# 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.

## Keywords

Finite Subset Finite Breadth Deterministic Process Continuous Lattice Denotational Semantic## Preview

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