<s>
In	O
mathematics	O
,	O
a	O
Riesz	B-Algorithm
space	I-Algorithm
,	O
lattice-ordered	B-Algorithm
vector	I-Algorithm
space	I-Algorithm
or	O
vector	B-Algorithm
lattice	I-Algorithm
is	O
a	O
partially	O
ordered	O
vector	O
space	O
where	O
the	O
order	O
structure	O
is	O
a	O
lattice	O
.	O
</s>
<s>
Riesz	B-Algorithm
spaces	I-Algorithm
are	O
named	O
after	O
Frigyes	O
Riesz	O
who	O
first	O
defined	O
them	O
in	O
his	O
1928	O
paper	O
Sur	O
la	O
décomposition	O
des	O
opérations	O
fonctionelles	O
linéaires	O
.	O
</s>
<s>
Riesz	B-Algorithm
spaces	I-Algorithm
have	O
wide-ranging	O
applications	O
.	O
</s>
<s>
They	O
are	O
important	O
in	O
measure	O
theory	O
,	O
in	O
that	O
important	O
results	O
are	O
special	O
cases	O
of	O
results	O
for	O
Riesz	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Riesz	B-Algorithm
spaces	I-Algorithm
have	O
also	O
seen	O
application	O
in	O
mathematical	O
economics	O
through	O
the	O
work	O
of	O
Greek-American	O
economist	O
and	O
mathematician	O
Charalambos	O
D	O
.	O
Aliprantis	O
.	O
</s>
<s>
A	O
preordered	O
vector	B-Algorithm
lattice	I-Algorithm
is	O
a	O
preordered	O
vector	O
space	O
in	O
which	O
every	O
pair	O
of	O
elements	O
has	O
a	O
supremum	O
.	O
</s>
<s>
More	O
explicitly	O
,	O
a	O
preordered	O
vector	B-Algorithm
lattice	I-Algorithm
is	O
vector	O
space	O
endowed	O
with	O
a	O
preorder	O
,	O
such	O
that	O
for	O
any	O
:	O
</s>
<s>
A	O
preordered	O
vector	O
space	O
is	O
a	O
preordered	O
vector	B-Algorithm
lattice	I-Algorithm
if	O
and	O
only	O
if	O
it	O
satisfies	O
any	O
of	O
the	O
following	O
equivalent	O
properties	O
:	O
</s>
<s>
A	O
Riesz	B-Algorithm
space	I-Algorithm
or	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
is	O
a	O
preordered	O
vector	B-Algorithm
lattice	I-Algorithm
whose	O
preorder	O
is	O
a	O
partial	O
order	O
.	O
</s>
<s>
Note	O
that	O
many	O
authors	O
required	O
that	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
be	O
a	O
partially	O
ordered	O
vector	O
space	O
(	O
rather	O
than	O
merely	O
a	O
preordered	O
vector	O
space	O
)	O
while	O
others	O
only	O
require	O
that	O
it	O
be	O
a	O
preordered	O
vector	O
space	O
.	O
</s>
<s>
We	O
will	O
henceforth	O
assume	O
that	O
every	O
Riesz	B-Algorithm
space	I-Algorithm
and	O
every	O
vector	B-Algorithm
lattice	I-Algorithm
is	O
an	O
ordered	O
vector	O
space	O
but	O
that	O
a	O
preordered	O
vector	B-Algorithm
lattice	I-Algorithm
is	O
not	O
necessarily	O
partially	O
ordered	O
.	O
</s>
<s>
If	O
is	O
an	O
ordered	O
vector	O
space	O
over	O
whose	O
positive	O
cone	O
(	O
the	O
elements	O
)	O
is	O
generating	O
(	O
that	O
is	O
,	O
such	O
that	O
)	O
,	O
and	O
if	O
for	O
every	O
either	O
or	O
exists	O
,	O
then	O
is	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
.	O
</s>
<s>
An	O
order	O
unit	O
of	O
a	O
preordered	O
vector	O
space	O
is	O
any	O
element	O
such	O
that	O
the	O
set	O
is	O
absorbing	B-Algorithm
.	O
</s>
<s>
If	O
a	O
space	O
is	O
ordered	O
then	O
its	O
order	B-Algorithm
bound	I-Algorithm
dual	I-Algorithm
is	O
a	O
vector	O
subspace	O
of	O
its	O
algebraic	O
dual	O
.	O
</s>
<s>
Finite-dimensional	O
Riesz	B-Algorithm
spaces	I-Algorithm
are	O
entirely	O
classified	O
by	O
the	O
Archimedean	B-Algorithm
property	I-Algorithm
:	O
</s>
<s>
Theorem	O
:	O
Suppose	O
that	O
is	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
of	O
finite-dimension	O
If	O
is	O
Archimedean	B-Algorithm
ordered	I-Algorithm
then	O
it	O
is	O
(	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
)	O
isomorphic	O
to	O
under	O
its	O
canonical	O
order	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
epi-mono	O
factorization	O
in	O
the	O
category	O
of	O
-vector	O
spaces	O
also	O
applies	O
to	O
Riesz	B-Algorithm
