B is a numeric scalar or vector of up to three items, specifying the ranks of the cells to which f should be applied. The most general form is a three item vector p q r, where:

• p specifies the rank of the argument cells when f is applied monadically
• q specifies the rank of the left argument cells when f is applied dyadically
• r specifies the rank of the right argument cells when f is applied dyadically

If B is a two item vector q r, it is implicitly extended to r q r. If B has a single item r, it is implicitly extended to r r r.

If an item k of B is zero or positive it selects k-cells of the corresponding argument. If it is negative, it selects (r+k)-cells where r is the rank of the corresponding argument. A value of ¯1 selects major cells. For further information, see Cells and Sub-arrays.

If X is omitted, f may be any monadic function that returns a result. Y may be any array. The Rank operator ⍤ applies function f successively to the sub-arrays in Y specified by p (i.e. the first item of B, as specified or implicitly extended).

If X is specified, it may be any array and f may be any dyadic function that returns a result. Y may be any array. In this case, the Rank operator applies function f successively between the sub-arrays in X specified by q and the sub-arrays in Y specified by r.

The sub-arrays of R are the results of the individual applications of f. If these results differ in rank or shape, they are extended to a common rank and shape in the manner of Mix. See Mix.

Notice that it is necessary to prevent the right operand k binding to the right argument. This can be done using parentheses e.g. (f⍤1)Y. The same can be achieved using ⊢ e.g. f⍤1⊢Y because ⍤ binds tighter to its right operand than ⊢ does to its left argument, and ⊢ therefore resolves to Identity.

Monadic Examples

Using enclose (⊂) as the left operand elucidates the workings of the rank operator.

      Y
36 99 20  5
63 50 26 10
64 90 68 98
           
66 72 27 74
44  1 46 62
48  9 81 22
      ⍴Y
2 3 4

      ⊂⍤2 ⊢Y
┌───────────┬───────────┐
│36 99 20  5│66 72 27 74│
│63 50 26 10│44  1 46 62│
│64 90 68 98│48  9 81 22│
└───────────┴───────────┘

      ⊂⍤1 ⊢Y
┌───────────┬───────────┬───────────┐
│36 99 20 5 │63 50 26 10│64 90 68 98│
├───────────┼───────────┼───────────┤
│66 72 27 74│44 1 46 62 │48 9 81 22 │
└───────────┴───────────┴───────────┘

The function {(⊂⍋⍵)⌷⍵} sorts a vector.

      {(⊂⍋⍵)⌷⍵} 3 1 4 1 5 9 2 6 5
1 1 2 3 4 5 5 6 9

The rank operator can be used to apply the function to sub-arrays; in this case to sort the 1-cells (rows) of a 3-dimensional array.

      Y
36 99 20  5
63 50 26 10
64 90 68 98
           
66 72 27 74
44  1 46 62
48  9 81 22

      ({(⊂⍋⍵)⌷⍵}⍤1)Y
 5 20 36 99
10 26 50 63
64 68 90 98
           
27 66 72 74
 1 44 46 62
 9 22 48 81

Dyadic Examples

      10 20 30 (+⍤0 1)3 4⍴⍳12
10 11 12 13
24 25 26 27
38 39 40 41

Using the function {⍺ ⍵} as the left operand demonstrates how the dyadic case of the rank operator works.

      10 20 30 ({⍺ ⍵}⍤0 1)3 4⍴⍳12
┌──┬─────────┐
│10│0 1 2 3  │
├──┼─────────┤
│20│4 5 6 7  │
├──┼─────────┤
│30│8 9 10 11│
└──┴─────────┘

Note that a right operand of ¯1 applies the function between the major cells (in this case elements) of the left argument, and the major cells (in this case rows) of the right argument.

      10 20 30 ({⍺ ⍵}⍤¯1)3 4⍴⍳12
┌──┬─────────┐
│10│0 1 2 3  │
├──┼─────────┤
│20│4 5 6 7  │
├──┼─────────┤
│30│8 9 10 11│
└──┴─────────┘