elementary embedding

Let $\tau$ be a signature and $𝒜$ and $ℬ$ be two structures for $\tau$ such that $f:𝒜\to ℬ$ is an embedding. Then $f$ is said to be elementary if for every first-order formula $\varphi \in F(\tau )$, we have

$$𝒜\models \varphi \mathit{\hspace{1em}}\text{iff}\mathit{\hspace{1em}}ℬ\models \varphi .$$

In the expression above, $𝒜\models \varphi$ means: if we write $\varphi =\varphi (x_1,\mathrm{\dots },x_n)$ where the free variables of $\varphi$ are all in $\{x_1,\mathrm{\dots },x_n\}$, then $\varphi (a_1,\mathrm{\dots },a_n)$ holds in $𝒜$ for any $a_i\in 𝒜$ (the underlying universe of $𝒜$).

If $𝒜$ is a substructure of $ℬ$ such that the inclusion homomorphism is an elementary embedding, then we say that $𝒜$ is an elementary substructure of $ℬ$, or that $ℬ$ is an elementary extension of $𝒜$.

Remark. A chain ${𝒜}_1\subseteq {𝒜}_2\subseteq \mathrm{\cdots }\subseteq {𝒜}_n\subseteq \mathrm{\cdots }$ of $\tau$-structures is called an elementary chain if ${𝒜}_i$ is an elementary substructure of ${𝒜}_{i+1}$ for each $i=1,2,\mathrm{\dots }$. It can be shown (Tarski and Vaught) that

is a $\tau$-structure that is an elementary extension of ${𝒜}_i$ for every $i$.