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,cvlad@pace.edu

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.

Let**X be an arbitrary nonempty set and L a lattice of subsets of X such that**∅,**X**∈**L. AL**denotes
the algebra generated by**L, and ML**denotes those nonnegative, finite, finitely additive measures
on**AL**. 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.

Let**X be an arbitrary nonempty set and L a lattice of subsets of X such that**∅,**X**∈**L. A**

lattice**L is a partially ordered set any two elements***x, y*of which have both sup*x, y*and

inf*x, y*.

**AL**denotes the algebra generated by**L;***σ***L**is the*σ*algebra generated by**L;***δ***L**

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

intersections of sets from**L;***ρ***L**is the smallest class closed under countable intersections

**2.1. Lattice Terminology**

**2.1. Lattice Terminology**

The lattice**L is called:**

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

**L**

implies *L* ∩*L _{n}*,

*n*1

*, . . . ,*∞,

*L*∈

_{n}**L**where prime denotes the complement;

*disjunctive if forx*∈**X and***L*1 ∈**L such that***x /*∈ *L*1there exists*L*2 ∈**L with***x* ∈*L*2

and*L*1∩*L*2 ∅*; separating*or**T1**if*x, y* ∈**X and***x /y*implies there exists*L* ∈ **L**

such that*x*∈*L*,*y /*∈ *L*;**T _{2}** if for

*x, y*∈

**X and**

*x /y*there exist

*L*1

*, L*2 ∈

**L such that**

*x*∈*L*

1,*y*∈*L*2, and*L*1∩*L*2∅*; normal if for anyL*1*, L*2∈**L with***L*1∩*L*2∅there exist

*L*3*, L*4∈**L with***L*1⊂*L*_{3},*L*2⊂*L*_{4}, and*L*_{3}∩*L*_{4} ∅*; compact if for any collection*{*Lα*}of

sets of**L with**∩* _{α}L_{α}* ∅, there exists a finite subcollection with empty intersection;

*countably compact if for any countable collection* {*L _{α}*} of sets of

**L with**∩

*∅,*

_{α}L_{α}there exists a finite subcollection with empty intersection.

**2.2. Measure Terminology**

**2.2. Measure Terminology**

**ML**denotes those nonnegative, finite, finitely additive measures on**AL**.

A measure*μ*∈**ML**is called:

**σ**-smooth on**L if for all sequences**{*L _{n}*}of sets of

**L with**

*L*↓ ∅,

_{n}*μL*→ 0;

_{n}**σ**-smooth on**AL**if for all sequences{*A _{n}*}of sets of

**AL**with

*A*↓ ∅,

_{n}*μA*→ 0,

_{n}that is, countably additive.

* L-regular if for anyA*∈

**AL**,

*μA* sup*μL*|*L*⊂*A, L*∈**L***.* 2.1

We denote by**M**_{R}**L**the set of**L-regular measures of ML**;**M**_{σ}**L**the set of*σ*-smooth

measures on**L, of ML**;**M***σ***L**the set of*σ*-smooth measures on**AL**of**ML**;**M***σ _{R}*

**L**the set

of**L-regular measures of M***σ***L**.

In addition,**IL**,**I**_{R}**L**,**I**_{σ}**L**,**I***σ***L**,**I***σ _{R}*

**L**are the subsets of the corresponding

**M’s**

which consist of the nontrivial zero-one valued measures.

Finally, let**X***,***Y be abstract sets and L1**a lattice of subsets of**X and L2**a lattice of subsets

of**Y. Let***μ*1∈**ML1**and*μ*2∈**ML2**.

*The product measureμ*1×*μ*2∈**ML1**×**L2**is defined by

*μ*1×*μ*2

*L*1×*L*2 *μ*1*L*1*μ*2*L*2 ∀*L*1∈**AL1***, L*2 ∈**AL2***.* 2.2

**2.3. Lattice-Measure Correspondence**

**2.3. Lattice-Measure Correspondence**

*The support ofμ*∈**ML**is*Sμ* ∩{*L*∈**L***/μL* *μX*}.

