| Version 4 |
Version 3 |
| \begin{conv} All structures share a common signature; the first-order |
Structures $\mathcal{M}$ and $\mathcal{N}$ for the first-order language $\mathcal{L}$ are \emph{elementarily equivalent}, $\mathcal{M}\equiv\mathcal{N}$, if and only if they satisfy the same sentences of $\mathcal{L}$, i.e., |
| language $\mathcal{L}$ is the language determined by that signature. |
\[\mathcal{M}\equiv\mathcal{N} \text{ if and only if } \mathcal{M}\vDash\phi \text{ if and only if } \mathcal{N}\vDash\phi\text{,}\] |
| \end{conv} |
for each sentence $\phi$ of $\mathcal{L}$. |
| \begin{definition}The \emph{theory} of a structure $\mathcal{M}\text{, }\theory(\mathcal{M})\text{,}$ |
|
| is the set, $\{\phi \mid \mathcal{M} \vDash \phi \}\text{,}$ of all sentences |
|
| of $\mathcal{L}$ that are true in $\mathcal{M}.$ |
|
| \end{definition} |
|
| \begin{definition}Structures $\mathcal{M}$ and $\mathcal{N}$ are \emph{elementarily equivalent}, |
|
| (in symbols: $\mathcal{M} \equiv \mathcal{N})$ if and only if |
|
| $\theory(\mathcal{M}) = \theory(\mathcal{N})$. |
|
| \end{definition} |
|
