Ideals of quasi-ordered sets. (Q1771903)

For a given collection of subsets \(\mathcal A\) of a quasi-ordered set or a poset \(Z\) the author defines the collection \(\mathcal I(\mathcal A)\) of \(\mathcal A\)-ideals of \(Z\). They form a closure system, and this leads to the definition of \(\mathcal A\)-ideal continuity of functions. The author shows the existence of a choice-free \(\mathcal A\)-ideal continuous imbedding of a poset into a \({\mathcal B}\)-join complete poset with an appropriate universal mapping property. Topological applications include the imbedding of Scott spaces and Alexandrov spaces into up-complete Scott spaces.