# type

Let $L$ be a first order language. Let $M$ be an http://planetmath.org/node/3384$L$-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 $\phi(x,\bar{b})$ with parameters $\bar{b}$ from $B$ so that $M\models\phi(a,\bar{b})$. A collection of $L$-formulas is a $n$-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 $\psi(x,\bar{b})$ 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:

• $p$ is the type of some sequence of $n$ elements $a$ over $B$ in some model $N\equiv M$

• $p$ 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 2013-03-22 13:22:45 Last modified on 2013-03-22 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