuous distributive complete lattices, the compact elements coincide with the hypercompact ones (see [10, Lemma 1.4]). A lattice L is said to be (super-, hyper-) compactly generated if the set of (super-, hyper-) compact elements is join-dense in L. If, in addition, L is complete then L is a (super-, hyper-) algebraic lattice. Similarly, a lattice L is atomistic if the set of atoms is join-dense in L. The atomistic supercompactly generated lattices are just the atomic Boolean lattices. For a thorough study of supercompactly generated (= principally separated) lattices and posets, see [6]. If a lattice and its dual are compactly generated then we speak of a bi-compactly generated lattice, etc. Diagram 2.1 Compact generation in lattices algebraic atomistic compactly atomistic superalgebraic atomistic hyperalgebraic Boolean superalgebraic atomistic atomistic hypercomp. supercomp. hyperalgebraic atomistic generated hypercomp. algebraic generated compactly generated Notice that a lattice L is supercompactly generated iff so is its dual L op, and that L is algebraic iff it is algebraic and completely distributive. For this and other characterizations of superalgebraic lattices, see [6, Lemma 4.2]. 7

8 The following topological descriptions of inaccessible and (hyper-) compact elements are immediate consequences of the involved definitions (for the last statement, apply 1.6). Lemma 2.2 Let y be an element of a lattice L. (1) y is inaccessible iff (y[= (y] {y} is closed in τ S (resp. in τ o ). (2) y is compact iff [y) is open in τ S (resp. in τ o ). (3) y is hypercompact iff [y) is open in τ u. (4) If L is complete, y is hypercompact iff [y) is open in τ i. The element ( 1, 0) of the (non-complete) lattice in Example 1.7 generates a τ i -open principal dual ideal. This element is compact, but it is not hypercompact (and not even strongly compact). Lemma 2.2 suggest the following definitions. Given a topology τ on a lattice L, we say an element y is τ-inaccessible if the deleted principal ideal (y[= (y]{y} is τ-closed, and y is τ-intranscessible if the principal dual ideal [y) is τ-open. (In order to prevent confusion with the topological notion of compactness, we avoid here the word τ-compact for τ-intranscessible ). In accordance with [12, II 1.16], we call a topology τ between τ u and τ S order-consistent. Thus, for any element of a lattice L and for any order-consistent topology τ on L, we have the following implications and equivalences: τ u -intranscessible hypercompact τ-intranscessible τ S -intranscessible compact and analogous implications hold for the diverse types of inaccessible elements. Notice that the set of all τ-intranscessible elements is closed under finite joins; in particular, this applies to compact and to hypercompact elements. The observation that compact elements are join-inaccessible, and that in meet-continuous lattices the converse is also true, admits the following generalization: Lemma 2.3 Let τ be any topology on a lattice L. (1) If τ is finer than τ u then every τ-intranscessible element is τ-inaccessible. (2) If y is a τ-inaccessible element of L and the unary meet operation y is τ-continuous then y is τ-intranscessible. Proof. (1) If (y] is τ-closed and [y) is τ-open then (y[ = (y] [y) is τ-closed. (2) If (y[ is τ-closed and y is τ-continuous then [y) = 1 y [L(y[ ] is τ-open. We call a lattice L τ-generated if for any two elements x, z L with x z, there is a τ-open principal dual ideal containing x but not z; in other words, if the τ-intranscessible elements form a join-base. In particular, τ S -generated means compactly generated and τ u -generated means hypercompactly generated. By definition, every τ-generated lattice with τ l τ is totally order-disconnected. 8

