IJMMS 28:10 (2001) 561–570
PII. S0161171201007256
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
NORMAL CHARACTERIZATIONS OF LATTICES
CARMEN D. VLAD
(Received 30 March 2001)
Abstract. 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 and I(ᏸ) denote those nontrivial,
zero-one valued, finitely additive measures on Ꮽ(ᏸ). 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.
1. Introduction. This paper presents a systematic study of various concepts pertaining to normal lattices. Normal lattices are an important class of lattices which have
been investigated both in an abstract setting and in a point-set framework by others
(see [1, 4, 8, 9]).
In Section 2, we give notations and definitions which is fairly standard (see [1, 2,
3, 10]) and a brief background consisting of several results pertaining to lattices,
measures, and filters.
In Section 3, we study normality properties of lattices in the point-set framework;
most of this material can be readily extended to general lattices. We discuss some of
the implications between normality properties and the associated lattice regular measures; in the case of zero-one valued lattice regular measures, these properties have
associated filter and topological consequences. Also, by choosing normal lattices in a
topological setting, we present interesting results on the associated outer measures
on both ᏸ and ᏸ .
In Section 4, we start a study of slightly, mildly, and strongly normal lattices and
we investigate them from a measure theoretic point of view and thereby extend some
results given in [6, 9].
This work is then carried out in Section 5 where we express some of these results in
terms of generalized Wallman spaces. Specific characteristic on regularity and normality are given for the various Wallman spaces associated with the considered measures
σ
of Iσ (ᏸ), IR
(ᏸ), and I(σ , ᏸ).
2. Background and notation. In this section, we introduce the notation and terminology that will be used throughout the paper. All is fairly standard and we include it
for the reader’s convenience.
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CARMEN D. VLAD
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 = ∩ Ln , n =
1, . . . , ∞, Ln ∈ ᏸ (where prime denotes the complement); disjunctive if for x ∈ X and
L1 ∈ ᏸ such that x ∉ L1 there exists L2 ∈ ᏸ with x ∈ L2 and L1 ∩ L2 = ∅; separating
(or T1 ) if x, y ∈ X and x ≠ y implies there exists L ∈ ᏸ such that x ∈ L, y ∉ L; T2 if
for x, y ∈ X and x ≠ y there exist L1 , L2 ∈ ᏸ such that x ∈ L1 , y ∈ L2 , and L1 ∩L2 = ∅;
normal if for any L1 , L2 ∈ ᏸ with L1 ∩L2 = ∅ there exist L3 , L4 ∈ ᏸ with L1 ⊂ L3 , L2 ⊂ L4 ,
and L3 ∩ L4 = ∅; compact if for any collection {Lα } of sets of ᏸ with ∩α Lα = ∅, there
exists a finite subcollection with empty intersection; countably compact if for any
countable collection {Lα } of sets of ᏸ with ∩α Lα = ∅, there exists a finite subcollection
with empty intersection; countably paracompact if for any sequence {An ∈ ᏸ}, An ↓ ∅,
there exists {Ln ∈ ᏸ}, such that An ⊆ Ln and Ln ↓ ∅; Lindelöf if for any collection
{Lα } of sets of ᏸ with ∩α Lα = ∅, 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). A measure µ ∈ M(ᏸ) is called: σ -smooth
on ᏸ if for all sequences {Ln } of sets of ᏸ with Ln ↓ ∅, µ(Ln ) → 0; σ -smooth on Ꮽ(ᏸ)
if for all sequences {An } of sets of Ꮽ(ᏸ) with An ↓ ∅, µ(An ) → 0, that is, countably
additive; strongly σ -smooth on ᏸ if and only if for any sequence {Ln ∈ ᏸ}Ln ↓ L
where L ∈ ᏸ, then µ(L) = inf µ(Ln ) = lim µ(Ln ); ᏸ-regular if for any A ∈ Ꮽ(ᏸ), µ(A) =
sup{µ(L) | L ⊂ A, L ∈ ᏸ}.
We denote by MR (ᏸ) the set of ᏸ-regular measures of M(ᏸ); Mσ (ᏸ) the set of
σ -smooth measures on ᏸ, of M(ᏸ); Mσ (ᏸ) the set of σ -smooth measures on Ꮽ(ᏸ) of
σ
M(ᏸ); MR
(ᏸ) the set of ᏸ-regular measures of Mσ (ᏸ); and M(σ , ᏸ) the set of strongly
σ -smooth measures on ᏸ.
σ
In addition, I(ᏸ), IR (ᏸ), Iσ (ᏸ), Iσ (ᏸ), IR
(ᏸ), and I(σ , ᏸ) are the subsets of the
corresponding M’s which consist of the nontrivial zero-one valued measures.
Finally, P(ᏸ) = {π , defined on ᏸ, nontrivial, monotone and π (A ∩ B) = π (A)π (B),
A, B, ∈ ᏸ} is the set of all premeasures on ᏸ; Pσ (ᏸ) is the set of all premeasures on ᏸ
which are σ -smooth on ᏸ.
