Erné, M. Adjunctions and standard constructions for partially ordered sets. (English) Zbl 0533.06001 Proc. Klagenfurt Conf. 1982, Contrib. Gen. Algebra 2, 77-106 (1983). [For the entire collection see Zbl 0512.00011.] This nice paper deals with functions \(Z\) assigning to each poset \(P\) a system \(Z(P)\) of lower ends which contains at least all principal ideals of \(P\). It develops an idea of a standard extension of B. Banaschewski [see Z. Math. Logik Grundlagen Math. 2, 117–130 (1956; Zbl 0073.269)]. Main examples are lower ends, Frink ideals, cuts (normal ideals) and Scott-closed sets. Among others, the author studies different maps between posets depending on \(Z\) and generalizes the passage from finitely generated lower ends to Frink ideals by assigning to \(Z\) an “opposite” \(\tilde Z\). The main result is an adjunction theorem which subsumes many known and unknown universal constructions for posets. Reviewer: J.Rosický Cited in 11 Documents MSC: 06A06 Partial orders, general 06B23 Complete lattices, completions Keywords:subset system; Z-complete posets; Z-continuous maps; monad; completion of posets; lower ends; principal ideals; standard extension; Frink ideals; cuts; adjunction theorem Citations:Zbl 0512.00011; Zbl 0073.269 PDFBibTeX XML