 Theory of torsion-free divisible Abelian groups allowing the trivial group

Remove the nonzero-model axiom from DAG. The inclusion $\{0\}\subseteq \mathbb{Q}$ is then an inclusion of models but is not a simple closure, because $x\ne 0$ has a witness only in the larger model. The theory does not have quantifier elimination: all closed group terms are zero, so no quantifier-free sentence distinguishes the two models.