# Rank (operator)

*This article is about the operator. See Rank for the number associated with every array. For numbers associated with a function specifying its argument rank, see function rank.*

Rank (`⍤`

) is a primitive dyadic operator which applies its left operand function to cells of its arguments specified by its right operand array.

## Contents

## Rank specification

The right operand specifies the rank of subarrays to which the left operand function is applied as follows:
For left argument `⍺`

and right argument `⍵`

,

```
⍤ c ⍝ Rank-c cells of ⍵ (monadic) or both arguments (dyadic)
⍤ b c ⍝ Rank-b cells of ⍺ and rank-c cells of ⍵ (dyadic)
⍤a b c ⍝ Rank-a cells of ⍵ (monadic), b-cells of ⍺ and c-cells of ⍵ (dyadic)
```

A non-negative right operand specifies the number of *final* axes to which the function applies. A negative right operand specifies *complementary* rank, i.e. the number of leading axes to be *excluded*. Negative rank can also be thought of as rank specification *relative to* the overall rank of the argument array.

Since a rank specification greater than the rank of the argument array means to apply the function to the whole array, `99`

, `(⌊/⍬)`

or `∞`

, depending on the implementation, is "rank infinity" and always specifies the whole argument array.

## Examples

Rotate rows in matrices of a 3D array:

```
⊖⍤2⊢3 2 4⍴⎕A
EFGH
ABCD
MNOP
IJKL
UVWX
QRST
```

Laminate scalars from arrays of differing ranks:

```
'ABCD',⍤0⍤1⊢2 4⍴⍳8
A 1
B 2
C 3
D 4
A 5
B 6
C 7
D 8
```

Flat outer product:

```
-⍤1⍤1 99⍨3 2⍴6 7 1 1 2 4 ⍝ ↑∘.-⍨↓
0 0
5 6
4 3
¯5 ¯6
0 0
¯1 ¯3
¯4 ¯3
1 3
0 0
```

## History

The rank operator was invented by Arthur Whitney in 1982 and first implemented in SHARP APL in 1983. It has been described as "a microcosm of APL history"^{[1]}, its evolution a progression from scalar extension, which has been in APL since its inception, through leading axis theory to a construct which is a generalisation of scalar extension, inner (matrix) product, outer product, maplist in LISP, map in modern functional programming languages and the broadcast facility in NumPy.