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 Pσ (ᏸ) =
{π ∈ P(ᏸ) | if π (Ln ) = 1 forall n where Ln ∈ ᏸ, then ∩ Ln ≠ ∅};
NORMAL CHARACTERIZATIONS OF LATTICES
563
(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 Iσ (ᏸ);
(v) ᏸ-ultrafilters and elements of IR (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 ;
σ
(ᏸ).
(vi) ᏸ-ultrafilters with countable intersection property and elements of IR
Definition 2.4 (lattice-measure correspondence). The support of µ ∈ I(ᏸ) is
S(µ) = ∩{L ∈ ᏸ | µ(L) = 1}.
With this notation and in light of the above correspondences, we now note that:
for any µ ∈ I(ᏸ), there exists ν ∈ IR (ᏸ) such that µ ≤ ν(ᏸ) (i.e., µ(L) ≤ ν(L) for all
L ∈ ᏸ). For any µ ∈ I(ᏸ), there exists ν ∈ IR (ᏸ ) such that µ ≤ (ᏸ ). ᏸ is compact
if and only if S(µ) ≠ ∅ for every µ ∈ IR (ᏸ) and ᏸ is countably compact if and only
σ
if IR (ᏸ) = IR
(ᏸ). ᏸ is normal if and only if for each µ ∈ I(ᏸ), there exists a unique
ν ∈ IR (ᏸ) such that µ ≤ ν(ᏸ). ᏸ is regular if and only if whenever µ1 , µ2 ∈ I(ᏸ) and
σ
µ1 ≤ µ2 (ᏸ), then S(µ1 ) = S(µ2 ). ᏸ is replete if and only if for any µ ∈ IR
(ᏸ), S(µ) ≠ ∅.
ᏸ is prime-complete if and only if for any µ ∈ Iσ (ᏸ), S(µ) ≠ ∅. ᏸ is Lindelöf if and
only if for any π ∈ Pσ (ᏸ), S(π ) ≠ ∅. Finally, if µx is the measure concentrated at
x ∈ X then µx ∈ IR (ᏸ), for all x ∈ X if and only if ᏸ is disjunctive.
For further results and related matters see [2, 3, 4].
3. 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.
Define G(ᏸ) = {L ∈ ᏸ | L ∩ A ≠ ∅ for all A ∈ F(ᏸ)}. Then G(ᏸ) is a prime ᏸ-filter if
and only if ᏸ is normal (actually G(ᏸ) is an ᏸ-ultrafilter).
Proof. (a) Suppose that ᏸ is normal. Let L ∈ G(ᏸ), B ∈ ᏸ and L ⊂ B. For any
A ∈ F(ᏸ), ∅ ≠ L ∩ A ⊂ B ∩ A, so G(ᏸ) is closed under supersets. Let L1 , L2 ∈ G(ᏸ)
such that L1 ∩ L2 ∉ G(ᏸ); then (L1 ∩ A) ∩ (L2 ∩ A) = ∅ for some A ∈ F(ᏸ). Since ᏸ is
normal, there exist L3 , L4 ∈ ᏸ, L1 ∩ A ⊂ L3 , L2 ∩ A ⊂ L4 , and L3 ∩ L4 = (L3 ∪ L4 ) = ∅,
that is, L3 ∪ L4 = X.
Suppose that L3 ∈ F(ᏸ); then L1 ∩A∩L3 = ∅, with A, L3 , A∩L3 ∈ F(ᏸ); contradiction,
since L1 ∈ G(ᏸ). Similarly, for L4 ∈ F(ᏸ). Now, suppose that L1 , L2 ∈ ᏸ and L1 ∪ L2 ∈
G(ᏸ) but L1 , L2 ∉ G(ᏸ). Then there exist A, B ∈ F(ᏸ) with L1 ∩ A = ∅, L2 ∩ B = ∅, and
(L1 ∪ L2 ) ∩ (A ∩ B) = ∅, contradiction. It follows that G(ᏸ) is a prime ᏸ-filter.
If G(ᏸ) ⊂ H(ᏸ), where H(ᏸ) is an ᏸ-filter and if L ∈ H(ᏸ), then for any A ∈ F(ᏸ) ⊂
G(ᏸ) ⊂ H(ᏸ), L ∩ A ≠ ∅, therefore G(ᏸ) is an ᏸ-ultrafilter.
(b) Now let µ ∈ I(ᏸ) corresponding to F(ᏸ) and suppose µ ≤ ν1 , ν2 (ᏸ) where ν1 , ν2 ∈
IR (ᏸ). Corresponding to ν1 and ν2 we have the ᏸ-ultrafilters F1 (ᏸ) and F2 (ᏸ), with
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CARMEN D. VLAD
F(ᏸ) ⊂ F1 (ᏸ), F2 (ᏸ) ⊂ G(ᏸ), hence F1 (ᏸ) = F2 (ᏸ), = G(ᏸ), that is, ν ∈ IR (ᏸ) must be
unique, therefore ᏸ is normal.
