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 Algebra Logika, 2007, Volume 46, Number 6, Pages 763–788 (Mi al325)

Degrees of presentability of structures. I

A. I. Stukachev

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: Presentations of structures in admissible sets, as well as different relations of effective reducibility between the structures, are treated. Semilattices of degrees of $\Sigma$-definability are the main object of investigation. It is shown that the semilattice of degrees of $\Sigma$-definability of countable structures agrees well with semilattices of $T$- and $e$-degrees of subsets of natural numbers. Also an attempt is made to study properties of the structures that are inherited under various effective reducibilities and explore how degrees of presentability depend on choices of different admissible sets as domains for presentations.

Keywords: admissible set, structure, semilattice of degrees of $\Sigma$-definability.

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English version:
Algebra and Logic, 2007, 46:6, 419–432

Bibliographic databases:

UDC: 510.5
Revised: 12.03.2007

Citation: A. I. Stukachev, “Degrees of presentability of structures. I”, Algebra Logika, 46:6 (2007), 763–788; Algebra and Logic, 46:6 (2007), 419–432

Citation in format AMSBIB
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This publication is cited in the following articles:
1. A. I. Stukachev, “Degrees of presentability of structures. II”, Algebra and Logic, 47:1 (2008), 65–74
2. I. Sh. Kalimullin, “Relations between algorithmic reducibilities of algebraic systems”, Russian Math. (Iz. VUZ), 53:6 (2009), 58–59
3. I. Sh. Kalimullin, “Uniform reducibility of representability problems for algebraic structures”, Siberian Math. J., 50:2 (2009), 265–271
4. V. G. Puzarenko, “A certain reducibility on admissible sets”, Siberian Math. J., 50:2 (2009), 330–340
5. A. I. Stukachev, “A jump inversion theorem for the semilattices of $\Sigma$-degrees”, Siberian Advances in Mathematics, 20:1 (2010), 68–74
6. V. G. Puzarenko, “Fixed points for the jump operator”, Algebra and Logic, 50:5 (2011), 418–438
7. Kalimullin I., “Algorithmic Reducibilities of Algebraic Structures”, J. Logic Comput., 22:4, SI (2012), 831–843
8. Montalban A., “Rice Sequences of Relations”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 370:1971, SI (2012), 3464–3487
9. A. I. Stukachev, “Properties of $s\Sigma$-reducibility”, Algebra and Logic, 53:5 (2014), 405–417
10. Fokina E.B. Harizanov V. Melnikov A., “Computable Model Theory”, Turing'S Legacy: Developments From Turing'S Ideas in Logic, Lecture Notes in Logic, 42, ed. Downey R., Cambridge Univ Press, 2014, 124–194
11. A. I. Stukachev, “Generalized hyperarithmetical computability over structures”, Algebra and Logic, 55:6 (2017), 507–526
12. Harrison-Trainor M., Melnikov A., Miller R., Montalban A., “Computable Functors and Effective Interpretability”, J. Symb. Log., 82:1 (2017), 77–97
13. A. I. Stukachev, “Processes and structures on approximation spaces”, Algebra and Logic, 56:1 (2017), 63–74
14. Dino Rossegger, “On functors enumerating structures”, Sib. elektron. matem. izv., 14 (2017), 690–702
15. R. R. Avdeev, V. G. Puzarenko, “A computable structure with non-standard computability”, Siberian Adv. Math., 29:2 (2019), 77–115
16. A. S. Morozov, “$\Sigma$-preorderings in ${\mathbb{HF}(\mathbb{R})}$”, Algebra and Logic, 58:5 (2019), 405–416
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