θ-continuity and Dθ-completion of posets

We introduce a new concept of continuity of posets, called θ-continuity. Topological characterizations of θ-continuous posets are put forward. We also present two types of dcpo-completion of posets which are Dθ-completion and Ds2-completion. Connections between these notions of continuity and dcpo-completions of posets are investigated. The main results are (1) a poset P is θ-continuous iff its θ-topology lattice is completely distributive iff it is a quasi θ-continuous and meet θ-continuous poset iff its Dθ-completion is a domain; (2) the Dθ-completion of a poset B is isomorphic to a domain L iff B is a θ-embedded basis of L; (3) if a poset P is θ-continuous, then the Dθ-completion Dθ(P) is isomorphic to the round ideal completion RI(P, ≪θ).

The Injective Hull and the -Compactification of a Continuous Poset

In [57] (2.12), D. S. Scott showed that the continuous lattices, invented by him in his study of a mathematical theory of computation [56], are precisely (when they are made into topological spaces via the Scott topology) the injective T0-spaces, i.e., the injective objects in the category T0 of T0-spaces and continuous maps with regard to the class of all embeddings. Moreover, the sort of morphisms between continuous lattices Scott considered are precisely the continuous maps with regard to the respective Scott topologies. These are fairly non-Hausdorff topologies. (Indeed, the Scott topology induces the partial order of the lattice L via x ≦ y if and only if x ∊ cl{j}, the “specialization order” of the topology; hence L is Hausdorff in the Scott topology if and only if L has at most one element.) In topological algebra, compact Lawson semilattices (= compact Hausdorff topological ∧-semilattices such that the ∧-preserving continuous maps into the unit interval, with its ordinary topology and the min-semilattice structure, separate the points) with a unit element 1 have attracted considerable interest.