International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID 920737,9pages
doi:10.1155/2011/920737
Research Article
Topological Aspects of the Product of Lattices
Carmen Vlad
Department of Mathematics, Pace University, Pleasantville, NY 10570, USA
Correspondence should be addressed to Carmen Vlad,[email protected]
Received 8 June 2011; Accepted 23 July 2011
Academic Editor: Alexander Rosa
Copyrightq2011 Carmen Vlad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
LetX be an arbitrary nonempty set and L a lattice of subsets of X such that∅,X∈L. 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∅,X∈L. 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
2.1. Lattice Terminology
The latticeL is called:
δ-lattice ifL is closed under countable intersections; complement generated ifL ∈ L
implies L ∩Ln, n 1, . . . ,∞, Ln ∈ Lwhere prime denotes the complement;
disjunctive if forx∈X andL1 ∈L such thatx /∈ L1there existsL2 ∈L withx ∈L2
andL1∩L2 ∅; separatingorT1ifx, y ∈X andx /yimplies there existsL ∈ L
such thatx∈L,y /∈ L;T2 if forx, y ∈X andx /ythere existL1, L2 ∈L such that
x∈L
1,y∈L2, andL1∩L2∅; normal if for anyL1, L2∈L withL1∩L2∅there exist
L3, L4∈L withL1⊂L3,L2⊂L4, andL3∩L4 ∅; compact if for any collection{Lα}of
sets ofL with∩αLα ∅, there exists a finite subcollection with empty intersection;
countably compact if for any countable collection {Lα} of sets ofL with∩αLα ∅,
there exists a finite subcollection with empty intersection.
2.2. Measure Terminology
MLdenotes those nonnegative, finite, finitely additive measures onAL.
A measureμ∈MLis 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,
that is, countably additive.
L-regular if for anyA∈AL,
μA supμL|L⊂A, L∈L. 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 measureμ1×μ2∈ML1×L2is defined by
μ1×μ2
L1×L2 μ1L1μ2L2 ∀L1∈AL1, L2 ∈AL2. 2.2
2.3. Lattice-Measure Correspondence
The support ofμ∈MLisSμ ∩{L∈L/μL μX}.
In caseμ∈ILthen the support isSμ ∩{L∈L/μL 1}.
With this notation and in light of the above correspondences, we now note:
For anyμ∈IL, there existsν∈IRLsuch thatμ≤νonLi.e.,μL≤νLfor all
L is compact if and only ifSμ/∅for everyμ∈IRL.L is countably compact if and
only ifIRL IσRL.L is normal if and only if for eachμ∈IL, there exists a unique
ν∈IRLsuch thatμ≤νonL. L is regular if and only if wheneverμ1, μ2∈ILand
μ1≤μ2onL, thenSμ1 Sμ2.L is replete if and only if for anyμ∈IσRL,Sμ/∅.
L is prime-complete if and only if for anyμ∈IσL,Sμ/∅.
Finally, ifμxis the measure concentrated atx∈X, thenμx ∈IRL, for allx∈X 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 μ∈IσRL/μL 1
, L∈L 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/L ∈ L} is a lattice
inIσRL, isomorphic toL under the mapL → WσL,L ∈L. WσLis replete and a base for
the closed setstWσL, all arbitrary intersections of sets ofWσL.
IfA ∈AL, thenWσA {μ ∈ IσRL/μA 1}and the following statements are
true:
WσA∪B WσA∪WσB,
WσA∩B WσA∩WσB,
WσAWσA,
A⊃BiffWσA⊃WσB,
AWσL WσAL.
2.4
The Induced Measure
Letμ∈IσRLand consider the induced measureμ∈IσRWσL, defined by
μWσA μA, A∈AL. 2.5
The mapμ → μis a bijection betweenIσRLandIσRWσL.
3. The Case of Finite Product of Lattices
3.1. Notations
LetX, Y be abstract sets and L1 a lattice of subsets ofX and L2 a lattice of subsets ofY. We denote:
1L∗L1×L2 {L1×L2/L1∈L1, L2 ∈L2},
2LLL∗, the lattice generated byL∗. ¡list-item¿¡label/¿
We have the following:
4AL∗ AL,
5SLμ SL∗μ,
6IσL∗ IσL,
7IRL∗ IRL.
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. ForA ∈ AL1×AL2 AL1×L2, we haveA i1n Ai1×Ai2, disjoint union and
Ai
1∈AL1,Ai2∈AL2.
Letμ∈IRL1andν∈IRL2and 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×νL1×L2 μL1νL2 1 andL1×L2∈L∗.
