Difference between revisions of "Identity element"

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Miraheze>Adám Brudzewsky
m (Text replacement - "{{APL programming language}}" to "{{APL features}}")
m (→‎Support: Identity of And should be 1 (was 0), Or should be 0 (was 1))
 
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Since the identity element preserves the ''other'' argument, it can be a left and/or a right identity. For example, [[Add]] (<source lang=apl inline>+</source>) has the left and right identity element <source lang=apl inline>0</source> because <source lang=apl inline>N≡N+0</source> and <source lang=apl inline>N≡0+N</source> for all arrays <source lang=apl inline>N</source> in the domain of <source lang=apl inline>+</source>. However, the identity of [[Divide]] (<source lang=apl inline>÷</source>), <source lang=apl inline>1</source>, is only a right identity because while <source lang=apl inline>N≡N÷1</source> is true for all <source lang=apl inline>N</source> in the domain of <source lang=apl inline>÷</source>, this isn't so for <source lang=apl inline>N≡1÷N</source>, and no alternative identity element value exists which would fulfil the condition.
 
Since the identity element preserves the ''other'' argument, it can be a left and/or a right identity. For example, [[Add]] (<source lang=apl inline>+</source>) has the left and right identity element <source lang=apl inline>0</source> because <source lang=apl inline>N≡N+0</source> and <source lang=apl inline>N≡0+N</source> for all arrays <source lang=apl inline>N</source> in the domain of <source lang=apl inline>+</source>. However, the identity of [[Divide]] (<source lang=apl inline>÷</source>), <source lang=apl inline>1</source>, is only a right identity because while <source lang=apl inline>N≡N÷1</source> is true for all <source lang=apl inline>N</source> in the domain of <source lang=apl inline>÷</source>, this isn't so for <source lang=apl inline>N≡1÷N</source>, and no alternative identity element value exists which would fulfil the condition.
 +
 +
If a function <source lang=apl inline>f</source> has both a left identity element and a right identity element (call them <source lang=apl inline>l</source> and <source lang=apl inline>r</source>), then they must be the same. This is because <source lang=apl inline>l f r</source> {{←→}} <source lang=apl inline>r</source>, since <source lang=apl inline>l</source> is a left identity, and <source lang=apl inline>l f r</source> {{←→}} <source lang=apl inline>l</source>, since <source lang=apl inline>r</source> is a right identity, so <source lang=apl inline>l</source> {{←→}} <source lang=apl inline>r</source>.
  
 
== Reduction over a length-0 axis ==
 
== Reduction over a length-0 axis ==
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| [[Power]] || <source lang=apl inline>*</source> || <source lang=apl inline>1</source> || {{No}} || {{Yes}} ||
 
| [[Power]] || <source lang=apl inline>*</source> || <source lang=apl inline>1</source> || {{No}} || {{Yes}} ||
 
|-
 
|-
| [[Circular]] || <source lang=apl inline>○</source> || <source lang=apl inline>¯9</source> || {{Yes}} || {{No}} ||  
+
| [[Circle function]] || <source lang=apl inline>○</source> || <source lang=apl inline>¯9</source> || {{Yes}} || {{No}} ||  
 
|-
 
|-
 
| [[Binomial]] || <source lang=apl inline>!</source> || <source lang=apl inline>1</source> || {{Yes}} || {{No}} ||
 
| [[Binomial]] || <source lang=apl inline>!</source> || <source lang=apl inline>1</source> || {{Yes}} || {{No}} ||
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| [[Root]] || <source lang=apl inline>√</source> || <source lang=apl inline>1</source> || {{Yes}} || {{No}} ||
 
| [[Root]] || <source lang=apl inline>√</source> || <source lang=apl inline>1</source> || {{Yes}} || {{No}} ||
 
|-
 
|-
| [[And]]/[[LCM]] || <source lang=apl inline>∧</source> || <source lang=apl inline>0</source> || {{Yes}} || {{Yes}} ||
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| [[And]]/[[LCM]] || <source lang=apl inline>∧</source> || <source lang=apl inline>1</source> || {{Yes}} || {{Yes}} ||
 
