substructureSubstructure2003-08-12 02:58:012003-08-12 02:58:01Definition% 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
\let \proves = \vdash
\let \implies = \rightarrow
\let \Implies = \Rightarrow
\let \iff = \leftrightarrow
\let \Iff = \Leftrightarrow
\let \iso = \cong
\newcommand{\eqclass}[1]{[\![#1]\!]}
\newcommand{\set}[1]{\{#1\}}
\newcommand{\tuple}[1]{\langle#1\rangle}
\newcommand{\gen}[1]{\langle\!\langle #1 \rangle\!\rangle}
\newcommand{\powerset}{\mathcal P}
\newcommand{\A}{\mathfrak{A}}
\newcommand{\B}{\mathfrak{B}}
\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\range}{range}
\DeclareMathOperator{\im}{im}Let $\Sigma$ be a fixed signature, and $\A$ and $\B$ structures for $\Sigma$. We say $\A$ is a \emph{substructure} of $\B$, denoted \(\A \subseteq \B\), if for all \(x \in \A\) we have \(x \in \B\), and the inclusion map $i\colon \A \to \B : x \mapsto x$ is an embedding.