Pub
.
Ma
.
UAB
N°
20
Se
.
1980
Ac es
VII
JMHL
ON
LATTICE-DILATIONSANDCONTRACTIONS
IN
-RINGS
Joan
T ias
Pai d
Dp
.
de
Ma emá iques
i
Es adís ica,
E.T
.S.A.B
.
Uni e si a
Poli bcnica
de
Ba celona
ABSTPACT
.
This
pape
spli s
b oadly
in o
wo
ela ed
pa s,
conce ned
espec-
i ely
wi h
gene alized
idempo en s(associa ed
o
supe uni ies
and
subuni ies)
andwi h
la ice-dila ions
and con ac ions
in
an
- ing
A
.
I
u
is
a
supe uuii-
y,
we
cha ac e ize
he
mappings
F
:A-->A
sa i ying
IF(x)-F(y)I
=
uIx-y1 (u-di-
la ions)as
he
mappings
o
he
o m
F(x)=
x(u-2e) b,
wi he
being
a
gene ali
.-
zed
idempo en ,
and ob ain
an
analogous
esul o
la ice-con ac ions
.
The
se
o
he
hcnageneous
ones(bo h
cases) a e
p o ed
obe
Boolean
algeb as
.
Ou
e minology
and
no a ions
a e
mos ly
s anda d,
and
ollowwidely
L1]
.
Recall
ha
inan
-e- ing
A
,
u
is
a
supe uni y
133
i ux
Axu
.>,,x
holds
o
e-
e y
x
>/
0,
and
s
is
a
subuni y
L6 ]
i
0
<
sx
<
x
,
0
<
xs
;
x

o
e e y
x>
0
.
Now,
i
ollows
a
summa y
o
esul swi hou
p oo s
.
1
.
Gene alized
ideaqo en s
.
In an
-¿- ing
A,
we
in oduce he ollowing
de i-
ni ions
:
a)
i
u
is
a
s
u
T
ne uni
y,
hen
eEA
is
a
u-
idempo en
i
eu= ue=
e2
.
b)
i
s
is
a
subuni y,
hen
e
E
A
is
an
s-
idempo en
i
es
=
se
=
e
2
.
The
es-
pec i e
se s
will
be
deno ed
by
I(u)
and
I(s)
.
We
i s
show
ha
in
an
- ing
A
,
I
(p)
=
{
x
E
A
1
x
A
(p-x)
=
0 }
i
p
is
a
subuni y
o
a
supe uni y,
and ob ain
as
a
consequence
:
Theo em
1 .I(p)
is
a
Boolean
algeb a
wi h
he
o
de ing
o
he
ing
.
2
.
Boolean
al

as
o
gene alized
idempo en s
.
On
accoun
o
I
(p)
ha ing
no
special
p ope y
whi
.c
h
an
a bi a y
Booleanalgeb a
needno
na e,
we
ha eana-
lized
i s
boolean
p ope ies
in
eonnec ion
wi h
la ice
and
algeb aic- heo e ic
p ope ies
o
he
ing
.
Since
I
(p)
is
closed
by
aking
a bi a y
sup ema
and
in ima,
i is
easy o
ela e
he
o de
oomple eness
p ope ies
o A wi h
~le e
ness
p ope ies
o
I(p)
.
We
pay
conside able
a en ion
o
p ojec able
- ings,
ha
is,
hose
o
which
a
l ®
-L-=
A
holds
o
e e y
a E
A
.
They
a e
specially
in e is ing
in
his
con ex
in
iew
o
he ollowing
esul
o
- ings
wi h a
supe uni y
u
:
Theo em
2
.
A
is
p ojec able
i
and
only
i
he
pola
o
e e y
elemen
is
he
pola
o a
unique
u-idempo en
.
Fo
he
"only
i "
pa o Th
.2,
i
su ices
p o ing
ha
i
a
E
A,
hen
al =

