Here is the question:

Just in case:

I have constructed a couple of monoids with homomorphisms, but I'm not getting anywhere. In addition I have a proof of

that does not use the inverses relationship

, so I don't see where this would enter the problem. Additionally the proof does not seem to give me any clues as to how

might not be equal to

.

The proof is short so I will include it. Perhaps someone may be able to see a loophole that is escaping me.

Proof:

Let G be a set with a binary relation * and let there be an element of G,

such that

for all g in G. Let H be another set with a binary relation

and let there be an element

such that

for all h in H. Further let f be a homeomorphism

.

Then

. Since by definition

for all h,

must be

.

-Dan