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)

O yes, good call. In that case there's a case for putting a result in place clarifying this. The fact that Group Homomorphism Preserves Identity is because the only idempotent element in a group is the identity, of course. So: if a monoid is cancellable, identity is automatically preserved? I will need to think about it ... --prime mover 10:10, 9 August 2012 (UTC)

In contemplating that a functor is characterised by only the morphism property (pres. of id. can be proved), I found the following reasoning:

$\phi(h)=\phi(e\circ h)=\phi(e)*\phi(h)$

and mutatis mutandis the other identity property. Thus the confusion arose only because we were too narrow-minded to contemplate different proofs for Group Homomorphism Preserves Identity. I will add a new proof for that one using this rule. Then we can in principle dispose of this page in favour of Group Hom. --Lord_Farin 07:14, 16 August 2012 (UTC)

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)

Yes, I had that realisation when I went down this alley myself on first posting up this material. There are plenty words to that effect in the various pages "Epimorphism Preserves ..." etc. When the thing you are morphing to is a structure with appropriately specified properties, e.g. a Group or a Ring etc., then of course you can state things about objects not in the image. Hence the care with declaring "group homomorphism" and so on - they specify both the domain and range as being groups.--prime mover 07:26, 16 August 2012 (UTC)