Proc.
Univ.
of Houston
L a t t i c e Theory C o n f . . H o u s t o n
1973
ORDERING UNIFORM COMPLETIONS OF PARTIALLY
ORDERED SETS
R. H.
Redfield
We give here a d i s c u s s i o n and summary of r e s u l t s whose p r o o f s
appear elsewhere
[6].
The f a m i l i a r construction of the r e a l numbers from the
numbers v i a Cauchy sequences was put i n t o i t s
In the 1 9 3 0 ' s , . W e i l
final
rational
form by Cantor
[ 8 ] - an a l t e r n a t e view i s T u k e y ' s
systems of d i f f e r e n t p o i n t s .
In p a r t i c u l a r ,
a H a u s d o r f f uniform space
mentioned above,
of the r e a l s
from the
Both the real numbers and the r a t i o n a l numbers are
in f a c t ,
neighbourhood
in
[ l ] can be "completed" by a process which mimics the Cauchy
sequence construction
and,
[2].
[7] - showed that
the essence of this process was a certain s i m i l a r i t y between
the sense of
will
rationals.
(additive)
groups,
the r a t i o n a l numbers are u s u a l l y considered as a subgroup
the real numbers.
I t i s w e l l known
(see,
for example,
[4])
that
of
this
e x t e n s i o n of the group operation can be done in the more general case of
Hausdorff uniform s p a c e s , provided that the uniform structure
and the group
structure
Hausdorff
are s u f f i c i e n t l y
intertwined.
group with continuous group o p e r a t i o n s ,
ties,
Specifically,
on'any
there are s e v e r a l n a t u r a l
uniformi-
at l e a s t one of whose completions can always be endowed w i t h a group
m u l t i p l i c a t i o n which extends
the o r i g i n a l
operation.
Now the real numbers and the r a t i o n a l numbers are not only
but t o t a l l y ordered groups as w e l l .
I t seems r e a s o n a b l e ,
for conditions whereby the order structure of a p a r t i a l l y
504
groups,
therefore,
to ask
ordered set with
Hausdorff uniformity may be extended to the uniform completion of the s e t .
As in the case of groups, i t seems i n t u i t i v e l y clear that some sort of
connection between the order structure and the uniform structure must be
assumed to ensure a satisfactory extension.
by the idea of a uniform ordered space.
Nachbin
That connection i s provided
This concept was introduced by
[5] who used i t to investigate the ramifications of complete
regularity on ordered sets with uniformities.
To define uniform ordered spaces, we need a l i t t l e notation and a
definition:
Let
X
be a s e t ;
as u s u a l ,
A(X) = { ( x , x )
be the diagonal of
XXX
X.
let
€ xxx | x ( X}
A semi-uniform structure on
X
is a f i l t e r
3
on
such that
(i)
for a l l
V € 7,
A (X) £ V ;
(ii)
for a l l
V € J",
there exists
U € 7
such that
Thus a semi-uniform structure is almost a uniformity:
symmetric base.
I t i s easy to see that
Let
7*
=
{u
fl v "
1
| u,
v
f
J}
.
i s a uniformity, which we c a l l the uniformity
3.
(P, 5)
be a p a r t i a l l y ordered s e t .
G(<) = { ( x , y )
be the graph
i t lacks only a
This requirement may be added by considering
7*
generated by
U°U £ V .
of
£ Pxp
Let
| x < y}
A nearly uniform ordered space i s a p a r t i a l l y ordered
505
set
(P, "S)
with Hausdorff uniformity
uniform structure
J
on
P
U
satisfying
such that there exists a semi-
J* = U
and
fl 3 £ G(5) .
A
uniform ordered space [5] is a nearly uniform ordered space with a semiuniform structure
3
s a t i s f y i n g the conditions for a nearly uniform
(7* = U, D 7 £ G ( < ) )
ordered space
D Jc
and additionally
G(5) .
Every nearly uniform ordered space is locally convex, and there e x i s t
nearly uniform ordered spaces which are not uniform ordered spaces.
the concepts of p a r t i a l l y ordered set
with Hausdorff uniformity,
uniform ordered space, and uniform ordered space are d i s t i n c t .
Thus
nearly
In the
totally ordered case, however, nearly uniform ordered spaces and uniform
ordered spaces are the same.
Since nets are usually easier than f i l t e r s
to use where relations on
sets are concerned, we w i l l express our ideas and results in terms of nets
rather than f i l t e r s .
