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pointwise_function_product [2014/03/21 11:11] 127.0.0.1 external edit |
pointwise_function_product [2015/04/17 15:28] nikolaj old revision restored (2015/04/17 15:27) |
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| ===== Pointwise function product ===== | ===== Pointwise function product ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| - | | @#55CCEE: context | @#55CCEE: $X$ | | + | | @#55CCEE: context | @#55CCEE: $S$ ... set | |
| - | | @#55CCEE: context | @#55CCEE: $M$...Magma | | + | | @#55CCEE: context | @#55CCEE: $\langle\!\langle M,* \rangle\!\rangle$ ... magma | |
| + | | @#FFBB00: definiendum | @#FFBB00: $\langle\!\langle M^S,\star \rangle\!\rangle$ | | ||
| + | | @#FF9944: definition | @#FF9944: $\star\in$ binary operation on $M^S$ | | ||
| + | | @#FF9944: definition | @#FF9944: $(f\star g)(s):=f(s)*g(s)$ | | ||
| - | Denote the binary operation on $M$ by $*$. | + | ----- |
| + | === Discussion === | ||
| + | Extends to groups, etc. | ||
| - | | @#FFBB00: definiendum | @#FFBB00: $ \star:(X\to M)\times(X\to M)\to (X\to M) $ | | + | >the following could be phrased more explicitly. |
| - | | $ f,g:X\to M $ | | + | 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]]. |
| - | | @#FFBB00: definiendum | @#FFBB00: $(f\star g)(x):=f(x)*g(x)$ | | + | === Reference === |
| + | Wikipedia: | ||
| + | [[http://en.wikipedia.org/wiki/Pointwise_product|Pointwise product]], | ||
| + | [[http://en.wikipedia.org/wiki/Magma_%28algebra%29|Magma]] | ||
| - | ==== Discussion ==== | + | ----- |
| - | A binary operation induces a binary operation on the function space of the respective magma. | + | |
| - | ==== Reference ==== | + | |
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Pointwise_product|Pointwise product]] | + | |
| - | ==== Parents ==== | + | |
| === Context === | === Context === | ||
| + | [[Magma]] | ||
| + | === Requirements* === | ||
| [[Magma]] | [[Magma]] | ||
