Talk:Extension Theorem for Distributive Operations

The question (paraphrased) is: "Why is it necessary for $\left({T, *}\right)$ to be an abelian group?"

The stipulation was made in Warner that all elements of $R$ are cancellable for $*$. This is necessary for $\left({T, *}\right)$ to be a group (if it's not then not all elements are invertible). Warner then claims that this is necessary for Extension Theorem for Homomorphisms to be applied:


 * "By Theorem 20.2 (which is Inverse Completion of Commutative Semigroup is Abelian Group) $\left({T, *}\right)$ is a commutative group. Therefore by Theorem 20.4 (which is Extension Theorem for Homomorphisms, every homomorphism from $\left({R, *}\right)$ into $\left({T, *}\right)$ is the restriction to $R$ of one and only one endomorphism of $\left({T, *}\right)$ and so if $g$ and $h$ are two endomorphisms of $T$ coinciding on $R$ (that is, whose restrictions to $R$ are the same function), then $g = h$."

But I can't see why it is necessary for $\left({T, *}\right)$ to be a group (by default abelian) for this to apply. I'm missing something obvious.

Anyone able to help here? --prime mover (talk) 16:00, 3 May 2015 (UTC)