definable type

Let $M$ be a first order structure. Let $A$ and $B$ be sets of parameters from $M$. Let $p$ be a complete $n$-type over $B$. Then we say that $p$ is an $A$-definable type iff for every formula $\psi (\overline{x},\overline{y})$ with ln$(\overline{x})=n$, there is some formula $d\psi (\overline{y},\overline{z})$ and some parameters $\overline{a}$ from $A$ so that for any $\overline{b}$ from $B$ we have $\psi (\overline{x},\overline{b})\in p$ iff $M\vDash d\psi (\overline{b},\overline{a})$.

Note that if $p$ is a type over the model $M$ then this condition is equivalent to showing that $\{\overline{b}\in M:\psi (\overline{x},\overline{b})\in M\}$ is an $A$-definable set.

For $p$ a type over $B$, we say $p$ is definable if it is $B$-definable.

If $p$ is definable, we call $d\psi$ the defining formula for $\psi$, and the function $\psi \mapsto d\psi$ a defining scheme for $p$.

Title	definable type
Canonical name	DefinableType
Date of creation	2013-03-22 13:29:26
Last modified on	2013-03-22 13:29:26
Owner	Timmy (1414)
Last modified by	Timmy (1414)
Numerical id	4
Author	Timmy (1414)
Entry type	Definition
Classification	msc 03C07
Classification	msc 03C45
Related topic	Type2
Defines	definable type
Defines	defining scheme