# Lattice Operators and Topologies

Authors: Eva Cogan
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

Working within a complete (not necessarily atomic) Boolean algebra, we use a sublattice to define a topology on that algebra. Our operators generalizecomplement on a lattice which in turn abstracts the set theoretic operator. Less restricted than those of Banaschewski and Samuel, the operators exhibit some surprising behaviors. We consider properties of such lattices and their interrelations. Many of these properties are abstractions and generalizations of topological spaces. The approach is similar to that of Bachman and Cohen. It is in the spirit of Alexandroff, Frolík, and Nöbeling, although the setting is more general. Proceeding in this manner, we can handle diverse topological theorems systematically before specializing to get as corollaries as the
topological results of Alexandroff, Alo and Shapiro, Dykes, Frolík, and Ramsay.


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## Body

## 1. Introduction

We begin withℒ, a sublattice of a complete (not necessarily atomic) Boolean algebra ℬ. If ℒ is closed under arbitrary meets, it abstracts the closed sets of a topological space. If not, we introduce a Kurotowski closure operator to define the associated topological lattice. The operators we define generalize complement on a lattice which in turn abstracts the set theoretic operator. Less restricted than those of Banaschewski [1] and Samuel [2], the operators exhibit some surprising behaviors. We consider certain properties of such lattices and the implications for the properties of one lattice from those of another, when one is a sublattice of the other. Many of these properties are abstractions and generalizations of topological spaces.The approach is similar to that of Bachman and Cohen [3, 4]. It is in the spirit of Alexandroff [5], Frolík [6], and Nöbeling [7], although the setting is more general. We generalize a variety of filter arguments used in paved space [8, 9] and in the theory of realcompactness [10, 11]. Proceeding in this manner, we can handle diverse topological theorems systematically before specializing to get as corollaries as the topological results of [5, 8, 12–14].Section2 provides some background material and generates a topology on an algebra by means of a sublattice. Section 3 defines operators and topological type properties for a lattice. Section 4 examines filter and measure behavior with respect to the operators. Section 5 looks at covering properties. Section 6 investigates the relationships between two lattices.


## 2. Background, Terminology, and Notation

We work within a complete Boolean algebraℬ with minimal element 0 and maximal element e. The usual operators are denoted by ∨, ⋀, and ′. ℬ is not necessarily atomic; equivalently, ℬ is not necessarily completely distributive [15].(i)
μ,μf denote finitely additive zero-one measures on ℬ.(ii)
ℒ,ℒ1,  and ℒ2 denote sublattices of ℬ containing 0 and e.(iii)
𝒜(ℒ) denotes the algebra generated by ℒ.(iv)
𝒫(S) is the power set of the set S.(v)
The indicesi,j,k, and n index countable (finite or countably infinite) collections, while α,β, and γ index arbitrary ones.Definition 2.1.
(i)F⊆ℒ is an ℒ-filter if and only if for all a,b∈ℒ:(a)
a,b∈F⇒a⋀b∈F,(b)
a∈F and a≤b⇒b∈F.
When there is no ambiguity, we simply say thatF is a filter.
(ii) Anℒ-filter F is a primeℒ-filter if and only if for all a,b∈ℒ,a∨b∈F⇒ either a∈F or b∈F.
(iii) Anℒ-filter F is an ℒ-ultrafilter (or ultra) if and only if F is a maximal ℒ-filter.
(iv) A filterF is fixed if and only if ⋀F≠0. Otherwise F is free.
(v) A filterF has cmp (countable meet property or countable intersection property) if and only if for any countable subset of F, ⋀iai≠0.Remark 2.2.
It follows that(i)
0∉F if and only if F≠ℒ,(ii)
every ultrafilter is prime.In this paper, 0 is not in any filter.Definition 2.3.
A measureμ on an algebra ℬ containing ℒ is ℒ-regular if and only if for all b∈ℬ,μ(b)=sup{μ(a):a∈ℒ,a≤b}.Lemma 2.4.
There exist one-to-one correspondences between(i)
zero-one measures onℒ and prime ℒ-filters,(ii)
ℒ-regular zero-one measures on 𝒜(ℒ), and ℒ-ultrafilters [3, 4].Thus, analogous measure theoretic results may easily be derived from our filter statements.We now topologizeℬ by means of a sublattice ℒ. The lattice elements themselves may not be sufficient to be used as open or closed sets. However, we will generate a topology.Definition 2.5.
Letℬ be an algebra, ℒ a sublattice of ℬ,and a,b∈ℬ.(i)
b¯=⋀{a∈ℒ:a≥b}.(ii)
b0=⋁{a′:a∈ℒanda′≤b}.(iii)
b∈ℬ is closed if and only if b=b¯.(iv)
τ(ℒ) denotes the set of closed elements of ℬ.Remark 2.6.
(i)b↦b¯ is a Kurotowski closure operator.
(ii)b↦b0 is an interior operator.
(iii)τ(ℒ)={b∈ℬ:b=⋀αaα,aα∈ℒ}.
(iv)ℒ⊆τ(ℒ).
(v)τ(ℒ) is a lattice which is closed under arbitrary meets.We observe thatτ(ℒ) is an obvious abstraction of the closed sets in a topological space.Example 2.7.
Letℒ be the lattice of zero sets in a T2 completely regular topological space. Then τ(ℒ) is the lattice of closed sets [11].Definition 2.8.
Letℳ⊆ℬ be a meet semilattice (i.e., a subset of ℬ closed under finite meets). Consider the generated lattice ℒ(ℳ)={⋁1n{ai:ai∈ℳ}:n∈ℕ}. All terminology remains the same except that F is a primeℳ-filter if and only if ⋁1nai∈F (for ai∈ℳ) implies that one of the ai∈F.Lemma 2.9.
(i) There exist one-to-one correspondences between the prime, ultra-, fixed, and free filters onℳ and those on ℒ(ℳ).
(ii)ℳ-filters with cmp correspond with ℒ-filters with cmp.Thus we lose no generality in “treating”ℳ like a lattice.Example 2.10.
Letℳ={b∈ℬ:b=(b¯)0}. ℳ is a meet semi-lattice since a⋀b¯≤a¯⋀b¯ implies that (a⋀b¯)0≤(a¯⋀b¯)0=(a¯)0⋀(b¯)0=a⋀b,foralla,b∈ℳ. “An open subset G in a topological space is regularly open if and only if G is the interior of its closure” [16]. Thus the regularly open sets in a topological space form a meet semi-lattice.Table1 summarizes the notation used in this paper.Table 1
Summary of notation.
ℬComplete Boolean algebra with zero (0) and unit (e)ℒ,ℒnLattices with0 and eF,Fn,G,HFilters without0𝒜(ℒ)The Algebra generated byℒμ,μfFinitely additive zero-one measures onℬτ(ℒ)Closed elements ofℬℳMeet semi-lattice𝒫(S)Power set of the setS


## 3. Lattice Operators and Properties

In this section we define certain operators and lattice properties. These properties reduce to the conventional topological ones when the operator is taken to becomplement.Definition 3.1.
Letℳ be a meet semi-lattice containing 0 and e. We define T to be a one-to-one operator on ℳ such that T(a∨b)=T(a)⋀T(b) and T(e)=0. We define a* = T(a) and ℒ*={a*:a∈ℒ}.Proposition 3.2.
(a)a≤b⇔T(a)≥T(b).
(b)⋁1nT(ai)≤T(⋀1nai).
(c)⋁1nT(ai)=e⇒⋀1nai=0.
(d)T(⋁1nai)=0⇒⋁1nai=e.Proof.
(a)a≤b⇔a∨b=b⇔T(a)⋀T(b)=T(b)⇔T(a)≥T(b),
(b) For allj,aj≥⋀1nai⇒ for all j,T(aj)≤T(⋀1nai)⇒⋁1nT(aj)≤T(⋀1nai).
The proof of (c) and (d) follows readily.From now on, we assume thatT is defined on a lattice ℒ. Then ℒ* is a meet semi-lattice. By Lemma 2.9, we may “treat” ℒ* like a lattice.Corollary 3.3.
WhenT is defined on τ(ℒ), one has the following. (a)
⋁αT(aα)≤T(⋀αaα).(b)
⋁αT(aα)=e⇒⋀αaα=0.Example 3.4.
LetS={1,2,3,4} and ℬ=𝒫(S) (the power set of S) with set union and intersection as the join and meet operations. Let ℒ={∅,a,b,S} with a={1,2} and b={3,4} as in Figure 1. The operator T is defined by T(S)=∅,T(a)={4},T(b)={2}, and T(∅)={2,4}. We have T(a∨b)=T(a)⋀T(b) and T(S)=∅. In addition T(a⋀b)=T(a)∨T(b) and T(a)⋀a=∅. Note that, unlike in [2], T(a) is not the maximal element disjoint from a, and although ℒ is complemented (that is, a∈ℒ⇒a′∈ℒ),T(a)≠a′.Figure 1
ℒ in Example 3.4.We now define various properties forℒ. They generalize some of the definitions in point set topology, reducing to the conventional properties when ℬ is the power set of a set X and T is complement.Definition 3.5.
(i)ℒ is compact if and only if every ℒ-filter is fixed.
(ii)ℒ is ℵ0-compact if and only if every ℒ-filter has cmp.
(iii)ℒ is an I-lattice (P-lattice) if and only if every (prime) ℒ-filter with cmp is contained in an ultrafilter with cmp.
(iv)ℒ is an R-lattice if and only if every filter which contains a fixed prime filter is also fixed.
(v) A topological space is anI-space if and only if the lattice of closed sets is an I-lattice [17].Proposition 3.6.
As an immediate consequence of Definition3.5, one has the following.(i)
ℒ is compact ⇒ℒ is ℵ0-compact ⇒ℒ is an I-lattice ⇒ℒ is a P-lattice.(ii)
The following are equivalent:
(a)
ℒ is compact (ℵ0-compact),(b)
every prime filter is fixed (has cmp),(c)
every ultrafilter is fixed (has cmp).Definition 3.7.
(i)ℒ is ℵ0-paracompact if and only if whenever there exists {an}⊆ℒwithan↓0, there exists {bn}⊆ℒsuch that an≤bn* and bn*↓0.
(ii) Whenℒ⊆τ(ℒ*), we say that ℒ is perfect.Proposition 3.8.
Every perfect lattice isℵ0-paracompact.Proof.
Assumeℒ is perfect and an↓0. For each an, there exists {bnα}⊆ℒ such that an=⋀αbnα*. Let cn*=⋀{bkα*:k≤n}. Then cn*≥an and cn*↓0.Example 3.9.
The zero sets in a topological space are perfect (i.e.,complement generated in the sense of [11]) and thus ℵ0-paracompact.Definition 3.10.
Leta,ai,b∈ℬ,andf,fi,g,gi∈ℒ. Then the following are given. (i)
ℒ is T1 if and only if for all a1≰a2 there exists g*∈ℒ* such that a2≤g* but a1≰g*.(ii)
ℒ is Hausdorff if and only if for all a1,a2≠0 with a1⋀a2=0 there exist g1*,g2*∈ℒ* such that a1⋀g1*≠0,a2⋀g2*≠0,andg1*⋀g2*=0.(iii)
ℒ is regular if and only if for all b∈ℬ,b≠0,f∈ℒ,b⋀f=0 there exist g1*,g2*∈ℒ* such that b⋀g1*≠0,f≤g2*, and g1*⋀g2*=0.The following proposition provides an example.Proposition 3.11.
Letℬ be the power set of a topological space X, let ℒ be the lattice of closed sets in X, and let T be complement. X is a T1 (Hausdorff, regular) space if and only if ℒ is T1 (Hausdorff, regular).Proof.
Letℒ be the lattice of closed sets in X. Suppose ℒ is a T1 lattice. Let a1≠a2 be atoms in ℬ. Since a1≰a2, there exists g2′∈ℒ′ such that a2<g2′ but a1≰g2′. By symmetry, there exists g1′∈ℒ′ such that a1≤g1′ but a2≰g1′. Thus X is a T1 space.
Now supposeX is a T1 topological space. Let b,c∈ℬ,b≰c. Then there exists an atom a≤b but a≰c. Thus for all atoms aα≤c,a≠aα, and there exists gα′∈ℒ′ such that aα≤gα′ but a≰gα′. Let g′=⋁αgα′. Then c≤g′,b≰g′, and thus ℒ is T1.
The proofs for Hausdorff and regular are similar.Proposition 3.12.
SupposeT(a)⋀a=0 and ℒ is regular. If F1 is prime and F1⊆F2, then ⋀F1=⋀F2.Proof.
Suppose there existsF1⊆F2such that ⋀F1≠⋀F2. Let a=⋀F1.a≰⋀F2 implies there exists f∈F2witha≰f. But then b=a⋀f′≠0 and b⋀f=0. By regularity, there exist c1*,c2*∈ℒ* such that b⋀c1*≠0,f≤c2*, and c1*⋀c2*=0. Now f≤c2* implies that c2∉F2⊇F1. From b⋀c1*≠0, it follows that a≰c1 and thus c1∉F1. But c1*⋀c2*=0 implies that c1∨c2=e, and thus F1 is not prime.Example 3.13.
The closed sets in a regular topological space form anR-lattice.Definition 3.14.
ℒ is normal if and only if for all f1,f2∈ℒ,f1⋀f2=0 there exist g1*,g2*∈ℒ* such that f1≤g1*, f2≤g2*, and g1*⋀g2*=0.The following proposition demonstrates an example of an application.Proposition 3.15.
