Free abelian lattice-ordered groups

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