Normal lattices and coseparation of lattices

Abstract

Let X be an arbitrary non-empty set, and let ℒ be a lattice of subsets of X such that ∅, X∈ℒ. We first summarize a number of known conditions which are equivalent to ℒ being normal. We then develop new equivalent conditions in terms of set functions associated with μ∈I(ℒ), the set of all non-trivial, zero-one valued finitely additive measures on the algebra generated-by ℒ′. We finally generalize all the above to the situation where ℒ1 and ℒ2 are a pair of lattices of subsets of X with ℒ′1⊂ℒ2, and where we obtain equivalent conditions for ℒ1 to coseparate ℒ2.