ElementarilyEquivalent 2002-08-28 23:28:58 2007-12-27 23:33:24 Definition theory sentence % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here %\PMlinkescapeword{theory} \newtheorem{definition}{Definition} \renewcommand{\thedefinition}{} \newtheorem{conv}{Conventions} \renewcommand{\theconv}{} \newcommand{\theory}{\operatorname{Th}} \begin{conv} All structures share a common signature; the first-order language $\mathcal{L}$ is the language determined by that signature. \end{conv} \begin{definition}The \emph{theory} of a structure $\mathcal{M}\text{, }\theory(\mathcal{M})\text{,}$ is the set 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}