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:

• $p$ 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