# Leading axis agreement

**Leading axis agreement**, sometimes called **prefix agreement**, is a conformability rule designed for leading axis theory and used by J and BQN. It states that a dyadic scalar function can be applied between two arrays only if one of their shapes is a prefix of the other. The shape of the result is that of the argument with higher rank.

## Examples

The following examples use J for demonstration purposes.

A scalar dyadic function works when the two arrays have the same shape:

```
]x =: 2 3 $ 10
10 10 10
10 10 10
]y =: 2 3 $ i.6
0 1 2
3 4 5
x + y
10 11 12
13 14 15
```

as well as when one is a scalar:

```
]x =: 10
10
]y =: 2 3 $ i.6
0 1 2
3 4 5
x + y
10 11 12
13 14 15
```

The two cases above are already supported in other APLs in the form of scalar extension. J goes one step further, allowing the lower-rank array argument to have nonzero rank, as long as the leading dimensions match:

```
]x =: 10 20
10 20
]y =: 2 3 $ i.6
0 1 2
3 4 5
x + y
10 11 12
23 24 25
```

In this case, `x`

has shape `2`

and `y`

has shape `2 3`

. Since the leading axes agree and the rank difference is 1, each atom (or 0-cell) of `x`

is matched with each row (or 1-cell) of `y`

, and the two rows in the result are the results of `10 + 0 1 2`

and `20 + 3 4 5`

, respectively.

## Model

In dialects that do not feature leading axis agreement, it can nevertheless be utilised by the introduction of an explicit operator:

```
_LA←{⍺ ⍺⍺⍤(-⍺⌊⍥(≢⍴)⍵)⊢⍵}
⊢x ← 10 20
10 20
⊢y ← 2 3 ⍴ ⍳ 6
0 1 2
3 4 5
x +_LA y
10 11 12
23 24 25
```

## Aligning axes using the Rank operator

When using the Rank operator for dyadic functions as in `X (f⍤m n) Y`

, the frames of `X`

and `Y`

are checked for conformability. Combined with leading axis agreement, the Rank operator can be used to align the axes to be matched.

```
NB. $x : 2|3
NB. $y : |3 2
NB. ------------------
NB. $x +"1 2 y : 2 3 2
]x =: 2 3 $ 10 20 30 40 50 60
10 20 30
40 50 60
]y =: 3 2 $ 1 2 3 4 5 6
1 2
3 4
5 6
x +"1 2 y
11 12
23 24
35 36
41 42
53 54
65 66
```

APL features [edit]
| |
---|---|

Built-ins | Primitives (functions, operators) ∙ Quad name |

Array model | Shape ∙ Rank ∙ Depth ∙ Bound ∙ Index (Indexing) ∙ Axis ∙ Ravel ∙ Ravel order ∙ Element ∙ Scalar ∙ Vector ∙ Matrix ∙ Simple scalar ∙ Simple array ∙ Nested array ∙ Cell ∙ Major cell ∙ Subarray ∙ Empty array ∙ Prototype |

Data types | Number (Boolean, Complex number) ∙ Character (String) ∙ Box ∙ Namespace |

Concepts and paradigms | Conformability (Scalar extension, Leading axis agreement) ∙ Scalar function (Pervasion) ∙ Identity element ∙ Complex floor ∙ Total array ordering ∙ Tacit programming (Function composition, Close composition) ∙ Glyph |

Errors | LIMIT ERROR ∙ RANK ERROR ∙ SYNTAX ERROR ∙ DOMAIN ERROR ∙ LENGTH ERROR ∙ INDEX ERROR ∙ VALUE ERROR |