The first topologies defined on a lattice directly from the lattice ordering (that is, Birkhoff's order topology and Frink's interval topology) involved “symmetrical” definitions – the topologies assigned to L and to Lop were identical. A guiding example was always the unit interval of real numbers in its natural order, which is of course a highly symmetrical lattice. The initial interest was in such questions as which lattices became compact and/or Hausdorff in these topologies. The Scott topology stands in strong contrast to such an approach. Indeed it is a “unidirectional” topology, since, for example, all the open sets are always upper sets; thus, for nontrivial lattices, the T0 separation axiom is the strongest it satisfies. Nevertheless, we saw in Chapter II that the Scott topology provides many links between domains and general topology in such classical areas as the theory of semicontinuous functions and in the study of lattices of closed (compact, convex) sets (ideals) in many familiar structures.

In this chapter we introduce a new topology, called the Lawson topology, which is crucial in linking continuous lattices and domains to topological algebra. Its definition is more in the spirit of the interval and order topologies, and indeed it may be viewed as a mixture of the two. However, it remains asymmetrical – the Lawson topologies on L and Lop need not agree. But, even if one is seeking an appropriate Hausdorff topology for continuous lattices, this asymmetry is not at all surprising in view of the examples we have developed.

In Chapter I we encountered the rich order theoretic structure of complete lattices and of continuous lattices. Wherever it was feasible to express statements on the level of generality of dcpos and domains we did so. Perhaps even more typical for these partially ordered sets is their wealth of topological structure. The aim of the present chapter is to introduce topology into the study – a program to be continued in Chapter III.

Section II-1 begins with a discussion of the Scott topology and its connection with the convergence given in order theoretic terms by lower limits, or liminfs. This leads to a characterization theorem for domains in terms of properties of their lattices of Scott open sets (II-1.14) – a type of theorem that will become a recurrent theme (see Chapter VII). One motivation for such considerations arises from the appearances of domain theory in theoretical computer science: one typically needs the generality of domains to model the structures and constructions under consideration, while continuous lattices enter the scene as their lattices of open sets.

In Section II-2 we determine that the functions continuous for the Scott topology are those preserving directed sups. We can thus express one and the same property of a function between dcpos either in topological or in order theoretical terms. The space [S → T] of all Scott-continuous functions between continuous lattices is itself a continuous lattice, and the category of continuous lattices proves to be cartesian closed.

Our final chapter is devoted to exploring further links between topological algebra and continuous lattice and domain theory. This theme has already played an important role: the Fundamental Theorem of Compact Semilattices (VI-3.4) is just one example. In this chapter, however, the methods of topological algebra occupy a more central role, while the methods of continuous lattices are somewhat less prominent.

Section VII-1 is devoted to somewhat technical results about certain non-Housdorff topological semilattices; they are included primarily to facilitate the proof of later results concerning separate continuity of semilattice and lattice operations implying joint continuity. Section VII-2 makes various observations about topological lattices and their topologies, with a particular focus on completely distributive lattices.

Section VII-3 introduces the class of continuous lattices for which the Lawson topology is equal to the interval topology: the hypercontinuous lattices. The distributive ones are paired with the quasicontinuous domains via the spectral theory of Chapter V. Thus several earlier themes are nicely rounded out.

Section VII-4 characterizes those meet continuous complete lattices which admit a compact semilattice topology as being exactly those lattices whose lattice of Scott open sets forms a continuous lattice; this augments II-1.14, which shows that the continuous lattices are exactly those complete lattices whose Scott open sets form a completely distributive lattice. The final part of Section VII-4 is devoted to a proof that a compact semitopological semilattice is in fact topological.

A mathematics book with six authors is perhaps a rare enough occurrence to make a reader ask how such a collaboration came about. We begin, therefore, with a few words on how we were brought to the subject over a ten-year period, during part of which time we did not all know each other. We do not intend to write here the history of continuous lattices but rather to explain our own personal involvement. History in a more proper sense is provided by the bibliography and the notes following the sections of the book, as well as by many remarks in the text. A coherent discussion of the content and motivation of the whole study is reserved for the introduction.

In October of 1969 Dana Scott was led by problems of semantics for computer languages to consider more closely partially ordered structures of function spaces. The idea of using partial orderings to correspond to spaces of partially defined functions and functionals had appeared several times earlier in recursive function theory; however, there had not been very sustained interest in structures of continuous functionals. These were the ones Scott saw that he needed. His first insight was to see that – in more modern terminology – the category of algebraic lattices and the (so-called) Scott-continuous functions is cartesian closed. Later during 1969 he incorporated lattices like the reals into the theory and made the first steps toward defining continuous lattices as “quotients” of algebraic lattices.

BACKGROUND. In 1980 we published A Compendium of Continuous Lattices. A continuous lattice is a partially ordered set characterized by two conditions: firstly, completeness, which says that every subset has a least upper bound; secondly, continuity, which says that every element can be approximated from below by other elements which in a suitable sense are much smaller, as for example finite subsets are small in a set theoretical universe. A certain degree of technicality cannot be avoided if one wants to make more precise what this “suitable sense” is: we shall do this soon enough. When that book appeared, research on continuous lattices had reached a plateau.

