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Initial segments of computable linear orders with additional computable predicates

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Algebra and Logic Aims and scope

We study computable linear orders with computable neighborhood and block predicates. In particular, it is proved that there exists a computable linear order with a computable neighborhood predicate, having a Π 01 -initial segment which is isomorphic to no computable order with a computable neighborhood predicate. On the other hand, every Σ 01 -initial segment of such an order has a computable copy enjoying a computable neighborhood predicate. Similar results are stated for computable linear orders with a computable block predicate replacing a neighborhood relation. Moreover, using the results obtained, we give a simpler proof for the Coles–Downey–Khoussainov theorem on the existence of a computable linear order with Π 02 -initial segment, not having a computable copy.

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Authors and Affiliations

  1. Chebotarev Institute of Mathematics and Mechanics, Kazan State University, ul. Prof. Nuzhina 17, Kazan, 420008, Russia

    M. V. Zubkov

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  1. M. V. Zubkov
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Correspondence to M. V. Zubkov.

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Supported by RFBR, project No. 05-01-00605.

Translated from Algebra i Logika, Vol. 48, No. 5, pp. 564-579, September-October, 2009.

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Zubkov, M.V. Initial segments of computable linear orders with additional computable predicates. Algebra Logic 48, 321–329 (2009). https://doi.org/10.1007/s10469-009-9068-7

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  • DOI: https://doi.org/10.1007/s10469-009-9068-7

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