This chapter contains material at the interface between order theory and computer science. It discusses domains – those CPOs in which each element is the supremum of its ‘finite’ approximations – paralleling the discussion given in Chapter 7 of the class of algebraic lattices lying within the class of complete lattices. These domains provide a setting for denotational semantics, in a way we outline in 9.33. An alternative approach to domains, via information systems, is also presented. The final section of the chapter returns to fixpoint theory, and shows how the theorems in Chapter 8 can be applied in the solution of recursive equations, and in particular domain equations.

Domains for computing

This section brings together the notions of directed joins and of finite approximations.

Definitions. Let S be a non-empty subset of an ordered set P. Then S is said to be consistent if, for every finite subset F of S, there exists z ∈ P such that z ∈ Fu.

Remarks. Non-consistency arises only in ordered sets without ⊤. A directed set is, of course, consistent. The difference between the two notions is in the location of upper bounds: for D to be directed we require every finite subset F of D to have some upper bound which is a member of D, but for consistency an upper bound in P suffices.