# Topological Aspects of the Product of Lattices

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International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID 920737,9pages

doi:10.1155/2011/920737

## Topological Aspects of the Product of Lattices

Department of Mathematics, Pace University, Pleasantville, NY 10570, USA

Received 8 June 2011; Accepted 23 July 2011

LetX be an arbitrary nonempty set and L a lattice of subsets of X such that∅,XL. ALdenotes the algebra generated byL, and MLdenotes those nonnegative, finite, finitely additive measures onAL. In addition, IL denotes the subset of ML which consists of the nontrivial zero-one valued measures. The paper gives detailed analysis of products of lattices, their associated Wallman spaces, and products of a variety of measures.

### 1. Introduction

It is well known that given two measurable spaces and measures on them, we can obtain the product measurable space and the product measure on that space. The purpose of this paper is to give detailed analysis of product lattices and their associated Wallman spaces and to investigate how certain lattice properties carry over to the product lattices. In addition, we proceed from a measure theoretic point of view. We note that some of the material presented here has been developed from a filter approach by Kost, but the measure approach lends to a generalization of measures and to an easier treatment of topological style lattice properties.

### 2. Background and Notations

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.

LetX be an arbitrary nonempty set and L a lattice of subsets of X such that∅,XL. A

latticeL is a partially ordered set any two elementsx, yof which have both supx, yand

infx, y.

ALdenotes the algebra generated byL;σLis theσalgebra generated byL;δL

is the lattice of all countable intersections of sets from L; τL is the lattice of arbitrary

intersections of sets fromL;ρLis the smallest class closed under countable intersections

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### 2.1. Lattice Terminology

The latticeL is called:

δ-lattice ifL is closed under countable intersections; complement generated ifLL

implies LLn, n 1, . . . ,∞, LnLwhere prime denotes the complement;

disjunctive if forxX andL1 ∈L such thatx /L1there existsL2 ∈L withxL2

andL1∩L2 ∅; separatingorT1ifx, yX andx /yimplies there existsLL

such thatxL,y /L;T2 if forx, yX andx /ythere existL1, L2 ∈L such that

xL

1,yL2, andL1∩L2∅; normal if for anyL1, L2∈L withL1∩L2∅there exist

L3, L4∈L withL1⊂L3,L2⊂L4, andL3L4; compact if for any collection{}of

sets ofL withαLα ∅, there exists a finite subcollection with empty intersection;

countably compact if for any countable collection {Lα} of sets ofL withαLα ∅,

there exists a finite subcollection with empty intersection.

### 2.2. Measure Terminology

MLdenotes those nonnegative, finite, finitely additive measures onAL.

A measureμMLis called:

σ-smooth onL if for all sequences{Ln}of sets ofL withLn↓ ∅,μLn → 0;

σ-smooth onALif for all sequences{An}of sets ofALwithAn↓ ∅,μAn → 0,

L-regular if for anyAAL,

μA supμL|LA, LL. 2.1

We denote byMRLthe set ofL-regular measures of ML;MσLthe set ofσ-smooth

measures onL, of ML;MσLthe set ofσ-smooth measures onALofML;MσRLthe set

ofL-regular measures of MσL.

In addition,IL,IRL,IσL,IσL,IσRLare the subsets of the correspondingM’s

which consist of the nontrivial zero-one valued measures.

Finally, letX,Y be abstract sets and L1a lattice of subsets ofX and L2a lattice of subsets

ofY. Letμ1∈ML1andμ2∈ML2.

The product measureμμ2∈ML1×L2is defined by

μμ2

LL2 μ1L1μ2L2 ∀L1∈AL1, L2 ∈AL2. 2.2

### 2.3. Lattice-Measure Correspondence

The support ofμMLis ∩{LL/μL μX}.

In caseμILthen the support is ∩{LL/μL 1}.

