<s>
Rank	O
is	O
a	O
generalization	O
of	O
looping	O
as	O
used	O
in	O
scalar	O
(	O
non-array-oriented	O
)	O
programming	O
languages	O
.	O
</s>
<s>
It	O
is	O
also	O
a	O
generalization	O
of	O
mapcar	O
in	O
the	O
language	O
Lisp	B-Language
and	O
map	O
in	O
modern	O
functional	B-Language
programming	I-Language
languages	I-Language
,	O
and	O
a	O
generalization	O
of	O
scalar	O
extension	O
,	O
inner	O
(	O
matrix	B-Architecture
)	O
product	O
,	O
and	O
outer	O
product	O
in	O
APL	O
\360	O
.	O
</s>
<s>
The	O
canonical	O
implementation	O
of	O
rank	O
may	O
be	O
the	O
language	O
J	B-Language
,	O
but	O
it	O
is	O
also	O
available	O
in	O
Dyalog	O
APL	O
,	O
the	O
International	O
Organization	O
for	O
Standardization	O
(	O
ISO	O
)	O
technical	O
standard	O
on	O
Extended	O
APL	O
,	O
and	O
NARS2000	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
concept	O
of	O
rank	O
is	O
used	O
to	O
treat	O
an	O
orthogonal	O
array	B-Data_Structure
in	O
terms	O
of	O
its	O
subarrays	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
two-dimensional	O
array	B-Data_Structure
may	O
be	O
dealt	O
with	O
at	O
rank	O
2	O
as	O
the	O
entire	O
matrix	B-Architecture
,	O
or	O
at	O
rank	O
1	O
to	O
work	O
with	O
its	O
implicit	O
one-dimensional	O
columns	O
or	O
rows	O
,	O
or	O
at	O
rank	O
0	O
to	O
work	O
at	O
the	O
level	O
of	O
its	O
individual	O
atoms	O
.	O
</s>
<s>
Understanding	O
rank	O
requires	O
knowing	O
some	O
very	O
basic	O
array-oriented	O
programming	O
concepts	O
.	O
</s>
<s>
In	O
most	O
array-based	O
languages	O
,	O
reduction	O
is	O
denoted	O
with	O
a	O
forward	O
slash	O
/	O
.	O
</s>
<s>
In	O
J	B-Language
,	O
the	O
slash	O
takes	O
a	O
left	O
argument	O
of	O
the	O
function	O
and	O
a	O
right	O
argument	O
of	O
the	O
array	B-Data_Structure
to	O
be	O
reduced	O
by	O
that	O
function	O
.	O
</s>
<s>
An	O
N-dimensional	O
integer	O
array	B-Data_Structure
can	O
also	O
be	O
created	O
with	O
i	O
.	O
which	O
takes	O
a	O
vector	O
of	O
integers	O
as	O
its	O
arguments	O
.	O
</s>
<s>
Now	O
let	O
's	O
reduce	O
a	O
two-dimensional	O
array	B-Data_Structure
by	O
addition	O
.	O
</s>
<s>
This	O
application	O
of	O
+	O
/	O
to	O
a	O
two-dimensional	O
array	B-Data_Structure
corresponds	O
to	O
the	O
C	O
code	O
fragment	O
:	O
</s>
<s>
We	O
can	O
do	O
this	O
in	O
J	B-Language
without	O
looping	O
simply	O
by	O
using	O
rank	O
.	O
</s>
<s>
To	O
illustrate	O
further	O
how	O
rank	O
works	O
in	O
J	B-Language
,	O
we	O
can	O
see	O
the	O
original	O
expression	O
is	O
rank	O
2	O
.	O
</s>
<s>
The	O
operator	O
is	O
mapped	O
at	O
the	O
highest	O
rank	O
to	O
the	O
array	B-Data_Structure
.	O
</s>
<s>
Nouns	O
,	O
in	O
J	B-Language
,	O
are	O
arrays	O
.	O
</s>
<s>
The	O
rank	O
of	O
a	O
noun	O
is	O
the	O
number	O
of	O
dimensions	O
of	O
that	O
array	B-Data_Structure
.	O
</s>
<s>
Verbs	O
,	O
in	O
J	B-Language
,	O
are	O
functions	O
which	O
take	O
noun	O
arguments	O
and	O
produce	O
noun	O
results	O
.	O
</s>
<s>
So	O
,	O
if	O
we	O
want	O
to	O
add	O
a	O
vector	O
to	O
each	O
vector	O
in	O
a	O
matrix	B-Architecture
:	O
</s>
