Join-independent and meet-independent sets in complete lattices
type: Item
instance of: scholarly article
title: Join-independent and meet-independent sets in complete lattices
zbMATH Open document ID: 1009.06005
DOI: 10.1023/A:1012265912445
published in: Order
publication date: 2002-08-21T00:00:00Z
review text: Let \(L\) be a complete lattice. A subset \(A\) of \(L\) is called join-independent if \(a\not \leq\sup A\setminus \{a\}\) for all \(a\in A\) (a meet-independent set is defined dually). For a set \(X\subseteq L\), let \(\psi (X)\) denote the supremum of \(X\) in \(L\). It is proved that \(A\subseteq L\) is join-independent if and only if for every \(x\in A\) and for every \(X\subseteq L\) we have \(x\in X\) whenever \(x\leq\sup X\), and it is equivalent to the fact that \(\psi\) is an order isomorphism and a complete join-embedding from the poset of all subsets of \(A\) into \(L\). Several properties of join-independent sets are given. Connections to closure spaces and incidence structures are discussed.
reviewed by: Václav Koubek
Mathematics Subject Classification ID: 06B23
zbMATH DE Number: 1785646
zbMATH Keywords: complete lattice
zbMATH Keywords: independent set
author: Frantisek Machala
MaRDI profile type: Publication
full work available at URL: a:1012265912445
OpenAlex ID: W86821443
description: scientific article; zbMATH DE number 1785646