With this notation and in light of the above correspondences, we now note:

For anyμIL, there existsνIRLsuch thatμνonLi.e.,μLνLfor all

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L is compact if and only ifSμ/∅for everyμIRL.L is countably compact if and

only ifIRL IσRL.L is normal if and only if for eachμIL, there exists a unique

νIRLsuch thatμνonL. L is regular if and only if wheneverμ1, μ2∈ILand

μ1≤μ2onL, then1 2.L is replete if and only if for anyμIσRL,Sμ/∅.

L is prime-complete if and only if for anyμIσL,Sμ/∅.

Finally, ifμxis the measure concentrated atxX, thenμxIRL, for allxX if

and only ifL is disjunctive.

For further results and related matters see1–3.

### 2.4. The General Wallman Space and Wallman Topology

The Wallman topology inIσRLis obtained by taking all

WσL μIσRL/μL 1

, LL 2.3

as a base for the closed sets in IσRL and then IσRL is called the general Wallman space

associated withX and L. Assuming L is disjunctive, WσL {WσL/LL} is a lattice

inIσRL, isomorphic toL under the mapLWσL,LL. WσLis replete and a base for

the closed setstWσL, all arbitrary intersections of sets ofWσL.

IfAAL, thenWσA {μIσRL/μA 1}and the following statements are

true:

WσAB WσAWσB,

WσAB WσAWσB,

WσAWσA,

ABiﬀWσAWσB,

AWσL WσAL.

2.4

### The Induced Measure

LetμIσRLand consider the induced measureμIσRWσL, defined by

μWσA μA, AAL. 2.5

The mapμμis a bijection betweenIσRLandIσRWσL.

### 3.1. Notations

LetX, Y be abstract sets and L1 a lattice of subsets ofX and L2 a lattice of subsets ofY. We denote:

1LL1×L2 {LL2/L1∈L1, L2 ∈L2},

2LLL∗, the lattice generated byL∗. ¡list-item¿¡label/¿

We have the following:

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4ALAL,

5SLμ SLμ,

6IσLIσL,

7IRLIRL.

### 3.2. Results

Theorem 3.1the finite product of lattices/regular measures. LetX, Y be abstract sets and let

L1, L2be lattices of subsets ofX and Y, respectively. Then IRL1×IRL2 IRL.

Proof. ForAAL1×AL2 AL1×L2, we haveA i1n AiAi2, disjoint union and

Ai

1∈AL1,Ai2∈AL2.

LetμIRL1andνIRL2and considerμ×νdefined onAL1×AL2.

Ifμ×νA 1, thenμ×νAi1×Ai2 1 for somei.

ThenμAi1νAi2 1, and sinceμandνare zero-one valued measures,μAi11 and

νAi

2 1. By the regularity ofμandνthere existL1 ⊂Ai1,L1∈L1withμL1 1 andL2⊂Ai2,

L2∈L2withνL2 1.

Thereforeμ×νLL2 μL1νL2 1 andLL2∈L∗.

If we letMLL2⊂Ai1×Ai2A, then

μ×νA supμ×νM/MA, ML∗⇒μ×νIRL. 3.1

Conversely, let μIRLIRLand define μ1 on AL1 byμ1A μA×Y,

AAL1. Since μis a zero-one measure on AL1 ×L2, it follows that μ1 is a zero-one

measure onAL1, that is,μ1∈IRL1.

Supposeμ1A μA×Y 1; there existsA×YLL2∈L∗such thatμLY 1

andμLL2 1. Thenμ1L1 μLY 1 andL1 ⊂Awhich shows thatμ1 ∈IRL1.

Similarly takeμ2onAL2defined byμ2B μX×B,BAL2.

Then, as beforeμ2is regular onL2.

Finally for anyAAL1and anyBAL2we haveμμ2A×B μ12B

μA×YμX×B μA×YX×B μAX×YB μA×Bwhich shows that

μμμ2, and thereforeIRL1×IRL2 IRL.

Theorem 3.2the product of lattices/σ-smooth regular measures. LetX, Y be abstract sets and

letL1, L2be lattices of subsets ofX and Y, respectively. Then IσRL1×IσRL2 IσRL1×L2.

