The Duality Theory of General Z-continuous Posets

In this paper, we research further into Z -predistributive and Z -precontinuous posets introduced by Erne. We focus on duality theorems based on the application of Galois connections whenever Z is a closed subset selection. For example, there is a duality between the categories Z - PD G and Z - PD D of all Z -predistributive posets with weakly Z △ -continuous maps which have a lower adjoint, and maps preserve Z -below relation that have an upper adjoint, respectively, as morphisms. We introduce the concept of Z 0 -approximating auxiliary relation, and have made a slight improvement on Z -precontinuity, so that there is a generalization of the classical equivalence between domains and auxiliary relations.