Geodesic
Definability of linear orders over negative equivalences
Algebra i logika, Tome 55 (2016) no. 1, pp. 37-57.

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We study linear orders definable over negative and positive equivalences and their computable automorphisms. Special attention is paid to equivalences like $\eta(\alpha)=\alpha^2\cup\mathrm{id}_\omega$, $\alpha\subseteq\omega$. In particular, we describe orders that have negative presentations over such equivalences for co-enumerable sets $\alpha$. Presentable and nonpresentable order types are exemplified for equivalences with various extra properties. We also give examples of negative orders with computable automorphisms whose inverses are not computable.
Keywords: linear order, negative equivalence, computable automorphism. 
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N. Kh. Kasymov; A. S. Morozov. Definability of linear orders over negative equivalences. Algebra i logika, Tome 55 (2016) no. 1, pp. 37-57. http://geodesic.mathdoc.fr/item/AL_2016_55_1_a2/

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