On normal and strongly normal lattices
type: Item
instance of: scholarly article
title: On normal and strongly normal lattices
zbMATH Open document ID: 0777.28003
DOI: 10.1155/S0161171293000407
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1993-08-25T00:00:00Z
full work available at URL: https://eudml.org/doc/46881
review text: Let \(X\) be a non-empty set, let \(\mathcal L\) be a lattice in \((2^ X,\subset)\) containing \(\emptyset\) and \(X\), let \(A({\mathcal L})\) be the Boolean subalgebra of \(2^ X\) generated by \(\mathcal L\) and let \(I({\mathcal L})\) be the set of all non-trivial zero-one valued finitely additive set functions on \(A({\mathcal L})\). A set function \(\pi: {\mathcal L}\to\{0,1\}\) is called a premeasure on \(\mathcal L\) if it is increasing, multiplicative and \(\pi(X)=1\). The set of all premeasures on \(\mathcal L\) will be denoted by \(\Pi({\mathcal L})\). Write \(\overline I({\mathcal L})=\{\pi\in\Pi({\mathcal L})\): if \(L_ 1,L_ 2\in{\mathcal L}\) and \(L_ 1\cup L_ 2=X\), then \(\pi(L_ 1)=1\) or \(\pi(L_ 2)=1\}\). We say that a) \({\mathcal L}\) is a normal lattice if, for every \(\mu\in I({\mathcal L})\) there exists a unique \({\mathcal L}\)-regular element \(\nu\) of \(I({\mathcal L})\) such that \(\mu\leq \nu\) on \(\mathcal L\). b) \(\mathcal L\) is a strongly normal lattice if, for every \(\mu,\mu_ 1,\mu_ 2\in I({\mathcal L})\) with \(\mu\leq \mu_ 1\) and \(\mu\leq \mu_ 2\) on \(\mathcal L\), then \(\mu_ 1\leq\mu_ 2\) or \(\mu_ 2\leq \mu_ 1\) on \(\mathcal L\). Then the main results of the paper under review can be stated as follows: Theorem A. \(\mathcal L\) is a normal lattice if and only if the set \(\{L\in{\mathcal L}: L\cap A\neq\emptyset\) for all \(A\in{\mathcal L}\) such that \(\pi(A)=1\) for all \(\pi\in\Pi({\mathcal L})\}\) is a \({\mathcal L}\)-ultrafilter. Theorem B. \(\mathcal L\) is a strongly normal lattice if and only if \(I({\mathcal L})=\overline I({\mathcal L})\).
Mathematics Subject Classification ID: 28A60
Mathematics Subject Classification ID: 28C15
Mathematics Subject Classification ID: 06B30
zbMATH DE Number: 179253
zbMATH Keywords: Lindelöf lattice
zbMATH Keywords: zero-one valued finitely additive set functions
zbMATH Keywords: premeasure
zbMATH Keywords: normal lattice
zbMATH Keywords: strongly normal lattice
author: El-Bachir Yallaoui
reviewed by: Pedro Morales
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 179253