9 With the help of 2.3 one easily detects many types of τ-generated lattices. For example, since any atom is τ-inaccessible if {0} is closed, it follows from 2.3 (2) that every atomistic lattice with a T 1 topology τ making the unary meet operations continuous is τ-generated. More generally, for any topology τ finer than the upper topology τ u, every -irreducible element y is τ-inaccessible (because (y[ is a principle ideal). Thus we have: Proposition 2.4 Every -irreducibly generated semitopological lattice with a T 1 topology τ is τ-generated. It is easy to see that a lattice is -irreducibly generated (i.e. each element is a join of -irreducibles) iff it has a minimal (equivalently a smallest) join-dense subset, namely the set of all -irreducible elements (cf. [6, 3.7 and 3.8]). Every atomistic lattice, but also every dually algebraic lattice and, in particular, every lattice satisfying the descending chain condition is -irreducibly generated (see [1, VII Thm. 16] or [3, 6.1]). In the case of order-consistent topologies,there are various interesting characterizations of τ-generated lattices. Theorem 2.5 The following statements on a lattice L and an order-consistent topology τ on L are equivalent: (a) L is τ-generated. (b) L is compactly generated and τ is the Scott topology. (c) τ has an interval base. (d) τ has a smallest base. (e) τ has a minimal base. (f) The lattice of open sets is supercompactly generated. (g) The lattice of closed sets is (super-)compactly generated. (h) (L, τ) is a topological lattice whose τ-inaccessible elements form a join-base. Proof. (a) (b): Suppose x U τ S. Since x is the directed join of all τ-intranscessible elements y x, one such element y must belong to U. Hence [y) is a τ-neighborhood of x contained in U, and U is τ-open. Thus, by orderconsistency, τ coincides with τ S. In particular, the compact elements are τ- intranscessible and form a join-dense subset of L. (b) (c): The principal dual ideals generated by compact elements form a base for τ = τ S. (c) (d) (e) (f) (g): Since τ is order-consistent, each τ-open set is an upper set, and consequently, any interval base for τ must consist of principal dual ideals; but these are just the cores with respect to τ, where the core of a point x in a topological space is the intersection of all neighborhoods of x. Hence condition (c) states that (L, τ) is a B-space in the sense of [7], that is, a space with a base of open cores. It has been shown in [5, Thm. 2.11] and in [9, Prop. 2.2 B] that each of the conditions (d) (g) characterizes B-spaces. (c) (h): For x L, the principal dual ideal [x) is the core of x, hence the intersection of τ-open cores, i.e. principal dual ideals. In other words, x is a directed join of τ-intranscessible elements, and these are τ-inaccessible, by 2.3 (1). The binary meet is trivially τ-continuous, as the inverse image of a 9

10 principal dual ideal [x) und is [x) [x). Concerning continuity of the binary join, assume x 1 x 2 U τ. Then U is τ S -open, so we find τ-intranscessible elements y 1, y 2 such that x 1 [y 1 ) τ, x 2 [y 2 ) τ and [y 1 ) [y 2 ) = [y 1 y 2 ) U. (h) (a): By 2.3 (2), each τ-inaccessible element is τ-intranscessible. As an application of this theorem, one obtains, for example, several common topological characterizations of compactly generated lattices, by taking for τ the Scott topology τ S ; similar results are obtained for hypercompactly generated lattices, by taking for τ the upper topology τ u. Specifically: Corollary 2.6 A lattice is compactly generated iff its Scott topology has an interval base, and a lattice is hypercompactly generated iff its upper topology has an interval base. Combining 2.5 with 1.4 and [12-III, 2.16], we arrive at the following characterizations of τ-generated complete lattices in terms of the topology τ = τ τ l : Theorem 2.7 For a lattice L and an order-consistent topology τ on L, the following statements are equivalent: (a) L is complete and τ-generated. (b) L is compactly generated, and τ is a compact T 2 topology. (c) L is algebraic, and τ is the Lawson topology. (d) (L, τ ) is a compact zero-dimensional topological meet-semilattice. Obviously, every τ-generated lattice is a totally order-disconnected topological meet-semilattice with respect to the topology τ, but the converse is not always true. For example, the lattice in Example 1.7 is discrete, hence certainly totally order-disconnected in the interval topology τ u, but it is not τ u -generated, since there is no τ u -open principal dual ideal containing the point ( 1, 0), except the whole lattice. This underscores the importance of the completeness hypothesis in the previous considerations. Concerning the last condition in 2.5 and 2.7, respectively, it should be mentioned that an algebraic lattice need not be a topological lattice with respect to the Lawson topology. Indeed, it follows from [12, VII 2.4], that a distributive algebraic lattice with that property must already be superalgebraic, a rather restrictive condition. An example of a distributive and but not superalgebraic lattice is sketched below. Its meet-operation is not continuous at the bottom. 10

