Theory of nontrivial torsion-free divisible Abelian groups ($\mathrm{DAG}$)

This first-order theory has as models the nonzero torsion-free divisible Abelian groups. Rational division equips them with the structure of vector spaces over the rational numbers. A rational divisible hull gives an algebraically prime extension of each nonzero torsion-free base, while the zero base uses a one-dimensional rational space.

In a torsion-free divisible Abelian group, division by a nonzero integer is unique. The rational scalar operations are therefore a definitional expansion. Formulas reduce to Boolean combinations of rational linear equations: eliminate an existential variable in a conjunction by solving any equation with nonzero coefficient; if none fixes the variable, the finitely many nontrivial disequalities exclude only finitely many choices in an infinite group. Equations with zero coefficient reduce to parameter conditions.

Thus over a finite parameter set a complete one-variable type either specifies an element of its linear span, or excludes every element of that span. An uncountable rational vector space realizes the latter type because a finite-parameter span is countable. It is consequently an omega-saturated model in the additive language.

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.