Extending orders to types
Lorenzo Luperi Baglini, Marcello Mamino, Rosario Mennuni, Mariaclara Ragosta, Boris Šobot
TL;DR
The paper develops a method to extend the ambient order to preorders on type spaces and proves a precise characterisation in the definably complete linear-order setting: $S_1(A)/\approx$ is isomorphic to the cut space $\mathrm{CC}(A)$. This framework is then applied to divisibility preorders on ultrafilters, showing that for a prime ultrafilter $p$, the associated order $\mathcal{E}_p$ is isomorphic to $\mathrm{CC}(\gamma)$ for $\gamma \models p$, and that the isomorphism status across nonprincipal primes is independent of ZFC. The analysis extends to finitely many primes, yielding a detailed classification of points in $\mathcal{E}_q$ (where $q$ is the type of an increasing $k$-tuple of primes) into five cases, with singleton-ness tied to lying on an antichain. The results connect model-theoretic type spaces with ultrafilter divisibility and open several directions, including higher-dimensional descriptions, interactions among $\mathcal{E}_q$, and the infinite-prime regime via patterns.
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.