11 3 Two-sided topologies with interval bases So far, we mainly have studied certain one-sided topologies like the Scott topology and the upper topology; these have the advantage that the order relation is encoded in the topological strucutre and can be reconstructed from the latter (via the specialization order). On the other hand, such topologies have the drawback to violate even weak separation axioms like T 1 in all non-trivial cases. Now, we turn our attention to two-sided topologies such as the interval topology and the order topology; these have better separation properties, but they forget the order structure. In this context we frequently have occasion to consider certain filterbases consisting of intervals. More specifically, given two subsets Y and Z of a lattice L, let β Y,Z denote the filterbase of all finite intersections formed by members of the system {[y) : y Y } {(z] : z Z}. We define a topology τ Y,Z on L by declaring this system as an open subbase for τ Y,Z. In other words, the interval filterbase β Y,Z is a base for the topology τ Y,Z. However, not every topology with an interval base has a strong interval base, that is, a base of the form β Y,Z. Example 3.1 Adjoining universal bounds to the disjoint union of the chain ω and a single extra element x, a complete lattice is obtained whose order topology coincides with the interval topology and has an interval base (in fact, all points except the top element are isolated). However, the only principal dual ideal which is a neighborhood of x is the whole lattice, so that the order topology does not possess any subbase consisting of principal ideals and principal dual ideals. Of course, such a failure cannot happen in semitopological lattices, because continuity of the unary join and meet operations ensures that for any topologically open interval [y, z], the principal ideal (z] and the principal dual ideal [y) are again open sets whose intersection is [y, z]. It is easy to see that a lattice with a topology of the form τ Y,Z is (semi-)topological iff the following condition and its dual hold: For all x, y L and z Z with x y z, there exist finite sets E, F Z such that x E, y F, and E F z. Hence we may note: Lemma 3.2 Every semitopological lattice with an interval base is already a topological lattice and has a strong interval base. Topologies with strong interval bases play an important role in the theory of compactly generated lattices and of orthomodular lattices (cf. [2], [11],[17]-[19]), being a helpful tool for the translation of algebraic properties into a topological language, and vice versa. Let us start with some necessary and sufficient conditions for topologies with strong interval bases to be totally order-disconnected. Proposition 3.3 The following statement on two subsets Y and Z of a lattice L and the topology τ = τ Y,Z are equivalent: 11

12 (a) Y is join-dense and Z is meet-dense in L. (b) τ is finer than τ o. (c) τ is finer than τ i (that is, (L, τ) is T 1 -ordered). (d) (L, τ) is totally order-disconnected. Each of these conditions implies that (L, τ) is a zero-dimensional Hausdorff space. Proof. (a) (b): For x U τ o, join-density of Y and meet-density of Z yield the equation x = (Y (x] = (Z [x)). Put Y x = { E : E is a finite subset of Y (x]}, Z x = { F : F is a finite subset of Z [x)}. (If or is not defined, omit it). Then Y x is directed, Z x is filtered, and Y x = Z x = x U, so we find elements y Y x and z Z x with [y, z] U. But [y, z] is an intersection of finitely many subbasic τ-open sets, and consequently, U is τ-open. (b) (c): The interval topology is always coarser than the order topology. (c) (d): The inequality x z means x L (z] τ i τ, so there are finite sets E Y and F Z with x [ E, F ] L (z]. Hence there is at least one y E with y x and y z; in other words, U = [y) is a τ-clopen upper set containing x yet not z. Since the sets [y) with y Y and (z] with z Z form a clopen subbase, τ is zero-dimensional. (d) (a): Suppose x z, and choose a τ-clopen upper set U with x U and z U. As before, we find finite sets