type
Let $L$ be a first order language. Let $M$ be an http://planetmath.org/node/3384$\mathrm{L}$structure^{}. Let $B\subseteq M$, and let $a\in {M}^{n}$. Then we define the type of $a$ over $B$ to be the set of $L$formulas^{} $\varphi (x,\overline{b})$ with parameters $\overline{b}$ from $B$ so that $M\vDash \varphi (a,\overline{b})$. A collection^{} of $L$formulas is a complete^{} $n$type over $B$ iff it is of the above form for some $B,M$ and $a\in {M}^{n}$.
We call any consistent collection of formulas $p$ in $n$ variables with parameters from $B$ a partial $n$type over $B$. (See criterion for consistency of sets of formulas.)
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 $\psi (x,\overline{b})$ over $B$ we have either $\psi (x,\overline{b})\in p$ or $\mathrm{\neg}\psi (x,\overline{b})\in p$. In fact, for every collection of formulas $p$ in $n$ variables the following are equivalent^{}:
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$p$ is a maximal consistent set of formulas.
For $n\in \omega $ we define ${S}_{n}(B)$ to be the set of complete $n$types over $B$.
Some authors define a collection of formulas $p$ to be a $n$type iff $p$ is a partial $n$type. Others define $p$ to be a type iff $p$ is a complete $n$type.
A type (resp. partial type/complete type) is any $n$type (resp. partial type/complete type) for some $n\in \omega $.
Title  type 
Canonical name  Type 
Date of creation  20130322 13:22:45 
Last modified on  20130322 13:22:45 
Owner  ratboy (4018) 
Last modified by  ratboy (4018) 
Numerical id  6 
Author  ratboy (4018) 
Entry type  Definition 
Classification  msc 03C07 
Related topic  Formula 
Related topic  DefinableType 
Related topic  TermsAndFormulas 
Defines  type 
Defines  complete type 
Defines  partial type 