nLab omega-categorical structure

Idea

$\omega$-categorical structures are “highly symmetric” first-order structures.

Definition

A countable structure $A$ is $\omega$-categorical if any other countable structure $B \models \mathbf{Th}(A)$ with the same first-order theory as $A$ is isomorphic to $A$.

Examples

• The countable dense linear orders without endpoints are $\omega$-categorical.

• The countable random graph is $\omega$-categorical.

• Many Fraisse limits are $\omega$-categorical.

• The canonical (orbit) structure induced by an oligomorphic (finitely many orbits in each power) permutation group on a countable set is $\omega$-categorical.

Remarks

• Conversely, the action $\Aut(A) \curvearrowright A$ is oligomorphic.

• The automorphism group of an $\omega$-categorical structure $A$ equipped with the topology of pointwise convergence comprises a complete set of invariants for $\mathbf{Th}(A)$ up to bi-interpretability: this is the Coquand-Ahlbrandt-Ziegler theorem.

References

• Silvia Barbina, Automorphism groups of $\omega$-categorical structures