Date: Sun, 23 Oct 1994 15:35:43 +0000 (GMT) From: "John G. Stell" <john@cs.keele.ac.uk>
What results of the form `A sub-X of a free X is free' are known? I am aware of groups, magmas (= set with binary operation), and modules over a principal ideal domain. I'd be interested to hear of structures for which this is either known to be true, or examples where it fails.
John Stell
It is true for abelian groups, but is false for both monoids and commutative monoids. The free (commutative) monoid on one element x has a submonoid consisting of 1,x^3,x^5,x^6 and all x^n for n > 7 is not free. Throwing away 1, you get counter-examples for semigroups and commutative semigroups. By taking the free abelian group on this example, you get counter-examples for rings and commutative rings. I am sure that it is a fairly rare occurrence. Michael Barr