## 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

### Citations:

Zbl 0512.00011; Zbl 0073.269