The lattice of completions of an ordered set
type: Item
instance of: scholarly article
title: The lattice of completions of an ordered set
zbMATH Open document ID: 0888.06003
DOI: 10.1023/A:1005805026315
author: J. B. Nation
author: Alex Pogel
published in: Order
publication date: 1998-05-13T00:00:00Z
review text: The join dense completions of an ordered set \(P\) form a complete lattice \(K(P)\). Its least element is \(\mathcal O(P)\), the lattice of all order ideals of \(P\), and its greatest element is \(\mathcal M(P)\), the Dedekind-MacNeille completion of \(P\). It is isomorphic to an ideal of the lattice of all closure operators on \(\mathcal O(P)\). Some local structural properties of lattices of closure operators on complete lattices are proved. These are inherited by \(K(P)\) as well. In particular, if \(K(P)\) is finite, then it is an upper semimodular lattice and an upper bounded homomorphic image of a free lattice, and hence meet semidistributive.
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 06A15
zbMATH DE Number: 1067849
zbMATH Keywords: join dense completion
zbMATH Keywords: closure operator
zbMATH Keywords: order ideal
reviewed by: Josef Niederle
MaRDI profile type: Publication
full work available at URL: a:1005805026315
OpenAlex ID: W160182017
description: scientific article; zbMATH DE number 1067849