Free abelian lattice-ordered groups

Let n be a positive integer and FAℓ(n) be the free abelian lattice-ordered group on n generators. We prove that FAℓ(m) and FAℓ(n) do not satisfy the same first-order sentences in the language L={+,−,0,∧,∨} if m≠n. We also show that Th(FAℓ(n)) is decidable iff n∈{1,2}. Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices.