Re: sub-structures of free structures

There is certainly a good bit more known than in Michael's response. The varieties in which subalgebras of free algebras are free are called Schreier varieties. A paper of mine in the early 70's -- Epimorphisms and dominions, V, probably in Alg. Univ. -- answered someone's question (Sabbagh?): Is every Schreier variety balanced? I remember the answer was No. John Isbell