LetX be a topological space. Let T be complement and let ℒ′ be the lattice of open sets. Then the following are equivalent.(a)
ℒ′ is normal.(b)
a′,b′∈ℒ′ and a′⋀b′=0⇒a′¯⋀b′¯=0.(c)
a′∈ℒ′⇒a′¯∈ℒ′ (i.e., X is extremally disconnected).Proof.
Consider the following. (i) (a) implies (b): leta',b'∈ℒ',a'⋀b'=0. Then there exist c,d∈ℒ such that a′≤c,b′≤d,andc⋀d=0 (by normality). But a'¯≤c and b'¯≤d implies that a'⋀b'=0.
(ii) (b) implies (c): leta'∈ℒ'. Let b=a'¯ so that a'⋀b'=0. Then a'¯⋀b'¯=0 (by hypothesis), so that b⋀b'¯=0. Then b≤(b'¯)'=b0, so that b=b0andb∈ℒ'. Therefore a'¯∈ℒ' by definition of b.
(iii) (c) implies (b): leta′,b′∈ℒ′,a′⋀b′=0, so that a′¯⋀b′=0. Since a′¯∈ℒ′, then a′¯⋀b′¯=0.
(iv) (b) implies (a):a′¯,b′¯ are elements of ℒ.Proposition 3.16.
LetT(a)⋀a=0. If ℒ is normal and F is a prime ℒ-filter contained in two ℒ-ultrafilters G and H, then G=H.Proof.
LetG and H be two distinct ultrafilters and let F⊆G∩H. Then there exist g∈G,h∈Hwith g⋀h=0. By normality, there exist a*,b*∈ℒ* such that g≤a*,h≤b*,anda*⋀b*=0. But g⋀a=0 implies that a∉G, and h⋀b=0 implies that b∉H, so that a,b∉G∩H⊇F, that is, a,b∉F. But a∨b=e implies that F is not prime.Definition 3.17.
ℒ is ℵ0-normal if and only if it is normal and ℵ0-paracompact.Example 3.18.
A normal topological space is countably paracompact if and only if the lattice of closed sets isℵ0-paracompact [18]. Willard [16] calls such a space binormal.Proposition 3.19.
LetT be complement in ℬ. If ℒ is ℵ0-normal and F1 is a prime filter with cmp, then F1⊆F2 implies that F2 has cmp.Proof.
LetF1⊆F2 where F1 is a prime ℒ-filter and F2 is an ℒ-filter without cmp. Then there exists {fk}⊆F2 such that ⋀k{fk}=0. Let gn=⋀1nfk.{gn}⊆F2 and gn↓0. Thus there exist {bn}⊆ℒ such that gn≤bn′ and bn′↓0. Now gn⋀bn=0, so there exist two sequences {cn},{dn} such that gn≤cn′,bn≤dn′,cn⋀dn=0, for all n. Since cn∨dn=eand F1 is prime, we have cn∈F1 or dn∈F1, for all n. But gn≤cn′ implies that cn∉F1, sodn∈F1, for all n. And since bn′≥dn implies that dn↓0, we have that F1 does not have cmp.Remark 3.20.
Ifℒ is a normal lattice that has the stronger property that whenever {an}⊆ℒ such that ⋁nan*=e there exists {bn}⊆ℒ such that ⋁nbn=e and for all n,an*≥bn, then we need only to assume that T(a)⋀a=0.Corollary 3.21.
LetT be complement in ℬ. If ℒ is ℵ0-normal, then ℒ is a P-lattice.


