Cohesive Powers of Linear Orders

Rumen Dimitrov, Valentina Harizanov, Andrey Morozov, Paul Shafer, Alexandra Soskova, Stefan Vatev

Published 2019-01-15Version 1

Cohesive powers of computable structures can be viewed as effective ultraproducts over effectively indecomposable sets called cohesive sets. We investigate the isomorphism types of cohesive powers $\Pi _{C}% \mathcal{L}$ for familiar computable linear orders $\mathcal{L}$. If $% \mathcal{L}$ is isomorphic to the ordered set of natural numbers $\mathbb{N}$ and has a computable successor function, then $\Pi _{C}\mathcal{L}$ is isomorphic to $\mathbb{N}+\mathbb{Q}\times \mathbb{Z}.$ Here, $+$ stands for the sum and $\times $ for the lexicographical product of two orders. We construct computable linear orders $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ isomorphic to $\mathbb{N},$ both with noncomputable successor functions, such that $\Pi _{C}\mathcal{L}_{1}\mathbb{\ }$is isomorphic to $\mathbb{N}+% \mathbb{Q}\times \mathbb{Z}$, while $\Pi _{C}\mathcal{L}_{2}$ is not$.$ While cohesive powers preserve all $\Pi _{2}^{0}$ and $\Sigma _{2}^{0}$ sentences, we provide new examples of $\Pi _{3}^{0}$ sentences $\Phi $ and computable structures $% \mathcal{M}$ such that $\mathcal{M}\vDash \Phi $ while $\Pi _{C}\mathcal{M}% \vDash \urcorner \Phi .$