The bi-embeddability relation for countable abelian groups

Calderoni, Filippo; Thomas, Simon

doi:10.1090/tran/7513

The bi-embeddability relation for countable abelian groups
HTML articles powered by AMS MathViewer

by Filippo Calderoni and Simon Thomas PDF

Trans. Amer. Math. Soc. 371 (2019), 2237-2254 Request permission

Abstract:

We analyze the complexity of the bi-embeddability relations for countable torsion-free abelian groups and for countable torsion abelian groups.

References

Jon Barwise and Paul Eklof, Infinitary properties of abelian torsion groups, Ann. Math. Logic 2 (1970/71), no. 1, 25–68. MR 279180, DOI 10.1016/0003-4843(70)90006-9
Peter Crawley and Alfred W. Hales, The structure of abelian $p$-groups given by certain presentations, J. Algebra 12 (1969), 10–23. MR 238947, DOI 10.1016/0021-8693(69)90014-3
Rod Downey and Antonio Montalbán, The isomorphism problem for torsion-free abelian groups is analytic complete, J. Algebra 320 (2008), no. 6, 2291–2300. MR 2437501, DOI 10.1016/j.jalgebra.2008.06.007
Solomon Feferman, Impredicativity of the existence of the largest divisible subgroup of an abelian $p$-group, Model theory and algebra (A memorial tribute to Abraham Robinson), Lecture Notes in Math., Vol. 498, Springer, Berlin, 1975, pp. 117–130. MR 0401446
Harvey Friedman and Lee Stanley, A Borel reducibility theory for classes of countable structures, J. Symbolic Logic 54 (1989), no. 3, 894–914. MR 1011177, DOI 10.2307/2274750
L. Fuchs, On the structure of abelian $p$-groups, Acta Math. Acad. Sci. Hungar. 4 (1953), 267–288 (English, with Russian summary). MR 60515, DOI 10.1007/BF02127586
László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
László Fuchs, Infinite abelian groups. Vol. II, Pure and Applied Mathematics. Vol. 36-II, Academic Press, New York-London, 1973. MR 0349869
Su Gao, Invariant descriptive set theory, Pure and Applied Mathematics (Boca Raton), vol. 293, CRC Press, Boca Raton, FL, 2009. MR 2455198
Greg Hjorth, Orbit cardinals: on the effective cardinalities arising as quotient spaces of the form $X/G$ where $G$ acts on a Polish space $X$, Israel J. Math. 111 (1999), 221–261. MR 1710740, DOI 10.1007/BF02810686
Greg Hjorth, Classification and orbit equivalence relations, Mathematical Surveys and Monographs, vol. 75, American Mathematical Society, Providence, RI, 2000. MR 1725642, DOI 10.1090/surv/075
Greg Hjorth and Alexander S. Kechris, Analytic equivalence relations and Ulm-type classifications, J. Symbolic Logic 60 (1995), no. 4, 1273–1300. MR 1367210, DOI 10.2307/2275888
Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
V. G. Kanoveĭ and M. Reeken, Some new results on the Borel irreducibility of equivalence relations, Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), no. 1, 59–82 (Russian, with Russian summary); English transl., Izv. Math. 67 (2003), no. 1, 55–76. MR 1957916, DOI 10.1070/IM2003v067n01ABEH000418
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
L. Ya. Kulikov, Generalized primary groups. I, Trudy Moskov. Mat. Obšč. 1 (1952), 247–326 (Russian). MR 0049188
Alain Louveau and Christian Rosendal, Complete analytic equivalence relations, Trans. Amer. Math. Soc. 357 (2005), no. 12, 4839–4866. MR 2165390, DOI 10.1090/S0002-9947-05-04005-5
D. A. Martin and R. M. Solovay, A basis theorem for $\sum _3^1$ sets of reals, Ann. of Math. (2) 89 (1969), 138–159. MR 255391, DOI 10.2307/1970813
Peter M. Neumann, On the structure of standard wreath products of groups, Math. Z. 84 (1964), 343–373. MR 188280, DOI 10.1007/BF01109904
Laurel A. Rogers, Ulm’s theorem for partially ordered structures related to simply presented abelian $p$-groups, Trans. Amer. Math. Soc. 227 (1977), 333–343. MR 442115, DOI 10.1090/S0002-9947-1977-0442115-X
Helmut Ulm, Zur Theorie der abzählbar-unendlichen Abelschen Gruppen, Math. Ann. 107 (1933), no. 1, 774–803 (German). MR 1512826, DOI 10.1007/BF01448919
J. Zapletal, Reducibility invariants in higher set theory (book in preparation).
Leo Zippin, Countable torsion groups, Ann. of Math. (2) 36 (1935), no. 1, 86–99. MR 1503210, DOI 10.2307/1968666