Pointwise function product
Set
| context | $S$ … set |
| context | $\langle\!\langle M,* \rangle\!\rangle$ … magma |
| definition | $\star\in$ binary operation on $M^S$ |
| definition | $(f\star g)(s):=f(s)*g(s)$ |
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: Pointwise product, Magma
Context
Subset of
Requirements*
