θ-continuity and Dθ-completion of posets

We introduce a new concept of continuity of posets, called theta-continuity. Topological characterizations of theta-continuous posets are put forward. We also present two types of dcpo-completion of posets which are D-theta-completion and D-s2-completion. Connections between these notions of continuity and dcpo-completions of posets are investigated. The main results are (1) a poset P is theta-continuous iff its theta-topology lattice is completely distributive iff it is a quasi theta-continuous and meet theta-continuous poset iff its D-theta-completion is a domain; (2) the D-theta-completion of a poset B is isomorphic to a domain L iff B is a theta-embedded basis of L; (3) if a poset P is theta-continuous, then the D-theta-completion D-theta(P) is isomorphic to the round ideal completion RI(P, <<(theta)).