Proof. LetμIσRL1andνIσRL2. Hence forA1n∈AL1withA1n↓ ∅we haveμA1n → 0

and forA2nAL2withA2n↓ ∅we haveνA2n → 0,n1,2, . . ..

Consider the sequence {Bn} of sets from AL1× AL2. As in Theorem 3.1Bn

k

i1Ai1n×Ai2n, disjoint union andAi1n∈AL1,Ai2n∈AL2.

Suppose thatBn↓ ∅, that is,Ai1n×Ai2n↓ ∅for alli. ThereforeA1n↓ ∅orA2n↓ ∅or both:

μ×νBn μ×ν

k

i1

Ai

1n×Ai2n

k

i1

μ×ν Ai

1n×Ai2n

k

i1

μ Ai 1n

ν Ai 2n

−→0, thereforeμ×νIσRL1×L2.

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Conversely, letμIσRL1×L2and defineμ1onAL1by

μ1A μA×Y, AAL1. 3.3

If{An}is a sequence of sets withAnAL1andAn ↓ ∅, thenAn×Y↓ ∅, and since

μIσL

1×L2it follows thatμAn×Y → 0.

Thereforeμ1∈IσL1.

Similarly, definingμ2onAL2byμ2B μX×B,BAL2we getμ2∈IσL2.

Henceμμμ2 ∈IσL1×IσL2 IσL1×L2.

Theorem 3.3product of supports of measures. LetX, Y be abstract sets and let L1, L2be lattices

of subsets ofX and Y, respectively. The following statements are true:

aifμμμ2∈IL1×IL2 IL1×L2thenSμ SμSμ2;

bif L1andL2are compact lattices thenL is compact.

Proof. We have

aSμ Sμμ2 ∩{LL2∈L1×L2/μLL2 μX×Y},

Sμ1

×Sμ2

LL2/L1∈L1, L2∈L2, μ1L1 μ1X, μ2L2 μ2Y

, 3.4

ButμLL2 μμ2LL2 μ1L1μ2L2andμX×Y μμ2X×Y

μ1Xμ2Y,

bSμ SμSμ2/∅, sinceSμi/∅,Libeing compact.

Theorem 3.4 product of Wallman spaces/Wallman topologies. Consider the spaces IRLi

with the Wallman topologiestWiLi,i1,2.

It is known that the topological spaces IRLi,tWiLi are compact and T1. Then the

topological spaceIRL1×IRL2, tW1L1×tW2L2is also compact andT1.

Proof. SinceIRLiare compact topological spaces,

SL1 μ

W1L1∈W1L1/μW1L1 1

/

,

SL2

νW2L2∈W2L2/νW2L2 1

/

.

3.5

We have

μW1A μA, μIRL1, AAL1, μIRW1L1,

νW2B νB, νIRL2, BAL2, νIRW2L2.

3.6

Therefore

μ×νW1A×W2B μ×νA×B μAνB,

μ×νW1A×W2B μW1AνW2B μAνB,

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so thatμ×ν μ×νIRW1L1×IRW2L2, and thenSL1×L2μ×ν SL1×L2μ×ν

SL1μ×SL2ν/∅ ⇒IRL1×IRL2is compact.

To show thatIRL1×IRL2is aT1-space, let μ, νIRL and supposeμ /ν. Since

μμμ2withμ1, μ2∈IRL1andννν2withν1, ν2∈IRL2we getμ11andμ22.

There existL1,L1∈L1andL2,L2 ∈L2with

μ1∈W1L1, ν1∈W1L1; ν1∈W1 L1

, μ1∈W1 L1

,

μ2∈W2L2, ν2∈W2L2; ν2∈W2 L2

, μ2∈W2 L2

.

3.8

Thereforeμ1L1 μ2L2 1,ν1L1 ν2L2 0,μ1L1 μ2L2 0,ν1L1 ν2L2 1

which impliesμWLL2,νWLL2;νWLL2,μWLL2

.

