\(\Delta_1\)-completions of a poset.
type: Item
instance of: scholarly article
title: \(\Delta_1\)-completions of a poset.
zbMATH Open document ID: 1317.06002
author: Mai Gehrke
author: Ramon Jansana
author: Alessandra Palmigiano
published in: Order
publication date: 2013-03-20T00:00:00Z
review text: 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.
reviewed by: Vinayak Joshi
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 06B23
zbMATH DE Number: 6147394
zbMATH Keywords: completions of posets
zbMATH Keywords: canonical extensions
zbMATH Keywords: closure systems
zbMATH Keywords: filters
zbMATH Keywords: ideals
zbMATH Keywords: MacNeille completion
zbMATH Keywords: compact completions
MaRDI profile type: Publication
full work available at URL: s11083-011-9226-0
OpenAlex ID: W2098708076
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DOI: 10.1007/S11083-011-9226-0
description: scientific article; zbMATH DE number 6147394