In case*μ*∈**IL**then the support is*Sμ* ∩{*L*∈**L***/μL* 1}.

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

For any*μ*∈**IL**, there exists*ν*∈**I***R***L**such that*μ*≤*ν*on**L**i.e.,*μL*≤*νL*for all

* L is compact if and only ifSμ/*∅for every

*μ*∈

**I**

*R*

**L**.

**L is countably compact if and**only if**I**_{R}**L** **I****σ**_{R}**L**.* L is normal if and only if for eachμ*∈

**IL**, there exists a unique

*ν*∈**I***R***L**such that*μ*≤*ν*on* L. L is regular if and only if wheneverμ*1

*, μ*2∈

**IL**and

*μ*1≤*μ*2on**L, then***Sμ*1 *Sμ*2.* L is replete if and only if for anyμ*∈

**I**

**σ**_{R}**L**,

*Sμ/*∅.

* L is prime-complete if and only if for anyμ*∈

**I**

**σ****L**,

*Sμ/*∅.

Finally, if*μx*is the measure concentrated at*x*∈**X, then***μx* ∈**I***R***L**, for all*x*∈**X if**

and only if**L is disjunctive.**

For further results and related matters see1–3.

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

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

*The Wallman topology in***I****σ**** _{R}L**is obtained by taking all

*WσL* *μ*∈**I****σ****RL***/μL* 1

*, L*∈**L** 2.3

as a base for the closed sets in **I****σ**** _{R}L** and then

**I**

**σ**

_{R}L*is called the general Wallman space*

associated with**X and L. Assuming L is disjunctive, W***σ***L** {*WσL/L* ∈ **L**} is a lattice

in**I****σ**_{R}**L**, isomorphic to**L under the map***L* → *WσL*,*L* ∈**L. W***σ***L**is replete and a base for

the closed sets**tW**_{σ}**L**, all arbitrary intersections of sets of**W**_{σ}**L**.

If*A* ∈**AL**, then*WσA* {*μ* ∈ **I****σ**_{R}L*/μA* 1}and the following statements are

true:

*WσA*∪*B* *WσA*∪*WσB,*

*WσA*∩*B* *WσA*∩*WσB,*

*WσAWσA,*

*A*⊃*B*iﬀ*W _{σ}A*⊃

*W*

_{σ}B,**AW****σ****L** **W****σ****AL***.*

2.4

*The Induced Measure*

Let*μ*∈**I****σ**_{R}L*and consider the induced measureμ*∈**I****σ**_{R}W**σ****L***, defined by*

*μWσA* *μA, A*∈**AL***.* 2.5

The map*μ* → *μ*is a bijection between**I****σ**** _{R}L**and

**I**

**σ**

_{R}W

_{σ}**L**.

**3. The Case of Finite Product of Lattices**

**3.1. Notations**

**3.1. Notations**

Let**X, Y be abstract sets and L1** a lattice of subsets of**X and L2** a lattice of subsets of* Y. We*
denote:

1**L**∗**L1**×**L2** {*L*1×*L*2*/L*1∈**L1***, L*2 ∈**L2**},

2**LLL**∗, the lattice generated by**L**∗. ¡list-item¿¡label/¿

We have the following:

4**AL**∗ **AL**,

5**SL***μ* **SL**∗*μ*,

6**I****σ****L**∗ **I****σ****L**,

7**IRL**∗ **IRL**.

**3.2. Results**

**3.2. Results**

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

**L1****, L****2***be lattices of subsets of X and Y, respectively. Then I*

**RL1**×

**IRL2**

**IRL**

*.*

*Proof. ForA* ∈ **AL1**×**AL2** **AL1**×**L2**, we have*A* *i1n* *Ai*1×*Ai*2, disjoint union and

*Ai*

1∈**AL1**,*Ai*2∈**AL2**.

Let*μ*∈**I _{R}L_{1}**and

*ν*∈

**I**and consider

_{R}L_{2}*μ*×

*ν*defined on

**AL**×

_{1}**AL**.