Theorem 3.5product of normal lattices. LetX, Y be abstract sets and let L1, L2 be normal

lattices of subsets ofX and Y, respectively. Then L is a normal lattice of subsets of X×Y.

Proof. LetμILandν, ρIRLsuch thatμν, ρonL.

Then, sinceμμμ2∈IL1×IL2,ννν2 ∈IRL1×IRL2andρρρ2 ∈

IRL1×IRL2, we obtainμiνi,ρionLi,i1,2.

Linormal lattices⇒νiρi; thereforeνν2 ρρ2, that is,νρ.

### 3.3. Examples

1LetX, Y be topological spaces and let O1,O2be the lattices of open sets ofX and Y,

respectively. Consider the product spaceX×Y with a base of open sets given by

{OO2/O1∈O1, O2∈O2}. 3.9

We have

OO2

x, yX×Y/x, y/OO2

x, y/x, yX×O 2

orx, yO1 ×Y

X×O 2

O

Y

X×F2∪FY.

3.10

Hence F tLF1 ×F2 where F1, F2 are the lattices of closed sets of X and Y,

respectively.

2Let X, Y be topological T3.5-spaces and let Z1, Z2 be the lattices of zero sets of

continuous functions ofX and Y, respectively. Then for the product space X×Y

we consider a base of open sets given by

Z

Z2/Z1∈Z1, Z2∈Z2

3.11

such that any open set fromX×Y is of the formO αZ×Z and any closed

set is

F O

αZ1ε×YX×Z2α∈tLZZ2

3.12

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### 4. The General Case of Product of Lattices

Let{Xα}α∈Λbe a collection of abstract setsΛan arbitrary index setand letLαbe the lattice

of subsets ofXαfor allα.

We denote L∗ α∈Λ Lα α∈Λ

Lα/LαLα, LαXα for almost allα

. 4.1

### 4.1. Results

Theorem 4.1the product of lattices/regular measures. One has

α∈Λ

IRLα IRL IR

α∈Λ

Lα

. 4.2

Proof. We note thatα∈ΛALα Aα∈ΛLα ALand thatα∈ΛALαis the collection of

all finite cylinder sets which means that ifBα∈ΛALαthenBis a cylinder set for which

there exists a nonempty finite subsetF {α1, α2, . . . , αn}ofΛand a subsetEFα∈FALα

such thatBP−1F EFwith

PF :

α∈Λ

Xα−→

α∈Λ

XαXα1×Xα2× · · · ×Xαn, :

α∈Λ

Xα−→Xα. 4.3

LetμαIRLαfor allα∈Λwithμα:ALαand define

μ

α∈Λ

μα

α∈Λ

IRLα, μ:

α∈Λ

ALα. 4.4

LetAα∈ΛALαwithμA 1. Thenα∈ΛμαA α∈ΛμαPF−1EF 1 that

is

α∈F

ALα P

−1

F

−−−→

α

ALα

αμα

−−−−−→ {0,1} 4.5

and forEFα∈FALαwe have

α∈Λ

μαPF−1

EF

α∈Λ

μα PF−1EF α∈F EF

μαμα2× · · · ×μαn

EF 1.

4.6

As in the finite case, we getEF2× · · ·LαnwhereLαiLαiandμαiLαi 1 for

(8)

which shows thatμA sup{μPF−11 ×2× · · ·Lαn/PF−11 ×2× · · ·LαnAand

P−1

F 2× · · ·Lαn

α∈ΛLαL∗}; henceμisL-regular.

Conversely, letμIRL IRα∈ΛLαand defineμαonALαby

μαA μ

A×

β∈Λ−{α}

Xβ

, AALα, that is,μαA μ P−1

α A

. 4.7

Since μ is a zero-one valued measure on Aα∈ΛLα it follows from the above

definition thatμαILα. IfμαA 1, thenμPα−1A 1, and sinceμisL-regular, there

existsβ∈Λsuch that−1Aβ∈ΛLβL∗andμβ∈ΛLβ 1.