## 4. Behavior of Filters and Measures UnderT

In this section we look at the behavior of filters and measures with respect toT. It is interesting to see an example where T does not “behave as nicely” as complement.Definition 4.1.
LetD⊆ℒ. D̃={a*∈ℒ*:a∉D}.Proposition 4.2.
LetF be a prime ℒ-filter. F̃ is a prime ℒ*-filter.Proof.
(a)0∉F̃ since e∈F.
(b)a*≤b*,a*∈F̃⇒a≥b and a∉F⇒b∉F⇒b*∈F̃.
(c) Leta*,b*∈F̃, so that a,b∉F; equivalently, a∨b∉F. But then (a∨b)*∈F̃, and so a*⋀b*∈F̃. Thus (by (a), (b), and (c)), F̃ is a filter.
(d) Leta*,b*∈ℒ*,a*∨b*≥c*∈F̃. (a⋀b)*≥a*∨b*≥c*∈F̃⇒a⋀b≤c∉F⇒a⋀b∉F⇒a∉F or b∉F⇒a*∈F̃ or b*∈F̃. Thus F̃ is prime.Definition 4.3.
A filterF is coultra if and only if F̃ is ℒ*-ultra. μF is coregular if and only if F is coultra.Proposition 4.4.
F̃ is ℒ*-ultra if and only if a∈F is equivalent to a*≤(b*)′ for some b*∈F̃ (i.e., a*∉F̃⇔ there exists b*∈F̃ such that a*≤(b*)′).Remark 4.5.
Proposition4.4 generalizes a theorem of [12] which we get by taking T to be complement.As demonstrated by the following example, we can associate measures with the prime filters onℒ as usual, but they may lack some of the properties to which we are accustomed.Example 4.6.
LetS={1,2,3,4}. Let ℬ=𝒫(S), with set union and intersection as the join and meet operations. Figure 2 defines the sublattice ℒ.Figure 2
ℒ in Example 4.6.DefineT on ℒ as follows:(4.1)T(e)=0,T(0)=e,T(a)=g,T(g)=a,T(b)=h,T(h)=b,T(c)=c,T(d)=d,T(f)=f.Incidentally,ℒ=ℒ* and for all a∈ℒ,T(T(a))=a. However a⋀T(a) does not necessarily = 0.NowT can be extended to all of 𝒜(ℒ)=ℬ by defining T(a∨b)=T(a)⋀T(b),T(a⋀b)=T(a)∨T(b),andT(a′)=T(a)′.LetF=Ff={x∈ℒ:x≥f}={e,b,f}.F̃={a*,d*,h*,c*,g*,0*}={g,d,b,c,a,e}=Fg. Since F̃ is an ℒ*-ultrafilter, F is a coultra filter. F is a prime filter so we have the associated measure μ=μF=μf, where(4.2)μ(x)={1ifx≥f,0otherwise.Now consider that{3}∈𝒜(ℒ).μ({2,3})=1,μ({2})=0, so that μ({3})=μ({2,3})-μ({2})=1. But {3}≱ any element of ℒ* whose measure is one, so μ is not ℒ*-regular even though it is ℒ-coregular.Note thatμ(a)=0andμ(a*)=μ(g)=0,μ(f)=1andμ(f*)=μ(f)=1,μ(h)=0butμ(h*)=μ(b)=1,μ(b)=1butμ(b*)=μ(h)=0.Thus,T is not necessarily measure inverting (μ(T(a))=1-μ(a)∀μ), preserving, increasing, or decreasing.Remark 4.7.
IfT(a)⋀a=0 and T is measure inverting, then every ℒ-coregular measure is ℒ*-regular. These concepts are equivalent when T is complement.From now on we assume thatT(a⋀b)=T(a)∨T(b). Now ℒ* is a lattice.Example 4.8.
See Examples4.6 and 3.4.Proposition 4.9.
F is a prime ℒ-filter if F̃ is a prime ℒ*-filter. (Please see Definition 4.1 and Proposition 4.2 for the converse.)Proof.
(a)0∉F since e∈F̃.
(b)a,b∈F⇒a*,b*∉F̃⇒a*∨b*∉F̃⇒(a⋀b)*∉F̃⇒a⋀b∈F.
(c)a≤banda∈F⇒b*≤a*anda*∉F̃⇒b*∉F̃⇒b∈F.
(d)a∨b∈F⇒(a∨b)*∉F̃⇒a*⋀b*∉F̃⇒a*∉F̃orb*∉F̃⇒a∈Forb∈F.Corollary 4.10.
(a)F is a prime ℒ-filter if and only if F̃ is a prime ℒ*-filter.
(b)F is a prime filter if F is a coultra filter.


