\(\Delta_1\)-completions of a poset. (English) Zbl 1317.06002

A \(\Delta_1\)-completion of a poset is a completion for which, simultaneously, each element is obtainable as a join of meets of elements of the original poset and as a meet of joins of elements from the original poset. In this paper, the authors completely classify \(\Delta_1\)-completions of a poset \(P\) in terms of certain polarities \((\mathcal F,\mathcal I,R)\) where \(\mathcal F\) is a closure system of up-sets of \(P\) and \(\mathcal I\) is a closure system of down-sets of \(P\) and \(R\) is a relation from \(\mathcal F\) to \(\mathcal I\) satisfying four conditions. The relation essentially specifies which meets of down-sets are below which joins of up-sets in the completion. Further, the authors prove that the compact \(\Delta_1\)-completions may be described just by a collection of filters and a collection of ideals, taken as parameters. The compact \(\Delta_1\)-completions of a poset include its MacNeille completion and all its join- and all its meet-completions. These completions also include the canonical extension of the given poset, a completion that encodes the topological dual of the poset when it has one. Finally, the authors use parametric description of \(\Delta_1\)-completions to compare the canonical extension to other compact \(\Delta_1\)-completions identifying its relative merits.


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