Abstract
We introduce the concept of KΣ-structure and prove the existence of a universal Σ-function in the hereditarily finite superstructure over this structure. We exhibit some examples of families of KΣ-structures of the theory of trees.
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Ershov Yu. L., Definability and Computability [Russian], Nauchnaya Kniga, Novosibirsk; Èkonomika, Moscow (2000) (Siberian School of Algebra and Logic).
Rogers H., Theory of Recursive Functions and Effective Computability, McGraw-Hill Book Comp., New York, St. Louis, San Francisco, Toronto, London, and Sydney (1967).
Rudnev V. A., “A universal recursive function on admissible sets,” Algebra and Logic, vol. 25, no. 4, 267–273 (1986).
Ershov Yu. L., Puzarenko V. G., and Stukachev A. I., “\(\mathbb{HF}\)-computability,” in: Computability in Context, Imp. Coll. Press, London, 2011, 169–242 (Computation and Logic in the Real World).
Morozov A. S. and Puzarenko V. G., “Σ-Subsets of natural numbers,” Algebra and Logic, vol. 43, no. 3, 162–178 (2004).
Kalimullin I. Sh. and Puzarenko V. G., “Computable principles on admissible sets,” Siberian Adv. in Math., vol. 15, no. 4, 1–33 (2005).
Puzarenko V. G., “Computability in special models,” Sib. Math. J., vol. 46, no. 1, 148–165 (2005).
Aleksandrova S. A., “The uniformization problem for Σ-predicates in a hereditarily finite list superstructure over the real exponential field,” Algebra and Logic, vol. 53, no. 1, 1–8 (2014).
Korovina M. V., “On a universal recursive function and abstract machines on reals with list superstructure,” in: Structural Algorithmic Properties of Computability (Vychisl. Sistemy; No. 156) [Russian], Novosibirsk, 1996, 24–43.
Stukachev A. I., “The uniformization theorem in hereditary finite superstructures,” in: Generalized Computability and Definability (Vychisl. Sistemy; No. 161) [Russian], Izdat. IM SO RAN, Novosibirsk, 1998, 3–14.
Khisamiev A. N., “On Σ-Subsets of naturals over abelian groups,” Sib. Math. J., vol. 47, no. 3, 574–583 (2006).
Khisamiev A. N., “Σ-Bounded algebraic systems and universal functions. I,” Sib. Math. J., vol. 51, no. 1, 178–192 (2010).
Khisamiev A. N., “Σ-Bounded algebraic systems and universal functions. II,” Sib. Math. J., vol. 51, no. 3, 537–551 (2010).
Khisamiev A. N., “Σ-Uniform structures and Σ-functions. I,” Algebra and Logic, vol. 50, no. 5, 447–465 (2011).
Khisamiev A. N., “Σ-Uniform structures and Σ-functions. II,” Algebra and Logic, vol. 51, no. 1, 89–102 (2012).
Khisamiev A. N., “On a universal Σ-function over a tree,” Sib. Math. J., vol. 53, no. 3, 551–553 (2012).
Khisamiev A. N., “Universal functions and almost c-simple models,” Sib. Math. J., vol. 56, no. 3, 526–540 (2015).
Khisamiev A. N., “A class of almost c-simple rings,” Sib. Math. J., vol. 56, no. 6, 1133–1141 (2015).
Khisamiev A. N., “Universal functions and unboundedly branching trees,” Algebra and Logic, vol. 57, no. 4, 309–319 (2018).
Acknowledgment
The author is grateful to S. S. Goncharov for stating the problem and giving useful advice.
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Dedicated to the outstanding mathematician Yuri Leonidovich Ershov on the occasion of his 80th birthday.
Russian Text © The Author(s), 2020, published in Sibirskii Matematicheskii Zhurnal, 2020, Vol. 61, No. 3, pp. 703–716.
The author was funded within the Government Task to the Sobolev Institute of Mathematics (Project 0314-2019-0003).
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Khisamiev, A.N. Universal Functions and KΣ-Structures. Sib Math J 61, 552–562 (2020). https://doi.org/10.1134/S0037446620030192
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DOI: https://doi.org/10.1134/S0037446620030192