##
**Continuous lattices and domains.**
*(English)*
Zbl 1088.06001

Encyclopedia of Mathematics and Its Applications 93. Cambridge: Cambridge University Press (ISBN 0-521-80338-1/hbk). xxxvi, 591 p. (2003).

The theory of continuous lattices provides truly interdisciplinary tools. It is extremely useful for order theory, algebra, topology, topological algebra, analysis, the theory of computing and computability.

In 1980, A compendium of continuous lattices [Berlin-Heidelberg-New York: Springer-Verlag (1980; Zbl 0452.06001)] was published by famous mathematicians G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott. This book was a comprehensive reference on continuous lattices and their generalizations. A number of monographs, proceedings and texts appeared in a steady stream following the Compendium. The present book presents a new edition containing the original information as well as reflecting developments of two decades of research in the large scope of domain theory.

The aim of the book is to present the mathematical foundations of the theory of continuous lattices and domains from the ingredients of order theory, topology and algebra, and to lay the groundwork for the numerous applications that domain theory has found in the areas of abstract theories of computation, semantics of programming languages, logic and lambda calculus, and in other branches of mathematics.

The book is divided into eight chapters, each of which consists of some sections. The history of continuous lattices is provided by the bibliography and the notes following the sections of the book, as well as by many remarks in the text. To aid the student, a few exercises with hints are included in each section. The book also contains a list of symbols, a list of categories and an index, providing a full (thorough) list of terms used.

In the initial chapter 0, “A primer on ordered sets and lattices”, fundamental concepts and notations of order-theoretical nature are discussed. The feature of this chapter is the formalism of Galois connections, explained in section 0-3. Meet-continuous lattices, of which both continuous lattices and complete Heyting algebras are subclasses, are discussed in section 0-4. The basic topological ideas which indicate how ordered structures arise out of topological ones are gathered in section 0-5.

Chapter I, “Order theory of domains”, introduces continuous lattices from the order-theoretic point of view. Section I-1 is devoted to the introduction of the “way-below” relation and of continuous lattices and domains. The paradigmatic examples of continuous lattices and domains are exhibited. Section I-2 gives an equational characterization of continuous lattices and discusses their variety-like properties. In this section various techniques of constructing new domains (continuous semilattices (lattices)) etc. from known ones are demonstrated. The constructions include forming direct products with the product order, taking subsets closed under appropriate operations and taking images under maps preserving appropriate operations. 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. Section I-3 introduces prime and irreducible elements and generalizations thereof and shows the plentiful supply of them in continuous lattices. The basic properties of algebraic lattices and some of their relationships with continuous lattices appear in section I-4. A parallel theory on the level of algebraic domains and their relation to continuous ones is also developed.

Chapter II, “The Scott topology”, defines the Scott topology and develops its applications to continuous lattices. Of fundamental importance in this chapter are the basic properties of the Scott topology and Scott-continuous functions appearing in sections II-1 and II-2. In section II-1 the Scott topology is defined and the liminf-convergence in the Scott-topology is characterized for continuous lattices. The function space aspect is amplified; the function space is described and exposed as a true generalization of the classical compact-open topology. Furthermore, the posets \(Q(X)\) of compact saturated sets on a \(T_0\)-space with respect to a partial order are discussed allowing a full treatment of the Hofmann-Mislove theorem and its various aspects. The feature of this chapter is the Characterization Theorem for domains in terms of properties of their lattices of Scott-open sets. Section II-2 gives the definition and characterizations of Scott-continuous functions. The space of all Scott-continuous functions between continuous lattices is itself a continuous lattice, and the category of continuous lattices proves to be Cartesian closed. Some other, more general, Cartesian closed categories of domains are also identified. In section II-3 it is shown that continuous lattices endowed with their Scott topologies form the “injectives” in the category of \(T_0\)-topological spaces. Section II-4 is concerned with function spaces (in particular the set of Scott-continuous functions between spaces and/or lattices) which carry a generalization of the compact-open topology, the Isbell topology, and an associated order, the order of specialization, and questions of the categorical notion of “Cartesian closedness”.