This convention w i l l be simplified by restricting
ourselves to a single domain as follows:
(Y, U)
uniform space.
Let
y Ç Y
{ y r | 6 € A } c y
o
~~
IJS
and let
converges to
Let
rv
<v
CP, U)
y
{x y
(Y, U)
be the completion of
be a Hausdorff
(Y, U)
U.
at
be a Cauchy net converging to
U
i s the set of symmetric entourages of
exists a Cauchy net
I
s
| U € U } £Y,
Let
y.
If
directed downwards, then there
with domain
and such t h a t , as subsets of
Y,
U
s
,
such that
(Xy)
(Xjj) £
(P, U)
be a nearly uniform ordered space, with completion
U.
We might reasonably expect the following d e f i n i t i o n to
at
extend the order on
net
Let
{Xy} £
p
converging to
P
to
converging to
^
y
such that
P ;
x < y
x,
i f and only i f for each Cauchy
there exists a Cauchy net
xy <
S
U Ç U .
for a l l
{y^j} £
However,
p
this
f>*
r e l a t i o n , which we call the strong order on
506
P,
does not necessarily
extend the original order:
the condition that
x^ <
Thus we are led to weaken this d e f i n i t i o n as follows:
uniform structure for
P
such that
i f and only i f for each Cauchy net
V £ J,
each
J* = U
fey}
£
there exists a Cauchy net
and
p
is too r e s t r i c t i v e .
Let
be a semi-
fl J" ^ G (S) .
converging to
{y^ £
7
p
Then
x,
converging to
x
y
and for
y
such
g
that
on
(x , y )
P.)
€ V
Thus, i f
for a l l
CP, U, 5)
semi-uniform structure for
x
y
nets
{y^} £ P
U ( U .
is a uniform ordered space, and i f
P
J*
such that
means that for every net
(pointwise)
(We call this relation the J-order
converging to
y
{x^} £ P
= U
and
1
0 7 = G(5) ,
converging to
x,
is a
then
we can find
which are as close as we want to being
greater than or equal to
{x^}.
For every nearly uniform ordered space, the strong order and the
orders are p a r t i a l orders on P .
Clearly
G CO £ G
.
Furthermore,
if
(P, U, <)
for any J-order
is a uniform ordered space, then
5 j.
Also,
any
G(5) = GC~jO fl C P X P)
«7-order makes the uniform completion
of a nearly uniform ordered space into a uniform ordered space.
Now the uniform completion of a Hausdorff uniform space
( X , (J)
is
in the sense that i t s a t i s f i e s the following universal mapping property
If
(Y, 10
is any complete Hausdorff uniform space, and i f
uniformly continuous function from
Cx, U)
a unique uniformly continuous function
(X, U)
of
(X, U), to
(Y, V)
f
to
(Y, V),
> C X , 0)
507
[l]:
i s any
then there exists
from the uniform completion
such that the diagram
CX, U)
f
"free"
commutes, where
i
i s the canonical embedding of
X
into
X.
In other
words, the uniform completion provides an adjoint to the functor which
embeds the category of complete Hausdorff uniform spaces in the
category
of Hausdorff uniform spaces.
I f we hope to achieve a similar property for the ordered case, we
must be able to pick out a p a r t i c u l a r semi-uniform structure w i t h which
to define an J'-order.
We can do this as f o l l o w s :
uniformity on a p a r t i a l l y ordered set
(P, 5 ) .
y(U) = {V Ç U | there e x i s t V ^
such that
V => GCS)
n —
Thus
U)
be a Hausdorff
Let
V2,
V =
and
U
Let
...
€ U
and for a l l
V
n+1
°V
^
c
n+1 —
v
n
n,
}.
consists of a l l those entourages which could conceivably be
members of a semi-uniform structure which generates
section contains the graph of
space,
J(U)
(D J(U)
£ G(<) ) H J(U)
Then for every
U
and whose inter-
(nearly)
i s the unique maximal semi-uniform structure
= G(<)
7(M) * = U.
and
uniform ordered
satisfying
Furthermore, i f
e
i s the
embedding functor of the category of complete uniform ordered spaces
the category of uniform ordered spaces, both with uniformly
order-preserving functions, then the functor
'V
ordered space
(P, U f <)
to
ê,
into
continuous
which takes a uniform
'Y
(P, U, S j ^ )
and a uniformly
continuous
function to i t s unique uniformly continuous extension, i s adjoint to
e.
