# Does the category Monoid of monoids have finite coproducts?

Does the category Monoid of monoids have finite coproducts?

-
I rolled this question back to its previous version. Someone edited it to simply read aaaaaaaaaaaaaaaaaaaaaaaaaaa. (Which is presumably why it attracted votes to close.) –  David Speyer Mar 13 '10 at 19:04
@David: I haven't (yet) voted to close but the question immediately reminded me of mathoverflow.net/questions/18026/… –  Yemon Choi Mar 13 '10 at 19:55
I should add that the present question is a bit more interesting than the one I linked to (IMHO). –  Yemon Choi Mar 13 '10 at 20:00
There's something a bit strange going on here. After I answered, someone else posted a one-"sentence" answer that sounded like a drunk person yelling nonsense. It must have been multiply flagged then deleted by the moderators. –  Tom Leinster Mar 13 '10 at 20:04
Yes, the drunken shouting was deleted. –  David Speyer Mar 13 '10 at 23:28

Explicitly, the initial monoid ("0-fold coproduct of monoids") is the one-element monoid. The coproduct $A * B$ of two monoids $A, B$ is constructed similarly to the coproduct of two groups (often called their "free product"). That is, it's the free monoid generated by all the elements of $A$ together with all the elements of $B$, quotiented out by all the relations that hold in $A$ and all the relations that hold in $B$.