Invariant universality for projective planes

• Autorzy: Paolini Gianluca

Identyfikatory

DOI

10.4467/20842589RM.23.002.18801

• Języki publikacji: EN

Abstrakty

EN

We continue the work of [1, 2, 3] by analyzing the equivalence relation of bi-embeddability on various classes of countable planes, most notably the class of countable non-Desarguesian projective planes. We use constructions of the author from [13] to show that these equivalence relations are invariantly universal, in the sense of [3], and thus in particular complete analytic. We also introduce a new kind of Borel reducibility relation for standard Borel G-spaces, which requires the preservation of stabilizers, and explain its connection with the notion of full embeddings commonly considered in category theory.

Słowa kluczowe

EN

invariant descriptive set theory; bi-embeddability relation; projective planes; invariant universality

• Wydawca: Wydawnictwo Uniwersytetu Jagiellońskiego
• Czasopismo: Reports on Mathematical Logic
• Rocznik: 2023
• Tom: Vol. 58
• Strony: 15--27
• Opis fizyczny: Bibliogr. 15 poz., rys.

Twórcy

autor

Paolini Gianluca

• Dipartimento di matematica “Giuseppe Peano”, Universitá di Torino, Via Carlo Alberto 10, 10121 Torino — Italy

Bibliografia

• [1] A. D. Brooke-Taylor, F. Calderoni, S. K. Miller, Invariant Universality for Quandles and Fields, Fund. Math. 251 (2020), 1-16.
• [2] F. Calderoni, L. Motto Ros, Universality of Group Embeddability, Proc. Amer. Math. Soc. 146 (2018), 1765-1780.
• [3] R. Camerlo, A. Marcone, L. Motto Ros, Invariantly Universal Analytic Quasi-Orders, Trans. Amer. Math. Soc. 365 (2013), no. 4, 1901-1931.
• [4] S.-D. Friedman, L. Motto Ros, Analytic Equivalence Relations and Bi-Embeddability, J. Symbolic Logic 76 (2011), no. 1, 243-266.
• [5] E. Fried, J. Sichler, Homomorphisms of Commutative Rings with Unit Element, Pacific J. Math. 45 (1973), 485-491.
• [6] S. Gao, Some Dichotomy Theorems for Isomorphism Relations of Countable Models, J. Symbolic Logic 66 (2001), no. 2, 902-922.
• [7] M. Hall, Projective Planes, Trans. Amer. Math. Soc. 54 (1943), 229-277.
• [8] A. S. Kechris, Classical descriptive set theory, volume 156 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1995.
• [9] A. Louveau, C. Rosendal, Complete Analytic Equivalence Relations, Trans. Amer. Math. Soc. 357 (1943), no. 12, 4839-4866.
• [10] D. R. Hughes, F. C. Piper, Projective Planes, Graduate Texts in Mathematics, Vol. 6. SpringerVerlag, New York-Berlin, 1973.
• [11] T. Hyttinen, G. Paolini, Beyond Abstract Elementary Classes: On The Model Theory of Geometric Lattices, Ann. Pure Appl. Logic 169 (2018), no. 2, 117-145.
• [12] M. Lupini, Polish Groupoids and Functorial Complexity, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6683-6723.
• [13] G. Paolini, The Class of Countable non-Desarguesian Projective Planes is Borel Complete, Proc. Amer. Math. Soc. 146 (2018), 4927-4936.
• [14] A. Pultr, V. Trnkov´a, Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories, North-Holland, 1980.
• [15] F. W. Stevenson, Projective Planes, W. H. Freeman and Co., San Francisco, Calif., 1972.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).