On the Smyth power domain

Abstract

This paper explores the connection between the Smyth power domain PS(D) of a domain D and the domain D itself. The Smyth power domain is the most prevalent of the three power domain constructions commonly used to model nondeterminism in the denotational semantics of high-level programming languages. One definition of the Smyth power domain PS(D) is as the set of all Lawson-closed upper sets X from the domain D, so there is the natural inclusion x ↦↑x: D → PS(D). On the other hand, the inf map X ↦ ∧ X: PS(D) → D is an upper adjoint to this inclusion, and we use this adjunction to obtain information about PS(D) from the domain D. If D is distributive, spectral theory implies that each element X of PS(D) satisfies ∧ X is the infimum of a unique set of primes minimal with respect to being contained in X. Results which characterize when a domain D does not contain a copy of 2^N are invoked to show that the set of such primes is finite in certain cases. We indicate how these results can be generalized to the case that D is locally distributive or semiprime. Our results are motivated by an interest in understanding the Smyth power domain PS(D) in terms of the domain D, and we feel they should have application to the semantics of high-level programming languages. An indication of some possible applications of these results is given at the end of the paper.