This special issue of Mathematical Structures in Computer Science contains six papers from the Workshop on Domains IX held at the University of Sussex (Brighton), on 22–24 September 2008. This was the ninth event in the long tradition of Domains workshops, which started in Darmstadt in 1994. Since then, workshops have been organised in Braunschweig (1996), Munich (1997), Siegen (1998), Darmstadt again (1999, 2004), Birmingham (2002) and Novosibirsk (2007).

We present a method for constructing from a given domain representation of a space X with underlying domain D, a domain representation of a subspace of compact subsets of X where the underlying domain is the Plotkin powerdomain of D. We show that this operation is functorial over a category of domain representations with a natural choice of morphisms. We study the topological properties of the space of representable compact sets and isolate conditions under which all compact subsets of X are representable. Special attention is paid to admissible representations and representations of metric spaces.

Bar recursion arises in constructive mathematics, logic, proof theory and higher-type computability theory. We explain bar recursion in terms of sequential games, and show how it can be naturally understood as a generalisation of the principle of backward induction that arises in game theory. In summary, bar recursion calculates optimal plays and optimal strategies, which, for particular games of interest, amount to equilibria. We consider finite games and continuous countably infinite games, and relate the two. The above development is followed by a conceptual explanation of how the finite version of the main form of bar recursion considered here arises from a strong monad of selections functions that can be defined in any cartesian closed category. Finite bar recursion turns out to be a well-known morphism available in any strong monad, specialised to the selection monad.

We introduce convex–concave duality for various models of non-deterministic choice, probabilistic choice and the two of them combined. This complements the well-known duality of stably compact spaces in a pleasing way: convex–concave duality swaps angelic and demonic choice, and leaves probabilistic choice invariant.

One natural way to generalise domain theory is to replace partially ordered sets by categories. This kind of generalisation has recently found application in the study of concurrency. An outline is given of the elegant mathematical foundations that have been developed. This is specialised to give a construction of cartesian closed categories of domains, which throws light on standard presentations of domain theory.

We prove that the category of -cocomplete separated -categories has bilimits of expanding sequences. This result generalises on various levels the well-known theorem of domain theory that guarantees the existence of bilimits in the category of directed-complete posets and Scott-continuous maps.

We investigate universal algebra over the category Nom of nominal sets. Using the fact that Nom is a full reflective subcategory of a monadic category, we obtain an HSP-like theorem for algebras over nominal sets. We isolate a ‘uniform’ fragment of our equational logic, which corresponds to the nominal logics present in the literature. We give semantically invariant translations of theories for nominal algebra and NEL into ‘uniform’ theories, and systematically prove HSP theorems for models of these theories.