  This paper presents new results on the identities satisfied by the sylvester
and Baxter monoids. We show how to embed these monoids, of any rank strictly
greater than 2, into a direct product of copies of the corresponding monoid of
rank 2. This confirms that all monoids of the same family, of rank greater than
or equal to 2, satisfy exactly the same identities. We then give a complete
characterization of those identities, and prove that the varieties generated by
the sylvester and the Baxter monoids have finite axiomatic rank, by giving a
finite basis for them.
