On a computable presentation of low linear orders

A.N. Frolov

Kazan Federal University, Kazan, 420008 Russia

E-mail:  andrey.frolov@kpfu.ru

Received September 12, 2017

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Abstract

R. Downey in the review paper of 1998 stated the research program on studying and des­cription of sufficient conditions of computable representability of linear orders, namely, the problem of des­cription of the order type P such that, for any low linear order L, from P(L) it follows that L has a computable presentation.

 This paper is a part of the program initiated by R. Downey. We have shown that each low linear order with η condensation and no infinite strongly η-like interval has a computable presentation via a 0"-computable isomorphism. The countable linear order is called η-like if there exists some natural number k such that each maximal block of the order has power no more than k.

 We have also proven that the above-discussed result does not hold for 0'-computable isomorphism instead of 0"-computable. Namely, we have constructed a low linear order L with η condensation and no infinite strongly η-like interval such that L is not 0'-computably isomorphic to a computable one.

Keywords: linear order, computable presentation, low degree, strongly η-like linear order

Acknowledgments. This study was supported by the Russian Foundation for Basic Research (project no. 16-31-60077).

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For citation: Frolov A.N. On a computable presentation of low linear orders. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 4, pp. 518–528. (In Russian)


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