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Identity element: Difference between revisions
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Miraheze>Adám Brudzewsky (Created page with "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...") |
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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}} || | ||
|- | |- | ||
| [[ | | [[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> | | [[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> | | [[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 | | [[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> || {{ | | [[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:Identity element|Identity element]] | ||
=== Documentation === | === Documentation === | ||
* [ | * [https://help.dyalog.com/latest/#Language/Primitive%20Operators/Reduce.htm Dyalog APL] | ||
* [https://www.ibm.com/downloads/cas/ZOKMYKOY#page=227 APL2] | * [https://www.ibm.com/downloads/cas/ZOKMYKOY#page=227 APL2] | ||
{{APL | {{APL features}}[[Category:Function characteristics]] | ||