Theorem 3.2. Let ᏸ be a lattice of subsets of X and define
J(ᏸ) = π ∈ P(ᏸ) | if X = L1 ∪ L2 , then π L1 = 1 or π L2 = 1, L1 , L2 ∈ ᏸ .
(3.1)
Let π ∈ J(ᏸ) and define
F(ᏸ) = A ∈ ᏸ | π (A) = 1 ,
G(ᏸ) = L ∈ ᏸ | L ∩ A ≠ ∅ ∀A ∈ F(ᏸ) .
(3.2)
If ᏸ is normal then G(ᏸ) is an ᏸ-ultrafilter.
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 L3 ∩ L4 ≠ ∅.
Definition 3.3 (associated outer measures). For µ ∈ I(ᏸ) and E ⊂ X define
µ (E) = inf µ L | E ⊂ L , L ∈ ᏸ .
(3.3)
For µ ∈ Iσ (ᏸ) and E ⊂ X define
µ Li | E ⊂ ∪i Li , Li ∈ ᏸ .
µ (E) = inf
(3.4)
i
Then µ is a finitely subadditive outer measure and µ is a countably additive outer
measure.
Furthermore, if µ ∈ Iσ (ᏸ) then
(a) µ ≤ µ ≤ µ (ᏸ).
(b) µ ≤ µ = µ(ᏸ ).
σ
(ᏸ) then µ = µ = µ (ᏸ) and µ = µ = µ(ᏸ ).
(c) If µ ∈ IR
(d) µ ∈ IR (ᏸ) if and only if µ = µ (ᏸ).
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]).
Theorem 3.4. The lattice ᏸ is normal if and only if for ν ∈ I(ᏸ) and µ ∈ IR (ᏸ) with
ν ≤ µ(ᏸ), then µ = ν (ᏸ).
Theorem 3.5. Let µ ∈ IR (ᏸ) and ν ∈ IR (ᏸ ) such that µ ≤ ν(ᏸ ). Then ᏸ is normal
if and only if µ = ν (ᏸ) for all such µ and ν.
Proof. Suppose µ ≤ ν(ᏸ ). Then ν ≤ µ(ᏸ) and since µ ∈ IR (ᏸ) it follows that
µ = µ (ᏸ). Then ν ≤ µ = µ ≤ ν (ᏸ). Suppose ᏸ is normal. Let A ∈ ᏸ with µ(A) = 0.
Since µ ∈ IR (ᏸ), there exists L ⊂ A , L ∈ ᏸ with µ(L) = 1. But A ∩ L = ∅ implies
that there exist C , D such that A ⊂ C , L ⊂ D , C ∩ D = ∅, C, D ∈ ᏸ. Then we
have A ⊂ C ⊂ D ⊂ L and ν(C ) ≤ ν(D) ≤ µ(D) ≤ µ(L ) = 0. Then ν (A) = 0 and
since A was arbitrary in ᏸ, µ = ν (ᏸ). Conversely, suppose that µ = ν (ᏸ) and let
µ ∈ I(ᏸ), µ1 , µ2 ∈ IR (ᏸ) with µ ≤ µ1 , µ2 (ᏸ). But µ ≤ ν ∈ IR (ᏸ ) on ᏸ , so we have
ν ≤ µ ≤ µ1 , µ2 (ᏸ). By the assumption, ν = µ1 = µ2 (ᏸ) and therefore we get that
µ1 = µ1 = ν = µ2 = µ2 , that is, ᏸ is normal.
NORMAL CHARACTERIZATIONS OF LATTICES
565
Theorem 3.6. Let ᏸ be a normal lattice and let µ ∈ Iσ (ᏸ) and ν ∈ IR (ᏸ) such that
σ
µ ≤ ν(ᏸ). Then ν ∈ Iσ (ᏸ ). If ᏸ is also countably paracompact then ν ∈ IR
(ᏸ).
Proof. Let Ln ↓ ∅, Ln ∈ ᏸ and suppose ν(Ln ) = 1 for all n. Since ν ∈ IR (ᏸ), there
exist L̃n ⊂ Ln , L̃n ∈ ᏸ, ν(L̃n ) = 1 for all n and L̃n ↓ ∅. Since ᏸ is normal, therefore there
exist An , Bn such that Ln ⊂ An , L̃n ⊂ Bn , An ∩ Bn = ∅ and An , Bn ∈ ᏸ. Consider the
sequence of inclusions L̃n ⊂ Bn ⊂ An ⊂ Ln . Since Ln ↓ ∅, we may assume, with no
loss of generality, that L̃n , Bn , and An ↓ ∅. µ ∈ Iσ (ᏸ) implies µ(Bn ) → 0. But µ(Bn ) ≥
ν(Bn ) = 1, contradiction. Next, let ᏸ be also countably paracompact and let ν ∈ Iσ (ᏸ ),
Bn ∈ ᏸ, Bn ↓ ∅. There exist An ∈ ᏸ, Bn ⊂ An ↓ ∅, and ν(Bn ) ≤ ν(An ) → 0. Therefore
σ
(ᏸ).
