In general, any abelian group can be expressed as a direct limit of its f.g. subgroups. For the case of $\ell$-group (lattice-ordered group) is that true or not? As an abelian group we do not have problem but the question is with the ordered structure.



Direct limit of lattice-ordered groups
I assume "$\ell$-group" abbreviates "lattice-ordered group"; actually, all I need is that it means algebras in some variety. In any variety, every algebra is the direct limit of its finitely generated subalgebras.


