# 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 $A$ type iff for every formula $\psi(\bar{x},\bar{y})$ with ln$(\bar{x})=n$, there is some formula $d\psi(\bar{y},\bar{z})$ and some parameters $\bar{a}$ from $A$ so that for any $\bar{b}$ from $B$ we have $\psi(\bar{x},\bar{b})\in p$ iff $M\models d\psi(\bar{b},\bar{a})$.

Note that if $p$ is a type over the model $M$ then this condition is equivalent to showing that $\{\bar{b}\in M:\psi(\bar{x},\bar{b})\in M\}$ 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 DefinableType 2013-03-22 13:29:26 2013-03-22 13:29:26 Timmy (1414) Timmy (1414) 4 Timmy (1414) Definition msc 03C07 msc 03C45 Type2 definable type defining scheme