ν ∈ IR
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 ∈ IR (ᏸ) such
that µ ≤ ν(ᏸ). Then ν ∈ I(σ , ᏸ ).
Proof. Let Ln ↓ L , Ln , L ∈ ᏸ and suppose that ν(Ln ) = 1 for all n but ν(L ) = 0.
Then, since ᏸ is normal, it follows that there exist An ∈ ᏸ, An ⊂ Ln , and µ(An ) = 1
(see [8]).
Now we may assume that An ↓ A ∈ ᏸ, since ᏸ is a δ-lattice and µ(A) = 1 since
µ ∈ I(σ , ᏸ). Then ν(A) = 1, but A ⊂ L , contradiction.
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.
Proof. Let F(ᏸ) = {L ∈ ᏸ | µ (L) = 1}. Suppose that ᏸ is normal. Clearly, ∅ ∉ F(ᏸ).
Let L ∈ F(ᏸ), B ∈ ᏸ with L ⊂ B; then 1 = µ (L) ≤ µ (B), so B ∈ F(ᏸ). Let L1 , L2 ∈ F(ᏸ)
but L1 ∩ L2 ∉ F(ᏸ). Then µ (L1 ∩ L2 ) = 0, that is, there exists L ⊃ L1 ∩ L2 , L ∈ ᏸ with
µ(L ) = 0. We have L ⊂ L1 ∪L2 and since ᏸ is normal it follows that L = L3 ∪L4 , L3 ⊂ L1 ,
and L4 ⊂ L2 . Hence µ(L3 ) = 0 or µ(L4 ) = 0, contradiction. Now let L1 , L2 ∈ ᏸ such that
L1 ∪ L2 ∈ F(ᏸ) but L1 , L2 ∉ F(ᏸ), that is, µ (L1 ∪ L2 ) = 1 but µ(L1 ) = µ(L2 ) = 0. We get
a contradiction, since µ (L1 ∪ L2 ) ≤ µ(L1 ) + µ(L2 ).
Conversely, suppose that µ determines a prime ᏸ-filter, F(ᏸ). If ᏸ is not normal,
then there exist µ ∈ I(ᏸ), µ1 , µ2 ∈ IR (ᏸ), µ1 ≠ µ2 and µ ≤ µ1 , µ2 ≤ µ (ᏸ). To µ1 and µ2
correspond the ᏸ-ultrafilters G(ᏸ) and H(ᏸ) and by the above we have G(ᏸ), H(ᏸ) ⊂
F(ᏸ); therefore G(ᏸ) = H(ᏸ) = F(ᏸ), which implies µ1 = µ2 which is a contradiction.
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 ∈ Iσ (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.
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CARMEN D. VLAD
Remark 3.10. Note that if Theorem 3.9(b) is true, then by Theorem 3.6, for µ ∈
σ
Iσ (ᏸ) and ν ∈ IR (ᏸ) such that µ ≤ ν(ᏸ) it follows that ν ∈ IR
(ᏸ) and then by Definition
3.3(c) we have ν = ν = ν (ᏸ). Therefore in this case, for both µ ∈ Iσ (ᏸ) and the dominating ν ∈ IR (ᏸ) we have the equality of µ , µ and ν , ν , respectively.
4. Normal, slightly normal, mildly normal, and strongly normal lattices. In this
section, we summarize some known results (see [6, 9]) on normal, slightly normal,
mildly normal, and strongly normal lattices and give slightly different proofs. First,
we recall some definitions.
Definition 4.1. The lattice ᏸ is slightly normal if for all µ ∈ Iσ (ᏸ ), there exists a
unique ν ∈ IR (ᏸ) such that µ ≤ ν(ᏸ).
Definition 4.2. The lattice ᏸ is mildly normal if for all µ ∈ Iσ (ᏸ), there exists a
unique ν ∈ IR (ᏸ) such that µ ≤ ν(ᏸ).
Theorem 4.3. If ᏸ is regular and Lindelöf then it is mildly and slightly normal.
Proof. Suppose there exist µ ∈ Iσ (ᏸ), ν1 , ν2 ∈ IR (ᏸ) such that ν1 ≠ ν2 and µ ≤
ν1 , ν2 (ᏸ). Then there exist L1 , L2 ∈ ᏸ such that L1 ∩ L2 = ∅, ν1 (L1 ) = ν2 (L2 ) = 1, and
ν1 (L2 ) = ν2 (L1 ) = 0. Since ᏸ is regular, S(µ) = S(ν1 ) ⊂ L1 and S(µ) = S(ν2 ) ⊂ L2 ,
therefore S(µ) ⊂ (L1 ∩ L2 ) = ∅. But S(µ) = S(ν1 ) = S(ν2 ) = ∩α {Lα | µ(Lα ) = 1} = ∅.
