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Identity element: Difference between revisions
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→Left and right identities: Prove left and right identities can't be distinct
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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 == | ||