# elementarily equivalent

###### Conventions

All structures^{} share a common signature^{}; the first-order
language $\mathrm{L}$ is the language^{} determined by that signature.

###### Definition

The *theory* of a structure $\mathrm{M}\mathit{}\mathit{\text{,}}\mathit{}\mathrm{Th}\mathit{}\mathrm{(}\mathrm{M}\mathrm{)}\mathit{}\mathit{\text{,}}$
is the set of all sentences^{}
of $\mathrm{L}$ that are true in $\mathrm{M}\mathrm{.}$

###### Definition

Structures $\mathrm{M}$ and $\mathrm{N}$ are *elementarily equivalent*,
(in symbols: $\mathrm{M}\mathrm{\equiv}\mathrm{N}\mathrm{)}$ if and only if
$\mathrm{Th}\mathit{}\mathrm{(}\mathrm{M}\mathrm{)}\mathrm{=}\mathrm{Th}\mathit{}\mathrm{(}\mathrm{N}\mathrm{)}$.

Title | elementarily equivalent |
---|---|

Canonical name | ElementarilyEquivalent |

Date of creation | 2013-03-22 13:00:26 |

Last modified on | 2013-03-22 13:00:26 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 8 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 03C99 |

Defines | theory |