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The largest Cartesian closed category of domains. (English) Zbl 0556.68017

Most of the studies in semantics of programming languages use ’domains’, i.e. $\omega$-algebraic cpo’s; the corresponding category $\omega$ ACPO should be closed under function-space formation, reasonably. This is not true however. One does obtain Cartesian closure (the technical name of what we want) by considering the category $\omega$ ACPO-CC of consistently complete domains, but now the powerdomain construction takes us outside the category.

Plotkin has conjectured that the category SFP which is an extension of consistently complete domains while still a subcategory of that of domains (and which is closed under powerdomain and function-space formation) is the largest category of domains closed under the constructions aforementioned. The paper under review proves this, making extensive use of the set of finite elements of a domain and of the set of minimal bounds of a poset. Finally some extensions are considered in case the notion of ’domain’ is modified either to effectively given domains or to continuous domains: the author conjectures some of the results to be still true.

Reviewer: M.Eytan

##### MSC:
 68Q99 Theory of computing 18D15 Closed categories 06B23 Complete lattices, completions 68Q55 Semantics