Internat. J. Math. & Math. Sci.
VOL. 14 NO. 2 (1991) 319-324
319
REMARKS ON SEMISEPARATION OF LATTICES
CARMEN D. VLAD
Department of Mathematics
Pace University
New York,N .Y. 10038,U .S .A.
(Received April 10, 1990 and in revised form November I, 1990)
ABSTRACT.
This paper is concerned primarily with conditions for semiseparation
separation of lattices. These conditions are expressed in terms of the general
and
Wall-
man space.
KEY WORDS AND PHRASES. Lattice, measure, filter.
1980 AMS SUBJECT CLASSIFICATION CODE. 28C15.
,
INTRODUCTION.
lattices of subsets of X such that
Let X be an abstract set and J’s and
A/C
implies there exists a C,C such that C)B, and
If A/B=, A (X,,
1.
,,cm
B
)
then { is said to semiseparate ,i. This notion is important in topological spaces,
specific lattices such as, for example, the zero-sets and the
and
where
closed sets.
We investigate this property in terms of associated measures and outer measures
associated with the respective lattices, and also with respective Wallman spaces. This
gives us new conditions for one lattice to semiseparate another, and gives additional
facts pertaining to the measures. These investigations are carried out in sections 3
and 4. In section 2 we give some background material, which ’is fairly standard by now,
and can be found in [i-3]. This material has been added mainly for the reader’s con-
are
venience.
,
BACKGROUND AND NOTATIONS.
a lattice of subsets of X. It is assumed that
Let X be an abstract set and
(:(.
We denote by (.(,) the algebra generated by, ;(’(.), the lattice of all
),X
countable intersections of sets from C.
isDEFINITION 2.1
delta lattice (--lattice) if Jis closed under countable intersections.
complement generated if L,I implies L=/’L n’ Ln,’( (where prime denotes comple2.
,
.
ment).
I(
xL
such that
disjunctive if for xX and L
I there exists L
and
L2=).
there exist
normal if for any
2 E.C with LI/L2=),
=).
and
of sets of. with
compact if for any collection
finite subcollection with empty intersection.
LI/
L3,L4:Z
LI,L
L2C L
L/ L
2(, with
L.
/k.(=),
with
x L2
L C.
there exists a
L1/2,
C.D. VLAD
320
IL.(lof
countably compact if for any countable collection
sets of
with (’IL.(:I,
there exists a finite subcollection with empty intersection.
Lindelf if for any collection
sets of
with
there exists a countable subcollection with empty intersection.
y(L and
T2-1attice if for x,ycX, x#y, there exist L 1,L 2( such that
Lof
#rL,(=IB,
xL,
DEFINITION 2.2 We give now some measure terminology which will be used throughout. M(,) denotes the set of finite valued bounded finitely additive non-trivial
measures on ((). Without loss of generality may assume throughout that all mea
sures are non-negative. A measure z(M(,C) is called"
-smooth on if for all sequences ILn| of sets of with
(Ln)---,O(’-smooth on ((,-.) if for all sequences
of sets of (.) with An),
(A n) --0 i.e. countably, additive measures on ()).
-regular if for any A(:(:(), /(A)=sup
L(,.
In addition we denote by MR(), the set of ,-regular measures of M(,C);
the set of -smooth measures on
of M(); M’(,I), the set of
-smooth measures on
(,) of M(.);
the set of ,-regular measures of MC).
are "the subsets of the corresponding M’s which con(,),
sist of the non-trivial zero-one valued measures.
DEFINITION 2.3 For
the support ofz is
( is replete iff for any
S(/)).
