NORMAL CHARACTERIZATIONS OF LATTICES

Let X be an arbitrary nonempty set and a lattice of subsets of X such that ∅, X ∈ . Let ( ) denote the algebra generated by and I( ) denote those nontrivial, zero-one valued, finitely additive measures on ( ). 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. 2000 Mathematics Subject Classification. 28A12, 28C15.

Let X be an arbitrary nonempty set and ᏸ a lattice of subsets of X such that ∅, X ∈ ᏸ.Let Ꮽ(ᏸ) denote the algebra generated by ᏸ; δ(ᏸ) be the lattice of all countable intersections of sets from ᏸ. Definition 2.1 (lattice terminology).The lattice ᏸ is called: δ-lattice if ᏸ is closed under countable intersections; complement generated if L ∈ ᏸ implies L = ∩L n , n = 1,...,∞, L n ∈ ᏸ (where prime denotes the complement); disjunctive if for x ∈ X and L 1 ∈ ᏸ such that x ∉ L 1 there exists L 2 ∈ ᏸ with x ∈ L 2 and L 1 ∩ L 2 = ∅; separating (or T 1 ) if x, y ∈ X and x ≠ y implies there exists L ∈ ᏸ such that x ∈ L, y ∉ L; T 2 if for x, y ∈ X and x ≠ y there exist L 1 ,L 2 ∈ ᏸ such that x ∈ L 1 , y ∈ L 2 , and L 1 ∩L 2 = ∅; normal if for any L 1 , L 2 ∈ ᏸ with L 1 ∩L 2 = ∅ there exist L 3 ,L 4 ∈ ᏸ with L 1 ⊂ L 3 , L 2 ⊂ L 4 , and L 3 ∩ L 4 = ∅; compact if for any collection {L α } of sets of ᏸ with ∩ α L α = ∅, there exists a finite subcollection with empty intersection; countably compact if for any countable collection {L α } of sets of ᏸ with ∩ α L α = ∅, there exists a finite subcollection with empty intersection; countably paracompact if for any sequence {A n ∈ ᏸ}, A n ↓ ∅, there exists {L n ∈ ᏸ}, such that A n ⊆ L n and L n ↓ ∅; Lindelöf if for any collection {L α } of sets of ᏸ with ∩ α L α = ∅, there exists a countable subcollection with empty intersection.Definition 2.2 (measure terminology).We denote by M(ᏸ) those nonnegative, finite, finitely additive measures on Ꮽ(ᏸ).
that is, countably additive; strongly σ -smooth on ᏸ if and only if for any sequence We denote by M R (ᏸ) the set of ᏸ-regular measures of M(ᏸ); M σ (ᏸ) the set of σ -smooth measures on ᏸ, of M(ᏸ); M σ (ᏸ) the set of σ -smooth measures on Ꮽ(ᏸ) of M(ᏸ); M σ R (ᏸ) the set of ᏸ-regular measures of M σ (ᏸ); and M(σ , ᏸ) the set of strongly σ -smooth measures on ᏸ.
In addition, I(ᏸ), I R (ᏸ), I σ (ᏸ), I σ (ᏸ), I σ R (ᏸ), and I(σ , ᏸ) are the subsets of the corresponding M's which consist of the nontrivial zero-one valued measures.

Definition 2.3 (filters-measures correspondence).
A filter in ᏸ is a subset of ᏸ, F(ᏸ), satisfying the conditions: ∅ ∉ F(ᏸ); F(ᏸ) is closed under finite intersections; A ∈ F(ᏸ), B ∈ ᏸ and A ⊂ B, then B ∈ F(ᏸ).An ultrafilter in ᏸ is a maximal filter (relative to the partial order on the collection of filters in ᏸ given by inclusion).An ᏸ-filter F(ᏸ) is prime if given A, B ∈ ᏸ such that A∪B ∈ F(ᏸ), then either A ∈ F(ᏸ) or

B ∈ F(ᏸ).
There exists a one-to-one correspondence between (i) ᏸ-filters F(ᏸ) and elements of P(ᏸ) defined by π (L) = 1 if and only if L ∈ F(ᏸ); (ii) ᏸ-filters F(ᏸ) with countable intersection property and elements of (iii) prime ᏸ-filters and elements of I(ᏸ) given by: for any µ ∈ I(ᏸ) associate the prime ᏸ-filter F(ᏸ) = {A ∈ ᏸ/µ(A) = 1}; (iv) prime ᏸ-filters with countable intersection property and elements of I σ (ᏸ); (v) ᏸ-ultrafilters and elements of I R (ᏸ) given by: for any ᏸ-ultrafilter F(ᏸ) associate the zero-one valued measure defined on Ꮽ(ᏸ) by: µ(E) = 1 if there exists A ∈ F(ᏸ), A ⊂ E, and µ(E) = 0 if there exists A ∈ F(ᏸ), A ⊂ E ; (vi) ᏸ-ultrafilters with countable intersection property and elements of I σ R (ᏸ).
Definition 2.4 (lattice-measure correspondence).The support of µ ∈ I(ᏸ) is With this notation and in light of the above correspondences, we now note that: for any µ ∈ I(ᏸ), there exists ν ∈ I R (ᏸ) such that µ ≤ ν(ᏸ) (i.e., µ(L) ≤ ν(L) for all L ∈ ᏸ).For any µ ∈ I(ᏸ), there exists ν ∈ I R (ᏸ ) such that µ ≤ (ᏸ ).ᏸ is compact if and only if S(µ) ≠ ∅ for every µ ∈ I R (ᏸ) and ᏸ is countably compact if and only if I R (ᏸ) = I σ R (ᏸ).ᏸ is normal if and only if for each µ ∈ I(ᏸ), there exists a unique ν ∈ I R (ᏸ) such that µ ≤ ν(ᏸ).ᏸ is regular if and only if whenever µ 1 ,µ 2 ∈ I(ᏸ) and For further results and related matters see [2,3,4].