The lattice ᏸ is Lindelöf, hence there exist a countable subcollection {Li } with ∩i {Li |
µ(Li ) = 1} = ∅ which give a contradiction since µ ∈ Iσ (ᏸ). Hence ᏸ is mildly normal.
Next, let µ ∈ Iσ (ᏸ ), ν1 , ν2 , ∈ IR (ᏸ) such that ν1 ≠ ν2 and µ ≤ ν1 , ν2 (ᏸ) as before.
Then S(µ) = ∅. Let π (L) = sup{µ(L̃ ) | L̃ ⊂ L, L ∈ ᏸ}, π ∈ P(ᏸ) and S(µ) = S(π ) since
ᏸ is regular. But µ ∈ Iσ (ᏸ ) implies π ∈ Pσ (ᏸ) and since ᏸ is Lindelöf, S(π ) ≠ ∅. Then
S(µ) ≠ ∅ which gives a contradiction. It follows that ᏸ is slightly normal.
Now we formulate a series of results omitting their proofs (see [6]). They will be
used in the next section.
Theorem 4.4. (a) If ᏸ is almost countably compact and mildly normal, then ᏸ is
normal.
(b) If ᏸ is complement generated, then ᏸ is slightly normal.
Definition 4.5. A lattice ᏸ is called strongly normal if for µ, µ1 , µ2 ∈ I(ᏸ) with
µ ≤ µ1 , µ2 (ᏸ) we have µ1 ≤ µ2 (ᏸ) or µ2 ≤ µ1 (ᏸ).
Theorem 4.6. Any strongly normal lattice ᏸ is normal.
Proof. Suppose that there exists a µ ∈ I(ᏸ) such that µ ≤ ν1 , ν2 (ᏸ) where ν1 , ν2
∈ IR (ᏸ). Since ᏸ is strongly normal ν1 ≤ ν2 (ᏸ) or ν2 ≤ ν1 (ᏸ). The regular measures ν1
and ν2 correspond to ᏸ-ultrafilters F1 (ᏸ), F2 (ᏸ) with F1 (ᏸ) ⊆ F2 (ᏸ) or F2 (ᏸ) ⊆ F1 (ᏸ)
which is a contradiction. Therefore ᏸ is normal.
Theorem 4.7. The lattice ᏸ is strongly normal if and only if I(ᏸ) = J(ᏸ).
Proof. Suppose that I(ᏸ) = J(ᏸ) but for arbitrary µ, µ1 , µ2 ∈ I(ᏸ) with µ ≤ µ1 ,
µ2 (ᏸ) we have µ1 µ2 (ᏸ) and µ2 µ1 (ᏸ). Then µ ≤ µ1 ∧ µ2 (ᏸ) where µ1 ∧ µ2 ∈
P(ᏸ), but ∈ I(ᏸ). Clearly, since I(ᏸ) = J(ᏸ), µ1 ∧ µ2 ∈ I(ᏸ) which is a contradiction.
NORMAL CHARACTERIZATIONS OF LATTICES
567
Conversely, suppose that ᏸ is strongly normal and let π ∈ J(ᏸ). We have then ν ≤
π = ∧µα (ᏸ), where π ≤ µα and ν, µα ∈ I(ᏸ). But {µα }α∈Λ is totally ordered since ᏸ is
strongly normal. Therefore, for any α, β ∈ Λ, να ≤ νβ (ᏸ) or νβ ≤ να (ᏸ). Suppose that
π (L1 ∪ L2 ) = 1, L1 , L2 ∈ ᏸ. Then µα (L1 ∪ L2 ) = 1 for all α ∈ Λ. Suppose that for some
λ ∈ Λ, µλ (L1 ) = 0; then µγ (L1 ) = 0 for all µγ ≤ µλ hence µγ (L2 ) = 1 for all µγ ≤ µλ (ᏸ).
But also µβ (L2 ) = 1 for all µβ ≥ µλ (ᏸ). Thus µα (L2 ) = 1 for all α ∈ Λ, so π (L2 ) = 1
which proves that π ∈ I(ᏸ).
5. Application to the Wallman spaces. Next, we briefly summarize some facts
σ
(ᏸ) and then proceed to consider some relations
(see [4]) about the Wallman space IR
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.
Definition 5.1 (general Wallman spaces and Wallman topologies). The Wallman
σ
topology in IR
(ᏸ) is obtained by taking all
σ
Wσ (L) = µ ∈ IR
(ᏸ) | µ(L) = 1 ,
L ∈ ᏸ,
(5.1)
σ
σ
as a base for the closed sets in IR
(ᏸ) and then IR
(ᏸ) is called the general Wallman
space associated with X and ᏸ. Assuming ᏸ is disjunctive, Wσ (ᏸ) = {Wσ (L) | L ∈ ᏸ} is
σ
a lattice in IR
(ᏸ), isomorphic to ᏸ under the map L → Wσ (L), L ∈ ᏸ. Wσ (ᏸ) is replete
and a base for the closed sets tWσ (ᏸ), all arbitrary intersections of sets of Wσ (ᏸ).
