Abstract

Let X be an arbitrary nonempty set and ℒ a lattice of subsets of X such that ∅, X∈ℒ. Let 𝒜(ℒ) denote the algebra generated by ℒ and I(ℒ) denote those nontrivial, zero-one valued, finitely additive measures on 𝒜(ℒ). In this paper, 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 σ-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.