E Y and F Z such that x [ E, F ] U. Since z does not belong to the upper set U, the inequality E z is impossible. Thus we find some y E Y such that y x but y z, proving join-density of Y ; dually, we get meet-density of Z. In the subsequent theorem, we list several necessary and sufficient criteria for the coincidence of the topology τ Y,Z with the order topology τ o, generalizing diverse results of Riečanov [17]-[19] from sets of atoms and coatoms in orthomodular lattices to arbitrary subsets of lattices without any restrictions (see also [11]). Theorem 3.4 The following statements on two subsets Y and Z of a lattice L and the topology τ = τ Y,Z are equivalent: (a) Y is a join-dense set of compact elements and Z is a meet-dense set of cocompact elements. (b) τ is the order topology. (c) τ-convergence agrees with order convergence. (d) (L, τ) is a (semi-)topological lattice, and τ is an o-coarse T 1 topology. Each of these conditions implies that L is a bi-compactly generated, hence totally order-disconnected order-topological lattice. Proof. By 3.3, join-density of Y together with meet-density of Z is equivalent to the inclusion τ o τ and also to total order-disconnectedness of τ. On the 12

13 other hand, we know from 2.1 (2) that an element y is compact iff [y) is τ o -open, and dually an element z is cocompact iff (z] is τ o -open. In other words, each element of Y is compact and each element of Z is cocompact iff τ τ o. This proves the equivalence of (a) and (b). Under hypothesis (a), L and its dual are compactly generated, so that by [4, Thm. 3], L is order-topological. This together with 3.3 establishes the implication (a) (d) and the final statement in 3.4. Again by 3.3, (d) implies (b), because any T 1 topology making the unary lattice operations continuous is finer than τ i. Finally, since τ has an interval base, it follows from 1.1 that o-convergence is topological on L if τ o = τ, whence (b) is equivalent to (c). Notice that cndition (a) in 3.4 may be replaced with (a ) L is o-continuous, Y is a join-dense set of join-inaccessible elements, and Z is a meet-dense set of meet-inaccessible elements. The prvious results immediately provide a common improvement of Theorem 3 in [4] and of Theorem 3.1 in [19]. Corollary 3.5 The following statements on a lattice L are equivalent: (a) L is bi-compactly generated. (b) L is o-continuous, and the order topology has an interval base. (c) The order topology has a strong interval base. (d) Order convergence agrees with τ Y,Z -convergence, where Y is a set of compact elements and Z is a set of cocompact elements. (e) L is a bi-inaccessibly generated totally order-disconnected order-topological lattice. The example of the rationals with the usual topology shows that a totally order-disconnected order-topological lattice need not possess any join- or meetinaccessible element. The crucial point is here the lack of enough joins and meets ensuring compactness of the space (compare Theorem 3.13). On the other hand, Example 3.1 shows that even a complete lattice whose order topology has an interval base need not be meet-continuous, hence neither o-continuous nor compactly generated. In the presence of enough irreducible elements, we can prove an interesting uniqueness statement for lattice topologies making the unary operations continuous: Proposition 3.6 Let (L, τ) be a T 1 -semitopological lattice, Y a set of -irreducible elements, and Z a set of -irreducible elements of L. Then the topology τ is finer than τ Y,Z. Hence, if Y is join-dense and Z is meet-dense in L then (L, τ) is totally order-disconnected, and τ is finer than the order topology. If, in addition, τ is o-coarse then τ = τ o = τ Y,Z. Proof. The same argument as for 2.3 shows that for each y Y, the principal dual ideal [y) is τ-open, and dually, each principal ideal (z] with z Z is τ-open, 13