σ
It is this topological space [IR
(ᏸ), tWσ (ᏸ)] and lattice Wσ (ᏸ) which we will consider
here and in subsequent sections.
Analogously, we also consider Iσ (ᏸ) and Vσ (ᏸ) = {Vσ (L) | L ∈ ᏸ} where Vσ (A) =
{µ ∈ Iσ (ᏸ) | µ(A) = 1}, A ∈ Ꮽ(ᏸ). Note that here we do not need the assumption
of disjunctiveness on ᏸ and that Vσ (ᏸ) is prime complete and a base for the closed
sets tVσ (ᏸ) of Iσ (ᏸ). Note also that in a similar way one can construct the Wallman
topological spaces [IR (ᏸ), tW (ᏸ)], (here ᏸ must be separating, too) and [I(ᏸ), tV (ᏸ)].
σ
(ᏸ) | µ(A) = 1} and the following statements are
If A ∈ Ꮽ(ᏸ), then Wσ (A) = {µ ∈ IR
true:
Wσ (A ∪ B) = Wσ (A) ∪ Wσ (B),
Wσ (A ∩ B) = Wσ (A) ∩ Wσ (B),
Wσ (A ) = Wσ (A) ,
(5.2)
A ⊃ B iff Wσ (A) ⊃ Wσ (B),
Ꮽ W σ (ᏸ ) = W σ Ꮽ (ᏸ ) .
σ
(ᏸ) and consider the induced
Definition 5.2 (the induced measure). Let µ ∈ IR
σ
measure µ ∈ IR (Wσ (ᏸ)), defined by
µ Wσ (A) = µ(A),
A ∈ Ꮽ(ᏸ).
σ
σ
The map µ → µ is a bijection between IR
(ᏸ) and IR
(Wσ (ᏸ)).
(5.3)
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CARMEN D. VLAD
σ
Theorem 5.3. (a) Consider IR
(ᏸ) and Wσ (ᏸ) with ᏸ disjunctive. Then Wσ (ᏸ) is
σ
regular if and only if for all µ1 , µ2 ∈ I(ᏸ) and ν ∈ IR
(ᏸ), if µ1 ≤ µ2 , ν(ᏸ) then µ2 ≤
ν(ᏸ).
(b) Consider Iσ (ᏸ) and Vσ (ᏸ). Then Vσ (ᏸ) is regular if and only if for all µ1 , µ2 ∈
I(ᏸ) and ν ∈ Iσ (ᏸ), if µ1 ≤ µ2 (ᏸ) and µ1 ≤ ν(ᏸ) then µ2 ≤ ν(ᏸ).
Proof. (a) Let µ1 , µ2 ∈ I(ᏸ) such that µ1 ≤ µ2 (ᏸ), there exist µ 1 , µ 2 ∈ I(Wσ (ᏸ)) and
µ 1 (Wσ (L)) = µ1 (L), µ 2 (Wσ (L)) = µ2 (L) for all L ∈ ᏸ. Moreover, µ1 (L) ≤ µ2 (L) implies
µ 1 ≤ µ 2 on Wσ (ᏸ). Suppose Wσ (ᏸ) is regular, then S(µ 1 ) = S(µ 2 ), where S(µ 1 ) =
σ
(ᏸ) with µ1 ≤ ν(ᏸ). We
∩{Wσ (L) ∈ Wσ (ᏸ) | µ 1 (Wσ (L)) = 1, L ∈ ᏸ}. Let now ν ∈ IR
σ
have ν ∈ IR (Wσ (ᏸ)) and µ 1 ≤ ν(Wσ (ᏸ)) therefore S(ν) ⊂ S(µ 1 ) = S(µ 2 ); hence µ 2 ≤
ν(Wσ (ᏸ)), that is, µ2 ≤ ν(ᏸ).
σ
(ᏸ) such that if µ1 ≤ µ2 (ᏸ) and µ1 ≤ ν(ᏸ)
Conversely, let µ1 , µ2 ∈ I(ᏸ) and ν ∈ IR
then µ2 ≤ ν(ᏸ). Let now λ1 , λ2 ∈ I(Wσ (ᏸ)) and assume λ1 ≤ λ2 (Wσ (ᏸ)). Then λ1 = µ 1
and λ2 = µ 2 where µ1 , µ2 ∈ I(ᏸ) and µ 1 ≤ µ 2 (Wσ (ᏸ)), that is, µ1 ≤ µ2 (ᏸ). Now S(µ 2 ) ⊂
σ
(ᏸ) and µ1 ≤ λ(ᏸ). Hence, by the assumption
S(µ 1 ); if λ ∈ S(µ 1 ), then clearly λ ∈ IR
µ2 ≤ λ(ᏸ) which implies λ ∈ S(µ 2 ).
Clearly, the proof of (b) is similar and we omit it.
Theorem 5.4. Consider Iσ (ᏸ) and Vσ (ᏸ). Then Vσ (ᏸ) is regular if and only if
σ
(ᏸ). (We note here that the assumption of regularity for Vσ (ᏸ) in Iσ (ᏸ) is
Iσ (ᏸ) = IR
very strong.)
