Domains at Darmstadt: Some history

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. It took about a year for the topological ideas to mature in his mind culminating in the paper published as \cite{scott72a}. Of course, a large part of Scott's work was devoted to a presentation of models for the type-free $\lambda$-calculus, but the search for such models was not the initial aim of the investigation of partially ordered structures; on the contrary, it was the avoiding of the formal and unmotivated use of lambda-calculus that prompted Scott to look more closely at the structures of the functions themselves, and it was only well after he began to see their possibilities that he realized there had to exist non-trivial T_0-spaces homeomorphic to their own function spaces.

Quite separately from this development, Karl Hofmann, Jimmie Lawson, Mike Mislove, and Al Stralka (among others) recognized the importance of compact semilattices as a central ingredient in the structure theory of compact semigroups. In his dissertation \cite{lawson67}, Lawson initiated the study of a class of compact semilattices distinguished by the property that each had enough continuous semilattice morphisms into the unit interval semilattice (in its natural order) to separate points. (Such a program had already been started by Nachbin for partially ordered spaces in \cite{nachbin65}.) Lawson characterized this class of compact semilattices as those which admitted a basis of subsemilattice neighborhoods at each point (small subsemilattices): the class proved to be of considerable theoretical interest and attracted the attention of other workers in the field. In fact, it was believed for some time that all compact semilattices were members of the class, partly because the theory was so satisfactory (for example, purely ''order-theoretic`` characterizations were discovered for the class by Lawson \cite{lawson73}), and because no natural counterexamples seemed to exist. However, Lawson found the first example of a compact semilattice which was not in the class, one in fact which admitted only constant morphisms into the unit interval \cite{lawson70}.

At about the same time, Klaus Keimel had been working on lattices and lattice-ordered algebras in pursuit of their spectral theory and their representation in sheaves. In his intensive collaboration with Gerhard Gierz on topological representations of non-distributive lattices, a spectral property emerged which turned out to be quite significant for compact semilattices with small subsemilattices.

The explanation for the fact that the topological algebra of Lawson's semilattices had been so satisfactory emerged clearly when Hofmann and Stralka gave a completely lattice-theoretical description \cite{hofmann76b}. It was Stralka who first recognized the relation of this class to Scott's continuous lattices, and this observation came about as follows. Two monographs on duality theories for lattices and topological structures emerged in the early seventies: One for topology and lattices by Hofmann and Keimel \cite{hofmann72}, and the other for compact zero-dimensional semilattices and lattices by Hofmann, Mislove, and Stralka \cite{hofmann74}. At the lattice theory conference in Houston in 1973, where such dualities were discussed, B. Banaschewski spoke on filters and mentioned Scott's work which was just about to appear in the Proceedings of the Dalhousie Category Theory Conference. Stralka checked out this hint, and while he and Hofmann were working on the algebraic theory of Lawson semilattices \cite{hofmann76b}, he realized the significance of this work as a link between the topological algebra of compact semilattices and the lattice theory of Scott's continuous lattices. This led to correspondence with Scott and much subsequent activity.

In the summer of 1976, Hofmann and Mislove spent some time collaborating with Keimel and Gierz at the Technische Hochschule in Darmstadt, and together they began a ''write-in`` seminar called the Seminar on Continuity in Semilattices, or SCS for short. The authors formed the core membership of the seminar, but their colleagues and students contributed greatly to the seminar by communicating their results, ideas, and problems. The seminar then convened in person for several lively and well-attended workshops. The first was hosted by Tulane University in the spring of 1977, the second by the Technische Hochschule Darmstadt in the summer of 1978, and the third by the University of California at Riverside in the spring of 1979. A fourth workshop was held at the University of Bremen in the fall of 1979. Participated in these seminars in particular H. Bauer, J. H. Carruth, Alan Day (who discovered an independent access to continuous lattices through the filter monad). Marcel Ern\'e, R.-E. Hoffmann, John Isbell, Jaime Ninio, A. R. Stralka, and 0. Wyler.

lt was at the Tulane Workshop that the idea of collecting together the results of research -- common and individual -- was first discussed. A preliminary version of the Compendium worked out primarily by Hofmann, Lawson, and Gierz was circulated among the participants of the Darmstadt Workshop.


Technische Universität Darmstadt
Fachbereich Mathematik
Arbeitsgruppe Logik

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