J’, satisfying the condi
DEFINITION 2.4 A filter in j is a subset of
tions"
’is closed under finite intersections; if A, B(,CL and A(::B
then B
relative to the partial order
is a maximal filter in
An ultrafilter in
on the collection of filters in
given by inclusion).
then either A-Cr
such that AUB
is prime if given A,B
An .’-filter
or B
There exists a one-to-one correspondence between -filters and elements of
(A)-(B), A,B-) defined by
defined on .LC monotone and (A/B)=
(L)=I iff L(. There exists a one-to-one correspondence between .’-filters
such
where
with countable intersection property and
There exists a one-to-one cothen
that if ’(Ln)=l all n where
rrespondence between all elements of IR(,C) and all .-ultrafilters. There exists a
and all .-ultrafilters
one-to-one correspondence between all elements of
with the countable intersection property. The correspondence is given by the follo
we associate the zero-one valued measure
wing rule" with each .-ultrafilter
defined on () by
if there exists A (, AC.E
Ln),
IAn
/(L)/L(A,
F()
M(,.),
(.)
I(,’),IR(J),
S(/w)=/{L/ZA(L)=z(X)
/w(M(,C),
/.I(.),
.
I;
,
.,
.
.
"()=
(,.,),
Ln,C
Ln# }.
(,:)= 7’(,)
I(J{)
..
/(E)=
AC.E’.
0 if there exists A
There exists a one-to-one correspondence between all elements of I(.() and all prime
-filters, given by the following rule" with each /((I() we associate the prime
This correspondence induces a one-to-one
-filter given by
,-filters
countable intersection property and
the
prime
with
correspondence between
= #A-//(A)=I
IS()
REMARK. It is not difficult to see in light of the above correspondences
such that
there exists a unique
is normal iff for each
/ml(J),
(i.e.
/(L)
(L)
for all L( ).
)IR(.C.)
that
/’ ()
SEPARATION OF LATTICES
3.
SEMISEPARATION.
DEFINITION 3.1
and define
’
(L’)/
/
IR(
E(ZX
The follo
-<(.’I.
IR(/)
/IR(l)
/(I().
__
/ =/Ac’ (=);
iff
and
L(.
EC-L’
be a lattice of subsets of X and let
THEOREM 3.1 Let
wing statements are true:
a) // is finitely subadditive;
b)
c)
I(,C)
be a lattice of subsets of X, let
Let
/w.’(E)=inf
321
is normal
such that
iff
and
/
PROOF. a)Since (I(=) it is clear that /x’(E)=inf[_.
/
and therefore /
is finitely subadditive.
b) For /IR(,C /(A)=inf
(L’)/ A(ZL’ L
c) Suppose
and since / IR(J() it follows that /=/(=C).
Then
Then </w
is normal, let m6. and suppose that /(A) =0.
Suppose
with (L):I. But AIL=)
Since
implies
there exists L(ZA’, L(;,C,
there exist C’,D’ such that AC’, L(ZD’, C’/’D’=), C,D,C. Then we have A(C’C_
cZD(ZL’
a was
(C’)=O,
and #(C’)<(D)(B)/(L’):O. So,
(,{) with
Conversely, suppose that
then
arbitrary in,v
on
with
as before. Let (I(,),
-/4z (,).
/ and
By
and ,Z///2 on Z
But /z
on .’/
so we have ,,’/// on
/
and therefore /z
the assumption, /z: and /Zx- on
/4a=i.e .’normal
be a lattice of subsets of X. The Wallman topology is
DEFINITION 3.2 Let
L( as a base for the closed sets
obtained by taking all W(L)=
with the Wallman topology is called the general Wallman space assoin
is disjunctive. Then if A(C(,{), let
We assume that
ciated with X and J
The following statements are trueW(A)=
Let
/= !(,)
and
for all such
,
(L.’.)/ECQL, Liff’["7--
[/
_<(Z)
,/
a ’..(’)
(IR(,C),
i.e.Z(A):O,
/z= ##
z:/
,
/fz}/(IR(,)
IR(, i
/-/(,),
z:
==
,
/(IR(,)//(L):II,
IR(). IR()
IIR(,W)//(a)=z I
W(AUB)= W(A)UW(B)
W(A#IB) W(A)IW(B)
w(a’) W(A)’
A:B iff W(A)Z)W(B)
a)
b)
c)
d)
e)
.
