Rank Operator (`f⍤k`)¶

Syntax¶

  (f⍤k) Y           ⍝ monadic: apply f to each k-cell of Y
X (f⍤k) Y           ⍝ dyadic: decompose X and Y, apply f to paired cells

Right operand (rank specification)¶

Form	Expansion	Meaning
`⍤c`	`c c c`	Same rank for monadic, left-dyadic, and right-dyadic
`⍤b c`	`c b c`	`b` = left rank, `c` = right and monadic rank
`⍤a b c`	as-is	`a` = monadic, `b` = left, `c` = right

Negative rank¶

Negative values are complementary: ⍤¯1 means cells of rank (array rank - 1). For a matrix, ⍤¯1 gives 1-cells (rows).

Cell decomposition¶

Given array rank r and cell rank k (after clamping to [0, r]):

Cell shape: the last k axes of the array.
Frame shape: the leading r-k axes.

The array is split into cells in row-major order of the frame.

Monadic use¶

      (⌽⍤1) 3 3⍴⍳9
3 2 1
6 5 4
9 8 7

Reverses each row (1-cell) of the matrix.

Dyadic use¶

      10 20 30 (+⍤0 1) 3 3⍴⍳9
11 12 13
24 25 26
37 38 39

Adds each scalar (0-cell) of the left argument to each row (1-cell) of the right argument.

Frame agreement¶

The frames of the left and right arguments must be identical, or one frame must be empty (scalar extension at the frame level).

Reassembly¶

Result cells are assembled back into a single array with shape frame_shape , max_cell_shape. If result cells differ in shape, shorter ones are padded with fill elements (0 for numeric, space for character).

Composing with reduce and scan¶

You can combine reduce or scan with rank:

      (+/⍤1) 3 3⍴⍳9
6 15 24

Row sums via +/ applied to each 1-cell.

      (+/⍤¯1) 3 3⍴⍳9
6 15 24

Equivalent using complementary rank.

First-axis reduce and scan¶

MARPLE does not implement ⌿ or ⍀. Use the rank operator instead:

f⌿ becomes (f/⍤¯1)
f⍀ becomes (f\⍤¯1)

Rank Operator (f⍤k)¶