Cohesive Powers of Linear Orders

Abstract

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 ΠCL for familiar computable linear orders L. If L is isomorphic to the ordered set of natural numbers N and has a computable successor function, then ΠCL is isomorphic to N+ Q× Z. Here, + stands for the sum and × for the lexicographical product of two orders. We construct computable linear orders L1 and L2 isomorphic to N, both with noncomputable successor functions, such that ΠCL1 is isomorphic to N+ Q× Z, while ΠCL2 is not. While cohesive powers preserve the satisfiability of all Π20 and Σ20 sentences, we provide new examples of Π30 sentences Φ and computable structures M such that M⊨ Φ while ΠCM⊨ ⌝ Φ.