σ
Proof. Suppose Iσ (ᏸ) = IR
(ᏸ). Then Vσ (ᏸ) = Wσ (ᏸ). Now let µ1 , µ2 ∈ I(ᏸ), ν ∈
σ
σ
(ᏸ), µ1 ∈ IR
(ᏸ) so µ1 = µ2
Iσ (ᏸ), and µ1 ≤ µ2 (ᏸ), µ1 ≤ ν(ᏸ). Then, since Iσ (ᏸ) = IR
and µ1 = ν. Conversely, suppose Vσ (ᏸ) is regular and let µ ∈ Iσ (ᏸ), there exists
ν ∈ IR (ᏸ) such that µ ≤ ν(ᏸ), that is, µ ≤ ν(Vσ (ᏸ)), where µ, ν ∈ Iσ (Vσ (ᏸ)). But
S(µ) = S(ν) since Vσ (ᏸ) is regular. Hence µ ∈ S(ν), that is, ν ≤ µ(ᏸ). It follows that
σ
µ = ν and then µ ∈ IR
(ᏸ).
Definition 5.5. The lattice ᏸ is almost countably compact if µ ∈ IR (ᏸ ) implies
µ ∈ Iσ (ᏸ).
Theorem 5.6. Suppose ᏸ is disjunctive, then we have the following:
σ
(a) Consider IR
(ᏸ) and Wσ (ᏸ) and suppose ᏸ is Lindelöf and satisfies the condition
σ
that for all µ1 , µ2 ∈ I(ᏸ) and ν ∈ IR
(ᏸ), if µ1 ≤ µ2 , ν(ᏸ) then µ2 ≤ ν(ᏸ).
Then Wσ (ᏸ) is slightly and mildly normal.
(b) If ᏸ is complement generated then Wσ (ᏸ) is slightly normal.
(c) If ᏸ is almost countably compact and mildly normal then Wσ (ᏸ) is normal.
Proof. (a) ᏸ disjunctive and Lindelöf implies Wσ (ᏸ) is Lindelöf. Also, by Theorem
5.3 it follows that Wσ (ᏸ) is regular. Now use Theorem 4.3 to conclude that Wσ (ᏸ) is
slightly and mildly normal.
(b) ᏸ is complement generated implies L = ∩n Ln , L and Ln ∈ ᏸ, for all n.
Wσ (L) = Wσ (∩n Ln ) = ∩n Wσ (Ln ) = ∩n [Wσ (Ln )] . Hence Wσ (ᏸ) complement generated which implies that Wσ (ᏸ) is slightly normal (by Theorem 4.4).
(c) By the assumption, for any µ ∈ IR (ᏸ ) it follows that µ ∈ Iσ (ᏸ) and then there
exists a unique ν ∈ IR (ᏸ) such that µ ≤ ν(ᏸ). Let µ ∈ I(ᏸ) such that µ ≤ λ(ᏸ ) with
NORMAL CHARACTERIZATIONS OF LATTICES
569
λ ∈ IR (ᏸ ). Also λ ∈ Iσ (ᏸ) and λ ≤ µ ≤ ν1 (ᏸ) with ν1 ∈ IR (ᏸ), unique. Therefore if
µ ≤ ν2 (ᏸ) with ν2 ∈ IR (ᏸ), then λ ≤ µ ≤ ν2 (ᏸ), and so ν1 = ν2 .
Hence ᏸ is normal and also Wσ (ᏸ) is normal.
As an immediate consequence, we get the following condition on Vσ (ᏸ) that will
be used later on, for Theorem 5.11.
Corollary 5.7. Consider Iσ (ᏸ) and Vσ (ᏸ) with ᏸ Lindelöf. If for all µ1 , µ2 ∈ I(ᏸ)
and ν ∈ Iσ (ᏸ) such that if µ1 ≤ µ2 (ᏸ) and µ1 ≤ ν(ᏸ), it follows that µ2 ≤ ν(ᏸ), then
Vσ (ᏸ) is slightly and mildly normal.
Theorem 5.8. (a) Let ᏸ be disjunctive, almost countably compact, and mildly normal
and let Wσ (ᏸ) be prime complete. Then ᏸ is countably compact.
(b) Let ᏸ be disjunctive, regular, Lindelöf, almost countably compact and let Wσ (ᏸ)
be prime complete. Then ᏸ is countably compact.
σ
(ᏸ). Let µ ∈ IR (ᏸ); we have µ ≤ ν(ᏸ )
Proof. (a) We must show that IR (ᏸ) = IR
where ν ∈ IR (ᏸ ). Since ᏸ is almost countably compact we have ν ≤ µ(ᏸ) with ν ∈
σ
(ᏸ) such
Iσ (ᏸ). But Wσ (ᏸ) is prime complete and by Theorem 5.3 there exists λ ∈ IR
that ν ≤ λ(ᏸ). ᏸ almost countably compact and mildly normal implies that ᏸ is normal
(by Theorem 4.4). By the normality of ᏸ, the ᏸ-regular measure µ such that ν ≤ µ must
σ
be unique, hence µ = λ ∈ IR
(ᏸ).
