The Lebesgue state of a unital Abelian lattice-ordered group, II

We characterize the Lebesgue state of a free finitely generated unital lattice-ordered abelian group G in terms of its value at each element of each basis of G. This significantly strengthens one of the main results of our previous paper (co-authored by D. Mundici) with the same title as the present one. As a consequence of independent interest, we obtain a state-theoretic characterization of free finitely generated objects in the category of unital lattice-ordered abelian groups and their unit-preserving lattice-group homomorphisms.