structureStructuresAndSatisfaction2002-06-03 19:25:102009-10-04 21:15:23Definitionstructureinterpretation% this is the default PlanetMath preamble. as your knowledge
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\end{array}\right)}Let $\tau$ be a signature.
A \emph{$\tau$-structure} $\mathcal{A}$ comprises of a set $A$, called the \emph{\PMlinkescapetext{universe}} (or \emph{\PMlinkescapetext{domain}}) of $\mathcal{A}$, and an \emph{interpretation} of the symbols of $\tau$ as follows:
\begin{itemize}
\item for each constant symbol $c\in\tau$, an
element $c^A\in A$;
\item for each $n$-ary function symbol $f\in\tau$,
a function $f^A:A^n\rightarrow A$;
\item for each $n$-ary relation symbol $R\in\tau$,
a subset $R^A\subseteq A^n$.
\end{itemize}
If $\mathcal{A}$ is a structure, then the \emph{cardinality} (or \emph{power}) of $\mathcal{A}$, $|\mathcal{A}|$, is the cardinality of its \PMlinkescapetext{universe} $A$.
If $\tau$ does not contain any function symbols, then a $\tau$-structure is called a relational structure. If $\tau$ does not contain any relation symbols, then a $\tau$-structure is called an algebraic structure.