|-
 
|-
| [[Or]]/[[GCD]] || <source lang=apl inline>∨</source> || <source lang=apl inline>1</source> || {{Yes}} || {{Yes}} ||
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| [[Or]]/[[GCD]] || <source lang=apl inline>∨</source> || <source lang=apl inline>0</source> || {{Yes}} || {{Yes}} || Non-negative reals only
 
|-
 
|-
 
| [[Less]] || <source lang=apl inline><</source> || <source lang=apl inline>0</source> || {{Yes}} || {{No}} || [[Boolean]]s only
 
| [[Less]] || <source lang=apl inline><</source> || <source lang=apl inline>0</source> || {{Yes}} || {{No}} || [[Boolean]]s only
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| [[Take]] || <source lang=apl inline>↑</source> || <source lang=apl inline>⍬</source> or <source lang=apl inline>⍴P</source>|| {{Yes}} || {{No}} ||
 
| [[Take]] || <source lang=apl inline>↑</source> || <source lang=apl inline>⍬</source> or <source lang=apl inline>⍴P</source>|| {{Yes}} || {{No}} ||
 
|-
 
|-
| [[Squad Indexing]] || <source lang=apl inline>⌷</source> || <source lang=apl inline>⍬</source> or <source lang=apl inline>⍳¨⍴P</source>|| {{Yes}} || {{No}} ||
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| [[Squad Index]] || <source lang=apl inline>⌷</source> || <source lang=apl inline>⍬</source> or <source lang=apl inline>⍳¨⍴P</source>|| {{Yes}} || {{No}} ||
 
|-
 
|-
| [[Without]] || <source lang=apl inline>~</source> || <source lang=apl inline>0⌿P</source> || {{Yes}} || {{Yes}} || <source lang=apl inline>1≤≢⍴Y</source>
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| [[Without]] || <source lang=apl inline>~</source> || <source lang=apl inline>0⌿P</source> || {{No}} || {{Yes}} || <source lang=apl inline>1≤≢⍴Y</source>
 
|-
 
|-
 
| [[Matrix Divide]] || <source lang=apl inline>⌹</source> || <source lang=apl inline>∘.=⍨⍳≢P</source> || {{No}} || {{Yes}} ||
 
| [[Matrix Divide]] || <source lang=apl inline>⌹</source> || <source lang=apl inline>∘.=⍨⍳≢P</source> || {{No}} || {{Yes}} ||
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== External links ==
 
== External links ==
  
* [[wikipedia:Identity element|Wikipedia]]
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* [[wikipedia:Identity element|Identity element]]
  
 
=== Documentation ===
 
=== Documentation ===
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* [https://www.ibm.com/downloads/cas/ZOKMYKOY#page=227 APL2]
 
* [https://www.ibm.com/downloads/cas/ZOKMYKOY#page=227 APL2]
  
{{APL features}}
+
{{APL features}}[[Category:Function characteristics]]

Latest revision as of 02:42, 2 June 2020

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 NN+0 and N0+N for all arrays N in the domain of +. However, the identity of Divide (÷), 1, is only a right identity because while NN÷1 is true for all N in the domain of ÷, this isn't so for N1÷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 r, since l is a left identity, and l f r l, since r is a right identity, so l 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:

      +/00
0
      +/0 20
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 , Pρ0,¯1ρP Yes No 1≤≢⍴Y
Rotate 0 or 0¯1↓⍴P Yes No
Rotate First 0 or 01↓⍴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 ~ 0P No Yes 1≤≢⍴Y
Matrix Divide ∘.=⍳≢P No Yes
Encode 0 No Yes
Union 0P 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

Documentation


APL features [edit]
Built-ins Primitive functionPrimitive operatorQuad name
Array model ShapeRankDepthBoundIndex (Indexing) ∙ AxisRavelRavel orderElementScalarVectorMatrixSimple scalarSimple arrayNested arrayCellMajor cellSubarrayEmpty arrayPrototype
Data types Number (Boolean, Complex number) ∙ Character (String) ∙ BoxNamespace
Concepts and paradigms Leading axis theoryScalar extensionConformabilityScalar functionPervasionGlyphIdentity elementComplex floorTotal array ordering
Errors LIMIT ERRORRANK ERRORSYNTAX ERRORDOMAIN ERRORLENGTH ERRORINDEX ERRORVALUE ERROR