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pointwise_function_product [2015/04/17 15:28]
nikolaj old revision restored (2015/04/17 15:19)
pointwise_function_product [2015/04/17 15:30] (current)
nikolaj
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 ===== Pointwise function product ===== ===== Pointwise function product =====
 ==== Set ==== ==== Set ====
-| @#55CCEE: context ​    | @#55CCEE: $X$ ... set | +| @#55CCEE: context ​    | @#55CCEE: $S$ ... set | 
-| @#55CCEE: context ​    | @#55CCEE: $\langle\!\langle M,​*\rangle\!\rangle$ ... magma | +| @#55CCEE: context ​    | @#55CCEE: $\langle\!\langle M,* \rangle\!\rangle$ ... magma | 
-| @#FF9944: definition ​ | @#FF9944: $ \star:(X\to M)\times(X\to M)\to (X\to M) $ | +| @#FF9944: definition ​ | @#FF9944: $\star\in$ binary operation on $M^S$ | 
-| @#FF9944: definition ​ | @#FF9944: $(f\star g)(x):=f(x)*g(x)$ |+| @#FF9944: definition ​ | @#FF9944: $(f\star g)(s):=f(s)*g(s)$ |
  
 ----- -----
 === Discussion === === Discussion ===
-A binary operation induces a binary operation on the function space of the respective magma+Extends to groups, etc. 
-==== Reference ​==== + 
-Wikipedia: [[http://​en.wikipedia.org/​wiki/​Pointwise_product|Pointwise product]]+>the following could be phrased more explicitly. 
 + 
 +Note that $M^S$ can is just another notation for ${\mathrm{Hom}}_{\bf{Set}}(S,​M)$. One of the main question of algebra is if a functor $F$ that maps into a a category of structures (like magmas) is representable,​ i.e. if there is a natural iso between $F$ and an internal [[Hom-functor]]. 
 + 
 +=== Reference === 
 +Wikipedia: ​ 
 +[[http://​en.wikipedia.org/​wiki/​Pointwise_product|Pointwise product]], 
 +[[http://​en.wikipedia.org/​wiki/​Magma_%28algebra%29|Magma]]
  
 ----- -----
 === Context === === Context ===
 +[[Magma]]
 +=== Subset of ===
 [[Magma]] [[Magma]]
 === Requirements* === === Requirements* ===
 [[Magma]] [[Magma]]
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