ThenPα−1−1AandμαLα μαPα−1 1.

ThereforeμαA sup{μαLα/LαA, LαLα}, that is,μαIRLα. Next, if B

L∗, we may considerB P−1

F 1 ×2× · · ·Lαnand then

α∈ΛμαB α∈ΛμαPF−11 ×

2 × · · ·Lαn

α∈FμαLα1 ×2 × · · ·Lαn μα1 ×μα2 × · · · ×μαnLα1 ×2 × · · ·Lαn

μα11μα22· · ·μαnLαn μPα−111· · ·μPα−1nLαn. If

α∈ΛμαB 1, thenμPα−1i Lαi

for alli; henceμni1Pα−1i Lαi 1 andμPF−12× · · ·Lαn μ

αLα 1, that is,

μB 1. Thusμ αμαIRL∗, and thenμαμαonαALα.

Theorem 4.2the product of normal lattices. LetLαbe a lattice of subsets ofXα. Then

aifμαμαIα∈ΛLα α∈ΛILαwe haveSμ α∈ΛSμα;

bifLα disjunctive for allα∈Λ, thenL Lα∈ΛLαis a disjunctive lattice of subsets of

α∈ΛXα;

csuppose thatLαis a normal lattice of subsets ofXαfor allα∈Λ; thenLLα∈ΛLαis

a normal lattice of subsets ofα∈ΛXα.

Proof. aWe haveSμα ∩{Lα/μαLα μαXα 1}andSμ Sα∈Λμα

∩{α∈Λα∈ΛLα/μαLα μαXα 1}. But μαLα 1 implies

αμαα∈ΛLα 1. ThenSμ α∈ΛSμα.

bLet x xαα∈Λα∈ΛXα Since α∈Λμxα

α∈ΛAα μxα∈ΛAα we get

α∈Λμxα μx.

Lαdisjunctive impliesμxαIRLαfor allα∈Λand then

α∈Λμxα

α∈ΛIRLα

IRα∈ΛLα; therefore μxIRα∈ΛLα which proves that L Lα∈ΛLα is

disjunctive.

cLetμILandν, ρIRLsuch thatμν, ρonL.

Butμ αμαα∈ΛILαand bothν α∈να,ρ α∈ραα∈ΛIRLαand

thenαμαα∈ναandαμαα∈ραonL withμαAα μPα−1,ναAα

νP−1

α forALα. By the previous work we getμαναandμαραonLα.

Since each Lα is normal it follows that να ρα for allα ∈ Λ, and therefore ν

ναα∈Λ ραα∈Λρwhich proves thatL is a normal lattice.

### 4.2. Examples

3LetXαbe a topologicalT3.5-spaces and letLαZαbe the replete lattices of zero sets

of continuous functions ofXαfor allαA.

(9)

Consider a latticeZ of subsets ofα∈AXαsuch that

α∈A

Zαt

α∈A

Zα

. 4.8

ThenZ is replete, andα∈AXαis realcompact.

4LetXαbe aT2and 0-dimensional space and letLαCαbe the replete lattice of clopen

sets for allαA. Then eachXα is said to beN-compact. Consider any lattice C of

subsets ofα∈AXαsuch thatα∈ACαtα∈ACαtα∈AZα F and C is replete

andα∈AXαisN-compact.

### Acknowledgment

The author is grateful to the late Dr. George Bachman for suggesting this research.

### References

1 G. Bachman and P. D. Stratigos, “On measure repleteness and support for lattice regular measures,”

International Journal of Mathematics and Mathematical Sciences, vol. 10, no. 4, pp. 707–724, 1987.

2 Z. Frol´ık, “Realcompactness is a Baire-measurable property,” Bulletin de l’Acad´emie Polonaise des

Sciences, vol. 19, pp. 617–621, 1971.

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