spaces	I-Algorithm
:	O
every	O
lattice-ordered	B-Algorithm
vector	I-Algorithm
space	I-Algorithm
injects	O
into	O
a	O
quotient	O
of	O
by	O
a	O
solid	B-Algorithm
subspace	I-Algorithm
.	O
</s>
<s>
Every	O
Riesz	B-Algorithm
space	I-Algorithm
is	O
a	O
partially	O
ordered	O
vector	O
space	O
,	O
but	O
not	O
every	O
partially	O
ordered	O
vector	O
space	O
is	O
a	O
Riesz	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Disjoint	O
complements	O
are	O
always	O
bands	B-Algorithm
,	O
but	O
the	O
converse	O
is	O
not	O
true	O
in	O
general	O
.	O
</s>
<s>
Every	O
Riesz	B-Algorithm
space	I-Algorithm
has	O
the	O
Riesz	O
decomposition	O
property	O
.	O
</s>
<s>
There	O
are	O
a	O
number	O
of	O
meaningful	O
non-equivalent	O
ways	O
to	O
define	O
convergence	O
of	O
sequences	O
or	O
nets	O
with	O
respect	O
to	O
the	O
order	O
structure	O
of	O
a	O
Riesz	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
A	O
sequence	O
in	O
a	O
Riesz	B-Algorithm
space	I-Algorithm
is	O
said	O
to	O
converge	O
monotonely	O
if	O
it	O
is	O
a	O
monotone	O
decreasing	O
(	O
resp	O
.	O
</s>
<s>
The	O
collection	O
of	O
each	O
kind	O
structure	O
in	O
a	O
Riesz	B-Algorithm
space	I-Algorithm
(	O
for	O
example	O
,	O
the	O
collection	O
of	O
all	O
ideals	O
)	O
forms	O
a	O
distributive	O
lattice	O
.	O
</s>
<s>
If	O
is	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
then	O
a	O
vector	O
sublattice	O
is	O
a	O
vector	O
subspace	O
of	O
such	O
that	O
for	O
all	O
belongs	O
to	O
(	O
where	O
this	O
supremum	O
is	O
taken	O
in	O
)	O
.	O
</s>
<s>
A	O
band	O
in	O
a	O
Riesz	B-Algorithm
space	I-Algorithm
is	O
defined	O
to	O
be	O
an	O
ideal	O
with	O
the	O
extra	O
property	O
,	O
that	O
for	O
any	O
element	O
for	O
which	O
its	O
absolute	O
value	O
is	O
the	O
supremum	O
of	O
an	O
arbitrary	O
subset	O
of	O
positive	O
elements	O
in	O
that	O
is	O
actually	O
in	O
-Ideals	O
are	O
defined	O
similarly	O
,	O
with	O
the	O
words	O
'	O
arbitrary	O
subset	O
 '	O
replaced	O
with	O
'	O
countable	O
subset	O
 '	O
.	O
</s>
<s>
The	O
intersection	O
of	O
an	O
arbitrary	O
family	O
of	O
bands	B-Algorithm
is	O
again	O
a	O
band	O
.	O
</s>
<s>
The	O
collection	O
of	O
all	O
projection	O
bands	B-Algorithm
in	O
a	O
Riesz	B-Algorithm
space	I-Algorithm
forms	O
a	O
Boolean	O
algebra	O
.	O
</s>
<s>
Some	O
spaces	O
do	O
not	O
have	O
non-trivial	O
projection	O
bands	B-Algorithm
(	O
for	O
example	O
,	O
)	O
,	O
so	O
this	O
Boolean	O
algebra	O
may	O
be	O
trivial	O
.	O
</s>
<s>
A	O
vector	B-Algorithm
lattice	I-Algorithm
is	O
complete	O
if	O
every	O
subset	O
has	O
both	O
a	O
supremum	O
and	O
an	O
infimum	O
.	O
</s>
<s>
A	O
vector	B-Algorithm
lattice	I-Algorithm
is	O
Dedekind	O
complete	O
if	O
each	O
set	O
with	O
an	O
upper	O
bound	O
has	O
a	O
supremum	O
and	O
each	O
set	O
with	O
a	O
lower	O
bound	O
has	O
an	O
infimum	O
.	O
</s>
<s>
An	O
order	B-Algorithm
complete	I-Algorithm
,	O
regularly	O
ordered	O
vector	B-Algorithm
lattice	I-Algorithm
whose	O
canonical	O
image	O
in	O
its	O
order	B-Algorithm
bidual	I-Algorithm
is	O
order	B-Algorithm
complete	I-Algorithm
is	O
called	O
minimal	O
and	O
is	O
said	O
to	O
be	O
of	O
minimal	O
type	O
.	O
</s>
<s>
A	O
sublattice	O
of	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
is	O
a	O
vector	O
subspace	O
of	O
such	O
that	O
for	O
all	O
belongs	O
to	O
(	O
importantly	O
,	O
note	O
that	O
this	O
supremum	O
is	O
taken	O
in	O
and	O
not	O
in	O
)	O
.	O
</s>
<s>
This	O
despite	O
being	O
an	O
order	B-Algorithm
complete	I-Algorithm
Archimedean	B-Algorithm
ordered	I-Algorithm
topological	B-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
and	O
is	O
a	O
solid	B-Algorithm
vector	O
subspace	O
of	O
then	O
defines	O
the	O
canonical	O
order	O
of	O
under	O
which	O
