Mathematical logic and foundations Definition

Elementary embedding

Builds on Structure Well-formed formula Function

Definition

Let $L$ be a signature and let $\mathcal{M}$ and $\mathcal{N}$ be $L$ -structures with domains $M$ and $N$ . A map $j \colon M \to N$ is an elementary embedding of $\mathcal{M}$ into $\mathcal{N}$ when, for every $L$ -formula $\varphi(x_1, \dots, x_n)$ and all $a_1, \dots, a_n \in M$ , $\mathcal{M} \models \varphi(a_1, \dots, a_n) \iff \mathcal{N} \models \varphi(j(a_1), \dots, j(a_n)).$

Remarks

An elementary embedding is injective: apply the condition to the formula $x_1 = x_2$ . It is an embedding of structures: apply the condition to the atomic formulas. Its image is the domain of an elementary substructure of $\mathcal{N}$ .

Example: the inclusion $\Q \to \R$ is an elementary embedding of $(\Q, <)$ into $(\R, <)$ , because $(\Q, <)$ is an elementary substructure of $(\R, <)$ .

Non-example: the map $j(n) = n + 1$ from $(\N, <)$ to $(\N, <)$ preserves $<$ and is injective. It is not elementary: for $\varphi(x) = \exists y \, (y < x)$ , $(\N, <) \models \neg\varphi(0)$ but $(\N, <) \models \varphi(j(0))$ .

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Function concept
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Elementary embedding definition

Tag	`9X5Z7DH5BF`
URI	`m2://9X5Z7DH5BF`
Kind	definition
Writing	W0 stub
MSC	03C07
Defines	Elementary embedding
Uses	Structure, Well-formed formula, Function
Changed	2 November 2026

Elementary embedding

Definition

Remarks

Reading path

Further reading

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