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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

MSC:

06A06 Partial orders, general
06B23 Complete lattices, completions