is	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
and	O
the	O
canonical	O
map	O
is	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
homomorphism	O
.	O
</s>
<s>
If	O
is	O
a	O
topological	B-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
and	O
is	O
a	O
closed	O
solid	B-Algorithm
sublattice	O
of	O
then	O
is	O
also	O
a	O
topological	B-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
.	O
</s>
<s>
If	O
and	O
are	O
vector	B-Algorithm
lattices	I-Algorithm
with	O
order	B-Algorithm
complete	I-Algorithm
and	O
if	O
is	O
the	O
set	O
of	O
all	O
positive	O
linear	O
maps	O
from	O
into	O
then	O
the	O
subspace	O
of	O
is	O
an	O
order	B-Algorithm
complete	I-Algorithm
vector	B-Algorithm
lattice	I-Algorithm
under	O
its	O
canonical	O
order	O
;	O
</s>
<s>
Suppose	O
that	O
and	O
are	O
preordered	O
vector	B-Algorithm
lattices	I-Algorithm
with	O
positive	O
cones	O
and	O
and	O
let	O
be	O
a	O
map	O
.	O
</s>
<s>
Then	O
is	O
a	O
preordered	O
vector	B-Algorithm
lattice	I-Algorithm
homomorphism	O
if	O
is	O
linear	O
and	O
if	O
any	O
one	O
of	O
the	O
following	O
equivalent	O
conditions	O
hold	O
:	O
</s>
<s>
A	O
pre-ordered	O
vector	B-Algorithm
lattice	I-Algorithm
homomorphism	O
that	O
is	O
bijective	O
is	O
a	O
pre-ordered	O
vector	B-Algorithm
lattice	I-Algorithm
isomorphism	O
.	O
</s>
<s>
A	O
pre-ordered	O
vector	B-Algorithm
lattice	I-Algorithm
homomorphism	O
between	O
two	O
Riesz	B-Algorithm
spaces	I-Algorithm
is	O
called	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
homomorphism	O
;	O
</s>
<s>
if	O
it	O
is	O
also	O
bijective	O
,	O
then	O
it	O
is	O
called	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
isomorphism	O
.	O
</s>
<s>
If	O
is	O
a	O
non-zero	O
linear	B-Algorithm
functional	I-Algorithm
on	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
with	O
positive	O
cone	O
then	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
is	O
a	O
surjective	O
vector	B-Algorithm
lattice	I-Algorithm
homomorphism	O
.	O
</s>
<s>
A	O
vector	B-Algorithm
lattice	I-Algorithm
homomorphism	O
from	O
into	O
is	O
a	O
topological	B-Algorithm
homomorphism	I-Algorithm
when	O
and	O
are	O
given	O
their	O
respective	O
order	B-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
There	O
are	O
numerous	O
projection	O
properties	O
that	O
Riesz	B-Algorithm
spaces	I-Algorithm
may	O
have	O
.	O
</s>
<s>
A	O
Riesz	B-Algorithm
space	I-Algorithm
is	O
said	O
to	O
have	O
the	O
(	O
principal	O
)	O
projection	O
property	O
if	O
every	O
(	O
principal	O
)	O
band	O
is	O
a	O
projection	O
band	O
.	O
</s>
<s>
Archimedean	B-Algorithm
property	I-Algorithm
if	O
,	O
for	O
every	O
pair	O
of	O
positive	O
elements	O
and	O
,	O
whenever	O
the	O
inequality	O
holds	O
for	O
all	O
integers	O
,	O
.	O
</s>
<s>
SDC	O
implies	O
DC	O
;	O
DC	O
implies	O
both	O
Dedekind	O
-completeness	O
and	O
the	O
projection	O
property	O
;	O
Both	O
Dedekind	O
-completeness	O
and	O
the	O
projection	O
property	O
separately	O
imply	O
the	O
principal	O
projection	O
property	O
;	O
and	O
the	O
principal	O
projection	O
property	O
implies	O
the	O
Archimedean	B-Algorithm
property	I-Algorithm
.	O
</s>
<s>
The	O
space	O
of	O
continuous	O
real	O
valued	O
functions	O
with	O
compact	O
support	O
on	O
a	O
topological	O
space	O
with	O
the	O
pointwise	O
partial	O
order	O
defined	O
by	O
when	O
for	O
all	O
is	O
a	O
Riesz	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Any	O
space	O
with	O
the	O
(	O
almost	O
everywhere	O
)	O
pointwise	O
partial	O
order	O
is	O
a	O
Dedekind	O
complete	O
Riesz	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
space	O
with	O
the	O
lexicographical	O
order	O
is	O
a	O
non-Archimedean	O
Riesz	B-Algorithm
space	I-Algorithm
.	O
</s>