If we letML1×L2⊂Ai1×Ai2⊂A, then
μ×νA supμ×νM/M⊂A, M∈L∗⇒μ×ν∈IRL∗. 3.1
Conversely, let μ ∈ IRL∗ IRLand define μ1 on AL1 byμ1A μA×Y,
A ∈ AL1. 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×Y⊃L1×L2∈L∗such thatμL1×Y 1
andμL1×L2 1. Thenμ1L1 μL1×Y 1 andL1 ⊂Awhich shows thatμ1 ∈IRL1.
Similarly takeμ2onAL2defined byμ2B μX×B,B∈AL2.
Then, as beforeμ2is regular onL2.
Finally for anyA∈AL1and anyB∈AL2we haveμ1×μ2A×B μ1Aμ2B
μA×YμX×B μA×Y∩X×B μA∩X×Y∩B μA×Bwhich shows that
μμ1×μ2, 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. Letμ∈IσRL1andν∈IσRL2. Hence forA1n∈AL1withA1n↓ ∅we haveμA1n → 0
and forA2n∈AL2withA2n↓ ∅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×ν∈IσRL1×L2.
Conversely, letμ∈IσRL1×L2and defineμ1onAL1by
μ1A μA×Y, A∈AL1. 3.3
If{An}is a sequence of sets withAn ∈AL1andAn ↓ ∅, thenAn×Y↓ ∅, and since
μ∈IσL
1×L2it follows thatμAn×Y → 0.
Thereforeμ1∈IσL1.
Similarly, definingμ2onAL2byμ2B μX×B,B∈AL2we getμ2∈IσL2.
Henceμμ1×μ2 ∈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:
aifμμ1×μ2∈IL1×IL2 IL1×L2thenSμ Sμ1×Sμ2;
bif L1andL2are compact lattices thenL is compact.
Proof. We have
aSμ Sμ1×μ2 ∩{L1×L2∈L1×L2/μL1×L2 μX×Y},
Sμ1
×Sμ2
∩L1×L2/L1∈L1, L2∈L2, μ1L1 μ1X, μ2L2 μ2Y
, 3.4
ButμL1×L2 μ1×μ2L1×L2 μ1L1μ2L2andμX×Y μ1×μ2X×Y
μ1Xμ2Y,
bSμ Sμ1×Sμ2/∅, 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
ν∩W2L2∈W2L2/νW2L2 1
/
∅.
3.5
We have
μW1A μA, μ∈IRL1, A∈AL1, μ∈IRW1L1,
νW2B νB, ν∈IRL2, B∈AL2, ν ∈IRW2L2.
3.6
Therefore
μ×νW1A×W2B μ×νA×B μAνB,
μ×νW1A×W2B μW1AνW2B μAνB,
so thatμ×ν μ×ν ∈ IRW1L1×IRW2L2, and thenSL1×L2μ×ν SL1×L2μ×ν
SL1μ×SL2ν/∅ ⇒IRL1×IRL2is compact.
To show thatIRL1×IRL2is aT1-space, let μ, ν ∈ IRL and supposeμ /ν. Since
μμ1×μ2withμ1, μ2∈IRL1andνν1×ν2withν1, ν2∈IRL2we getμ1/ν1andμ2/ν2.
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 impliesμ∈WL1×L2,ν∈WL1×L2;ν∈WL1×L2,μ∈WL1×L2
.
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. Letμ∈ILandν, ρ∈IRLsuch thatμ≤ν, ρonL.
Then, sinceμμ1×μ2∈IL1×IL2,νν1×ν2 ∈IRL1×IRL2andρρ1×ρ2 ∈
IRL1×IRL2, we obtainμi≤νi,ρionLi,i1,2.
Linormal lattices⇒νiρi; thereforeν1×ν2 ρ1×ρ2, 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
{O1×O2/O1∈O1, O2∈O2}. 3.9
We have
O1×O2
x, y∈X×Y/x, y/∈O1×O2
x, y/x, y∈X×O 2
orx, y∈O1 ×Y
X×O 2
∪O
1×Y
X×F2∪F1×Y.
