| Version 10 |
Version 9 |
| Let $\tau$ be a signature. |
Let $\tau$ be a signature. |
| A \emph{$\tau$-structure} $\A$ consists of a set $A$, called the \emph{\PMlinkescapetext{universe}} (or \emph{\PMlinkescapetext{domain}}) of $\A$, together with |
A \emph{$\tau$-structure} $\A$ consists of a set $A$, called the \emph{\PMlinkescapetext{universe}} (or \emph{\PMlinkescapetext{domain}}) of $\A$, together with |
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| \begin{enumerate} |
\begin{enumerate} |
| \item For each constant symbol $c\in\tau$, an |
\item For each constant symbol $c\in\tau$, an |
| element $c^A\in A$. |
element $c^A\in A$. |
| \item For each $n$-ary function symbol $f\in\tau$, |
\item For each $n$-ary function symbol $f\in\tau$, |
| a function $f^A:A^n\rightarrow A$. |
a function $f^A:A^n\rightarrow A$. |
| \item For each $n$-ary relation symbol $R\in\tau$, |
\item For each $n$-ary relation symbol $R\in\tau$, |
| a subset $R^A\subseteq A^n$. |
a subset $R^A\subseteq A^n$. |
| \end{enumerate} |
\end{enumerate} |
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| For notational convenience, when the context makes it clear in which structure we are working, we use the elements of $\tau$ to stand for the corresponding constant/function/relation. When $\tau$ is understood, we call $\A$ a \emph{structure}, instead of a $\tau$-structure. Also, it is common to write $a\in\A$ instead of $a\in A$. |
For notational convenience, when the context makes it clear in which structure we are working, we use the elements of $\tau$ to stand for the corresponding constant/function/relation. When $\tau$ is understood, we call $\A$ a \emph{structure}, instead of a $\tau$-structure. Also, it is common to write $a\in\A$ instead of $a\in A$. |
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| Let $FO(\tau)$ denote the first order language over $\tau$, and let $T$ be a \PMlinkname{theory}{FirstOrderTheories} from $FO(\tau)$. If $\A$ satisfies all the sentences of $T$, then $\A$ is called a \emph{\PMlinkname{model}{Model}} of $T$. |
Let $FO(\tau)$ denote the first order language over $\tau$, and let $T$ be a \PMlinkname{theory}{FirstOrderTheories} from $FO(\tau)$. If $\A$ satisfies all the sentences of $T$, then $\A$ is called a \emph{\PMlinkname{model}{Model}} of $T$. |
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| If $\A$ is a structure, then the \emph{cardinality} (or \emph{power}) of $\A$, which we denote $|\A|$, is the cardinality of its \PMlinkescapetext{universe} $A$. |
If $\A$ is a structure, then the \emph{cardinality} (or \emph{power}) of $\A$, which we denote $|\A|$, is the cardinality of its \PMlinkescapetext{universe} $A$. The number of possible structures for a given signature is at most $2^{|\A|}$ when $A$ is infinite.
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