On normal and strongly normal lattices (Q1210462)

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 \(I({\mathcal L})=\overline I({\mathcal L})\).