<s>
In	O
mathematics	O
,	O
specifically	O
in	O
order	O
theory	O
and	O
functional	B-Application
analysis	I-Application
,	O
two	O
elements	O
x	O
and	O
y	O
of	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
X	O
are	O
lattice	B-Algorithm
disjoint	I-Algorithm
or	O
simply	O
disjoint	O
if	O
,	O
in	O
which	O
case	O
we	O
write	O
,	O
where	O
the	O
absolute	O
value	O
of	O
x	O
is	O
defined	O
to	O
be	O
.	O
</s>
<s>
We	O
say	O
that	O
two	O
sets	O
A	O
and	O
B	O
are	O
lattice	B-Algorithm
disjoint	I-Algorithm
or	O
disjoint	O
if	O
a	O
and	O
b	O
are	O
disjoint	O
for	O
all	O
a	O
in	O
A	O
and	O
all	O
b	O
in	O
B	O
,	O
in	O
which	O
case	O
we	O
write	O
.	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>
If	O
A	O
is	O
a	O
subset	O
of	O
X	O
such	O
that	O
exists	O
,	O
and	O
if	O
B	O
is	O
a	O
subset	O
lattice	O
in	O
X	O
that	O
is	O
disjoint	O
from	O
A	O
,	O
then	O
B	O
is	O
a	O
lattice	B-Algorithm
disjoint	I-Algorithm
from	O
.	O
</s>
