Extending orders to types

Abstract

Given an ordered structure, we study a natural way to extend the order to preorders on type spaces. For definably complete, linearly ordered structures, we give a characterisation of the preorder on the space of $1$-types. We apply these results to the divisibility preorder on the space of ultrafilters on the set of natural numbers, giving an independence result about the suborder consisting of ultrafilters with only one fixed prime divisor, as well as a classification of ultrafilters with finitely many prime divisors.