_{2}If*μ*×*νA* 1, then*μ*×*νAi*_{1}×*Ai*_{2} 1 for some*i*.

Then*μAi*_{1}*νAi*_{2} 1, and since*μ*and*ν*are zero-one valued measures,*μAi*_{1}1 and

*νAi*

2 1. By the regularity of*μ*and*ν*there exist*L*1 ⊂*Ai*1,*L*1∈**L1**with*μL*1 1 and*L*2⊂*Ai*2,

*L*2∈**L2**with*νL*2 1.

Therefore*μ*×*νL*1×*L*2 *μL*1*νL*2 1 and*L*1×*L*2∈**L**∗.

If we let*ML*1×*L*2⊂*Ai*_{1}×*Ai*_{2}⊂*A*, then

*μ*×*νA* sup*μ*×*νM/M*⊂*A, M*∈**L**∗⇒*μ*×*ν*∈**I _{R}L**∗

*.*3.1

Conversely, let *μ* ∈ **IRL**∗ **IRL**and define *μ*1 on **AL1** by*μ*1*A* *μA*×**Y**,

*A* ∈ **AL1**. Since *μ*is a zero-one measure on **AL1** ×**L2**, it follows that *μ*1 is a zero-one

measure on**AL1**, that is,*μ*1∈**IRL1**.

Suppose*μ*1*A* *μA*×**Y** 1; there exists*A*×**Y**⊃*L*1×*L*2∈**L∗**such that*μL*1×**Y** 1

and*μL*1×*L*2 1. Then*μ*1*L*1 *μL*1×**Y** 1 and*L*1 ⊂*A*which shows that*μ*1 ∈**IRL1**.

Similarly take*μ*2on**AL2**defined by*μ*2*B* *μ***X**×*B*,*B*∈**AL2**.

Then, as before*μ*2is regular on**L2.**

Finally for any*A*∈**AL _{1}**and any

*B*∈

**AL**we have

_{2}*μ*1×

*μ*2

*A*×

*B*

*μ*1

*Aμ*2

*B*

*μA*×**Y***μ***X**×*B* *μA*×**Y**∩**X**×*B* *μA*∩**X**×**Y**∩*B* *μA*×*B*which shows that

*μμ*1×*μ*2, and therefore**IRL1**×**IRL2** **IRL**.

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

*let***L1****, L****2***be lattices of subsets of X and Y, respectively. Then Iσ*

**×**

_{R}L1**I**

**σ**

_{R}L2**I**

**σ****×**

_{R}L1**L2**

*.*

*Proof. Letμ*∈**I****σ**** _{R}L1**and

*ν*∈

**I**

**σ****. Hence for**

_{R}L2*A*1n∈

**AL1**with

*A*1n↓ ∅we have

*μA*1n → 0

and for*A*_{2n}∈**AL _{2}**with

*A*

_{2n}↓ ∅we have

*νA*

_{2n}→ 0,

*n*1

*,*2

*, . . .*.

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

_{k}

*i1Ai*1n×*Ai*2n, disjoint union and*Ai*1n∈**AL1**,*Ai*2n∈**AL2**.

Suppose that*Bn*↓ ∅, that is,*Ai*_{1n}×*Ai*2n↓ ∅for all*i*. Therefore*A*1n↓ ∅or*A*2n↓ ∅or both:

*μ*×*νBn* *μ*×*ν*

_{}_{k}

*i1*

*Ai*

1n×*Ai*2n

*k*

*i1*

*μ*×*ν* *Ai*

1n×*Ai*2n

*k*

*i1*

*μ* *Ai*
1n

*ν* *Ai*
2n

−→0*,* therefore*μ*×*ν*∈**I****σ**** _{R}L1**×

**L2**

*.*

Conversely, let*μ*∈**I****σ**** _{R}L1**×

**L2**and define

*μ*1on

**AL1**by

*μ*1*A* *μA*×**Y***, A*∈**AL1***.* 3.3

If{*An*}is a sequence of sets with*An* ∈**AL1**and*An* ↓ ∅, then*An*×**Y**↓ ∅, and since

*μ*∈**I****σ**_{}_{L}

**1**×**L2**it follows that*μAn*×**Y** → 0.

Therefore*μ*1∈**I****σ****L1**.

