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Cohesive Powers of Linear Orders

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Computing with Foresight and Industry (CiE 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11558))

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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 \(\varPi _{C} \mathcal {L}\) 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 \(\varPi _{C}\mathcal {L}\) 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 \(\varPi _{C}\mathcal {L}_{1}\mathbb {\ }\)is isomorphic to \(\mathbb {N}+\mathbb {Q}\times \mathbb {Z}\), while \(\varPi _{C}\mathcal {L}_{2}\) is not. While cohesive powers preserve the satisfiability of all \(\mathrm {\Pi }_{2}^{0}\) and \(\mathrm {\Sigma } _{2}^{0}\) sentences, we provide new examples of \(\mathrm {\Pi }_{3}^{0}\) sentences \(\varPhi \) and computable structures \(\mathcal {M}\) such that \(\mathcal {M}\vDash \varPhi \) while \(\varPi _{C}\mathcal {M} \vDash \urcorner \varPhi \).

The first three and the last two authors acknowledge partial support of the NSF grant DMS-1600625. The second author acknowledges support from the Simons Foundation Collaboration Grant, and from CCFF and Dean’s Research Chair GWU awards. The last two authors acknowledge support from BNSF, MON, DN 02/16. The fourth author acknowledges the support of the Fonds voor Wetenschappelijk Onderzoek – Vlaanderen Pegasus program.

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Dimitrov, R., Harizanov, V., Morozov, A., Shafer, P., Soskova, A., Vatev, S. (2019). Cohesive Powers of Linear Orders. In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science(), vol 11558. Springer, Cham. https://doi.org/10.1007/978-3-030-22996-2_15

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  • DOI: https://doi.org/10.1007/978-3-030-22996-2_15

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