Dcpo-completion of posets

We introduce a new type of dcpo-completion of posets, called D-completion. For any poset P , the D-completion exists, and P and its Dcompletion have the isomorphic Scott closed set lattices. This completion is idempotent. A poset P is continuous (algebraic) if and only if its D-completion is continuous(algebraic). Using the D-completion, we construct the local dcpocompletion of posets, that revises the one given by Mislove. In the last section, we define and study bounded sober spaces. The dcpo-completion of a poset P is usually taken to be the ideal completion Id(P ) consisting of all ideals of P . As has been pointed out by several authors [13], [5], the ideal completion is not an idempotent completion. Furthermore, the assignment of Id(P ) to poset P cannot be extended to a functor from the category POSd of posets and Scott continuous mappings to the full subcategory DCPO of directed complete posets. On the other hand, given a poset P , the set Γ(P ) of all Scott closed sets forms a complete lattice. Different classes of posets yield different Scott closed set lattices. One of the most eminent classical results in domain theory is that a dcpo P is continuous if and only if Γ(P ) is a completely distributive lattice(see [6],[11] ). Banaschewski characterized the Scott open set lattices of continuous lattices as the stably supercontinuous lattices [2]. In [7], [3], characterizations of the Scott closed set lattices of complete lattices as well as bounded complete posets are also obtained. However, very little is known about the properties of Scott closed set lattices for more general posets. One of the basic problems on Scott closed set lattices is : Do posets and dcpos define the same class of Scott closed set lattices? In this paper we introduce a new type of dcpo-completion of posets, called the D-completion. This completion has the following properties: (i) it is idempotent and can be extended to a functor from POSd to DCPO which is left adjoint to the forgetful functor; (ii) every poset P and its D-completion have isomorphic Scott closed set lattices, which gives a positive answer to the above problem; (iii) a poset P is continuous if and only if its D-completion is continuous; (iv) if P is a continuous noncomplete lattice then its D-completion is a complete lattice. In [13], Mislove defined a local dcpo BSpec(P ) for each poset P , and claimed that 2000 Mathematics Subject Classification. 06B30,06B35, 68055.