## 5. Covering Properties

In this section we define some covering properties forℒ and show that they are analogous to the topological ones. In particular, when T is taken to be complement, we get topological results as corollaries.Definition 5.1.
ℒ is comax compact if and only if every coultra filter is fixed. ℒ is comaxℵ0-compact if and only if every coultra filter has cmp.Proposition 5.2.
Every comax compactR-lattice is compact.Proof.
LetF be a prime ℒ-filter. Form F̃ and extend it to G̃, an ℒ*-ultrafilter. G is a prime ℒ-filter and fixed. G⊆F implies that F is fixed since ℒ is an R-lattice. Thus ℒ is compact by Proposition 3.6.Corollary 5.3.
IfT(a)⋀a=0 and ℒ is a comax compact regular lattice, then ℒ is compact.Definition 5.4.
ℒ is (prime, max, comax) complete if and only if every (prime, ultra-, coultra) filter with cmp is fixed.We have the implications in Figure3.Figure 3
Lattice Implications.Proposition 5.5.
Ifℒ is (comax-)ℵ0-compact, then it is (comax) complete if and only if it is (comax) compact.Proposition 5.6.
Ifℒ is an R-lattice, then ℒ is prime complete if and only if it is comax complete.Proof.
In the proof of Proposition5.2, let F have cmp.Proposition 5.7.
Ifℒ is a max complete I-lattice, then ℒ is complete.Proof.
LetF be a filter with cmp. Since ℒ is an I-lattice, F can be extended to G, an ultrafilter with cmp. G is fixed because ℒ is max complete. Hence F is fixed and ℒ is complete.Corollary 5.8.
If L is a max complete P-lattice, thenℒ is prime complete.Proof.
In the proof of Proposition5.7, take F to be a prime filter.Remark 5.9.
WhenT is defined on τ(ℒ) (as when ℒ=τ(ℒ)) and T(⋀αaα)=⋁αT(aα)∀aα∈ℒ, our definitions coincide with the conventional topological ones. (See Examples 3.4 and 4.6.) In particular T may be taken to be complement.Proposition 5.10.
LetT be defined on τ(ℒ) with T(⋀αaα)=⋁αT(aα) for all aα∈ℒ.(a)
ℒ is compact if and only if e=⋁αaα*⇒e=⋁1naαi*.(b)
ℒ is complete if and only if e=⋁αaα*⇒e=⋁1∞aαi*.(c)
ℒ is ℵ0-compact if and only if e=⋁1∞ai*⇒e=⋁j=1naij*.Proof.
We will prove only part (b). Parts (a) and (c) have similar proofs.
Suppose the condition holds. LetF={fα} be a free ℒ-filter. Then ⋀αfα=0 implies that ⋁αfα*=e, so that there exists {fαi}⊆{fα} such that ⋁ifαi*=e. But then ⋀ifα*=0 and F does not have cmp, so ℒ is complete.
Conversely, suppose that the condition does not hold. Then there exists{aα} such that ⋁αaα*=e but ⋁iaαi*≠e for any countable subset. Then 0≠⋀iaαi and {aα} is a subbase for a filter F. F has cmp since {fj}⊆F implies that ⋀jfj≥⋀jaαj>0.F is free since e=⋁αaα* implies that ⋀αaα=0, and therefore, ℒ is not complete.Corollary 5.11.
LetX be a topological space. Take ℒ to be the closed sets and take T to be complement. Then X is compact (Lindelöf, countably compact) if and only if ℒ is compact (complete, ℵ0-compact).Corollary 5.12.
X is a realcompact I-space if and only if it is a Lindelöf space [14].Corollary 5.13.
IfX is regular, countably paracompact, and almost realcompact, then X is realcompact [6].


