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*

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