202602102312

Tags : Model Theory

Elementary Equivalence of Models

---

Two structures $M$ and $N$ are said to be elementarily equivalent iff they have the same theory.

$N$ is called an Elementary embedding or an elementary substructure if $N$ is an embedding/ substructure of $M$ and it is elementarily equivalent to $M$.

The Tarski-Vaught test gives a necessary and sufficient condition to check if given a substructure, it is elementary.

---

References

• Tarski-Vaught test