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**1 - 3**of**3**### Cartesian closed stable categories q

, 2004

"... The aim of this paper is to establish some Cartesian closed categories which are between the two Cartesian closed categories: SLP (the category of L-domains and stable functions) and DI (the full subcategory of SLP whose objects are all dI-domains). First we show that the exponentials of every full ..."

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The aim of this paper is to establish some Cartesian closed categories which are between the two Cartesian closed categories: SLP (the category of L-domains and stable functions) and DI (the full subcategory of SLP whose objects are all dI-domains). First we show that the exponentials of every full subcategory of SLP are exactly the spaces of stable functions. Then we prove that the full subcategories SDMBC, SDCBC and SDABC of SLP which contain DI are all Cartesian closed, where the objects of SDMBC (resp., SDCBC, SDABC) are all distributive bc-domains which are meet-continuous (resp., continuous, algebraic). We also obtain many non-Cartesian closed full subcategories of SLP and present some reflective relations between those categories concerned.

### Under consideration for publication in Math. Struct. in Comp. Science The Largest Cartesian Closed Category of Domains, Considered Constructively

, 2002

"... A conjecture of Smyth is discussed which says that if D and [D → D] are effectively algebraic directed-complete partial orders with least element (cpo’s), then D is an effectively strongly algebraic cpo, where it was not made precise what is meant by an effectively algebraic and an effectively stron ..."

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A conjecture of Smyth is discussed which says that if D and [D → D] are effectively algebraic directed-complete partial orders with least element (cpo’s), then D is an effectively strongly algebraic cpo, where it was not made precise what is meant by an effectively algebraic and an effectively strongly algebraic cpo. Notions of an effectively strongly algebraic cpo and an effective SFP domain are introduced and shown to be (effectively) equivalent. Moreover, the conjecture is shown to hold if instead of being effectively algebraic, [D → D] is only required to be ω-algebraic and D is forced to have a completeness test, that is a procedure which decides for any two finite sets X and Y of compact cpo elements whether X is a complete set of upper bounds of Y. As a consequence, the category of effective SFP objects and continuous maps turns out to be the largest Cartesian closed full subcategory of the category of ω-algebraic cpo’s that have a completeness test. It is then studied whether such a result also holds in a constructive framework, where one considers categories with constructive domains as objects, that is, domains consisting only of the constructive (computable) elements of an indexed ω-algebraic cpo, and computable maps as morphisms. This is indeed the case: the category of constructive SFP domains is the largest constructively Cartesian closed weakly indexed effectively full subcategory of the category of constructive domains that have a completeness test and satisfy a further effectivity requirement.

### 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 effectively strongly algebraic cpo, where it was left open what exactly is meant by an effectively algebraic and an effec ..."

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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 effectively strongly algebraic cpo, where it was left open what exactly is meant by an effectively algebraic and an effectively strongly algebraic cpo. First, notions of an effectively strongly algebraic cpo and an effective 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.