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Zbl 0556.68017
Smyth, Michael B.
The largest Cartesian closed category of domains.
(English)
[J] Theor. Comput. Sci. 27, 109-119 (1983). ISSN 0304-3975

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. \par 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.
[M.Eytan]
MSC 2000:
*68Q99 Theory of computing
18D15 Closed categories
06B23 Complete lattices
68Q55 Semantics

Keywords: omega-algebraic cpo; complete partial order; semantics of programming languages; function-space formation; Cartesian closure; consistently complete domains; powerdomain; finite elements of a domain; minimal bounds of a poset; effectively given domains; continuous domains

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