## 6. Lattice Interrelations

In this section we investigate the implications between the properties of two lattices when one is a sublattice of the other.Proposition 6.1.
Letℒ1⊆ℒ2⊆τ(ℒ1)(a)
Ifℒ1 is complete, then ℒ2 is complete.(b)
Ifℒ1 is prime complete, then ℒ2 is prime complete.(c)
Ifℒ1 is a P-lattice, then that ℒ1 is max complete implies that ℒ2 is prime complete.Proof.
(a) LetF be an ℒ2-filter with cmp, G=F∩ℒ1, and b∈F. There exists {aα}⊆ℒ1 such that b=⋀αaα. As b≤aα,for all α, we have aα∈F∩ℒ1(which=G),for all α. Now let {bβ}⊆F.⋀βbβ=⋀α,βaα,β≠0, since G is fixed. Thus F is fixed and ℒ2 is complete.
(b) LetF in (a) be prime. Then G is prime.
(c) Sinceℒ1 is a P-lattice, G may be extended to an ℒ1-ultrafilter H with cmp. Since ℒ1 is max complete, H is fixed, and hence G and F are fixed.Corollary 6.2.
Letℒ1⊆ℒ2⊆τ(ℒ1) and let ℒ1 be a P-lattice. If ℒ1 is max complete, then ℒ2 is max and comax complete.Corollary 6.3.
Letℒ1⊆ℒ2⊆τ(ℒ1) and let ℒ1 be ℵ0-normal. If ℒ1 is max complete, then ℒ2 is max and comax complete.Corollary 6.4.
IfX is a normal, countably paracompact space, then Z-replete implies F-replete and realcompact implies α-complete [13].The following proposition generalizes two results of Alexandroff [5].Proposition 6.5.
Letℒ1⊆ℒ2⊆τ(ℒ1). (a)
Ifℒ1 is compact, then ℒ2 is compact.(b)
Ifℒ1 is compact and normal, then ℒ2 is normal.Proof.
(a) LetF be an ℒ2-filter. Let G=F∩ℒ1. The proof follows as in Proposition 6.1.
(b) Leta1,a2∈ℒ2,a1⋀a2=0, where a1=⋀βbβ,a2=⋀γcγ, and of course bβ,cγ∈ℒ1.a1≤bβanda2≤cγ,for all β,γ. Now there must exist bβ0,cγ0 such that bβ0⋀cγ0=0. (If not, {bβ,cγ} would form a subbase for a free filter in a compact space.) By normality of ℒ1, there exist d1*,d2*∈ℒ1* such that d1*≥bβ0,d2*≥cγ0, and d1*⋀d2*=0. But then d1*≥a1 and d2*≥a2 and so ℒ2 is normal.Proposition 6.6.
Letδ(ℒ) denote the smallest set containing ℒ and closed under countable meets. Let ℒ1⊆ℒ2⊆δ(ℒ1). If ℒ1 is ℵ0-compact, then ℒ2 is ℵ0-compact.Proposition 6.7.
Letℒ1⊆ℒ2. If ℒ2 is complete, then ℒ1 is complete.Proof.
LetF be an ℒ1-filter with cmp. Let G={a∈ℒ2:a≥fforsomef∈F}.G has cmp since ⋀gi≥⋀fi>0,gi∈G,  and fi∈F.ℒ2 is complete so G is fixed. Thus F is fixed and ℒ1 is complete.Corollary 6.8.
Letℒ1⊆ℒ2 where ℒ2 is an I-lattice.(a)
Ifℒ2 is max complete, then ℒ1 is complete.(b)
Ifℒ2 is a comax complete R-lattice, then ℒ1 is complete.Proof.
(a) Sinceℒ2 is an I-lattice, ℒ2 that is max complete implies that ℒ2 is complete, which implies by Proposition 6.7 that ℒ1 is complete.
(b) Sinceℒ2 is an R-lattice, its being comax complete implies that it is prime complete. Since ℒ2 is an I-lattice, ℒ2 is complete and thus ℒ1 is complete by Proposition 6.7.Definition 6.9.
ℒ2 is an ℒ1-P-lattice if and only if every ℒ1 prime filter with cmp is contained in an ℒ2 ultrafilter with cmp.Definition 6.10.
(i)ℒ2 is ℒ1-normal if and only if for all f1,f2∈ℒ2 with f1⋀f2=0 there exist g1*,g2*∈ℒ1* such that f1≤g1*,f2≤g2*, and g1*⋀g2*=0.
(ii)ℒ2 is ℒ1-ℵ0-paracompact if and only if for each {an}⊆ℒ2 such that an↓0 there exists a sequence {bn}⊆ℒ1 such that an≤bn* and bn*↓0.
(iii)ℒ2 is ℒ1-ℵ0-normal if and only if ℒ2 is ℒ1-normal and ℒ1-ℵ0-paracompact.Proposition 6.11.
LetT be complement. If ℒ2 is ℒ1-ℵ0-normal and F1 is a prime ℒ1-filter with cmp contained in an ℒ2-filter F2, then F2 has cmp.Proof.
Similar to the proof of Proposition3.19.Corollary 6.12.
LetT be complement. (a)
Thatℒ2 is ℒ1-ℵ0-normal implies that ℒ2 is an ℒ1-P-lattice.(b)
Ifℒ2 is ℵ0-paracompact and ℒ1 separates ℒ2, then ℒ2 is ℒ1-ℵ0-paracompact.Corollary 6.13.
(a) Letℒ2 be an ℒ1-P-lattice. That  ℒ2 is max complete implies that ℒ1 is prime complete.
(b) If in additionℒ2 is an R-lattice, then that ℒ2 is comax complete implies that ℒ1 is prime complete.We get Frolík's [8] theorems as our final corollary.Corollary 6.14.
(a) Ifℒ1⊆ℒ2 and ℒ2 is ℒ1-ℵ0-normal and prime complete, then ℒ1 is prime complete.
(b) LetX be a normal T2 space. If X is almost realcompact and countably paracompact, then X is realcompact.



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