(b) ᏸ regular and Lindelöf implies ᏸ mildly normal and by the above it follows that
ᏸ is countably compact.
Finally, we consider another general Wallman topological space, consisting of
strongly σ -smooth measures and analyze the relevant lattice in terms of normality:
I(σ , ᏸ), tV (σ , ᏸ) ,
(5.4)
where
V (σ , ᏸ) = V (σ , L) | L ∈ ᏸ
with
V (σ , L) = µ ∈ I(σ , ᏸ) | µ(L) = 1, L ∈ ᏸ . (5.5)
We recall a few statements on σ -smoothness that will be used throughout this
section for the reader’s convenience (see [7]).
(a) Iσ (ᏸ) ⊂ I(σ , ᏸ) ⊂ Iσ (ᏸ);
σ
(b) ᏸ normal and complement generated implies I(σ , ᏸ) ⊂ IR
(ᏸ);
σ
(c) IR (ᏸ) ⊂ I(σ , ᏸ).
Theorem 5.9. Consider I(σ , ᏸ) and V (σ , ᏸ). Then V (σ , ᏸ) is regular if and only if
for all µ1 , µ2 ∈ I(ᏸ) and ν ∈ I(σ , ᏸ), if µ1 ≤ µ2 , ν(ᏸ) then µ2 ≤ ν(ᏸ).
Proof. For µ1 , µ2 ∈ I(ᏸ) we have µ 1 , µ 2 ∈ I(Wσ (ᏸ)) and then µ 1 , µ 2 ∈ I(V (σ , ᏸ)).
If V (σ , ᏸ) is regular then S(µ 1 ) = S(µ 2 ), where S(µ 1 ) = ∩{V (σ , L) ∈ V (σ , ᏸ) |
µ 1 (V (σ , L)) = 1, L ∈ ᏸ}. Let ν ∈ I(σ , ᏸ); ν ∈ I(σ , V (σ , ᏸ)) and µ 1 ≤ ν(V (σ , ᏸ)). Then
ν ∈ S(µ 2 ), that is, µ2 ≤ ν(ᏸ).
Conversely, suppose that µ1 , µ2 ∈ I(ᏸ) and ν ∈ I(σ , ᏸ), such that if µ1 ≤ µ2 , ν(ᏸ)
then µ2 ≤ ν(ᏸ). Let λ1 , λ2 ∈ I(V (σ , ᏸ)) and λ1 ≤ λ2 (V (σ , ᏸ)). Then λ1 = µ 1 and λ2 =
µ 2 with µ1 , µ2 ∈ I(ᏸ). Thus µ 1 ≤ µ 2 (V (σ , ᏸ)) which implies that µ1 ≤ µ2 (ᏸ), hence
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CARMEN D. VLAD
S(µ 2 ) ⊂ S(µ 1 ). If λ ∈ S(µ 1 ) then clearly λ ∈ I(σ , ᏸ) and µ1 ≤ λ(ᏸ). By assumption,
µ2 ≤ λ(ᏸ) and then λ ∈ S(µ 2 ). Hence S(µ 2 ) = S(µ 1 ) and V (σ , ᏸ) is regular.
Theorem 5.10. Consider I(σ , ᏸ) and V (σ , ᏸ). If V (σ , ᏸ) is regular, then I(σ , ᏸ) =
σ
IR
(ᏸ).
Proof. Let µ ∈ I(σ , ᏸ), there exists ν ∈ IR (ᏸ) such that µ ≤ ν(ᏸ), hence µ ≤
ν(V (σ , ᏸ)) where µ ∈ I(σ , V (σ , ᏸ)) and ν ∈ IR (V (σ , ᏸ)). V (σ , ᏸ) is regular, therefore
S(µ) = S(ν), hence ν ≤ µ(ᏸ). It follows that µ = ν(ᏸ) and since ν ∈ IR (ᏸ), I(σ , ᏸ) ⊂
σ
(ᏸ).
Iσ (ᏸ). Therefore µ ∈ IR (ᏸ), Iσ (ᏸ) and then µ ∈ IR
Theorem 5.11. Consider I(σ , ᏸ) and V (σ , ᏸ) with ᏸ Lindelöf. If for all µ1 , µ2 ∈ I(ᏸ)
and ν ∈ I(σ , ᏸ), such that if µ1 ≤ µ2 , ν(ᏸ) then µ2 ≤ ν(ᏸ), it follows that V (σ , ᏸ) is
slightly and mildly normal.
Proof. By Theorem 5.9, V (σ , ᏸ) is regular. As in Corollary 5.7, we show that
V (σ , ᏸ) is Lindelöf and then it is also slightly and mildly normal.
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Carmen D. Vlad: Department of Mathematics, Pace University, New York, NY
10038, USA
E-mail address: cvlad@pace.edu