Similarly, defining*μ*2on**AL2**by*μ*2*B* *μ***X**×*B*,*B*∈**AL2**we get*μ*2∈**I****σ****L2**.

Hence*μμ*1×*μ*2 ∈**I****σ****L1**×**I****σ****L2** **I****σ****L1**×**L2**.

**Theorem 3.3**product of supports of measures*. Let X, Y be abstract sets and let L*

**1**

**, L****2**

*be lattices*

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

a*ifμμ*1×*μ*2∈**IL1**×**IL2** **IL1**×**L2***then***S***μ* **S***μ*1×**S***μ*2*;*

b*if* **L _{1}**

*and*

**L**

_{2}*are compact lattices then*

**L is compact.***Proof. We have*

a**S***μ* **S***μ*1×*μ*2 ∩{*L*1×*L*2∈**L1**×**L2***/μL*1×*L*2 *μ***X**×**Y**},

**S***μ*1

×**S***μ*2

∩*L*1×*L*2*/L*1∈**L1***, L*2∈**L2***, μ*1*L*1 *μ*1**X***, μ*2*L*2 *μ*2**Y**

*,* 3.4

But*μL*1×*L*2 *μ*1×*μ*2*L*1×*L*2 *μ*1*L*1*μ*2*L*2and*μ***X**×**Y** *μ*1×*μ*2**X**×**Y**

*μ*1**X***μ*2**Y**,

b**S***μ* **S***μ*1×**S***μ*2*/*∅, since**S***μi/*∅,**L***i*being compact.

**Theorem 3.4** product of Wallman spaces/Wallman topologies*. Consider the spaces* **IRL***i*

*with the Wallman topologies***tW**_{i}**L*** _{i},i*1

*,2.*

*It is known that the topological spaces* **IRL***i,***tW***i***L***i* *are compact and* **T1***. Then the*

*topological space***IRL1**×**IRL2***,* **tW1L1**×**tW2L2***is also compact and***T1***.*

*Proof. Since***I _{R}L**

*are compact topological spaces,*

_{i}**SL1** *μ*

∩*W*1*L*1∈**W1L1***/μW*1*L*1 1

*/*

∅*,*

**SL2**

*ν*∩*W*2*L*2∈**W2L2***/νW*2*L*2 1

*/*

∅*.*

3.5

We have

*μW*1*A* *μA,* *μ*∈**IRL1***,* *A*∈**AL1***,* *μ*∈**IRW1L1***,*

*νW*2*B* *νB,* *ν*∈**IRL2***,* *B*∈**AL2***,* *ν* ∈**IRW2L2***.*

3.6

Therefore

*μ*×*νW*1*A*×*W*2*B* *μ*×*νA*×*B* *μAνB,*

*μ*×*νW*1*A*×*W*2*B* *μW*1*AνW*2*B* *μAνB,*

so that*μ*×*ν* *μ*×*ν* ∈ **IRW1L1**×**IRW2L2**, and then*SL*1×L2*μ*×*ν* *SL*1×L2*μ*×*ν*

*SL*1*μ*×*SL*2*ν/*∅ ⇒**IRL1**×**IRL2***is compact.*

To show that**I _{R}L_{1}**×

**I**is a

_{R}L_{2}**T**

_{1}*-space, let*

*μ, ν*∈

**I**and suppose

_{R}L*μ /ν*. Since

*μμ*1×*μ*2with*μ*1*, μ*2∈**IRL1**and*νν*1×*ν*2with*ν*1*, ν*2∈**IRL2**we get*μ*1*/ν*1and*μ*2*/ν*2.