Chapter III, “The Lawson topology”, introduces the second important topology for continuous lattices, the Lawson topology. Like the Scott topology, it is defined in an order-theoretical fashion and is crucial in linking continuous lattices and domains to topological algebra. In section III-1 it is shown that the Lawson topology is compact and \(T_1\) for every complete lattice and compact Hausdorff for continuous lattices. Indeed, in section III-2 it is shown that for the meet-continuous complete lattices the Lawson topology is Hausdorff if and only if the lattice is continuous. In fact, continuous lattices equipped with the Lawson topology give compact Hausdorff topological semilattices which have a basis of subsemilattices – the most important class of semilattices in topological algebra. This interplay culminates in the Fundamental Theorem, which equates the two classes. Sections III-1 and III-2 are the basic sections of this chapter. Section III-3 characterizes convergence of nets in the Lawson topology. In this section, a class of posets containing the class of domains properly is introduced, and its members are called quasicontinuous domains. On a quasicontinuous domain, the Scott topology is locally compact and sober, and the Lawson topology is regular and Hausdorff, and indeed much of the theory of domains can be recovered in this more general setting. A notion of liminf-convergence is introduced, which is shown to be equivalent to topological convergence in the Lawson topology for (quasicontinuous) domains. Section III-4 generalizes the notion of a basis of a topology to continuous lattices and derives properties thereof. It is proved that the existence of a countable domain basis, of a countable basis of the Scott topology, and of a countable basis of the Lawson topology are all equivalent. Section III-5, entitled “Compact domains”, is largely devoted to the question of when the Lawson topology on a domain is compact and wraps up with the theory of the Isbell topology in the context of function space topologies.

Chapter IV, “Morphisms and functors”, presents a full treatment of the Lawson duality of domains, which parallels Pontryagin duality – notably when it is restricted to the category of continuous semilattice morphisms; in this form it is a veritable characterization theory for domains. The Lawson duality of continuous semilattices allows rounding off the complex of the Hofmann-Mislove theorem, which was presented in chapter II. A sort of geometric aspect of the duality between two domains is exposed in chapter V, because it realizes a pair of dual domains as the spectrum and the cospectrum of a completely distributive complete lattice. The section on projective limits in chapter IV is formulated for the category of domains and morphisms appearing as a pair of a Galois adjunction.

Chapter IV considers various important categories of continuous lattices together with certain categorical constructions. Section IV-1 is an important one; it presents important duality theorems for the study of continuous lattices. Section IV-2 introduces the category of domains and presents its elegant and important self-duality theory. Section IV-3 contains important results that a continuous lattice has sufficiently many semilattice homomorphisms of the right kind into the unit interval to separate points. Some of the exercises exemplify applications which were made possible on this level of generality. The next sections give general categorical construction for obtaining continuous lattices which are “fixed points” with respect to some self-functor of the category. This process is needed for the construction of self-theoretical models of the lambda calculus. The concept of projective limits in the relevant categories is analyzed in section IV-4. In section IV-5 the question of which functors preserve projective limits is treated. In section IV-6 general categorical criteria for fixed-point constructions for functors are worked out. This machinery is applied to the specific study of minimal solutions of domain equations in section IV-7. The chapter closes with an introduction to the important topic of powerdomains, including the extended probabilistic powerdomain, and gives constructions for the basic powerdomains, namely the Hoare, Smyth, and Plotkin powerdomains. The domain-theoretic version of the space of Borel measures is studied in section IV-9.

