nLab
omega-categorical structure

Contents

Idea

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

Definition

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

Examples

Remarks

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

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

References

Created on March 8, 2017 at 02:39:11. See the history of this page for a list of all contributions to it.