e
l
,
being
e
he
p ojec ion
o u
on o
á"
«
L
.
Some cceple eness
and
p ojec ion p ope ies
:L-,ply
ha
he - ing
A
be
p ojec able
.
Fo
ins ance,
hose
o
he
main
inclusion
heo em
[4J
,
and
o-
he s
.
We
ha e
p o ed
he e
ha
i
I(u)
is
eon ex,
hen
A
is
p ojec able
.
F cm Th
.2
we
ob ain
ha i
A
is
p ojec able
and non
o ally
o de ed,
hen
I(u)
is
non
i ial
.
Sane
esul s sugges
he
in e es
o
s udying
he
subse
Pu
(A)
=
e
-
'
-
1
eE
I(u)}
o
all he pola s
P(A)
o
he
ing
.
In his
connexion
we
p o e
:
Theo em
3
.
Wi h
he
o de ing
o
P(A),
Pu
(A)
is
a
Boolean
subalgeb a
o
P(A),
iscmo phic
o
he
algeb a
I(u)
.
Mo eo e ,
Pu
(A)
is
a
sub-la ice
o
PP(A), he
la ice
o
all
p incipal
pola s,
and a
sualgeb a
o
he
Boolean
algeb a
o
di ec
sum
:nands
.
I
P(A)
is
supe a onic, hen
I(u)
is
ini e
.
I
u'
is
ano he
supe uni y,
hen
I(u)
and
I(u')
a e
isom phic
in
he o-
llowing
cases
:
a)
A
is
Dedekind-cade e
;
b) I(u)
and
I(u')
a e
comple e
Boolean
algeb as
and a e
con ex
in
A
.
The
p oo
o
TM
uses
mainly
he
decomposi ion
A
=
e
l ®
(u-e) 1
i
eeI(u),
and
he
ac
ha i
e
l
,
e
2
e
I(u)
and
el
=
e2

hen
el
=e
2
.
The
isom
phism
o
he s a emen
is
gi en
by

I(u)-->
P
u
(A)
,
e~
(u-e)
1
.
By
using
Th
.3
in
he
p ojec able
case,
we
ob ain
:
Theo em
4
.
Le A be a
p ojec able
- ing
.
a)
I(u)
is
a omic
in he
ollo-
wing
cases
:
1)
P(A) is
a cenic
;
2)
A
is
basic
o
~le ely
dis ibu i e
.
b)
I
A
has
a
ini e
basis,
hen
I(u)
is
ini e
.
3
.
Boolean
al

as
o
la ice
-
dila ions
.
I
u
is
a
cen al
supenini y
o
he
- ing
A,
we
gene alize
he
no ion
o

¿-isone y
(
[21
,
[5'1,
[6
1)
by
conside ing
he
mappings
F
:
A->A
ha
sa is y
IF(x)-F(y)j
=
uIx-y1
o
e e y
x,yE
A
.
They
will
be
called
la ice
-
dila ions
o
u-
dila ions
,
since
IF(x)-
F(y)I>
1x-y1
.
We
deno e
by
H
u
(A)
he
'se
o
all
hcmogeneous
u-dila ions, ha
is,
hose
o
which
F(0)
=
0
.
On
he
o he
hand,
i
e E
I(u),
we
conside
he
mapping
c
e
:A
-~
A
,
~
e
(x)=
x(u-2e)
and se
Z
u
(A)
_
{ '
e
leeI(u)1
The
undamen al
esul
we
ha eob ained
ncw
ollows
:
Theo em
5
.
a)
Hu
(A)=
o~)
u
(A)
.
b)
E e y
u-dila ion
F
is o
he
o m
F(x)=
x(u-2e)T
b,
being
b
e
A
and
eE
I(u)
.
c)
E e y
Fe
Hu
(A)
is
a
ham ecy
o
a-
io
a,
wi h
I
a
I
= u
Pa
a)'o
Th
.5
has
been
p o ed
by
means
o
a
sui able
ep esen a ion
o
A
as
a
subdi ec
p oduc
o
o ally
o de ed
ings,
and
he
ac ha
o 'a
o allyo de ed
ing
wi h
a
supe uni y
u,
he
only
u-dila ions
a e
O'
6
(x)=
ux
o
e e y
x,
and
a-
u
(x)=
-ux
o
e e y
x
.
Pa
c)
ollows
on
accoun
o
e
E
I(u)
being
a
componen
o
u
.
1
Now, i
we
conside
he
Boolean
ing
s uc u e
o
he
Boolean
algeb a
I(u), hen
Th
.5
enables
us o
endow
Hu
(A)
wi h
a
Boolean
s uc u e
:
Theo em
6
.
Wi h
he
ope a ions

(a
-
e
®
é,)
(x)=
x(u-21e-e'I)

and
(oy
e
,Y
a
e'
)
(
x
)
=x
(u-2
(e
n
e'
)),

(H
u
(A),

,
IK
)
is
a
Boolean
ing
wi h
uni y,
isomo phic
o
he
Boolean
ing
I(u)
.
Nowin iewo
he
isceno phism
Hu(A)='I(u)
and he
heo ems
3
and
4,
we
i
ha e
de i ed
he
co esponding
p ope ies
o
he
Booleanalgeb a
Hu
(A),
bu
we
shall
no
explici ly
men ion
hemhe e
.
~ e ,
i
.
iswo h
no ing
ha
i
A
is
p ojec able
and non
o ally
o de ed, hen
he e
exis
non
i ial
u-iila ions
.
The
u-dila ions
'
o
and
o
-
u
a e
in e es ing
since
we
ha e
:
Theo em
7
.
I
F
E
Hu
(A)
,
hen
he e
exis s
a
unique
decanposi ion
A=
B$C,
wi h
-
B,C
being
¿-ideals,
o
which
FIB
'7
0
and
FI
C
Qu
.
Indeed,
by
Th
.5