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
1×Z2/Z1∈Z1, Z2∈Z2
3.11
such that any open set fromX×Y is of the formO αZ1α×Z2α and any closed
set is
F O
αZ1ε×Y∪X×Z2α∈tLZ1×Z2
3.12
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/Lα∈Lα, 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 ifB∈α∈ΛALαthenBis a cylinder set for which
there exists a nonempty finite subsetF {α1, α2, . . . , αn}ofΛand a subsetEF ∈α∈FALα
such thatBP−1F EFwith
PF :
α∈Λ
Xα−→
α∈Λ
XαXα1×Xα2× · · · ×Xαn, Pα:
α∈Λ
Xα−→Xα. 4.3
Letμα∈IRLαfor allα∈Λwithμα:ALαand define
μ
α∈Λ
μα∈
α∈Λ
IRLα, μ:
α∈Λ
ALα. 4.4
LetA∈α∈ΛALαwithμA 1. Thenα∈ΛμαA α∈ΛμαPF−1EF 1 that
is
α∈F
ALα P
−1
F
−−−→
α
ALα
αμα
−−−−−→ {0,1} 4.5
and forEF ∈α∈FALαwe have
α∈Λ
μαPF−1
EF
α∈Λ
μα PF−1EF α∈F EF
μα1×μα2× · · · ×μαn
EF 1.
4.6
As in the finite case, we getEF ⊃Lα1×Lα2× · · ·LαnwhereLαi ∈LαiandμαiLαi 1 for
which shows thatμA sup{μPF−1Lα1 ×Lα2× · · ·Lαn/PF−1Lα1 ×Lα2× · · ·Lαn ⊂ Aand
P−1
F Lα1×Lα2× · · ·Lαn∈
α∈ΛLαL∗}; henceμisL-regular.
Conversely, letμ∈IRL IRα∈ΛLαand defineμαonALαby
μαA μ ⎛
⎝A×
β∈Λ−{α}
Xβ ⎞
⎠, A∈ALα, 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 thatμα ∈ ILα. IfμαA 1, thenμPα−1A 1, and sinceμisL-regular, there
existsβ∈ΛLβsuch thatPα−1A⊃β∈ΛLβ∈L∗andμβ∈ΛLβ 1.
ThenPα−1Lα⊂Pα−1AandμαLα μαPα−1Lα 1.
ThereforeμαA sup{μαLα/Lα ⊂ A, Lα ∈ Lα}, that is,μα ∈ IRLα. Next, if B ∈
L∗, we may considerB P−1
F Lα1 ×Lα2× · · ·Lαnand then
α∈ΛμαB α∈ΛμαPF−1Lα1 ×
Lα2 × · · ·Lαn
α∈FμαLα1 ×Lα2 × · · ·Lαn μα1 ×μα2 × · · · ×μαnLα1 ×Lα2 × · · ·Lαn
μα1Lα1μα2Lα2· · ·μαnLαn μPα−11Lα1· · ·μPα−1nLαn. If
α∈ΛμαB 1, thenμPα−1i Lαi
for alli; henceμni1Pα−1i Lαi 1 andμPF−1Lα1×Lα2× · · ·Lαn μ
αLα 1, that is,
μB 1. Thusμ αμα∈IRL∗, and thenμαμαonαALα.
Theorem 4.2the product of normal lattices. LetLαbe a lattice of subsets ofXα. Then
aifμαμα∈Iα∈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 ∈Lα/μαLα μαXα 1}andSμ Sα∈Λμα
∩{α∈ΛLα ∈ α∈ΛLα/μαLα μαXα 1}. But μαLα 1 implies
αμαα∈ΛLα 1. ThenSμ α∈ΛSμα.
bLet x xαα∈Λ ∈ α∈ΛXα Since α∈Λμxα
α∈ΛAα μxα∈ΛAα we get
α∈Λμxα μx.
Lαdisjunctive impliesμxα ∈IRLαfor allα∈Λand then
α∈Λμxα ∈
α∈ΛIRLα
IRα∈ΛLα; therefore μx ∈ IRα∈ΛLα which proves that L Lα∈ΛLα is
disjunctive.
cLetμ∈ILandν, ρ∈IRLsuch thatμ≤ν, ρonL.
Butμ αμα ∈ α∈ΛILαand bothν α∈να,ρ α∈ρα ∈ α∈ΛIRLαand
thenαμα ≤α∈ναandαμα≤α∈ραonL withμαAα μPα−1Aα,ναAα
νP−1
α AαforAα∈ALα. By the previous work we getμα≤ναandμα≤ραonLα.
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 allα∈A.
Consider a latticeZ of subsets ofα∈AXαsuch that
α∈A
Zα⊂t
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 allα∈ A. Then eachXα is said to beN-compact. Consider any lattice C of
subsets ofα∈AXαsuch thatα∈ACα⊂tα∈ACα⊂tα∈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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