Abstract
Studying the elementary properties of free projective planes of finite rank, we prove that for m > n, an arbitrary ∀∃∀-formula Φ(ȳ) and a tuple ū of elements of the free projective plane \(\mathfrak{F}_{n}\) if Φ(ū) holds on the plane \(\mathfrak{F}_{m}\) then Φ(ū) holds on the plane \(\mathfrak{F}_{n}\) too. This implies the coincidence of the ∀∃-theories of free projective planes of different finite ranks.
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Russian Text © The Author(s), 2020, published in Sibirskii Matematicheskii Zhurnal, 2020, Vol. 61, No. 1, pp. 120–136.
The author was supported by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-5913.2018.1), the Russian Foundation for Basic Research (Grant 17-01-00247), and the Program of Basic Scientific Research of the Siberian Branch of the Russian Academy of Sciences (Grant I.1.1, Project 0314-2019-0002).
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Kogabaev, N.T. On the ∀∃-Theories of Free Projective Planes. Sib Math J 61, 95–108 (2020). https://doi.org/10.1134/S0037446620010085
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DOI: https://doi.org/10.1134/S0037446620010085