14 whence τ Y,Z is contained in τ. The other assertions are immediate consequences of 3.3. We call a lattice bi-irreducible generated if each element is a join of - irreducibles and a meet of -irreducibles. Thus, for example, every bi-atomistic lattice is bi-irreducibly generated. The second part of 3.6 tells us that on a bi-irreducibly generated lattice, any T 1 topology τ making the unary lattice operations continuous is totally order-disconnected and finer than the order topology. Our next aim is to establish a topological description of bi-algebraic lattices, which may be regarded as a two-sided analogue of 2.7. For this and other purposes, it will be convenient to call a collection of subsets of a topological space (X, τ)τ-cocompact if each member U of this collection is the complement of a τ-compact set. Though being rather suggestive, this nomenclature conflicts a bit with the usual convention to call the topology τ compact if so is the space (X, τ): in fact, the latter means that the topology τ is a τ-cocompact collection of sets. A beautiful theorem due to Lawson [15] states that the topology of any compact T 2 -topological lattice is uniquely determined, having a base of convex τ o -open sets. In case of topologies with strong interval bases, it turns out that τ i - cocompactness is a suitable relaxation of the compactness hypothesis in order to prove similar uniqueness statements for certain non-compact situations. Crucial for this purpose is the following Lemma 3.7 Let Y and Z be arbitrary subsets of a lattice L. Then the base β Y,Z { } is τ i -cocompact iff L is complete and the topology τ Y,Z is o-coarse. Proof. By 1.3 L is complete iff { } is τ i -cocompact, and by 3.7, τ Y,Z is o-coarse whenever β Y,Z is τ i -cocompact. Conversely, assume that L is complete and τ Y,Z is o-coarse. It remains to verify that for each U = [y, z] β Y,Z and for any interval filterbase β = β W,X with β U, there is some B β with B U. By completeness, β is the interval [ W, X]. The inclusion β Y,Z τ Y,Z τ o ensures that y is compact and z is cocompact (2.1). Hence y W entails y E for some finite E W, and dually, X z entails F z for some finite F X. But then B = [ E, F ] is a member of β with B U. Now to the announced topological characterization of bi-algebraic lattices (cf. 2.5 and 2.7): Theorem 3.8 The following statements on a lattice L and a topology τ on L are equivalent: (a) L and its dual are τ-generated complete lattices, and τ is o-coarse. (b) L is bi-algebraic, and τ is the order topology. (c) (L, τ) is a bi-irreducibly generated complete T 1 -topological lattice, and τ is o-coarse. (d) (L, τ) is a totally order-disconnected space with a τ i -cocompact strong interval base. 14

15 Proof. For the implication (a) (b), proceed as in proof of 2.5. (b) (c): By the Birkhoff-Frink Theorem [1, VIII Thm. 16] and its dual, L is bi-irreducibly generated, and by 3.5, (L, τ o ) is a totally order-disconnected topological lattice. (c) (d): See 3.6 and 3.8. (d) (a): By 3.7, the topology τ is o-coarse, and by hypothesis (d), it is of the form τ Y,Z. Hence, by 3.3, total order-disconnectedness ensures that τ is the order topology, and by 3.4, Y is a join-dense set of τ-intranscessible elements, and Z has the dual property. Since τ Y,Z is totally order-disconnected, the empty set is a member of the base β Y,Z unless L is a singleton. Thus 3.8 yields completeness of L. As in similar constellations, completeness of the underlying lattice is essential in 3.9. Example 3.9 Let Z denote the chain of integers. Then Z Z with the product order is a lattice in which every element is compact and cocompact, while no element is - or -irreducible. Here the order topology is discrete and has a strong interval base, but no interval has a τ i -compact complement. Nevertheless, the implication (a) (b) in 3.9 can be extended to noncomplete lattices, replacing (co)compact elements with strongly (co)compact elements and using the MacNeille completion (which is bi-algebraic iff the original lattice is strongly bicompactly generated; cf. [11]). On the other hand, similar arguments as before show that a bi-irreducibly generated T 1 -topological lattice with an o-coarse topology is always bi-compactly generated (because its -irreducible elements are compact and its -irreducible elements are cocompact) and carries the order topology. But Example 1.7 shows that such a topological lattice is not necessarily strongly bi-compactly generated, even if its topology is discrete. For an algebraic description of compact totally disconnected topological lattices possessing interval bases, the following observation will be helpful (cf. [12, O-4.4]: Lemma 3.10 If (L, τ) is a compact T 1 -semitopological lattice then τ i τ τ o, and L is an o-continuous complete