# Identity element

The identity element for a dyadic function is a value inherent to that function. It is defined as the value which would preserve the other argument of the dyadic function application, possibly only for a well-defined subset of the function's domain.

## Left and right identities

Since the identity element preserves the other argument, it can be a left and/or a right identity. For example, Add (`+`) has the left and right identity element `0` because `N≡N+0` and `N≡0+N` for all arrays `N` in the domain of `+`. However, the identity of Divide (`÷`), `1`, is only a right identity because while `N≡N÷1` is true for all `N` in the domain of `÷`, this isn't so for `N≡1÷N`, and no alternative identity element value exists which would fulfil the condition.

If a function `f` has both a left identity element and a right identity element (call them `l` and `r`), then they must be the same. This is because `l f r` $\Leftrightarrow$ `r`, since `l` is a left identity, and `l f r` $\Leftrightarrow$ `l`, since `r` is a right identity, so `l` $\Leftrightarrow$ `r`.

## Reduction over a length-0 axis

If a reduction (using one of `/`, `⌿`, `\`, or `⍀`) is performed over an axis of length 0, the resulting array is filled with identity elements. For example, the sum of an empty list is `0`, while the columnar sum of a two-column matrix with no rows is `0 0`:

```      +/0⍴0
0
+/0 2⍴0
0 0
```

## Support

Dialects differ in their support for such reductions. All define identity elements for most scalar primitives, and some stop there (e.g. SAX), while others (e.g. Dyalog APL and APL2) define identity elements for various mixed functions.

The identity element value for each function is defined in terms of the prototype `P` of the array `Y`:

Function name Glyph Value Left Right Notes
Add `+` `0` Yes Yes
Subtract `-` `0` No Yes
Multiply `×` `1` Yes Yes
Divide `÷` `1` No Yes
Residue `|` `0` Yes No
Minimum `⌊` `∞` Yes Yes the maximum representable number
Maximum `⌈` `-∞` Yes Yes the minimum representable number
Power `*` `1` No Yes
Circle function `○` `¯9` Yes No
Binomial `!` `1` Yes No
Root `√` `1` Yes No
And/LCM `∧` `1` Yes Yes
Or/GCD `∨` `0` Yes Yes Non-negative reals only
Less `<` `0` Yes No Booleans only
Less Or Equal `≤` `1` Yes No Booleans only
Equal to `=` `1` Yes Yes Booleans only
Greater Or Equal `≥` `1` No Yes Booleans only
Greater `>` `0` No Yes Booleans only
Not Equal `≠` `0` Yes Yes Booleans only
Reshape `⍴` `⍴P` Yes No
Catenate `,` `P⍴⍨ρ∘⊂⍨0,⍨¯1↓ρP` Yes No `1≤≢⍴Y`
Rotate `⌽` `0` or `0⍴⍨¯1↓⍴P` Yes No
Rotate First `⊖` `0` or `0⍴⍨1↓⍴P` Yes No
Transpose `⍉` `⍳≢⍴P` Yes No
Pick `⊃` `⍬` Yes No
Drop `↓` `⍬` or `0×⍴P` Yes No
Take `↑` `⍬` or `⍴P` Yes No
Squad Index `⌷` `⍬` or `⍳¨⍴P` Yes No
Without `~` `0⌿P` No Yes `1≤≢⍴Y`
Matrix Divide `⌹` `∘.=⍨⍳≢P` No Yes
Encode `⊤` `0` No Yes
Union `∪` `0⌿P` Yes Yes `1≤≢⍴Y`
Replicate `/` `1` Yes No `1≤≢⍴Y`
Expand `\` `∘.=⍨⍳≢P` Yes No `1≤≢⍴Y`
Inner products `+.×`
`∨.∧`
`∘.=⍨⍳≢P` Yes Yes
Inner product `∧.∨` `∘.≠⍨⍳≢P` Yes Yes