There exist*L*1*,L*1∈**L1**and*L*2*,L*2 ∈**L2**with

*μ*1∈*W*1*L*1*,* *ν*1∈*W*1*L*1; *ν*1∈*W*1 *L*1

*,* *μ*1∈*W*1 *L*1

_{}

*,*

*μ*2∈*W*2*L*2*,* *ν*2∈*W*2*L*2; *ν*2∈*W*2 *L*2

*,* *μ*2∈*W*2 *L*2

_{}

*.*

3.8

Therefore*μ*1*L*1 *μ*2*L*2 1,*ν*1*L*1 *ν*2*L*2 0,*μ*1*L*1 *μ*2*L*2 0,*ν*1*L*1 *ν*2*L*2 1

which implies*μ*∈*WL*1×*L*2,*ν*∈*WL*1×*L*2;*ν*∈*WL*1×*L*2,*μ*∈*WL*1×*L*2

.

**Theorem 3.5**product of normal lattices*. Let X, Y be abstract sets and let L*

_{1}

**, L**

_{2}*be normal*

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

**Y.***Proof. Letμ*∈**IL**and*ν, ρ*∈**IRL**such that*μ*≤*ν, ρ*on**L.**

Then, since*μμ*1×*μ*2∈**IL1**×**IL2**,*νν*1×*ν*2 ∈**IRL1**×**IRL2**and*ρρ*1×*ρ*2 ∈

**IRL1**×**IRL2**, we obtain*μi*≤*νi*,*ρi*on**L***i*,*i*1*,*2.

**L***i*normal lattices⇒*νiρi*; therefore*ν*1×*ν*2 *ρ*1×*ρ*2, that is,*νρ*.

**3.3. Examples**

**3.3. Examples**

1Let**X, Y be topological spaces and let O**1,**O**2*be the lattices of open sets of***X and Y,**

respectively. Consider the product space**X**×**Y with a base of open sets given by**

{*O*1×*O*2*/O*1∈**O**1*, O*2∈**O**2}*.* 3.9

We have

*O*1×*O*2

*x, y*∈**X**×**Y***/x, y/*∈*O*1×*O*2

*x, y/x, y*∈**X**×*O*
2

or*x, y*∈*O*_{1} ×**Y**

**X**×*O*
2

∪*O*

1×**Y**

**X**×*F*2∪*F*1×**Y***.*

3.10

Hence **F** **tLF**1 ×**F**2 where **F**1*,* **F**2 *are the lattices of closed sets of* **X and Y,**

respectively.

2Let **X, Y be topological T _{3.5}**-spaces and let

**Z**1,

**Z**2

*be the lattices of zero sets of*

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

we consider a base of open sets given by

*Z*

1×*Z*2*/Z*1∈**Z**1*, Z*2∈**Z**2

3.11

such that any open set from**X**×**Y is of the form***O* _{α}Z_{1α}×*Z*_{2α} and any closed

set is

*F* *O*_{}

*αZ*1ε×*Y*∪*X*×*Z*2α∈**tLZ**1×**Z**2

3.12

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

Let{**X*** _{α}*}

*be a collection of abstract setsΛan arbitrary index setand let*

_{α∈Λ}**L**

*be the lattice*

_{α}of subsets of**X*** _{α}*for all

*α*.

We denote
**L∗**_{}
*α∈Λ*
**L***α*
*α∈Λ*

*Lα/Lα*∈**L***α, Lα***X***α* for almost all*α*

*.* 4.1

**4.1. Results**

**4.1. Results**

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

*α∈Λ*

**IRL***α* **IRL** **IR**

*α∈Λ*

**L***α*

*.* 4.2

*Proof. We note that _{α∈Λ}*

**AL**

_{α}**A**

_{α∈Λ}**L**

_{α}**AL**and that

_{α∈Λ}**AL**

*is the collection of*

_{α}all finite cylinder sets which means that if*B*∈_{α∈Λ}**AL***α*then*B*is a cylinder set for which

there exists a nonempty finite subset*F* {*α*1*, α*2*, . . . , αn*}ofΛand a subset*EF* ∈*α∈F***AL***α*

such that*B*P−1_{F}*E _{F}*with

*PF* :

