# substructure

Let $\mathrm{\Sigma}$ be a fixed signature^{}, and $\U0001d504$ and $\U0001d505$ structures^{} for $\mathrm{\Sigma}$. We say $\U0001d504$ is a *substructure* of $\U0001d505$, denoted $\U0001d504\subseteq \U0001d505$, if for all $x\in \U0001d504$ we have $x\in \U0001d505$, and the inclusion map^{} $i:\U0001d504\to \U0001d505:x\mapsto x$ is an embedding^{}.

When $\U0001d504$ is a substructure of $\U0001d505$, we also say that $\U0001d505$ is an *extension* of $\U0001d504$.

A *submodel* $\U0001d504$ of a model $\U0001d505$ of a (first-order) language^{} $\mathcal{L}$ if $\U0001d504$ is a model of $\mathcal{L}$ and $\U0001d504$ is a substructure of $\U0001d505$.

Title | substructure |
---|---|

Canonical name | Substructure |

Date of creation | 2013-03-22 13:50:32 |

Last modified on | 2013-03-22 13:50:32 |

Owner | almann (2526) |

Last modified by | almann (2526) |

Numerical id | 6 |

Author | almann (2526) |

Entry type | Definition |

Classification | msc 03C07 |

Synonym | submodel |

Related topic | StructuresAndSatisfaction |

Defines | extension |