Normal lattices, filters, and outer measures.
In this section, we present a number of theorems on the normality properties of lattices.We consider the interplay between normal lattices, ᏸ-regular or σ -smooth measures, associated outer measures and filters.Many of the results derived in this part are not new.Their inclusion is justified by the necessity of enumerating the known facts in this field and the wish to make the paper self-contained.Theorem 3.1.Let ᏸ be a lattice of subsets of X and let F(ᏸ) be a prime ᏸ-filter.
If G(ᏸ) ⊂ H(ᏸ), where H(ᏸ) is an ᏸ-filter and if L ∈ H(ᏸ), then for any (b) Now let µ ∈ I(ᏸ) corresponding to F(ᏸ) and suppose µ ≤ ν 1 ,ν 2 (ᏸ) where ν 1 ,ν 2 ∈ I R (ᏸ).Corresponding to ν 1 and ν 2 we have the ᏸ-ultrafilters F 1 (ᏸ) and F 2 (ᏸ), with Theorem 3.2.Let ᏸ be a lattice of subsets of X and define Let π ∈ J(ᏸ) and define Proof.Clearly, F(ᏸ) is an ᏸ-filter, not a prime ᏸ-filter.The proof of the fact that G(ᏸ) is a prime ᏸ-filter follows the corresponding proof of Theorem 3.1 since we only need to show L 3 ∩ L 4 ≠ ∅.Definition 3.3 (associated outer measures).For µ ∈ I(ᏸ) and E ⊂ X define Then µ is a finitely subadditive outer measure and µ is a countably additive outer measure.
Furthermore, if Various lattice topological properties have been characterized in terms of the outer measures µ and µ .We note here Theorem 3.4 without proof (see [7]).
Proof.Let L n ↓ ∅, L n ∈ ᏸ and suppose ν(L n ) = 1 for all n.Since ν ∈ I R (ᏸ), there exist Ln ⊂ L n , Ln ∈ ᏸ, ν( Ln ) = 1 for all n and Ln ↓ ∅.Since ᏸ is normal, therefore there exist A n , B n such that L n ⊂ A n , Ln ⊂ B n , A n ∩ B n = ∅ and A n , B n ∈ ᏸ.Consider the sequence of inclusions Ln ⊂ B n ⊂ A n ⊂ L n .Since L n ↓ ∅, we may assume, with no loss of generality, that Ln , B n , and Next, let ᏸ be also countably paracompact and let ν ∈ I σ (ᏸ ), Consider now the following theorem which is similar to Theorem 3.6 in the case of strongly σ -smoothness.Theorem 3.7.Let ᏸ be a δ normal lattice and let µ ∈ I(σ , ᏸ) and ν ∈ I R (ᏸ) such that µ ≤ ν(ᏸ).Then ν ∈ I(σ , ᏸ ).
Proof.Let L n ↓ L , L n , L ∈ ᏸ and suppose that ν(L n ) = 1 for all n but ν(L ) = 0.Then, since ᏸ is normal, it follows that there exist A n ∈ ᏸ, A n ⊂ L n , and µ(A n ) = 1 (see [8]).

Now we may assume that
The next theorem relates the notions of normal lattice, outer measure and filter.
Theorem 3.8.The lattice ᏸ is normal if and only if for each µ ∈ I(ᏸ), µ determines a prime ᏸ-filter.
The following theorem on the equality of two outer measures µ and µ depending on the normality of ᏸ is well known (see [7]) and we just state it without proof.Theorem 3.9.Let µ ∈ I σ (ᏸ), then µ = µ (ᏸ) if one of the following conditions is satisfied: (a) ᏸ is normal and δ-lattice; or (b) ᏸ is normal and countably paracompact.

Application to the Wallman spaces.
Next, we briefly summarize some facts (see [4]) about the Wallman space I σ R (ᏸ) and then proceed to consider some relations between the measures and the induced measures on the Wallman spaces.A few of the properties to be considered have been investigated in [5], we give slightly different proofs, and include some of them for completeness.Specific characterizations are given for the various Wallman spaces associated with the considered zero-one valued measures concerning normality and related questions studied in the previous section.
as a base for the closed sets in I σ R (ᏸ) and then I σ R (ᏸ) is called the general Wallman space associated with X and ᏸ.Assuming ᏸ is disjunctive, W σ (ᏸ) = {W σ (L) | L ∈ ᏸ} is a lattice in I σ R (ᏸ), isomorphic to ᏸ under the map L → W σ (L), L ∈ ᏸ.W σ (ᏸ) is replete and a base for the closed sets tW σ (ᏸ), all arbitrary intersections of sets of W σ (ᏸ).It is this topological space [I σ R (ᏸ), tW σ (ᏸ)] and lattice W σ (ᏸ) which we will consider here and in subsequent sections.

Definition 5 . 1 (
general Wallman spaces and Wallman topologies).The Wallman topology in I σ R (ᏸ) is obtained by taking all