*α∈Λ*

**X***α*−→

*α∈Λ*

**X***α***X***α***1**×**X***α***2**× · · · ×**X***α***n***,* *Pα*:

*α∈Λ*

**X***α*−→**X***α.* 4.3

Let*μα*∈**IRL***α*for all*α*∈Λwith*μα*:**AL***α*and define

*μ*

*α∈Λ*

*μα*∈

*α∈Λ*

**IRL***α,* *μ*:

*α∈Λ*

**AL***α.* 4.4

Let*A*∈_{α∈Λ}**AL*** _{α}*with

*μA*1. Then

_{α∈Λ}μ_{α}A*−1*

_{α∈Λ}μ_{α}P_{F}*E*1 that

_{F}is

*α∈F*

*ALα* *P*

−1

*F*

−−−→

*α*

*ALα*

*αμα*

−−−−−→ {0*,*1} 4.5

and for*E _{F}* ∈

_{α∈F}**AL**

*we have*

_{α}_{}

*α∈Λ*

*μαP _{F}*−1

*EF*

_{}

*α∈Λ*

*μα* *P _{F}*−1

*EF*

_{}

*α∈F*

*EF*

*μα*1×*μα*2× · · · ×*μαn*

*EF* 1*.*

4.6

As in the finite case, we get*EF* ⊃*Lα*1×*Lα*2× · · ·*Lαn*where*Lαi* ∈**L***αi*and*μαiLαi* 1 for

which shows that*μA* sup{*μP _{F}*−1

*Lα*1 ×

*Lα*2× · · ·

*Lαn/PF*−1

*Lα*1 ×

*Lα*2× · · ·

*Lαn*⊂

*A*and

*P*−1

*F* *Lα*1×*Lα*2× · · ·*Lαn*∈

*α∈Λ***L***α***L**∗}; hence*μ*is**L-regular.**

Conversely, let*μ*∈**IRL** **IR***α∈Λ***L***α*and define*μα*on**AL***α*by

*μαA* *μ*
⎛

⎝_{A}_{×}

*β∈Λ−{α}*

**X***β*
⎞

⎠_{, A}_{∈}_{A}_{}_{L}_{α}_{}_{,}_{that is,}_{μ}_{α}_{}_{A}_{μ}* _{P}*−1

*α* *A*

*.* 4.7

Since *μ* is a zero-one valued measure on **A**_{α∈Λ}**L*** _{α}* it follows from the above

definition that*μα* ∈ **IL***α*. If*μαA* 1, then*μPα*−1*A* 1, and since*μ*is**L-regular, there**

exists* _{β∈Λ}Lβ*such that

*Pα*−1

*A*⊃

*β∈ΛLβ*∈

**L∗**and

*μβ∈ΛLβ*1.

Then*P _{α}*−1

*Lα*⊂

*Pα*−1

*A*and

*μαLα*

*μαPα*−1

*Lα*1.

Therefore*μ _{α}A* sup{

*μ*⊂

_{α}L_{α}/L_{α}*A, L*∈

_{α}**L**

*}, that is,*

_{α}*μ*∈

_{α}**I**

_{R}L*. Next, if*

_{α}*B*∈

**L∗**_{, we may consider}_{B}_{} * _{P}*−1

*F* *Lα*1 ×*Lα*2× · · ·*Lαn*and then

*α∈ΛμαB* *α∈ΛμαP _{F}*−1

*Lα*1 ×

*Lα*2 × · · ·*Lαn*

*α∈FμαLα*1 ×*Lα*2 × · · ·*Lαn* *μα*1 ×*μα*2 × · · · ×*μαnLα*1 ×*Lα*2 × · · ·*Lαn*

