The Duality Theory of General Z-continuous Posets

In this paper, we research further into Z-predistributive and Z-precontinuous posets introduced by Erné. 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-PDG and Z-PDD 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 Z0-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.