Some Properties Of Generalized Countably Approximating Posets

One of important aspects of domain theory is to carry the theory of contin-uous lattices to that of posets as much as possible. Continuous posets,countably approximating posets and quasicontinuous posets are some kinds of generalization of continuous lattices.As quasicontinuous posets is a kind of generalization of contin-uous posets,generalized countably approximating posets is a kind of generalization of countably approximating posets. In this thesis, we further discuss some proper-ties of generalized countably approximating posets.The main results in this thesis are following. We give the definition of general-ized σ-ideals on countably directed complete posets by using finite subsets on posets and prove that a poset P with countably directed joins is a generalized countably approximating poset if and only if the mapping induced by the generalized σ-ideals on P has a lower adjoint, and the finitely generated upper sets on P is countably approximating poset under the order of reverse inclusion. We define a minimal set of a generalized countably directed and prove that each element of a generalized countably approximating posets has a a minimal set of generalized countably di-rected and we give some equivalent characterizations of the mapping of preserving generalized countably directional minimal set. Basing on these results, we obtain two expansions theorem on generalized countably approximating posets.