example of definable type
Consider $$ as a structure^{} in a language^{} with one binary relation^{}, which we interpret as the order. This is a universal^{}, ${\mathrm{\aleph}}_{0}$categorical structure (see example of universal structure).
The theory of $$ has quantifier elimination^{}, and so is ominimal. Thus a type over the set $\mathbf{Q}$ is determined by the quantifier free formulas over $\mathbf{Q}$, which in turn are determined by the atomic formulas over $\mathbf{Q}$. An atomic formula in one variable over $B$ is of the form $$ or $x>b$ or $x=b$ for some $b\in B$. Thus each 1type over $\mathbf{Q}$ determines a Dedekind cut over $\mathbf{Q}$, and conversely a Dedekind cut determines a complete type over $\mathbf{Q}$. Let $D(p):=\{a\in \mathbf{Q}:x>a\in p\}$.
Thus there are two classes of type over $\mathbf{Q}$.

1.
Ones where $D(p)$ is of the form $(\mathrm{\infty},a)$ or $(\mathrm{\infty},a]$ for some $a\in \mathbf{Q}$. It is clear that these are definable from the above discussion.

2.
Ones where $D(p)$ has no supremum in $\mathbf{Q}$. These are clearly not definable by ominimality of $\mathbf{Q}$.
Title  example of definable type 

Canonical name  ExampleOfDefinableType 
Date of creation  20130322 13:29:43 
Last modified on  20130322 13:29:43 
Owner  aplant (12431) 
Last modified by  aplant (12431) 
Numerical id  5 
Author  aplant (12431) 
Entry type  Example 
Classification  msc 03C07 
Related topic  ExampleOfUniversalStructure 
Related topic  DedekindCuts 