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Universal Profinite Domains
 Information and Computation
, 1987
"... . We introduce a bicartesian closed category of what we call profinite domains. Study of these domains is carried out through the use of an equivalent category of preorders in a manner similar to the information systems approach advocated by Dana Scott and others. A class of universal profinite dom ..."
Abstract

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. We introduce a bicartesian closed category of what we call profinite domains. Study of these domains is carried out through the use of an equivalent category of preorders in a manner similar to the information systems approach advocated by Dana Scott and others. A class of universal profinite domains is defined and used to derive sufficient conditions for the profinite solution of domain equations involving continuous operators. As a special instance of this construction, a universal domain for the category SFP is demonstrated. Necessary conditions for the existence of solutions for domain equations over the profinites are also given and used to derive results about solutions of some equations. A new universal bounded complete domain is also demonstrated using an operator which has bounded complete domains as its fixed points. 1 Introduction. For our purposes a domain equation has the form X ΒΈ = F (X) where F is an operator on a class of semantic domains (typically, F is an endof...
The Largest Cartesian Closed Category of Domains, Considered Constructively
, 2000
"... A conjecture of Smyth [10] is discussed which says that if D and [D # D] are effectively algebraic directedcomplete partial orders with least element (cpo's), then D is an e#ectively strongly algebraic cpo, where it was left open what exactly is meant by an effectively algebraic and an e#ectively ..."
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A conjecture of Smyth [10] is discussed which says that if D and [D # D] are effectively algebraic directedcomplete partial orders with least element (cpo's), then D is an e#ectively strongly algebraic cpo, where it was left open what exactly is meant by an effectively algebraic and an e#ectively strongly algebraic cpo. First, notions of an e#ectively strongly algebraic cpo and an e#ective SFP object are introduced. The effective SFP objects are just the constructive (computable) objects in the effectively given category [9] of indexed # algebraic cpo's. Theorem Every effective SFP object is an effectively strongly algebraic cpo, and vice versa. Moreover, this equivalence holds effectively. This shows that the given notion of an effective SFP object is stable. In e#ectivity considerations of # algebraic cpo's it is usual to require that the partial order be decidable on the compact elements. Here, we use a stronger assumption. Theorem If D is an indexed #algebraic cpo that has a comp...