<s>
In	O
mathematics	O
,	O
specifically	O
in	O
order	O
theory	O
and	O
functional	B-Application
analysis	I-Application
,	O
an	O
ordered	O
vector	O
space	O
is	O
said	O
to	O
be	O
regularly	B-Algorithm
ordered	I-Algorithm
and	O
its	O
order	O
is	O
called	O
regular	O
if	O
is	O
Archimedean	B-Algorithm
ordered	I-Algorithm
and	O
the	O
order	B-Algorithm
dual	I-Algorithm
of	O
distinguishes	O
points	O
in	O
.	O
</s>
<s>
Being	O
a	O
regularly	B-Algorithm
ordered	I-Algorithm
vector	O
space	O
is	O
an	O
important	O
property	O
in	O
the	O
theory	O
of	O
topological	B-Algorithm
vector	I-Algorithm
lattices	I-Algorithm
.	O
</s>
<s>
Every	O
ordered	O
locally	O
convex	O
space	O
is	O
regularly	B-Algorithm
ordered	I-Algorithm
.	O
</s>
<s>
The	O
canonical	O
orderings	O
of	O
subspaces	O
,	O
products	O
,	O
and	O
direct	O
sums	O
of	O
regularly	B-Algorithm
ordered	I-Algorithm
vector	O
spaces	O
are	O
again	O
regularly	B-Algorithm
ordered	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
regularly	B-Algorithm
ordered	I-Algorithm
vector	B-Algorithm
lattice	I-Algorithm
then	O
the	O
order	B-Algorithm
topology	I-Algorithm
on	O
is	O
the	O
finest	O
topology	O
on	O
making	O
into	O
a	O
locally	O
convex	O
topological	B-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
.	O
</s>
