In late August 2004, some 60 mathematicians and computer scientists gathered in Darmstadt for the seventh Workshop Domains, to mark the 65th birthday of Professor Klaus Keimel and his retirement from his position at the Technical University Darmstadt. The papers in this volume were selected from submissions that were received in response to a call issued to participants during the meeting, and to a wider community afterwards.

We propose compactly generated monotone convergence spaces as a well-behaved topological generalisation of directed-complete partial orders (dcpos). The category of such spaces enjoys the usual properties of categories of ‘predomains’ in denotational semantics. Moreover, such properties are retained if one restricts to spaces with a countable pseudobase in the sense of E. Michael, a fact that permits connections to be made with computability theory, realizability semantics and recent work on the closure properties of topological quotients of countably based spaces (qcb spaces). We compare the standard domain-theoretic constructions of products and function spaces on dcpos with their compactly generated counterparts, showing that these agree in important cases, though not in general.

This paper studies modified bar recursion, a higher type recursion scheme, which has been used in Berardi et al. (1998) and Berger and Oliva (2005) for a realisability interpretation of classical analysis. A complete clarification of its relation to Spector's and Kohlenbach's bar recursion, the fan functional, Gandy's functional $\Gamma$ and Kleene's notion of S1–S9 computability is given.

This paper surveys what we have learned during the last ten years about the lattice $\lambda \mathcal{T}$ of all $\lambda$-theories (= equational extensions of untyped $\lambda$-calculus), via the sets $\lambda \mathcal{C}$ consisting of the $\lambda$-theories that are representable in a uniform class $\mathcal{C}$ of $\lambda$-models. This includes positive answers to several questions raised in Berline (2000), as well as several independent results, the state of the art on the long-standing open questions concerning the representability of $\lambda _{\beta},\lambda _{\beta\eta}$, $H$ as theories of models, and 22 open problems.

We will focus on the class $\mathcal{G}$ of graph models, since almost all the existing semantic proofs on $\lambda \mathcal{T}$ have been, or could be, more easily, obtained via graph models, or slight variations of them. But in this paper we will also give some evidence that, for all uniform classes $\mathcal{C},\mathcal{C}^{\prime}$ of proper $\lambda$-models living in functional semantics, $\lambda \mathcal{C}-\lambda \mathcal{C}^{\prime}$ should have cardinality $2^{\omega }$, provided $ \mathcal{C}$ is not included in $\mathcal{C}^{\prime}.$

In this note we show that quotients of countably based spaces (qcb spaces) and topological predomains, as introduced by M. Schröder and A. Simpson, are not closed under sobrification. As a consequence, replete topological predomains need not be sober, that is, in general, repletion is not given by sobrification. Our counterexample also shows that a certain tentative ‘equaliser construction’ of repletion fails for qcb spaces.

Our results also extend to the more general class of core compactly generated spaces.

We present a Cartesian closed category ELOC of equilocales, which contains the category LOC of locales as a reflective full subcategory. The embedding of LOC into ELOC preserves products and all exponentials of exponentiable locales.

In this paper, we describe three distinct monoids over domains, each with a commutative analog, which define bag domain monoids. Our results were inspired by work by Varacca (Varacca 2003), and they lead to a constructive approach to his Hoare indexed valuations over a continuous poset $P$. We use our constructive approach to describe an analog of the probabilistic power domain, and the laws that characterise it, that forms a Scott-closed subset of Varacca's construct. We call these the Hoare random variables over$P$.

We show that the extensional ordering of the sequential functionals of pure type 3, for example, as defined via game semantics (Abramsky et al. 1994; Hyland and Ong 2000), is not cpo-enriched. This shows that this model does not equal Milner's (Milner 1977) fully abstract model for PCF.

A useful separation lemma for partial cm-lattices is proved equivalent to PIT, the Prime Ideal Theorem. The relation of various versions of the Lemma to each other and to PIT is also explored.

We give a domain-theoretic analogue of the classical Banach–Alaoglu theorem, showing that the patch topology on the weak$*$ topology is compact. Various theorems follow concerning the stable compactness of spaces of valuations on a topological space. We conclude with reformulations of the patch topology in terms of polar sets or Minkowski functionals, showing, in particular, that the ‘sandwich set’ of linear functionals is compact.

The object-calculus is an imperative and object-based programming language in which every object comes equipped with its own method suite. Consequently, methods need to reside in the store (‘higher-order store’), which complicates the semantics. Abadi and Leino defined a program logic for this language enriching object types by method specifications. We present a new soundness proof for their logic using denotational semantics. It turns out that denotations of store specifications are predicates defined by mixed-variant recursion. A benefit of our approach is that derivability and validity can be kept distinct. Moreover, it reveals which of the limitations of Abadi and Leino's logic are incidental design decisions and which follow inherently from the use of a higher-order store. We discuss the implications for the development of other, more expressive, program logics.

In this article, we characterise all continuous posets that are partially metrisable in their Scott topology. We present conditions for pmetrisability, which are both necessary and sufficient, in terms of measurements, domain-theoretic bases and, in a more general setting, in terms of radially convex metrics. These conditions, together with their refinements and generalisations, set a natural hierarchy on the class of partially metrised posets. We locate the class of countably-based continuous dcpos within this hierarchy.