The set of axioms proved itself to be very reasonable from many viewpoints; at all of these aspects we looked carefully. The theory of continuous lattices and its consequences were extremely satisfying for order theory, algebra, topology, topological algebra, and analysis. In all of these fields, applications of continuous lattices were highly successful. Continuous lattices provided truly interdisciplinary tools.

Major areas of application were the theory of computing and computability, as well as the semantics of programming languages. Indeed, the order theoretical foundations of computer science had been, some ten years earlier, the main motivation for the creation of the unifying theory of continuous lattices. Already the Compendium of Continuous Lattices itself contained signals pointing future research toward more general structures than continuous lattices.

Here we enter into the discussion of our principal topics. Continuous lattices and domains exhibit a variety of different aspects, some are order theoretical, some are topological, some belong to topological algebra and some to category theory – and indeed there are others. We shall contemplate these aspects one at a time, and this chapter is devoted entirely to the order theory surrounding our topic.

Evidently we have first to define continuous lattices and domains. As we shall see from hindsight, there are numerous equivalent conditions characterizing them. We choose the one which is probably the simplest, but it does involve the consideration of an auxiliary transitive relation, definable in every poset, by which one can say that an element x is “way below” an element y. We will write this as x « y. We devote Section I-1 to the introduction of the way-below relation and of continuous lattices and domains. We demonstrate that the occurrence of this particular additional ordering is not accidental and explain its predominant role in the theory. We exhibit the paradigmatic examples of continuous lattices and domains; in due course we shall see many more.

In Section I-2 we show that continuous lattices have a characterization in terms of (infinitary) equations. This gives us the important information that the class of continuous lattices, as an equational class, is closed under the formation of products, subalgebras, and homomorphic images – provided we recognize from the equations which maps ought to be considered as homomorphisms.

This introductory chapter serves as a convenient source of reference for certain basic aspects of complete lattices needed in what follows. The experienced reader may wish to skip directly to Chapter I and the beginning of the discussion of the main topic of this book: continuous lattices and domains.

Section O-1 fixes notation, while Section O-2 defines complete lattices, complete semilattices and directed complete partially ordered sets (dcpos), and lists a number of examples which we shall often encounter. The formalism of Galois connections is presented in Section 3. This not only is a very useful general tool, but also allows convenient access to the concept of a Heyting algebra. In Section O-4 we briefly discuss meet continuous lattices, of which both continuous lattices and complete Heyting algebras (frames) are (overlapping) subclasses. Of course, the more interesting topological aspects of these notions are postponed to later chapters. In Section O-5 we bring together for ease of reference many of the basic topological ideas that are scattered throughout the text and indicate how ordered structures arise out of topological ones. To aid the student, a few exercises have been included. Brief historical notes and references have been appended, but we have not tried to be exhaustive.

Generalities and Notation

Partially ordered sets occur everywhere in mathematics, but it is usually assumed that the partial order is antisymmetric. In the discussion of nets and directed limits, however, it is not always so convenient to assume this property.

Background and Plan of the Work

The purpose of this monograph is to present a fairly complete account of the development of the theory of continuous lattices as it currently exists. An attempt has been made to keep the body of the text expository and reasonably self-contained; somewhat more leeway has been allowed in the exercises. Much of what appears here constitutes basic, foundational or elementary material needed for the theory, but a considerable amount of more advanced exposition is also included.

Background and Motivation

The theory of continuous lattices is of relatively recent origin and has arisen more or less independently in a variety of mathematical contexts. We attempt a brief survey in the following paragraphs in the hope of pointing out some of the motivation behind the current interest in the study of these structures. We first indicate a definition for these lattices and then sketch some ways in which they arise.

A DEfiNITION. In the body of the Compendium the reader will find many equivalent characterizations of continuous lattices, but it would perhaps be best to begin with one rather straightforward definition – though it is not the primary one employed in the main text. Familiarity with algebraic lattices will be assumed for the moment, but even if the exact details are vague, the reader is surely familiar with many examples: the lattice of ideals of a ring, the lattice of subgroups of a group.

With the exception of certain developments in Chapter II, notably Sections II-2 and II-4, we largely refrained from using category-theoretic language (even when we used its tools in the context of Galois connections). Inevitably, we have to consider various types of functions between continuous lattices, and this is a natural point in our study to use the framework of category theory.

In Section IV-1 we discuss a duality based on the formalism of Galois connections between the categories DCPOG and DCPOD of all dcpos with upper and lower adjoints, respectively, as morphisms. We discuss in particular the categories INF and SUP, whose objects are complete lattices (in both cases) and whose morphisms are functions preserving arbitrary infs (respectively, sups). These categories are dual (IV-1.3). We saw as early as I-2.10 ff. that maps preserving arbitrary infs and directed sups play an important role in our theory. This leads us to consider the subcategory INF↑ of INF. Its dual under the INF–SUP duality is denoted by SUP0; its morphisms are precisely characterized in IV-1.4(1)–(2), but as a category in itself, SUP0 plays a minor role. More important, however, are the full subcategories AL ⊆ CL ⊆ INF↑ and ALop ⊆ CLop ⊆ SUP0, which consist of algebraic and continuous lattices, respectively.