lattice. Proof. For the first claim, apply 3.7. For the second, observe that 1.3, L is complete; now, given x L, any directed set Y L is contained in the inverse image I of the principal ideal ( (x Y )] under the unary operation x. By the inclusion τ i τ τ o and continuity of x, the set I is τ o -closed, and it follows that Y I, hence x Y = (x Y ). This proves meet-continuity, and join-continuity is obtained dually. Proposition 3.11 The following statement on two subsets Y, Z of a lattice L and the topology τ = τ Y,Z are equivalent: (a) Y is a join-dense set of hypercompact elements, Z is a meet-dense set of hypercompact elements, and L is complete. 15

16 (b) Y is a join-dense set of compact elements, Z is a meet-dense set of cocompact elements, and (L, τ i ) is a compact T 2 space. (c) (L, τ) is a compact space with τ = τ i = τ o. (d) (L, τ) is a compact T 1 -ordered space. (e) (L, τ) is a compact totally (order-)disconnected topological lattice. Proof. (a) (b): We show that (L, τ i ) is totally order-disconnected, in particular T 2 : indeed, for x z, we may choose a hypercompact element y Y such that y x and y z. Hence, by 2.1(4), [y) is a τ i -clopen upper set containing x but not z. By 1.3, (L, τ i ) is compact. (b) (c) (e): Apply 1.2 and 3.4. (e) (d): See (d) (a): By 1.3, L is complete, and by 3.3, Y is join-dense and Z is meet-dense in L. Hence, for each y Y L [y) = {(z] : z Z, y z} is a τ-open covering of the τ-closed (hence compact) set L [y). Thus we find a finite set F Z such that L [y) = {(z] : z F }. This shows that Y consists of hypercompact elements, and a dual argument applies to Z. Parts of the subsequent characterization theorem for compact totally orderdisconnected topological lattices belong to the folklore of algebraic and topological lattice theorey (cf. [12], [14] and [16]), but we have not been able to pursue all traces to thier origins. Theorem 3.12 The following statements on a lattice L are equivalent: (a) L is bi-hyperalgebraic. (b) L is bi-algebraic, and the interval topology is T 2. (c) (L, τ) is a compact topological lattice for some T 2 topology τ with an interval base. (d) (L, τ) is a compact T 1 -ordered space for some topology τ with a strong interval base. (e) (L, τ) is a compact totally order-disconnected topological lattice for some topology τ. If such a topology τ exists then it agrees with τ i, τ o, and τ Y,Z, where Y is the set of all compact elements and Z is the set of all cocompact elements. Proof. For the implication (a) (b) (c) (d) (a) (e), use 3.12 and 3.2. (e) (b): By the Fundamental Theorem for Compact Totally Disconnected Semilattices (see [12, VI 3.13]), L is bi-algebraic, and by [12, VII 2.8], the interval topology is T 2. Furthermore, by 3.11, we have τ i τ τ 0. Hence, by 1.2 and 3.5, it follows that τ = τ i = τ 0 = τ Y,Z. As remarked earlier, in condition (e) totally order-disconnected may be substituted by totally disconnected and T 2, totally separated, or zero- 16

17 dimensional and T o. Moreover topological may be replaced with semitopological. Notice also that a lattice L is bihyperalgebraic iff it is linked bialgebraic (see [16]), i.e. L and L op are algebraic lattices whose Lawson topologies coincide (cf. the notion of bi-continuous lattices [12, VII 2.5]). The algebraic counterparts of compact totally order-disconnected topological lattices under the Priestley duality are the so-called catalytic distributive lattices (see [16]). Two additional remarks are opportune. First, look at Example 1.7 once more in order to see that a non-complete lattice whose interval topology has a strong interval base (and is, moreover, discrete) need not be hypercompactly generated. Second, Example 3.1 shows that even a complete lattice whose interval topology has an interval base need not be meet-continuous, all the less (hyper-)compactly generated. In the presence of enough atoms and coatoms, the study of semitopological lattices becomes particularly nice. Let τ A denote the topology τ Y,Z where Y is the set of all atoms and Z is the set of all coatoms. Corollary 3.13 If L is a bi-atomistic compact semitopological lattice with closed universal bounds for some topology τ, then it is a totally order-disconnected order-topological complete lattice with τ = τ i = τ o = τ A. Moreover, the atoms are hypercompact, the coatoms