## 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