The freeness of a group based on a distributive lattice

Abstract:

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 $(a \vee b) + (a \wedge b) = a + b$ 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.