are hypercocompact, and consequently, L is bihyperalgebraic. Proof. By 3.3 L is bi-atomistic iff τ o τ A iff τ A is totally order-disconnected. By 2.3 (2), each atom a generates a τ-open the principal dual ideal [a), because (a[= {0} is τ-closed. This and the dual observation yield the inclusion τ A τ, and by 3.7, we have τ τ o. Thus τ = τ o = τ A is a compact topology finer than τ i, and 3.13 applies. This result has interesting consequences for the theory of geometric lattices (in the sense of Grätzer [13, IV 3]; these are algebraic, atomistic and semimodular lattices (while in [1] and [3], the name geometric is reserved for finite-dimensional lattices). Corollary 3.14 Let L be a modular, complemented and atomic lattice, endowed with a compact T 1 topology τ making the unary lattice operations continuous. Then: (1) L is a geometric lattice in which every atom and every coatom has only a finite number of complements. (2) (L, τ) is a compact totally order-disconnected topological lattice. (3) τ agrees with the order topology, the interval topology, and with τ A. Proof. L is relatively complemented (being modular and complemented) and atomic, hence atomistic. Using modularity and relative complements once more, one verifies easily that L is also coatomistic, and then one may refer to Geometric lattices arising from usual geometries rarely admit a compact T 1 topology making the (unary) lattice operations continuous: indeed, in geometric lattices like the subspace lattice of a finite-dimensional real vector space, 17

18 every atom and every coatom has an infinite number of complements, and by 3.15, this excludes the existence of a compact topology with the desired properties. References [1] G. Birkhoff, Lattice Theory (third ed.). Amer. Math. Soc. Coll. Publ. 25, Providence, R.I. (1973). [2] T.H. Choe, R.J. Greechie, Profinite orthomodular lattices. Preprint. Mc Master University, Hamilton and Kansas State University, Mahattan (1990). [3] P. Crawley, R.P. Dilworth, Algebraic Theory of Lattices. [4] M. Erné, Order-topological lattices. Glasgow Math. J. 21 (1980), [5] M. Erné, Scott convergence and Scott topology in partially ordered sets II. In: B. Banaschewski and R.-E. Hoffmann (eds.) Continuous Lattices, Bremen 1979, Lecture Notes in Math. 871 Springer Verlag, Berlin Heidelberg New York (1981), [6] M. Erné, Bigeneration in complete lattice and principal separation of ordered sets. Order 8 (1991), [7] M. Erné, The ABC of order and topology. In: H. Herrlich and H.- E. Porst (eds.), Category Theory at Work. Heldermann Verlag, Berlin (1991), [8] M. Erné, S. Weck, Order convergence in lattices. Rocky Mountain J. Math. 10 (1980), [9] M. Erné, H. Gatzke,Convergence and continuoity in partially ordered sets and semilattices. In: R.-E. Hoffmann and K.H. Hoffmann (eds.) Continuous Lattices and Their Applications, Bremen Lecture Notes in Pure and Appl. Math. 109, Marcel Dekker, New York (1985), [10] M. Erné, R. Kopperman, Natural continuity space structure on dual Heyting algebras. Fund. Math. 136 (1990), [11] M. Erné, Z. Riečanová,Order-topological complete orthomodular lattices. Topology and its Applications (to appear). [12] G. Giertz, K.H. Hoffmann, K. Keimel, J.D. Lawson, M. Misloe, D.S. Scott, A Compendium of Continuous Lattices. Springer Verlag, Berlin-Heidelberg-New York (1980). [13] G. Grätzer, General Lattice Theory. Birkhäuser, Basel (1978). [14] P.T. Johnstone, Stone Spaces. Cambridge University Press, Cambridge (1982). [15] J.D. Lawson, Intrinsic topologies in totpological lattices and semilattices. Pacific J. Math. 44 (1974), [16] H.A. Priesley, Algebraic lattices as dual spaces of distributive lattice. In: R.-E.Hoffmann and K.H. Hoffmann (eds.), Continious Lattices and Their Applications, Bremen Lecture Notes in Pure and Appl. Math. 109, Marcel Dekker, New York (1985),

19 [17] Z. Riečanová, Topologies in atomic quantum logics. Acta Univ. Carolinae, Math. & Phys. 30 (1989), [18] Z. Riečanová, Applications of topological methods to the completion of atomic orthomodular lattices. Demonstratio Mathematica 24 (1991), [19] Z. Riečanová, Topological and order-topological orthomodular lattices. Bull. Austral. Math. Soc. 47 (1993), to appear. [20] L.A. Steen, J.A. Seebach jr., Counterexamples in Topology. Holt, Rinehart and Winston, Inc., New York (1970). 19