Free abelian lattice-ordered groups

Abstract

Let $n$ be a positive integer and $FA\ell \left(n\right)$ be the free abelian lattice-ordered group on $n$ generators. We prove that $FA\ell \left(m\right)$ and $FA\ell \left(n\right)$ do not satisfy the same first-order sentences in the language $\mathcal{L}=\{+,-,0,\wedge ,\vee \}$ if $m\ne n$. We also show that $\mathrm{Th}\left(FA\ell \left(n\right)\right)$ is decidable iff $n\in \{1,2\}$. Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices.