Question

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

Answer 1

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.

Answer 2

Andreas is right that structures defined by finitary operations (such as here, where $l$-groups are defined by group operations and lattice operations $\wedge$, $\vee$) have this property: any structure is a directed colimit (in fact the union) of its finitely generated substructures. (Beware that "direct limit" just means "colimit"; "directed colimit" means a colimit of a functor $J \to C$ where $J$ is a directed poset.)

Anticipating another question, this property holds even for partially ordered groups where the partial order is not a lattice. A catchphrase is that the category of partially ordered groups is locally finitely presentable.

One difference between $l$-groups and partially ordered groups is that $l$-group homomorphisms that are bijections (at the underlying set level) are $l$-group isomorphisms. This is true for any variety (any class of structures defined by finitary operations). But for partially ordered groups, it need not be true that bijective homomorphisms are isomorphisms.