Chapter V, “Spectral theory of continuous lattices”, deals with the spectral theory of continuous lattices, a section of domain theory from that branch of analysis dealing largely with Polish spaces. The chapter begins with an important lemma which plays a vital role in the spectral theory of continuous lattices. It states that a “finitely prime” element of a continuous lattice is also “compactly prime” with respect to the Lawson topology. The topologically generated sets are investigated in section V-2. In particular, it is shown that a distributive continuous lattice always has a unique smallest closed order-generating set, namely, the closure of the set of nonidentity primes (the irreducible elements). Considering elements in this closure leads to generalizations of the notion of prime and irreducible in section V-3. Section V-4 introduces the subject of the spectral theory of lattices in general, and section V-5 considers that of continuous lattices. In section V-4 the principal topic of the chapter begins, the spectral theory of frames in which the primes order generate. The set of nonidentity primes with the hull-kernel topology is called the spectrum; this given lattice is isomorphic to the lattice of open subsets of this space. In this fashion the duality between the category of frames with points and lattice morphisms preserving arbitrary sups on the one hand, and sober spaces and continuous maps on the other, are recorded. This prepares the way for the specific spectral theory of continuous lattices, which is discussed in section V-5: the spectrum of a continuous lattice is locally compact and sober, and all locally compact sober spaces are obtained in this way. The category of continuous frames is dual to the category of locally compact sober spaces and continuous maps. There a duality is set up between the category of all distributive continuous lattices and all locally compact sober spaces (with appropriate morphisms for each category). Here, probably sections V-1, V-4 and V-5 would be of greatest interest to the general reader. A good deal of supplementary information is collected in the exercises.

Chapter VI, “Compact posets and semilattices”, begins the study of continuous lattices as certain compact semilattices, that is, from the point of view of topological algebra. The sections of primary interest are VI-1 and VI-3. In section VI-1 the most basic and useful properties of pospaces and topological semilattices are given. This is followed by an order-theoretic description of the topology of a compact semilattice in section VI-2. In section VI-3, the principal part of the chapter, the Fundamental Theorem of compact semilattices is stated and proved, establishing the equivalence between the category of compact semilattices with small semilattices and the category of continuous lattices. It is proved that complete continuous semilattices are exactly the compact semilattices with small semilattices in the Lawson topology. The only two known examples of compact semilattices which are not complete continuous semilattices are presented in section VI-4. Section VI-5 develops the important theory of stably compact spaces, which may be viewed as \(T_0\)-variants of compact pospaces. In chapter VI, in section VI-6, “Stably compact spaces”, the concept of compact spaces, which emulate in the wide class of \(T_0\) spaces as many properties as seem reasonable of classical compact \(T_2\) spaces, is discussed. These spaces have a partner topology, called the co-compact topology; and the common refinement, called the patch topology, is a compact Hausdorff topology. The most prominent example of a stably compact space is a domain with the Scott topology such that the Lawson topology is compact; in this example the patch topology is the Lawson topology. Chapter VI ends with the spectral theory of these spaces. Section VI-7 develops the spectral theory of compact pospaces and continuous monotone maps, on the one hand, and a category of stably continuous frames, on the other. Several important applications of the spectral theory of continuous lattices are given in section VI-7 and chapter VII. The rest of chapter VI and chapter VII center on more specialized topics in topological algebra.

Chapter VII, “Topological algebra and lattice theory, applications”, is devoted to exploring further links between topological algebra and continuous lattice and domain theory. In section VII-1 topological semilattices in which every open set is an upper set are considered. It is shown that under rather mild restrictions this topology must be the Scott topology. Section VII-2 takes up the topic of topological lattices, lattices in which both operators are continuous, and their topologies, with 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. Hypercontinuous lattices are a special class of continuous lattices for which, among several diverse characterizations, the interval topology is Hausdorff. They stand in spectral duality to the quasicontinuous domains equipped with the Scott topology. In section VII-4 a lattice-theoretic characterization of compact topological semilattices is given, and it is shown that in such a setting separate continuity of the meet operator implies joint continuity. 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. The final part of section VII-4 is devoted to prove that a compact semitopological semilattice is in fact topological.

In sum, this book presents a fairly complete account of the development of the theory of continuous lattices as it currently exists. The book is a welcome contribution to the existing literature on the theory of continuous lattices and domains as well as a window on the latest developments in the field. Of special interest is also the bibliography of this book, which is one of the most complete guides to the literature in lattice and domain theory up to date, listing nearly 900 articles, books and monographs, dissertations and master’s theses, and memos circulated in the seminar on continuity in semilattices. I highly recommend this book to every specialist in the field of topological algebra and topological order theory.

In 1980, A compendium of continuous lattices [Berlin-Heidelberg-New York: Springer-Verlag (1980; Zbl 0452.06001)] was published by famous mathematicians G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott. This book was a comprehensive reference on continuous lattices and their generalizations. A number of monographs, proceedings and texts appeared in a steady stream following the Compendium. The present book presents a new edition containing the original information as well as reflecting developments of two decades of research in the large scope of domain theory.

The aim of the book is to present the mathematical foundations of the theory of continuous lattices and domains from the ingredients of order theory, topology and algebra, and to lay the groundwork for the numerous applications that domain theory has found in the areas of abstract theories of computation, semantics of programming languages, logic and lambda calculus, and in other branches of mathematics.

The book is divided into eight chapters, each of which consists of some sections. The history of continuous lattices is provided by the bibliography and the notes following the sections of the book, as well as by many remarks in the text. To aid the student, a few exercises with hints are included in each section. The book also contains a list of symbols, a list of categories and an index, providing a full (thorough) list of terms used.

In the initial chapter 0, “A primer on ordered sets and lattices”, fundamental concepts and notations of order-theoretical nature are discussed. The feature of this chapter is the formalism of Galois connections, explained in section 0-3. Meet-continuous lattices, of which both continuous lattices and complete Heyting algebras are subclasses, are discussed in section 0-4. The basic topological ideas which indicate how ordered structures arise out of topological ones are gathered in section 0-5.

Chapter I, “Order theory of domains”, introduces continuous lattices from the order-theoretic point of view. Section I-1 is devoted to the introduction of the “way-below” relation and of continuous lattices and domains. The paradigmatic examples of continuous lattices and domains are exhibited. Section I-2 gives an equational characterization of continuous lattices and discusses their variety-like properties. In this section various techniques of constructing new domains (continuous semilattices (lattices)) etc. from known ones are demonstrated. The constructions include forming direct products with the product order, taking subsets closed under appropriate operations and taking images under maps preserving appropriate operations. 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. Section I-3 introduces prime and irreducible elements and generalizations thereof and shows the plentiful supply of them in continuous lattices. The basic properties of algebraic lattices and some of their relationships with continuous lattices appear in section I-4. A parallel theory on the level of algebraic domains and their relation to continuous ones is also developed.

Chapter II, “The Scott topology”, defines the Scott topology and develops its applications to continuous lattices. Of fundamental importance in this chapter are the basic properties of the Scott topology and Scott-continuous functions appearing in sections II-1 and II-2. In section II-1 the Scott topology is defined and the liminf-convergence in the Scott-topology is characterized for continuous lattices. The function space aspect is amplified; the function space is described and exposed as a true generalization of the classical compact-open topology. Furthermore, the posets \(Q(X)\) of compact saturated sets on a \(T_0\)-space with respect to a partial order are discussed allowing a full treatment of the Hofmann-Mislove theorem and its various aspects. The feature of this chapter is the Characterization Theorem for domains in terms of properties of their lattices of Scott-open sets. Section II-2 gives the definition and characterizations of Scott-continuous functions. The space of all Scott-continuous functions between continuous lattices is itself a continuous lattice, and the category of continuous lattices proves to be Cartesian closed. Some other, more general, Cartesian closed categories of domains are also identified. In section II-3 it is shown that continuous lattices endowed with their Scott topologies form the “injectives” in the category of \(T_0\)-topological spaces. Section II-4 is concerned with function spaces (in particular the set of Scott-continuous functions between spaces and/or lattices) which carry a generalization of the compact-open topology, the Isbell topology, and an associated order, the order of specialization, and questions of the categorical notion of “Cartesian closedness”.

Chapter III, “The Lawson topology”, introduces the second important topology for continuous lattices, the Lawson topology. Like the Scott topology, it is defined in an order-theoretical fashion and is crucial in linking continuous lattices and domains to topological algebra. In section III-1 it is shown that the Lawson topology is compact and \(T_1\) for every complete lattice and compact Hausdorff for continuous lattices. Indeed, in section III-2 it is shown that for the meet-continuous complete lattices the Lawson topology is Hausdorff if and only if the lattice is continuous. In fact, continuous lattices equipped with the Lawson topology give compact Hausdorff topological semilattices which have a basis of subsemilattices – the most important class of semilattices in topological algebra. This interplay culminates in the Fundamental Theorem, which equates the two classes. Sections III-1 and III-2 are the basic sections of this chapter. Section III-3 characterizes convergence of nets in the Lawson topology. In this section, a class of posets containing the class of domains properly is introduced, and its members are called quasicontinuous domains. On a quasicontinuous domain, the Scott topology is locally compact and sober, and the Lawson topology is regular and Hausdorff, and indeed much of the theory of domains can be recovered in this more general setting. A notion of liminf-convergence is introduced, which is shown to be equivalent to topological convergence in the Lawson topology for (quasicontinuous) domains. Section III-4 generalizes the notion of a basis of a topology to continuous lattices and derives properties thereof. It is proved that the existence of a countable domain basis, of a countable basis of the Scott topology, and of a countable basis of the Lawson topology are all equivalent. Section III-5, entitled “Compact domains”, is largely devoted to the question of when the Lawson topology on a domain is compact and wraps up with the theory of the Isbell topology in the context of function space topologies.

Chapter IV, “Morphisms and functors”, presents a full treatment of the Lawson duality of domains, which parallels Pontryagin duality – notably when it is restricted to the category of continuous semilattice morphisms; in this form it is a veritable characterization theory for domains. The Lawson duality of continuous semilattices allows rounding off the complex of the Hofmann-Mislove theorem, which was presented in chapter II. A sort of geometric aspect of the duality between two domains is exposed in chapter V, because it realizes a pair of dual domains as the spectrum and the cospectrum of a completely distributive complete lattice. The section on projective limits in chapter IV is formulated for the category of domains and morphisms appearing as a pair of a Galois adjunction.

Chapter IV considers various important categories of continuous lattices together with certain categorical constructions. Section IV-1 is an important one; it presents important duality theorems for the study of continuous lattices. Section IV-2 introduces the category of domains and presents its elegant and important self-duality theory. Section IV-3 contains important results that a continuous lattice has sufficiently many semilattice homomorphisms of the right kind into the unit interval to separate points. Some of the exercises exemplify applications which were made possible on this level of generality. The next sections give general categorical construction for obtaining continuous lattices which are “fixed points” with respect to some self-functor of the category. This process is needed for the construction of self-theoretical models of the lambda calculus. The concept of projective limits in the relevant categories is analyzed in section IV-4. In section IV-5 the question of which functors preserve projective limits is treated. In section IV-6 general categorical criteria for fixed-point constructions for functors are worked out. This machinery is applied to the specific study of minimal solutions of domain equations in section IV-7. The chapter closes with an introduction to the important topic of powerdomains, including the extended probabilistic powerdomain, and gives constructions for the basic powerdomains, namely the Hoare, Smyth, and Plotkin powerdomains. The domain-theoretic version of the space of Borel measures is studied in section IV-9.

Chapter V, “Spectral theory of continuous lattices”, deals with the spectral theory of continuous lattices, a section of domain theory from that branch of analysis dealing largely with Polish spaces. The chapter begins with an important lemma which plays a vital role in the spectral theory of continuous lattices. It states that a “finitely prime” element of a continuous lattice is also “compactly prime” with respect to the Lawson topology. The topologically generated sets are investigated in section V-2. In particular, it is shown that a distributive continuous lattice always has a unique smallest closed order-generating set, namely, the closure of the set of nonidentity primes (the irreducible elements). Considering elements in this closure leads to generalizations of the notion of prime and irreducible in section V-3. Section V-4 introduces the subject of the spectral theory of lattices in general, and section V-5 considers that of continuous lattices. In section V-4 the principal topic of the chapter begins, the spectral theory of frames in which the primes order generate. The set of nonidentity primes with the hull-kernel topology is called the spectrum; this given lattice is isomorphic to the lattice of open subsets of this space. In this fashion the duality between the category of frames with points and lattice morphisms preserving arbitrary sups on the one hand, and sober spaces and continuous maps on the other, are recorded. This prepares the way for the specific spectral theory of continuous lattices, which is discussed in section V-5: the spectrum of a continuous lattice is locally compact and sober, and all locally compact sober spaces are obtained in this way. The category of continuous frames is dual to the category of locally compact sober spaces and continuous maps. There a duality is set up between the category of all distributive continuous lattices and all locally compact sober spaces (with appropriate morphisms for each category). Here, probably sections V-1, V-4 and V-5 would be of greatest interest to the general reader. A good deal of supplementary information is collected in the exercises.

