Geodesic
On the lattices of continuous functions
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 15 (2015) no. 1, pp. 3-20 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study some new applications of the generalized method of interpretations to prove decidability of the theories of some popular structures in analysis. Earlier we proved decidability of the theory of a real continuous functions lattice by this method. In this work we generalize the lattice of real continuous functions to an algebraic structure of continuous functions over a perfectly normal space. By the generalized method of interpretations we prove decidability of the theory of this structure under some conditions.
Keywords: elementary theory, lattice of continuous functions, decidability of theories, perfectly normal space, generalized method of interpretations. 
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V. S. Amstislavskiy. On the lattices of continuous functions. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 15 (2015) no. 1, pp. 3-20. http://geodesic.mathdoc.fr/item/VNGU_2015_15_1_a0/

[1] Grzegorczyk A., “Undecidability of Some Topological Theories”, Fundamenta Mathematicae, 38 (1951), 137–152 | MR

[2] M. O. Rabin, “Decidable theories”, Handbook of Mathematical Logic, ed. J. Barwise, North Holland Publishing Company, Amsterdam–N. Y.–Oxford, 1999, 595–630 | MR | MR

[3] Flum J., Ziegler M., Topological Model Theory, Lecture Notes in Mathematics, 769, eds. A. Dold, B. Eckmann, Springer, Berlin, 1986 | MR

[4] V. S. Amstislavskiy, “Elementary Theories of Continuous Functions Spaces”, Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 12:3 (2012), 22–34 (in Russian) | Zbl

[5] Soare R. I., Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets, Perspectives in Mathematical Logic, Springer Science and Business Media, 1987 | DOI | MR

[6] Ehrenfeucht A., “Decidability of the Theory of Linear Ordering Relation”, Notices Amer. Math. Soc., 6 (1959), 268–269

[7] Yu. L. Ershov, Problems of Decidability and Constructive Models, Nauka, M., 1980 (in Russian)

[8] Yu. L. Ershov, I. A. Lavrov, A. D. Taimanov, M. A. Taitslin, “Elementary theories”, Rus. Mat. Surveys, 20:4 (1965), 35–105 | DOI | MR | Zbl

[9] A. I. Kokorin, A. G. Pinus, “Decidability problems of extended theories”, Rus. Mat. Surveys, 33:2 (1978), 53–96 | DOI | MR | Zbl

[10] Hodges W., Model Theory, Cambridge University Press, Cambridge, 1993 | MR | Zbl

[11] C. C. Chang, H. K. Keisler, Model Theory, Studies in Logic and the Foundations of Mathematics, North Holland Publishing Company, Amsterdam–N. Y.–Oxford, 1990 | MR | Zbl

[12] Rabin M., “A Simple Method for Undecidability Proofs and Some Applications”, Logic, Methodology and Philosophy of Science, v. 2, ed. Y. Bar-Hillel, North-Holland, Amsterdam, 1965, 58–68 | MR

[13] Rabin M., “Decidability of Second Order Theories and Automata on Infinite Trees”, Trans. Amer. Math. Soc., 141 (1969), 1–35 | MR | Zbl

[14] Tarski A., A Decision Method for Elementary Algebra and Geometry, University of California Press, Berkeley, 1951 | MR | Zbl

[15] R. Engelking, General Topology, Panstwowe Wydawnictwo Naukowe, Warszawa, 1977 | MR

[16] Ostaszewski A. J., “On Countably Compact, Perfectly Normal Spaces”, J. of the London Math. Soc., 14:2 (1976), 505–516 | DOI | MR | Zbl

[17] Ishiu T., “A Fine Structure Construction of a Perfectly Normal Non-Realcompact Space”, Topological Proc., 30:2 (2006), 535–545 | MR

[18] Rosenlicht M., Introduction to Analysis, $1^\text{st}$ ed., Dover Publications, 1985 | MR

[19] Heinonen J., “Nonsmooth calculus”, Bull. Amer. Math. Soc., 44 (2007), 163–232 | DOI | MR | Zbl

[20] P. S. Alexandrov, Introduction in Set Theory and General Topology, Nauka, M., 1977 (in Russian)