Cartesian Product

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Cartesian Product ({, or @ in SAX), All, or Catalogue is a primitive monadic function that returns an array of all combinations of choices from its argument, much like Outer Product. It appears in Rationalized APL, A Dictionary of APL, SAX, and J. It shares a glyph with the dyadic function Select (like bracket indexing), which uses the same all-combinations pattern in the way it handles indices.

The argument to Cartesian Product is a boxed (nested) vector giving a list of arrays. It returns a boxed result whose shape is found by catenating all the argument array shapes, and where each result element contains one element from each argument array. In the following example from A Dictionary of APL, vectors of shapes 2, 2, and 3 (combined with Link) give a result of shape 2 2 3.

   { 'ht'⊃'ao'⊃'gtw'
|¯¯¯||¯¯¯||¯¯¯|
|hag||hat||haw|
|___||___||___|
|¯¯¯||¯¯¯||¯¯¯|
|hog||hot||how|
|___||___||___|

|¯¯¯||¯¯¯||¯¯¯|
|tag||tat||taw|
|___||___||___|
|¯¯¯||¯¯¯||¯¯¯|
|tog||tot||tow|
|___||___||___|

Documentation

• J NuVoc, Dictionary