# elementary embedding

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^{} $\varphi \in F(\tau )$, we have

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

In the expression above, $\mathcal{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 $\mathcal{A}$ 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 ${\mathcal{A}}_{1}\subseteq {\mathcal{A}}_{2}\subseteq \mathrm{\cdots}\subseteq {\mathcal{A}}_{n}\subseteq \mathrm{\cdots}$ of $\tau $-structures is called an *elementary chain* if ${\mathcal{A}}_{i}$ is an elementary substructure of ${\mathcal{A}}_{i+1}$ for each $i=1,2,\mathrm{\dots}$. It can be shown (Tarski and Vaught) that

$$ |

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

Title | elementary embedding |
---|---|

Canonical name | ElementaryEmbedding |

Date of creation | 2013-03-22 13:00:29 |

Last modified on | 2013-03-22 13:00:29 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 5 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 03C99 |

Synonym | elementary monomorphism |

Defines | elementary substructure |

Defines | elementary extension |

Defines | elementary chain |