The strong order is clearly e a s i e r to work w i t h than any of the X orders; so i t i s of interest to enquire when the strong order i s
to some J"-order (and thus to the
JiU) -order) .
508
I t turns out that
equivalent
if
(P, U,
<)
is a nearly uniform ordered space which s a t i s f i e s
(M) for all V Ç U, there exists W € U
such that W°G(<) £ g ( 5 ) o v ,
then the strong order is equivalent to the 7(U)-order.
This condition
(M)
w i l l in fact be s a t i s f i e d for certain semilattices, l a t t i c e s , and ^-groups.
We would hope that the extension procedure outlined thus far would take
lattices to lattices and ^-groups to ^-groups.
As long as the various opera-
tions are sufficiently well-behaved with respect to the uniform structure,
this does indeed happen.
( L , V)
S p e c i f i c a l l y , we say that a join-semilattice
with Hausdorff uniformity
(J
i s a j-uni form semi lattice i n case
i s uniformly continuous with respect to
U.
A uniform lattice
i s defined similarly, with both operations uniformly continuous.
j-uniform semilattice
condition
(M),
(L, V, U)
(L, V, A, LI)
Then a
is a uniform ordered space satisfying
and furthermore, the strong order on its uniform completion
iv
/v
L
i s the unique p a r t i a l order on
L
i s a join-subsemilattice of
ous.
V
rv
L
L,
such that
L
and the join on
is a join-semilattice,
L
is uniformly continu-
A similar result holds for uniform lattices.
The case for ^-groups answers most of a question raised by Conrad in
[3].
(It was this question which led the present author to consider the
general problem in the f i r s t p l a c e . )
group and lattice topology
of
B
B
3.
Consider an abelian £-group with Hausdorff
Is the strong order on the completion
at the usual uniformity the minimal lattice order on
is an abelian £-group, the positive cone of
cone of
B,
and the lattice operations on
509
B
B
B
B
such that
contains the positive
are continuous?
With one
assumption, some of our previous results may be used to give an affirmative
answer in the following more general s i t u a t i o n :
Hausdorff group and l a t t i c e topology
3".
Let
the usual uniformities associated with
if
Let
B
be an £-group with
B
be i t s completion at one of
at
least one of the
lattice
«V
operations on
B
is uniformly continuous,
the minimal lattice order on
j o i n is continuous.
B
then the strong order on
whose graph contains that of
Furthermore, i f
B
i s a group
lattice operations is uniformly continuous) , then
<
and lattice topology
(i)
V
3>,
In f a c t ,
is
and whose
(at least one of whose
(B,0
i s an &-group.
The assumption of uniform continuity of at least one of the
operations is not at a l l stringent.
B
for an &-group
B
lattice
with group
the following statements are e q u i v a l e n t :
i s uniformly continuous with respect to the r i g h t ,
l e f t or
two-sided uniformity;
(ii)
A
i s uniformly continuous with respect to the r i g h t ,
l e f t or
two-sided uniformity;
(iii)
the topology on
B
is locally
convex.
In the non-locally convex c a s e , i t is d i f f i c u l t to see i n t u i t i v e l y how to
order the completion, and thus a requirement of local convexity does not
really r e s t r i c t the cases that one might expect to consider.
510
REFERENCES
1.
N. Bourbaki, General Topology, Addison-Wesley Pub. C o . , Don M i l l s ,
Ontario,
1966
(translated from the French, Hermann,
2.
G. Cantor, Gesammelte Abhandlunqen, Springer, B e r l i n ,
3.
P.
Conrad,
Paris).
1932.
"The topological completion and the linearly compact h u l l
of an abelian £-group",
4.
J.
5.
L. Nachbin, Topology and Order, D. Van Nostrand C o . ,
6.
R. H. R e d f i e l d , "Ordering uniform completions of p a r t i a l l y ordered s e t s " ,
(to appear).
J . W. Tukey, Convergence and Uniformity in Topology, Ann. Math. Studies 2 ,
Princeton University Press, 1940.
7.
8.
L. K e l l e y ,
(to appear).
General Topology, D. Van Nostrand C o . ,
I n c . , New York,
Inc.,
Princeton,
A . W e i l , Sur les Espaces à Structure 'Uniforme et sur la Topologie
Act. S c i e n t , et I n d . , no. 5 5 1 , Hermann, P a r i s , 1937.
Simon Fraser University
Burnaby 2 , B r i t i s h
Columbia
Canada
511
1955.
1965
Generale