# Definition talk:Monoid Homomorphism

If $S$ and $T$ are monoids, then any homomorphism between the two will always preserve the identity. This will need to be proved as a result (same lines as Group Homomorphism Preserves Identity, basically), but it means that there should be no need to specify the required properties of the homomorphism that require it to be a "monoid homomorphism". --prime mover 09:38, 9 August 2012 (UTC)
Except that a monoid need not satisfy Cancellation Laws. I thought about it; this is genuinely necessary. Otherwise, the identity could in principle be mapped to any idempotent $a$ with $a \circ a = a$. --Lord_Farin 09:43, 9 August 2012 (UTC)
$\phi(h)=\phi(e\circ h)=\phi(e)*\phi(h)$
Argh, cancel that. It can only be shown in that way for epimorphisms, because we can't say anything about things not in the image of $\phi$... --Lord_Farin 07:18, 16 August 2012 (UTC)