W(a (,()).
(.(W(,I))
It is known that W() is disjunctive and that the topologicai space
is normal.
is disjunctive it is T 2 iff
is compact and T I and if
be two lattices of subsets of X. Suppose that
THEOREM 3.2 Let
is normal and consider the restriction map
disjunctive and
"IR(l)
,,
,
Then:
W2(L 2)
a)
’(W2(L2)):/WI(LI)/
b)
!
kl(=
(IR(,),tW()))
’
(z
where
k2
L2C_.LI=
are basic closed sets with respect to the Wallman topologies.
(m.
W2(L2)) is
C is
In(,
WI(L M)
and
semiseparates
it is compact and since k’ is
closed in
and
is compact and T 2
is normal, so
continuous, (W2(L2)) is compact.
since
and
where
Then
closed.
is
therefore (W2(L2))
for all
is disjunctive, L _.LI=
2
Then W2(L2)/W2(LI):), which implies
with
b) Let L2(( and L
and
if (:W() with "#(W2(L 2) then (L2)=1
if
For
eWl(L I)
’(W2(L2))WI(LI)=).
contradiction. Thus (W2(L2))(WI(LI):). By a) we have then
and
Since WI() is compact it follows
(’
/
PROOF. a) Since
W2((),
C
’
l.
I
)(LI):(LI)=I,
L2ILI:).
,
/ WI(LIK) L2LI,(, kl=(
IR()
(W2(L2))=C,, WI(L]w,),
WI(LI):).
Ll(=l
322
that
C.I).
LIXg(’,IWI(LI)=}. Then L2/.ILI
WI(LIwz)/L2(ZLldg
ALI=COROLLARY 3.1
,
VIAI)
=A
and
semiseparates
be a lattice of subsets of X. Then the following statements
which proves that
Let
are equivalent:
a) I()=IR(
semi separates ()
b)
c)
d)
PROOF. a) b)" () is disjunctive.
’
IR/)=IR()
we have
"
(().
IR(())=I(()) IR(
=I().
Consider the restriction map
By Theorem 3.2 it follows that
semi separates
then L’
().
AL=,
i.e.
AL’
() there
semi separates
Since
Therefore L’AcL’
(I()=IR()
is normal, since for
(). b)c): Let L@;
A=L’
i.e.
L’cA
exists A@
and
so
a)" Let I() and IR(),
() and
suppose that (A)=O, (A)=l, A. But
therefore there exists L’CA,
L wth P(L’)=I, or V(L)=O. Then ( =0 and since A’L, (A’)(L)=O
d)
c)
d)
clearly,
(IR(),
=
(A’)=O,
I()=IR().
contradiction. It follows that
i.e.
be two lattices of subsets of X, with
and
normal
disjunctive. Consider
and its restriction
Then
semi separates
iff
PROOF. Clearly
and suppose
Then
always Let L
and
on
there
semi
exists
By
separati
z
Then
and
and 9 =(,). Conversely
so
i.e.
since
If
suppose
therefore
then
L
V
i.e.
which by Theorem 3.1 implies
It follows by Theorem 3.2 that
semiseparates
DEFINITION 3.3 Let
be a lattice of subsets of X and define
i.e.
COROLLARY 3.2 Let
@IR(z)
z
z
L2(
,
1
(L)=O.
LIL
LICL,
=’()
IR(Z).
=/()
PROOF.
z
iff
.
(L)/ EL, L@
Cbe two lattices of subsets
(E)=inf
THEOREM 3.3 Let
semi separates
,
L2L L.
(L)/Therefore
L2CL
WZ(Ll)=O, I(
P(LI)=O
z
LICL
#I)=0"andlLl=.
L@,
(L1)=O
(LI):O
=(,). (L1)=O
=’(). So,’(LI)=O
’=’()
@I().
L
ECX.
of X and let
@IR().
