Geodesic
Ranks and approximations for families of order theories
Matematičeskie zametki, Tome 116 (2024) no. 4, pp. 531-551 Cet article a éte moissonné depuis la source Math-Net.Ru

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Rank values for various families of order theories are described as depending on the languages under consideration; a description of $\mathrm{e}$-total transcendence in terms of these languages is also given. Approximations of order theories are studied, including approximations by finite and countably categorical orders. Closures are studied and ranks are described for families of order theories, including the families of o-minimal and weakly o-minimal theories of various signatures, as well as theories of pure linear orders with various constraints on the discrete parts.
Keywords: rank, approximation, family of theories, ordered theory. 
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B. Sh. Kulpeshov; In. I. Pavlyuk; S. V. Sudoplatov. Ranks and approximations for families of order theories. Matematičeskie zametki, Tome 116 (2024) no. 4, pp. 531-551. http://geodesic.mathdoc.fr/item/MZM_2024_116_4_a3/

[1] A. Pillay, C. Steinhorn, “Definable sets in ordered structures. I”, Trans. Amer. Math. Soc., 295:2 (1986), 565–592 | DOI | MR

[2] H. D. Macpherson, D. Marker, C. Steinhorn, “Weakly o-minimal structures and real closed fields”, Trans. Amer. Math. Soc., 352:12 (2000), 5435–5483 | DOI | MR

[3] M. Dickmann, “Elimination of quantifiers for ordered valuation rings”, J. Symbolic Logic, 52:1 (1987), 116–128 | DOI | MR

[4] L. Van Den Dries, A. H. Lewenberg, “$T$-Convexity and tame extensions”, J. Symbolic Logic, 60:1 (1995), 74–102. | DOI | MR

[5] S. V. Sudoplatov, “Ranks for families of theories and their spectra”, Lobachevskii J. Math., 42:12 (2021), 2959–2968 | DOI | MR

[6] S. V. Sudoplatov, “Approximations of theories”, Sib. elektron. matem. izv., 17 (2020), 715–725 | DOI

[7] S. V. Sudoplatov, “Formulas and properties, their links and characteristics”, Mathematics, 9:12 (1391); 9:12 (2021)

[8] In. I. Pavlyuk, S. V. Sudoplatov, “Formulas and properties for families of theories of Abelian groups”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 36 (2021), 95–109 | DOI | MR

[9] N. D. Markhabatov, S. V. Sudoplatov, “Ranks for families of all theories of given languages”, Eurasian Math. J., 12:2 (2021), 52–58 | DOI | MR

[10] N. D. Markhabatov, S. V. Sudoplatov, “Algebras for definable families of theories”, Sib. elektron. matem. izv., 16 (2019), 600–608 | DOI | MR

[11] N. D. Markhabatov, “Ranks for families of permutation theories”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 28 (2019), 85–94 | DOI | MR

[12] In. I. Pavlyuk, S. V. Sudoplatov, “Ranks for families of theories of abelian groups”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 28 (2019), 95–112 | DOI | MR

[13] B. Sh. Kulpeshov, S. V. Sudoplatov, “Properties of ranks for families of strongly minimal theories”, Sib. elektron. matem. izv., 19:1 (2022), 120–124 | DOI | MR

[14] J. G. Rosenstein, “$\aleph_0$-categoricity of linear orderings”, Fund. Math., 64 (1969), 1–5 | DOI | MR

[15] E. Rosen, “Some aspects of model theory and finite structures”, Bull. Symbolic Logic, 8:3 (2002), 380–403 | DOI | MR

[16] J. Väänänen, “Pseudo-finite model theory”, Mat. Contemp., 24 (2003), 169–183 | MR

[17] G. Cherlin, E. Hrushovski, Finite Structures with Few Types, Ann. of Math. Stud., 152, Princeton Univ. Press, Princeton, NJ, 2003 | MR

[18] H. D. Macpherson, Ch. Steinhorn, “Definability in classes of finite structures”, Finite and Algorithmic Model Theory, London Math. Soc. Lecture Note Ser., 379, Cambridge Univ. Press, Cambridge, 2011, 140–176 | MR

[19] S. V. Sudoplatov, “Closures and generating sets related to combinations of structures”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 16 (2016), 131–144 | MR

[20] S. V. Sudoplatov, “Combinations of structures”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 24 (2018), 82–101 | DOI | MR

[21] E. A. Palyutin, “Spektr i struktura modelei polnykh teorii”, Spravochnaya kniga po matematicheskoi logike. Ch. 1. Teoriya modelei, eds. Dzh. Barvais, Yu. L. Ershov, E. A. Palyutin, A. D. Taimanov, Nauka, M., 1982

[22] N. D. Markhabatov, S. V. Sudoplatov, “Definable families of theories, related calculi and ranks”, Sib. elektron. matem. izv., 17 (2020), 700–714 | DOI | MR

[23] B. Sh. Kulpeshov, “Weakly o-minimal structures and some of their properties”, J. Symbolic Logic, 63 (1998), 1511–1528 | DOI | MR

[24] B. Sh. Kulpeshov, “Rang vypuklosti i ortogonalnost v slabo o-minimalnykh teoriyakh”, Izvestiya NAN RK. Ser. Fiz.-matem., 227 (2003), 26–31

[25] B. Sh. Kulpeshov, “Countably categorical quite o-minimal theories”, J. Math. Sci. (N.Y.), 188:4 (2013), 387–397 | DOI | MR

[26] B. Sh. Kulpeshov, S. V. Sudoplatov, “Vaught's conjecture for quite o-minimal theories”, Ann. Pure Appl. Logic, 168:1 (2017), 129–149 | DOI | MR

[27] D. Yu. Emelyanov, B. Sh. Kulpeshov, S. V. Sudoplatov, “On algebras of distributions of binary isolating formulas for quite o-minimal theories”, Algebra Logic, 57:6 (2019), 429–444 | DOI | MR