The Freeness of a Group Based on a Distributive Lattice1 P. Hill and H. Subramanian

Let L be a distributive lattice and G the abelian group with the following presentation. The generators of G are the elements of the lattice L, and the relations are (a V b) + (a A 6) = a + b where a and b are arbitrary elements of L, It is shown that G is free abelian. In particular, G is torsion free. The latter statement answers affirmatively a question posed several years ago by E. Weinberg. This brief note is in the nature of an addendum to an earlier paper by one of the authors [2]. Our purpose is to prove that an abelian group based on a distributive lattice remains free under the relations ia V b) + ia A b) = a + b. This result is basically a corollary of the two theorems in [2], but the application was previously overlooked. In fact, the second-named author initially proved our result, without reference to [2], using a theorem of G. Nöbeling [4], However, it is the more recent results of [l] and [2] that yield the short proof that follows. Theorem. Let L be an arbitrary distributive lattice and let G be the abelian group generated by the elements of L with relations ia V b) + ia A b) = a + b for all a, b £ L. Then G is free. Proof. Denote by F the free abelian group based on the distributive lattice L (with no relations at all on F). Then F is isomorphic to the additive group of the semigroup ring Z[S] where S is the semigroup associated with the set L and the operation A alone. The advantage of this identification is that F is now endowed, in a natural way, with multiplication and a ring structure. Indeed, F is a commutative ring generated by idempotents. We shall denote the additive structure of F by F and, more generally, follow the notation of [2]. Received by the editors June 10, 1974. AMS (MOS) subject classifications (1970). Primary 20K05, 20K15, 20K20, 20K25; Secondary 16A32, 06A35.