Then
By semi separati on
there
=0.
2
2
there exists B,
and (A)=O and since
Then there exists A(,
So
Therefore Lz(B’ and (B’)=O, hence
AB’ and
does not semiseparafor all
If
Conversely, suppose that
/
but
such that
then there exist
L
tes
exists
there
therefore
property,
intersection
has
the
finite
2,
and LI)L 2 and
for all
z) such that
separates
semi
contradiction. Hence
be a lattice of subsets of X, let (I() and EX and
DEFINITION 3.4
exists
’(L2)=inf
LI(
with
(B’)=O.
LIL LI}
(IR(
butj’(L2)=O,
define
@IR(),
LI
(LI)=I
Let
z
be a lattice of subsets of
"0.
X,
let/I()
(L ),ECOL
i’ L
are outer measures on P(X),clearly.
" z,
Both
then
THEOREM 3.4 Let
()
=IL1
(LI)=I. Thus(L2)=l
"(E)=inf
REMARK
(L
=’().
’(L2)=O.
@IR().
=’()
,
I(,
L2LI=
L2@
DEFINITION 3.5 Let
define
(E)=inf
parates
(L
L2CLIL.
L2A
(L 2) Now suppose
and
If
Suppose that
(IB()
and
ECX and
IfeI()
but
"().
Similar remarks for
then
be two lattices of subsets of X such that
Then
and let
is
@ I().
"=, ().
semise-
SEPARATION
PROOF.
Clearly,
II"L_’. Let ol--lattice
so that
IR(I). Then /z(L’)=/Z((../ L.)’)=/(L.’.)_</(L.’.)
’
’ ’"
By,,Theorm
#-</ ().
A i,
L=L L’and
ad.ditive;
let
i,
countably
/_/z
Ai’
Bi ~and
Hence
since
3.3 it follows /<"=z’=/(2). But since
everySu ppose z(k 2)-0 with L 2(:, but "(L 2)=I.
Then
and ,(Ai )-O all i. By the ,,-regularity of /. we have AiC_.B
U i,
and # (L2)=O, contradiction.
/(Bi)=O. Therefore
therefore /’=/j’.
where, we get
L2
323
()F I,ATTICES
L2(S:B /c(Bi)=O
# ="(Y.).
Further related material can be found in [4-6].
I-LATTICES
DEFINITION 4.1 A lattice
is called an l-lattice if every ,-filter with the
countable intersection property is contained in an -ultrafilter with the countable
intersection property (i.e. for
there exists
such that
THEOREM 4.1 If
is an l-lattice and replete then
is Lindelf.
PROOF. Let
There exists
I’-_/ (). replete implies
therefore S(/w.)S(I )#). But
and then
)=I
4.
J()
76’(Z).
#’
,
z(l().,
zIR)with
S(zz)=L,K//(L
(eZ/(L)=I} I i.e. is kindelf.
THEOREM 4.2 If
is a countably compact lattice then
PROOF’. Let IL,< o,, be a collection of subsets of X such
=
,
_/e(J)
(,)
is an I-lattice.
that
I. Since
and then
is a filter base which generates
Z countably compact,
an -filter with the countable intersection property,
We enlarge it to
an -ultrafilter with the countable intersection property. To ," it corresponds uniquely
and
THEOREM 4.3 If Z is disjunctive and Lindelt}f then
is an l-lattice.
PROOF. Let
and let
be a family of subsets of X. Then
o
IIF)I" Let xS(/F) and
is
I and since
disjunctive,
is
consider /z x. Clearly,
since
and
and #x(
,I
Therefore /Xx(
DEFINITION 4.2 Let ,Z be a disjunctive lattice of subsets of X and let
A((C(=) where
on
Define
by
Clearly, for A,B((J() the properties a)-e)
(A)=I and
that we stated in section 3 are still valid. Note that (l) is a disjunctive lattice.