Chapter VI, “Compact posets and semilattices”, begins the study of continuous lattices as certain compact semilattices, that is, from the point of view of topological algebra. The sections of primary interest are VI-1 and VI-3. In section VI-1 the most basic and useful properties of pospaces and topological semilattices are given. This is followed by an order-theoretic description of the topology of a compact semilattice in section VI-2. In section VI-3, the principal part of the chapter, the Fundamental Theorem of compact semilattices is stated and proved, establishing the equivalence between the category of compact semilattices with small semilattices and the category of continuous lattices. It is proved that complete continuous semilattices are exactly the compact semilattices with small semilattices in the Lawson topology. The only two known examples of compact semilattices which are not complete continuous semilattices are presented in section VI-4. Section VI-5 develops the important theory of stably compact spaces, which may be viewed as \(T_0\)-variants of compact pospaces. In chapter VI, in section VI-6, “Stably compact spaces”, the concept of compact spaces, which emulate in the wide class of \(T_0\) spaces as many properties as seem reasonable of classical compact \(T_2\) spaces, is discussed. These spaces have a partner topology, called the co-compact topology; and the common refinement, called the patch topology, is a compact Hausdorff topology. The most prominent example of a stably compact space is a domain with the Scott topology such that the Lawson topology is compact; in this example the patch topology is the Lawson topology. Chapter VI ends with the spectral theory of these spaces. Section VI-7 develops the spectral theory of compact pospaces and continuous monotone maps, on the one hand, and a category of stably continuous frames, on the other. Several important applications of the spectral theory of continuous lattices are given in section VI-7 and chapter VII. The rest of chapter VI and chapter VII center on more specialized topics in topological algebra.

Chapter VII, “Topological algebra and lattice theory, applications”, is devoted to exploring further links between topological algebra and continuous lattice and domain theory. In section VII-1 topological semilattices in which every open set is an upper set are considered. It is shown that under rather mild restrictions this topology must be the Scott topology. Section VII-2 takes up the topic of topological lattices, lattices in which both operators are continuous, and their topologies, with 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. Hypercontinuous lattices are a special class of continuous lattices for which, among several diverse characterizations, the interval topology is Hausdorff. They stand in spectral duality to the quasicontinuous domains equipped with the Scott topology. In section VII-4 a lattice-theoretic characterization of compact topological semilattices is given, and it is shown that in such a setting separate continuity of the meet operator implies joint continuity. 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. The final part of section VII-4 is devoted to prove that a compact semitopological semilattice is in fact topological.

In sum, this book presents a fairly complete account of the development of the theory of continuous lattices as it currently exists. The book is a welcome contribution to the existing literature on the theory of continuous lattices and domains as well as a window on the latest developments in the field. Of special interest is also the bibliography of this book, which is one of the most complete guides to the literature in lattice and domain theory up to date, listing nearly 900 articles, books and monographs, dissertations and master’s theses, and memos circulated in the seminar on continuity in semilattices. I highly recommend this book to every specialist in the field of topological algebra and topological order theory.

Reviewer: Oleg V. Gutik (Lviv)

### MSC:

06-02 | Research exposition (monographs, survey articles) pertaining to ordered structures |

06B30 | Topological lattices |

06B35 | Continuous lattices and posets, applications |

06D20 | Heyting algebras (lattice-theoretic aspects) |

06D22 | Frames, locales |

06F30 | Ordered topological structures |

22A26 | Topological semilattices, lattices and applications |

54B30 | Categorical methods in general topology |

54H12 | Topological lattices, etc. (topological aspects) |