# strongly minimal

Let $L$ be a first order language and let $M$ be an $L$-structure^{}. Let $S$, a subset of the domain of $M$ be a definable infinite set^{}. Then $S$ is minimal^{} iff every definable $C\subseteq S$ we have either $C$ is finite or $S\setminus C$ is finite. We say that $M$ is minimal iff the domain of $M$ is a strongly minimal set.

We say that $M$ is strongly minimal iff for every $N\equiv M$, we have that $N$ is minimal. Thus if $T$ is a complete^{} $L$ theory then we say $T$ is strongly minimal if it has some model (equivalently all models) which is strongly minimal.

Note that $M$ is strongly minimal iff every definable subset of $M$ is quantifier free definable in a language^{} with just equality. Compare this to the notion of o-minimal structures.

Title | strongly minimal |
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Canonical name | StronglyMinimal |

Date of creation | 2013-03-22 13:27:13 |

Last modified on | 2013-03-22 13:27:13 |

Owner | Timmy (1414) |

Last modified by | Timmy (1414) |

Numerical id | 5 |

Author | Timmy (1414) |

Entry type | Definition |

Classification | msc 03C07 |

Classification | msc 03C10 |

Classification | msc 03C45 |

Related topic | OMinimality |

Defines | strongly minimal |

Defines | minimal |