Normal lattices and coseparation of lattices

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 ℒ 1 and ℒ 2 are a pair of lattices of subsets of X with ℒ ′ 1 ⊂ ℒ 2 , and where we obtain equivalent conditions for ℒ 1 to coseparate ℒ 2 .