Degrees of autostability for linear orderings and linearly ordered Abelian groups
Algebra i logika, Tome 55 (2016) no. 4, pp. 393-418.

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It is proved that every computable ordinal has a degree of autostability. We construct new examples of degrees of autostability in the class of linear orderings and in the class of linearly ordered Abelian groups.
Keywords: autostability, computable categoricity, index set, linear ordering, autostability spectrum, categoricity spectrum, degree of autostability, degree of categoricity, ordered Abelian group.
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N. A. Bazhenov. Degrees of autostability for linear orderings and linearly ordered Abelian groups. Algebra i logika, Tome 55 (2016) no. 4, pp. 393-418. http://geodesic.mathdoc.fr/item/AL_2016_55_4_a0/

[1] E. B. Fokina, I. Kalimullin, R. Miller, “Degrees of categoricity of computable structures”, Arch. Math. Logic, 49:1 (2010), 51–67 | DOI | MR | Zbl

[2] B. F. Csima, J. N. Y. Franklin, R. A. Shore, “Degrees of categoricity and the hyperarithmetic hierarchy”, Notre Dame J. Form. Log., 54:2 (2013), 215–231 | DOI | MR | Zbl

[3] S. S. Goncharov, “Stepeni avtoustoichivosti otnositelno silnykh konstruktivizatsii”, Algoritmicheskie voprosy algebry i logiki, K 80-letiyu so dnya rozhd. akad. S. I. Adyana, Tr. MIAN, 274, MAIK, M., 2011, 119–129 | MR

[4] D. R. Hirschfeldt, B. Khoussainov, R. A. Shore, A. M. Slinko, “Degree spectra and computable dimensions in algebraic structures”, Ann. Pure Appl. Logic, 115:1–3 (2002), 71–113 | DOI | MR | Zbl

[5] R. Miller, “$d$-computable categoricity for algebraic fields”, J. Symb. Log., 74:4 (2009), 1325–1351 | DOI | MR | Zbl

[6] R. Miller, A. Shlapentokh, “Computable categoricity for algebraic fields with splitting algorithms”, Trans. Am. Math. Soc., 367:6 (2015), 3955–3980 | DOI | MR | Zbl

[7] B. Anderson, B. Csima, “Degrees that are not degrees of categoricity”, Notre Dame J. Form. Log., 57:3 (2016), 389–398 | MR | Zbl

[8] E. Fokina, A. Frolov, I. Kalimullin, “Categoricity spectra for rigid structures”, Notre Dame J. Form. Log., 57:1 (2016), 45–57 | DOI | MR | Zbl

[9] N. A. Bazhenov, “O $\Delta^0_2$-kategorichnosti bulevykh algebr”, Vestn. NGU. Ser. matem., mekh., inform., 13:2 (2013), 3–14 | Zbl

[10] S. S. Goncharov, V. D. Dzgoev, “Avtoustoichivost modelei”, Algebra i logika, 19:1 (1980), 45–58 | MR | Zbl

[11] J. B. Remmel, Recursively categorical linear orderings, 83:2 (1981), 387–391 | MR | Zbl

[12] C. McCoy, “$\Delta^0_2$-categoricity in Boolean algebras and linear orderings”, Ann. Pure Appl. Logic, 119:1–3 (2003), 85–120 | DOI | MR | Zbl

[13] Ch. F. D. Mak-Koi, “O $\Delta^0_3$-kategorichnosti dlya lineinykh poryadkov i bulevykh algebr”, Algebra i logika, 41:5 (2002), 531–552 | MR | Zbl

[14] C. J. Ash, “Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees”, Trans. Am. Math. Soc., 298:2 (1986), 497–514 | DOI | MR | Zbl

[15] S. S. Goncharov, Yu. L. Ershov, Konstruktivnye modeli, Sibirskaya shkola algebry i logiki, Nauchnaya kniga 1999, Novosibirsk

[16] C. J. Ash, J. F. Knight, Computable structures and the hyperarithmetical hierarchy, Stud. Logic Found. Math., 144, Elsevier Sci. B. V., Amsterdam etc., 2000 | MR | Zbl

[17] J. C. Rosenstein, Linear orderings, Pure Appl. Math., 98, Academic Press, New York etc., 1982 | MR | Zbl

[18] R. G. Downey, “Computability theory and linear orderings”, Handbook of recursive mathematics, v. 2, Stud. Logic Found. Math., 139, Recursive algebra, analysis and combinatorics, eds. Y. L. Ershov et al., Elsevier Science B. V., Amsterdam, 1998, 823–976 | DOI | MR | Zbl

[19] Kh. Rodzhers, Teoriya rekursivnykh funktsii i effektivnaya vychislimost, Mir, M., 1972 | MR

[20] C. Ash, J. Knight, M. Manasse, T. Slaman, “Generic copies of countable structures”, Ann. Pure Appl. Logic, 42:3 (1989), 195–205 | DOI | MR | Zbl

[21] C. J. Ash, J. F. Knight, “Pairs of recursive structures”, Ann. Pure Appl. Logic, 46:3 (1990), 211–234 | DOI | MR | Zbl

[22] N. A. Bazhenov, “Stepeni kategorichnosti superatomnykh bulevykh algebr”, Algebra i logika, 52:3 (2013), 271–283 | MR | Zbl

[23] S. S. Goncharov, Dzh. Nait, “Vychislimye strukturnye i antistrukturnye teoremy”, Algebra i logika, 41:6 (2002), 639–681 | MR | Zbl

[24] W. M. White, “On the complexity of categoricity in computable structures”, Math. Log. Q., 49:6 (2003), 603–614 | DOI | MR | Zbl

[25] C. J. Ash, “Categoricity in hyperarithmetical degrees”, Ann. Pure Appl. Logic, 34:1 (1987), 1–14 | DOI | MR | Zbl

[26] N. A. Bazhenov, “Spektry avtoustoichivosti bulevykh algebr”, Algebra i logika, 53:6 (2014), 764–769 | MR

[27] A. I. Kokorin, V. M. Kopytov, Lineino uporyadochennye gruppy, Nauka, M., 1972 | MR

[28] A. G. Melnikov, “Computable ordered abelian groups and fields”, Programs, proofs, processes, 6th conf. comput. Europe, CiE 2010 (Ponta Delgada, Azores, Portugal, June 30 – July 4, 2010), Lect. Notes Comp. Sci., 6158, eds. F. Ferreira et al., Springer, Berlin, 2010, 321–330 | DOI | MR | Zbl

[29] A. G. Melnikov, Effektivnye svoistva vpolne razlozhimykh abelevykh grupp, Dis. kand. fiz.-matem. nauk, NGU, Novosibirsk, 2011