===== Pointwise function product =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $S$ ... set |
| @#55CCEE: context     | @#55CCEE: $\langle\!\langle M,* \rangle\!\rangle$ ... magma |
| @#FF9944: definition  | @#FF9944: $\star\in$ binary operation on $M^S$ |
| @#FF9944: definition  | @#FF9944: $(f\star g)(s):=f(s)*g(s)$ |

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=== Discussion ===
Extends to groups, etc.

>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]]

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=== Context ===
[[Magma]]
=== Subset of ===
[[Magma]]
=== Requirements* ===
[[Magma]]