An Introduction to the Theory of Continuous Lattices

Abstract

The theory of continuous lattices traces its beginnings to the work of Dana Scott and of Jimmie Lawson. In the late 1960’s, Scott was looking for mathematical models of the λ-calculus of Church and Curry. Since this logic is type free, any such model would have to allow elements to represent functions as well as sets. This led Scott to look for cartesian closed categories, and for objects in such a category which are isomorphic to their own function space of self-maps. He found such a category and such an object by first considering certain T ₀-spaces, which he called injective. A T ₀-space I is injective if, given any pair of T ₀-spaces X ⊂ Y and any continuous function f : X → I, there is an extension f′ : Y → I for f. (Here, X ⊂ Y means X is a subspace of Y.) An example of an injective space is the so-called Sierpinski space 2 = {0,1} endowed with the topology making {1} open, but {0} not open. In fact, the injective spaces are characterized by the fact that they are precisely the retracts of products of copies of the Sierpinski space. It turns out that the category of injective spaces and continuous maps between them is exactly the category for which Scott was searching, but to see this more clearly, we first translate this situation into an order-theoretic setting. This is accomplished by defining the specialization order on a topological space X, whereby x ≤ y if and only if every open set containing x must also contain y. Said another way, x ≤ y iff y ∈ {x}¯. While this order is always reflexive and transitive, it is antisymmetric precisely when X is a T ₀-space. Now, under this order, the injective spaces correspond to objects which Scott called continuous lattices, and the continuous functions between the injective spaces can be characterized in completely algebraic terms. We clarify this aspect below.

At about the same time that Scott was searching for his models of the λ-calculus, Jimmie Lawson was beginning his investigations into a certain class of locally compact topological semilattices. He called these semilattices semilattices with small semilattices because his hypothesis was that each had a neighborhood basis of subsemilattices at each point. Due to his extensive work in the area, such semilattices have come to be called Lawson semilattices. That these two rather disparate lines of research should merge into one theory came about as follows.

In 1974, Karl Hofmann, Al Stralka and the author wrote a monograph [HMS] exploring in detail the Pontryagin duality between compact zero-dimensional semilattices, on the one hand, and discrete semilattices on the other. From our viewpoint, the crucial result in this work is the fact that each compact zero-dimensional semilattice is actually an algebraic lattice endowed with a unique, algebraically determined topology. Moreover, each algebraic lattice becomes a compact zero-dimensional semi- lattice when endowed with this topology. Since a compact zero-dimensional semilattice is, in particular, a Lawson semilattice, it was only natural that Hofmann and Stralka should endeavor to extend this result to the more general class of compact Lawson semilattices. They accomplished this in [HS], where they gave a completely algebraic description of compact Lawson semilattices and the continuous semilattice homomorphisms between them. Stralka then realized that the objects which Scott defined in [S] to be continuous lattices were exactly the same as those objects which algebraically describe compact Lawson semilattices in [HS]. This led to a great deal of activity in the area, centering around a write-in seminar (known as SCS), and culminated in the writing of the comprehensive treatise [C] about the theory of continuous lattices.

In order to give an exact definition of continuous lattices, we begin by recalling the definition of an algebraic lattice. First, we call a subset D of a complete lattice L directed if, given x,y ∈ D there is some z ∈ D with x,y ≤ z. An element x of L is then called compact if, for any directed set D, x ≤ sup D implies x ≤ d for some d ∈ D. For example, the zero of L is compact. If we denote the set of compact elements of L by K(L), then we say that the complete lattice L is algebraic if x = sup{y ∈ K (L) : y ≤ x} holds for each x ∈ L. Algebraic lattices arise naturally in algebra; for example, the lattice of ideals of a ring is algebraic, where the compact elements are exactly the finitely generated ideals.

Now, the unique topology referred to above relative to which each algebraic lattice becomes a compact zero-dimensional semi-lattice is easily described: A subbasis is the family {↑k : k ∈ K(L)} ∪ {L\↑x : x ∈ L}, where ↑x = {y ∈ L : x ≤ y}. Since each element of this basis is a subsemilattice, it is easy to see that L is a Lawson semilattice when endowed with this topology. It is also true that the topology so defined is compact and zero-dimensional. That each compact zero-dimensional semilattice is actually an algebraic lattice endowed with this topology follows from two observations: First, any such semilattice S has enough continuous semilattice homomorphisms φ : S → 2 into the two-point semilattice (with the discrete topology) to separate the points, and for each such homomorphism, inf φ⁻¹ (1) is a compact element of S. Thus, for each x ∈ S, we have x = sup{Λφ⁻¹ (1) : φ(x) = 1}, thus realizing each element as the sup of compact elements.