*μα*1*Lα*1*μα*2*Lα*2· · ·*μαnLαn* *μPα*−11*Lα*1· · ·*μPα*−1*nLαn*. If

*α∈ΛμαB* 1, then*μPα*−1*i* *Lαi*

for all*i*; hence*μn _{i1}P_{α}*−1

_{i}*Lαi*1 and

*μPF*−1

*Lα*1×

*Lα*2× · · ·

*Lαn*

*μ*

*αLα* 1, that is,

*μB* 1. Thus*μ* * _{α}μα*∈

**IRL∗**, and then

*μαμα*on

*αALα.*

**Theorem 4.2**the product of normal lattices*. Let***L**_{α}be a lattice of subsets of**X**_{α}. Then

a*ifμ _{α}μα*∈

**I**

*α∈Λ*

**L**

*α*

*α∈Λ*

**IL**

*αwe have*

**S**

*μ*

*α∈Λ*

**S**

*μα;*

b_{}*if***L**_{α}*disjunctive for allα*∈Λ*, then***L** **L**_{α∈Λ}**L**_{α}is a disjunctive lattice of subsets of

*α∈Λ***X***α;*

c*suppose that***L**_{α}is a normal lattice of subsets of**X*** _{α}for allα*∈Λ

*; then*

**LL**

_{α∈Λ}**L**

_{α}is*a normal lattice of subsets of _{α∈Λ}*

**X**

_{α}.*Proof.* aWe have**S***μα* ∩{*Lα* ∈**L***α/μαLα* *μαXα* 1}and**S***μ* **S***α∈Λμα*

∩{* _{α∈Λ}Lα* ∈

*α∈Λ*

**L**

*α/μαLα*

*μαXα*1}. But

*μαLα*1 implies

*αμαα∈Λ***L***α* 1. Then**S***μ* *α∈Λ***S***μα*.

bLet *x* *xαα∈Λ* ∈ *α∈ΛXα* Since *α∈Λμxα*

*α∈ΛAα* *μxα∈ΛAα* we get

*α∈Λμxα* *μx*.

**L***α*disjunctive implies*μxα* ∈**IRL***α*for all*α*∈Λand then

*α∈Λμxα* ∈

*α∈Λ***IRL***α*

**IR***α∈Λ***L***α*; therefore *μx* ∈ **IR***α∈Λ***L***α* which proves that **L** **L***α∈Λ***L***α* is

disjunctive.

cLet*μ*∈**IL**and*ν, ρ*∈**I***R***L**such that*μ*≤*ν, ρ*on**L.**

But*μ* * _{α}μα* ∈

*α∈ΛILα*and both

*ν*

*α∈να*,

*ρ*

*α∈ρα*∈

*α∈ΛIRLα*and

then* _{α}μα* ≤

*α∈να*and

*αμα*≤

*α∈ρα*on

**L with**

*μαAα*

*μPα*−1

*Aα*,

*ναAα*

*νP*−1

*α* *Aα*for*Aα*∈*ALα*. By the previous work we get*μα*≤*να*and*μα*≤*ρα*on**L***α*.

Since each **L***α* is normal it follows that *να* *ρα* for all*α* ∈ Λ, and therefore *ν*

*ναα∈Λ* *ραα∈Λρ*which proves that**L is a normal lattice.**

**4.2. Examples**

**4.2. Examples**

3Let**X*** _{α}*be a topological

**T**-spaces and let

_{3.5}**L**

_{α}**Z**

_{α}be the replete lattices of zero setsof continuous functions of**X***α*for all*α*∈*A*.

Consider a lattice**Z of subsets of**_{α∈A}**X***α*such that

*α∈A*

**Z***α*⊂*t*

_{}

*α∈A*

**Z***α*

*.* 4.8

Then**Z is replete, and**_{α∈A}**X***αis realcompact.*

4Let**X***α*be a*T*2and 0-dimensional space and let**L***α***C***αbe the replete lattice of clopen*

sets for all*α*∈ *A*. Then each**X*** _{α}* is said to be

**N-compact. Consider any lattice C of**subsets of_{α∈A}**X***α*such that*α∈A***C***α*⊂*tα∈A***C***α*⊂*tα∈A***Z***α* **F and C is replete**

and_{α∈A}**X***α*is**N-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´ı*k, “Realcompactness is a Baire-measurable property,” Bulletin de l’Acad´emie Polonaise des*

*Sciences, vol. 19, pp. 617–621, 1971.*

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