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**1 - 1**of**1**### 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 ..."

Abstract
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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...