The more general notion of a continuous lattice arises from a weakening of the notion of a compact element. For a complete lattice L, we say the element y is compact in the element x if, for any directed set D with x ≤ sup D, there is some d ∈ D with y ≤ d. We also say y is way-below x in this case, and we denote this relation by y ≪ x. Thus, the compact elements of L are exactly those elements which are way-below themselves. We then define the complete lattice L to be a continuous lattice if x = sup{y ∈ L : y ≪ x} for every x ∈ L. Clearly any algebraic lattice is a continuous lattice, since a compact element is way-below any element above it. However, there are many more continuous lattices than just the algebraic lattices; for example, the unit interval (or any complete chain, for that matter) is a continuous lattice, where y ≪ x exactly when y < x or else y = 0. That every continuous lattice has a unique compact Hausdorff topology relative to which it is a Lawson semilattice follows by generalizing the topology indicated above for algebraic lattices in the natural way; namely, we take as subbasis the family , where . This topology is called the Lawson topology. While it is not true that is a subsemilattice in general, nonetheless L becomes a compact Lawson semilattice when endowed with this topology. Furthermore, the continuous semilattice homomorphisms between compact Lawson semilattices are characterized as being those functions between continuous lattices which preserve all infima and directed suprema.

To close this circle of ideas, we point out how injective spaces arise in this context. Indeed, it is clear that the fundamental notion for continuous lattices is the idea that directed nets should converge to their suprema. We can define the weakest topology on a continuous lattice L for which this holds by letting ∪ ⊂ L be open iff ∪ = ↑∪ and, for every directed set D in L, if sup D ∈ ∪, then $D \cap U \ne \not 0$. This is the Scott topology on L, and sets such as ∪ above are called Scott open sets. Clearly the Lawson topology is a refinement of the Scott topology (well, it isn’t actually clear, but it does follow from the definition of a continuous lattice). In any event, it turns out that any continuous lattice, when endowed with the Scott topology is an injective T ₀-space, and every injective T ₀-space so arises. Moreover, the continuous functions between injective spaces are precisely those functions which preserve directed sups! Thus, the appropriate category for Scott’s construction of models of the λ-calculus is precisely the category of continuous lattices and Scott continuous functions. As it happens, this category is cartesian closed, and using fairly straightforward techniques (which are nonetheless rather clever) Scott constructs a continuous lattice which is isomorphic in a natural way to its space of Scott continuous self-maps. While he shows this can be done starting with any continuous lattice whatsoever — and, as a result, it follows that any T ₀-space is embeddable in an injective space which is isomorphic to its space of continuous self-maps — his application to the mathematical theory of computing uses the two point lattice as the building block (see [S-2] for the details).

Continuous lattice theory has made considerable contributions to general topology via spectral theory, and we want to close this introduction by indicating how this has come about. If we recall Stone’s basic result about Boolean algebras, it says that each Boolean algebra arises as the algebra of compact-open subsets of some compact zero-dimensional space, and, for a given Boolean algebra, the space in question is the space of maximal ideals of the algebra endowed with a suitable topology (see [St]). This spectral theory generalizes to a category of continuous lattices in the following way. If one examines Stone’s theorem carefully, it becomes clear that two facts are crucial to the proof. First, that the maximal ideals are the prime elements in the lattice of all ideals of the algebra, and second that the lattice of ideals of the algebra is a distributive algebraic lattice. This led Hofmann and Lawson to consider the space Spec L of all non-identity primes of an arbitrary distributive lattice L. Recalling that the lower topology on the lattice L has the sets of the form L\↑x for a subbasis, it turns out that Spec L, when endowed with the relative lower topology from L, becomes a locally quasicompact sober space (a space is locally quasicompact if each point has a neighborhood basis of quasicompact sets; it is sober if each closed subset which cannot be written as the union of two proper closed subsets has a unique dense point). Moreover, the map sending each point x of L to the open subset (Spec L)\↑x of Spec L is an isomorphism of the distributive continuous lattice L onto the lattice 0(Spec L) of open subsets of Spec L. On the other hand, for any locally quasicompact sober space X, the lattice 0(X) of open subsets of X is a distributive continuous lattice, and if f : X → Y is a continuous proper map between such spaces, then the map f ⁻¹ : 0(Y) → 0(X) preserves all sups, finite infs, and the way-below relation. The details of this duality are contained in [HL]. This theory also leads to many new and interesting questions about the theory of continuous lattices and its applications to other areas of mathematics. For example, a general theory of local compactness (in the absence of Hausdorff separation) has been developed in [HM], and this in turn has led to the laying of the foundations for a general theory of k-spaces (again without Hausdorff separation) in [SCS-HL].