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	∧	0	Yes	Yes
Or/GCD	∨	1	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

External links

• Identity element

Documentation

• Dyalog APL

• APL2