Linear Approximation On Bounded Complete Dcpos

The main objects in domain theory are partially ordered sets (posets for short) and categories formed by them. Bounded complete domain is a very important class of order structure since the first important cartesian closed category to be found is the one of algebraic bounded complete domains (or Scott domain). In this paper, instead of traditional Scott continuous maps, we take maps preserving all existing suprema (linear maps) as morphism, and investigate the properties of the category of bounded complete domains. We obtain several important results as follows:At first, we construct the conditioning lower powerdomain of a continuous domain with a least element, and we prove that the Scott continuous maps between domains can be lifted to the ones between bounded complete domains which preserve all existing suprema. Second, we define the first approximation—linear FS-domains with respect to maps preserving all existing suprema. Note that map's preserving all suprema is equivalent to having an upper adjoint for complete lattices, but it's not necessarily the case for bounded complete dcpos. So we consider another approximation—strongly linear FS-domains with respect to maps having upper adjoints.We prove that every linear FS-domain is a continuous domain; linear FSdomains are closed under the linear function space L→_l L and the Scott continuous function space [L→L]; the category of linear FS-domains with Scott continuous maps as morphism is cartesian closed; to any strongly linear FS-domain, the Scott topology and the upper topology are equal. And we illustrate the relation among strongly linear FS-domains, linear FS-domains and linear FS-lattices. Any linear FS-lattice is equivalent to a strongly linear FS-domain with a top el- ement. If the top element of a linear FS-lattice is compact, then it becomes a strongly linear FS-domain after removing the top element. The category of linear FS-lattices is a reflective subcategory of strongly linear FS-domains with suitably defined morphism. The category of strongly linear FS-domains with maps having upper adjoints as morphism is isomorphic to some proper subcategory of linear FS-lattices. And we give an example to show that there's a dcpo which is a linear FS-domain but not a strongly linear FS-domain.Furthermore, we define the distributivity of bounded complete dcpos, and obtain several equivalent characterizations for completely distributive bounded complete dcpos as follows: A bounded complete dcpo is completely distributive if and only if any of its principal ideal is a completely distributive lattice if and only if it is isomorphic to some subset of a completely distributive lattice, if and only if the bounded complete dcpo is itself a distributive linear FS-domain.