Geodesic
The theory of projective planes is complete with respect to degree spectra and effective dimensions
Algebra i logika, Tome 54 (2015) no. 5, pp. 599-627.

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove that the theory of Pappian projective planes is complete with respect to degree spectra of automorphically nontrivial structures, effective dimensions, degree spectra of relations, categoricity spectra, and automorphism spectra. Therefore, for every natural $n\ge2$, there exists a computable Pappian projective plane with computable dimension $n$.
Keywords: projective plane, Pappian projective plane, computable structure, degree spectrum of structure, degree spectrum of relation, categoricity spectrum, automorphism spectrum.
Mots-clés : computable dimension 
@article{AL_2015_54_5_a3,
     author = {N. T. Kogabaev},
     title = {The theory of projective planes is complete with respect to degree spectra and effective dimensions},
     journal = {Algebra i logika},
     pages = {599--627},
     publisher = {mathdoc},
     volume = {54},
     number = {5},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2015_54_5_a3/}
}
TY  - JOUR
AU  - N. T. Kogabaev
TI  - The theory of projective planes is complete with respect to degree spectra and effective dimensions
JO  - Algebra i logika
PY  - 2015
SP  - 599
EP  - 627
VL  - 54
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2015_54_5_a3/
LA  - ru
ID  - AL_2015_54_5_a3
ER  - 
%0 Journal Article
%A N. T. Kogabaev
%T The theory of projective planes is complete with respect to degree spectra and effective dimensions
%J Algebra i logika
%D 2015
%P 599-627
%V 54
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2015_54_5_a3/
%G ru
%F AL_2015_54_5_a3
N. T. Kogabaev. The theory of projective planes is complete with respect to degree spectra and effective dimensions. Algebra i logika, Tome 54 (2015) no. 5, pp. 599-627. http://geodesic.mathdoc.fr/item/AL_2015_54_5_a3/

[1] L. Richter, “Degrees of structures”, J. Symb. Log., 46:4 (1981), 723–731 | DOI | MR | Zbl

[2] J. F. Knight, “Degrees coded in jumps of orderings”, J. Symb. Log., 51:4 (1986), 1034–1042 | DOI | MR | Zbl

[3] T. A. Slaman, “Relative to any nonrecursive set”, Proc. Am. Math. Soc., 126:7 (1998), 2117–2122 | DOI | MR | Zbl

[4] S. Wehner, “Enumerations, countable structures, and Turing degrees”, Proc. Am. Math. Soc., 126:7 (1998), 2131–2139 | DOI | MR | Zbl

[5] I. Sh. Kalimullin, “Spektry stepenei nekotorykh algebraicheskikh struktur”, Algebra i logika, 46:6 (2007), 729–744 | MR | Zbl

[6] A. Frolov, I. Kalimullin, V. Harizanov, O. Kudinov, R. Miller, “Spectra of highn and non-lown degrees”, J. Log. Comput., 22:4 (2012), 755–777 | DOI | MR | Zbl

[7] S. S. Goncharov, “Problema chisla neavtoekvivalentnykh konstruktivizatsii”, Algebra i logika, 19:6 (1980), 621–639 | MR

[8] S. S. Goncharov, A. V. Molokov, N. S. Romanovskii, “Nilpotentnye gruppy konechnoi algoritmicheskoi razmernosti”, Sib. matem. zh., 30:1 (1989), 82–88 | MR | Zbl

[9] B. Khoussainov, R. A. Shore, “Computable isomorphisms, degree spectra of relations, and Scott families”, Ann. Pure Appl. Logic, 93:1–3 (1998), 153–193 | DOI | MR | Zbl

[10] P. Cholak, S. Goncharov, B. Khoussainov, R. A. Shore, “Computably categorical structures and expansions by constants”, J. Symb. Log., 64:1 (1999), 13–37 | DOI | MR | Zbl

[11] B. M. Khusainov, R. A. Shor, “O reshenii problemy Goncharova–Esha i problemy spektra v teorii vychislimykh modelei”, Dokl. RAN, 371:1 (2000), 30–31 | MR | Zbl

[12] V. S. Harizanov, “Uncountable degree spectra”, Ann. Pure Appl. Logic, 54:3 (1991), 255–263 | DOI | MR | Zbl

[13] V. S. Harizanov, “Some effects of Ash-Nerode and other decidability conditions on degree spectra”, Ann. Pure Appl. Logic, 55:1 (1991), 51–65 | DOI | MR | Zbl

[14] V. S. Harizanov, “The possible Turing degree of the nonzero member in a two element degree spectrum”, Ann. Pure Appl. Logic, 60:1 (1993), 1–30 | DOI | MR | Zbl

[15] D. R. Hirschfeldt, “Degree spectra of relations on structures of finite computable dimension”, Ann. Pure Appl. Logic, 115:1–3 (2002), 233–277 | DOI | MR | Zbl

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

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

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

[19] V. Kharizanova, R. Miller, A. S. Morozov, “Prostye struktury so slozhnoi simmetriei”, Algebra i logika, 49:1 (2010), 98–134 | MR | Zbl

[20] 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

[21] R. Miller, B. Poonen, H. Schoutens, A. Shlapentokh, A computable functor from graphs to fields, submitted

[22] A. I. Shirshov, A. A. Nikitin, “K teorii proektivnykh ploskostei”, Algebra i logika, 20:3 (1981), 330–356 | MR | Zbl

[23] A. I. Shirshov, A. A. Nikitin, Algebraicheskaya teoriya proektivnykh ploskostei, Novosibirskii gos. un-t, Novosibirsk, 1987

[24] D. R. Hughes, F. C. Piper, Projective planes, Grad. Texts Math., 6, Springer-Verlag, New York–Heidelberg–Berlin, 1973 | MR | Zbl

[25] N. T. Kogabaev, “Nevychislimost klassov pappovykh i dezargovykh proektivnykh ploskostei”, Sib. matem. zh., 54:2 (2013), 325–335 | MR | Zbl