I am trying to solve the following problem and what I need is to split a function on the square of a finite set into tensor of two functions:
$R(\Gamma)$ is a k-algebra of maps from $\Gamma$ to k. $\Delta$ is a functor acting from $R(\Gamma)$ to $R(\Gamma) \otimes R(\Gamma)$: $(\Delta f)(g,h)=f(gh)$
If $\Gamma$ is a finite set, then $\Delta$ maps $R(\Gamma)$ into $R(\Gamma) \otimes R(\Gamma)$.
My question is how to split $(\Delta f)$ into tensor of two functions? I have no idea how to do that since $f$ has no special property at all.
(This is my first time to use stackexchange. I am sorry for possible mistakes.)