F=
o
e
,
o
sane
e
e
I(u),
and
i
su ices
aking
B=
e
l
and
C=
(u-e)
l
.
Theo em
7
enables
us
o
gi e
sane
geome ic
in e p e a ion
o
ha ogeneous
u-dila icns,
especially
byneanso
he
coneep
o
la iee
axial
sycm y
.
Recall
can [ 6
]
ha
i
aeA
,
hen
:A-s
A
.
is
a
la ice
axial
symme y
o
axis
a
i
:
1)
is
a
g oup
hamam phism
:
2)
A=<aj®
aL
and
3)
La>
I
,
1
a1=
-I
,
wi h
I
being
he
iden i y
mapping
.
Then
we
can
pa ially
eph a-
se
TM
:
E e y
hcuegeneous
u-dila ion
é
is
a
homo ecy
o
a io
u on
o hogonal
di ec ions,
ollowed
by
a
la ice
axial
symme y
wi h
espec
o
oneo
ha
di ec ions(o
axis
u-e),
o ,
which
is
he
same
:
i is
a
la ice
axial
symme y ollowed
by
a
hano ecy
o
a io
u
.
Con e sely,
e e y
la ice
axial
symme y,
ollowed
bya
ha o ecy
o
a io
u
is
a
hanogeneous
u-dila-
ion
.
In
he cou se
o
ou
s udy
he
se
B o
all squa e
oo s
o u
2
has
na u a-
lly
a isen
.
We
ha e
pM ed
ha
B
=La¡
la
¡=
u}
.
Mo eo e ,
Theo em
8
.
Wi h
he
same
o de ing
o
he
ing,
B is
a
Boolean
algeb a,
ha
is
isamo phic
o
he
Booleanalgeb a
I(u)
.
Hence
isamo phic
o
Hu
(A)
.
The
p eceding
iscmo phism
is
gi en
by
I(u)
-B
,
e
-~2e-u
.
I
is
pos-
sible
now
o
ans e
o
Bmanyo
he
p ope ies
ha
could
be
asse ed
o
he
Booleanalgeb a
I(u)
.
4
.
La ice
-
oon ac i e
mappings
.
I
s
is
a
cen al
subuni y
o
he
- ing
A,
we
can
de ine
"mu a is
mu andis"
he
concep
o
la ice
-
con ac ion
(s-
con ac ion
)
and
hoimgeneous
s-con ac ion
by
only
in e changing
u
by
s
in he
de ini ion
.
Wi h
ce ainaddi ional
assump ions
on
A
,
i is
possible
o
de elop
a
heo y o s-con ac ions,
ha
is
pa allel
o
ha
o
u-dila-
ions,
hough
less
sa is ac o y
in some
aspec s
.
The
di icul yappea s
when
se me
o
he
p ope ies
ha
a e
alid
o
u-idemoo en s
do
no
hold
o
s-idempo en s
.
Fo
ins ance,
he
decomposi ich
A=
el e
(u-e)'
L
is no
mo e
a-
lid o
e e ye
E
I(s)
.
I
emains
alid
howe e
i
A
is
Dedekind-comple e
o
i
s
is
a
o mal
uni y
.
O he
p ope ies
s ill
hold
in
absence
o
nonze o
nilpo en
elemen s
.
REFERENCES
~1]
Biga d,A
.,
Keimel,K
.,Wol ens ein,S
. :
G oupes
e
anneaux
é iculés
.
Lec u e
N
.
in
Ma h
.
608
.
Be lin-Heidelbe g
N
.Y
.,
1978
.
12]
G ané,J
. :
Sob e
las
isome ias
de
los g upos
y
los
anillos
e iculados
.
Pub
.
Uni
.
d
e
Ba celona,
1978
.
3]
Hen iksen,M
.,Isbell,J
.R
. :
La ice-O de ed
Rings
and
Func ion
Rings
.
Paci ic
Ma h
.
J
.
12(533-565),
1962
.
4]
Luxembu g,W
.,Zaanen,A
.
:
Riesz
Spaces
I
.
Ams e dam,1971
.
15]
Swamy,
K
.L.N
.
:
Isome ies
in
Au ome ized
La ice
O de ed
G oups
Algeb a
Uni e salis
8(59-64),
1978
.
[61
T ias,J
. :
Con ibución
al
es udio
de
los
anillos
e iculados
y
-ani-,
llos
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ol
III,
no-2
(45-69),
1979
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[
7
1
Vulikh,B
.Z
.
:
In oduc ion
o
he
Theo y
o
Pa ially
O de ed
Spaoes
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G oningen,1967
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