typeType22003-01-22 20:00:382006-09-01 10:48:04Definitiontypecomplete typepartial type% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $L$ be a first order language.
Let $M$ be an \PMlinkid{$L$-structure}{3384}.
Let $B \subseteq M$, and let $a \in M^{n}$.
Then we define the {\em type of $a$ over $B$} to be the set of $L$-formulas $\phi(x,\bar{b})$ with parameters $\bar{b}$ from $B$ so that $M \models \phi(a,\bar{b})$.
A collection of $L$-formulas is a {\em complete} $n$-type over $B$ iff it is of the above form for some $B,M$ and $a \in M^{n}$.
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We call any consistent collection of formulas $p$ in $n$ variables with parameters from $B$ a {\em partial} $n$-type over $B$. (See criterion for consistency of sets of formulas.)
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Note that a complete $n$-type $p$ over $B$ is consistent so is in particular a partial type over $B$. Also $p$ is maximal in the sense that for every formula $\psi(x,\bar{b})$ over $B$ we have either $\psi(x,\bar{b}) \in p$ or $\lnot \psi(x,\bar{b}) \in p$.
In fact, for every collection of formulas $p$ in $n$ variables the following are equivalent:
\begin{itemize} \item $p$ is the type of some sequence of $n$ elements $a$ over $B$ in some model $N \equiv M$
\item $p$ is a maximal consistent set of formulas.\end{itemize}
For $n \in \omega$ we define $S_{n}(B)$ to be the set of complete $n$-types over $B$.
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Some authors define a collection of formulas $p$ to be a {\em $n$-type} iff $p$ is a partial $n$-type. Others define $p$ to be a {\em type} iff $p$ is a complete $n$-type.
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A type (resp. partial type/complete type) is any $n$-type (resp. partial type/complete type) for some $n \in \omega$.