Normal characterizations of lattices (Q1607779)

Summary: Let \(\mathbf{X}\) be an arbitrary nonempty set and \(\mathcal{L}\) a lattice of subsets of \(\mathbf{X}\) such that \(\emptyset\), \(\mathbf{X} \in \mathcal{L}\). Let \(\mathcal{A(L)}\) denote the algebra generated by \(\mathcal{L}\) and \(\mathbf{I} (\mathcal{L})\) denote those nontrivial, zero-one valued, finitely additive measures on \(\mathcal{A(L)}\). We discuss some of the normal characterizations of lattices in terms of the associated lattice regular measures, filters and outer measures. We consider the interplay between normal lattices, regularity or \(\sigma\)-smoothness properties of measures, lattice topological properties and filter correspondence. Finally, we start a study of slightly, mildly and strongly normal lattices and express then some of these results in terms of the generalized Wallman spaces.