The bi-embeddability relation for countable abelian groups. (English) Zbl 1472.03050
Summary: We analyze the complexity of the bi-embeddability relations for countable torsion-free abelian groups and for countable torsion abelian groups.
MSC:
| 03E15 | Descriptive set theory |
| 20K10 | Torsion groups, primary groups and generalized primary groups |
| 20K20 | Torsion-free groups, infinite rank |
| 03E57 | Generic absoluteness and forcing axioms |
References:
| [1] | Barwise, Jon; Eklof, Paul, Infinitary properties of abelian torsion groups, Ann. Math. Logic, 2, 1, 25-68 (1970/1971) · Zbl 0222.02014 · doi:10.1016/0003-4843(70)90006-9 |
| [2] | Crawley, Peter; Hales, Alfred W., The structure of abelian \(p\)-groups given by certain presentations, J. Algebra, 12, 10-23 (1969) · Zbl 0174.31103 · doi:10.1016/0021-8693(69)90014-3 |
| [3] | Downey, Rod; Montalb\'an, Antonio, The isomorphism problem for torsion-free abelian groups is analytic complete, J. Algebra, 320, 6, 2291-2300 (2008) · Zbl 1156.03042 · doi:10.1016/j.jalgebra.2008.06.007 |
| [4] | Feferman, Solomon, 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, 117-130. (1975), Springer, Berlin · Zbl 0316.02058 |
| [5] | Friedman, Harvey; Stanley, Lee, A Borel reducibility theory for classes of countable structures, J. Symbolic Logic, 54, 3, 894-914 (1989) · Zbl 0692.03022 · doi:10.2307/2274750 |
| [6] | Fuchs, L., On the structure of abelian \(p\)-groups, Acta Math. Acad. Sci. Hungar., 4, 267-288 (1953) · Zbl 0052.02203 · doi:10.1007/BF02127586 |
| [7] | Fuchs, L\'aszl\'o, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, xi+290 pp. (1970), Academic Press, New York-London · Zbl 0209.05503 |
| [8] | Fuchs, L\'aszl\'o, Infinite abelian groups. Vol. II, Pure and Applied Mathematics, Vol. 36-II (1973), Academic Press, New York-London · Zbl 0257.20035 |
| [9] | Gao, Su, Invariant descriptive set theory, Pure and Applied Mathematics (Boca Raton) 293, xiv+383 pp. (2009), CRC Press, Boca Raton, FL · Zbl 1154.03025 |
| [10] | Hjorth, Greg, 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, 221-261 (1999) · Zbl 0945.03069 · doi:10.1007/BF02810686 |
| [11] | Hjorth, Greg, Classification and orbit equivalence relations, Mathematical Surveys and Monographs 75, xviii+195 pp. (2000), American Mathematical Society, Providence, RI · Zbl 0942.03056 |
| [12] | Hjorth, Greg; Kechris, Alexander S., Analytic equivalence relations and Ulm-type classifications, J. Symbolic Logic, 60, 4, 1273-1300 (1995) · Zbl 0847.03023 · doi:10.2307/2275888 |
| [13] | Jech, Thomas, Set theory, Springer Monographs in Mathematics, xiv+769 pp. (2003), Springer-Verlag, Berlin · Zbl 1007.03002 |
| [14] | Kanove\u\i, V. G.; Reeken, M., Some new results on the Borel irreducibility of equivalence relations, Izv. Ross. Akad. Nauk Ser. Mat.. Izv. Math., 67 67, 1, 55-76 (2003) · Zbl 1068.03036 · doi:10.1070/IM2003v067n01ABEH000418 |
| [15] | Kechris, Alexander S., Classical descriptive set theory, Graduate Texts in Mathematics 156, xviii+402 pp. (1995), Springer-Verlag, New York · Zbl 0819.04002 · doi:10.1007/978-1-4612-4190-4 |
| [16] | Kulikov, L. Ya., Generalized primary groups. I, Trudy Moskov. Mat. Ob\v s\v c., 1, 247-326 (1952) · Zbl 0053.21001 |
| [17] | Louveau, Alain; Rosendal, Christian, Complete analytic equivalence relations, Trans. Amer. Math. Soc., 357, 12, 4839-4866 (2005) · Zbl 1118.03043 · doi:10.1090/S0002-9947-05-04005-5 |
| [18] | Martin, D. A.; Solovay, R. M., A basis theorem for \(\sum_3^1\) sets of reals, Ann. of Math. (2), 89, 138-159 (1969) · Zbl 0176.27603 · doi:10.2307/1970813 |
| [19] | Neumann, Peter M., On the structure of standard wreath products of groups, Math. Z., 84, 343-373 (1964) · Zbl 0122.02901 · doi:10.1007/BF01109904 |
| [20] | Rogers, Laurel A., Ulm’s theorem for partially ordered structures related to simply presented abelian \(p\)-groups, Trans. Amer. Math. Soc., 227, 333-343 (1977) · Zbl 0357.20032 · doi:10.2307/1997464 |
| [21] | Ulm, Helmut, Zur Theorie der abz\"ahlbar-unendlichen Abelschen Gruppen, Math. Ann., 107, 1, 774-803 (1933) · Zbl 0006.15003 · doi:10.1007/BF01448919 |
| [22] | zap J. Zapletal, Reducibility invariants in higher set theory (book in preparation). · Zbl 0988.03077 |
| [23] | Zippin, Leo, Countable torsion groups, Ann. of Math. (2), 36, 1, 86-99 (1935) · Zbl 0011.10401 · doi:10.2307/1968666 |
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