Let $\tau$ be a signature and $\mathcal{A}$ and $\mathcal{B}$ be two structures for $\tau$ such that $f:\mathcal{A}\to \mathcal{B}$ is an embedding. Then $f$ is said to be elementary if for every first-order formula $\phi \in F(\tau)$ , we have $$\mathcal{A}\vDash\phi \quad \mbox{iff} \quad \mathcal{B}\vDash \phi.$$ In the expression above, $\mathcal{A}\vDash\phi$ means: if we write $\phi=\phi(x_1,\ldots,x_n)$ where the free variables of $\phi$ are all in $\lbrace x_1,\ldots,x_n\rbrace$ , then $\phi(a_1,\ldots,a_n)$ holds in $\mathcal{A}$ for any $a_i\in \mathcal{A}$ (the underlying universe of $\mathcal{A}$ ).

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

Remark. A chain $\mathcal{A}_1\subseteq \mathcal{A}_2\subseteq \cdots \subseteq \mathcal{A}_n \subseteq \cdots$ of $\tau$ -structures is called an elementary chain if $\mathcal{A}_i$ is an elementary substructure of $\mathcal{A}_{i+1}$ for each $i=1,2,\ldots$ . It can be shown (Tarski and Vaught) that $$\bigcup_{i<\omega} \mathcal{A}_i$$ is a $\tau$ -structure that is an elementary extension of $\mathcal{A}_i$ for every $i$ .