The following theorem follows directly from the definitions:
iff /x
(More generally"
THEOREM 4.4 If
then
iff
then
if
is an l-lattice iff
is disjunctive then
THEOREM 4.5 If
IL,.I,
IL,(,’)
().
z IR()
IL:<//T(L,()=II
X((a)
./#Zx(,l
’
.(,,#’I L.(
,
IL,<},((a
LindelSf,,a/’L.(/l-(L, )=If=S(
,I)
/@IR(,
z IR()
W((A)=#I()/
O(We())=W((.,,)) Z’(W@.(A))=/.(A),
W@(.)= W((A)/A(C2_()}.
#_IRG’()
’(I%(W(,{)).
#’;I(W@()) andzIR()
/’IR(W))).
/IR(,’)
/I(,)
w.
(IR(,(),tW())
is Lindel6f.
T_I().
WO()
is an l-lattice. Let
PROOF. Necessity- first we show that
There exists
with Z_/ (). Hence by Theorem 4.4 we have
is disjunctive, Wo() is replete
Since
and
with
implies that
and by Theorem 4.1 it follows that WB() is LindelSf.
Lindel6f implies that WI(,v) Lindelf and
twe(,IC) is Lindelf. Sufficiencyis an l-lattice.
is disjunctive, by Theorem 4.3 it follows that
since
/
(;I
To
such that
Therefore for
there exists /
z I(:’)
/’(W.()).
(W((())
z’
,
WI(,)__tW.(,)
tW(()
W()
W(()
’
7#/(w(,’)):
’(W,_())
R (w())
and /’ correspond z) and /IR) such that -/().
r
is o and
THEOREM 4.6 Let =C be two lattices of subsets of X such that
an l-lattice and
W’#.
,
is disjunctive. Consider that the restriction
is closed with respect to Wallman topologies. Then
semiseparates
"IR(,Iz)---IR)
’_
3’2/4
C. 1). VIAI)
here
([2
and
(L1):I.
@(LI) WLI):.
But then
(L2):l
and
,I(),<)s
9(L1)=l,
contradiction
snce
Hence, since
is an I-lattice and disjunctive (because
dsjunctve), by lheorem 4.5
kndelf. ,ow
L I):B and then
where
L
which island
Hence
semi
separates
2.
Here we give conditions which guarantee that
is basically closed.
[INIIION 4.3
s countably bounded
lattice f for
n=l,2
and
there exists
n1,2
th
and
I[OR[N 4.7 ket
be to lattices of subsets of X such that
semsearates
-countably bounded and
lhen the restriction
is basically closed.
PROOF. To show that
Clearly
k2kl,
snce
.e. for.any
e have
,kl
No let
but
lherefore
snce
and
")
2
countably bounded,
So, (k,)=l all kuk, but (k2)=O. Snce
then (k)=l and
there exists k,k,, k,ea ,,)=1.
ton there exists
and kMkl=, gut k2k and (k2)=l implies
t fo1]OS
(kl)=1 an snce
(kM)l all
(klkl)=l hch
:W L
LgDL
LI.LI:,
z
n,
gn,
ngn
n
,
z, zs
a=t
() I(Z,)
k2)=n,k,
z=t
k2kl
k2z
,
=) (kl
’
().
k2
k2=kl
e()R
.
I(z),
klff k2k
contradicts that
also
k
kl
k
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I.
2.
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VLAD, C. Direct Product of Measures, Libertas Mathematica, Univ. of Texas,
5.
Vol. VII(1987), 71-84.
VLAD, C. Lattice Separation and Propertieof Wallman Type Spaces, Annali di
Matematica Pura ed Applicata
to appear).
HUERTA, C. Measure Requirements on Distributive Lattices for Boolean Algebras
and Topological Applications, Proc. Amer. Math.Soc. 106, no.2,1989,307-309.
EID, G.M. On Normal Lattices and Wallman Spaces, Internat.J.Math & Math. Sci.
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FROLIK,
3.
4.
13(I), (1990), 31-38.
Z.
553-575.
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