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