Skip to main content Accessibility help




We prove that for every uncountable cardinal κ such that κ = κ, the quasi-order of embeddability on the κ-space of κ-sized graphs Borel reduces to the embeddability on the κ-space of κ-sized torsion-free abelian groups. Then we use the same techniques to prove that the former Borel reduces to the embeddability relation on the κ-space of κ-sized R-modules, for every $\mathbb{S}$ -cotorsion-free ring R of cardinality less than the continuum. As a consequence we get that all the previous are complete $\Sigma _1^1$ quasi-orders.



Hide All
[1]Andretta, A. and Ros, L. M., Classifying uncountable structures up to bi-embeddability, preprint, 2016, arXiv:1609.09292v1.
[2]Baer, R., Abelian groups without elements of finite order. Duke Mathematical Journal, vol. 3 (1937), no. 1, pp. 68122.
[3]Calderoni, F., Mildenberger, H., and Ros, L. M., Uncountable structures are not classifiable up to bi-embeddability, submitted.
[4]Calderoni, F. and Thomas, S., The bi-embeddability relation for countable abelian groups. Transactions of the American Mathematical Society, to appear.
[5]Corner, A. L. S., Every countable reduced torsion-free ring is an endomorphism ring. Proceedings of the London Mathematical Society. Third Series, vol. 13 (1963), pp. 687710.
[6]Downey, R. and Montalbán, A., The isomorphism problem for torsion-free abelian groups is analytic complete. Journal of Algebra, vol. 320 (2008), no. 6, pp. 22912300.
[7]Friedman, S.-D., Hyttinen, T., and Kulikov, V., Generalized descriptive set theory and classification theory. Memoirs of the American Mathematical Society, vol. 230 (2014), no. 1081, pp. vi+80.
[8]Fuchs, L., Infinite Abelian Groups. Vol. I, Pure and Applied Mathematics, vol. 36, Academic Press, New York, London, 1970.
[9]Fuchs, L., Abelian Groups, Springer Monographs in Mathematics, Springer, Cham, 2015.
[10]Göbel, R. and Przeździecki, A. J., An axiomatic construction of an almost full embedding of the category of graphs into the category of R-objects. Journal of Pure and Applied Algebra, vol. 218 (2014), no. 2, pp. 208217.
[11]Göbel, R. and Trlifaj, J., Approximations and Endomorphism Algebras of Modules. Volume 2, extended ed., de Gruyter Expositions in Mathematics, vol. 41, Walter de Gruyter GmbH & Co. KG, Berlin, 2012, Predictions.
[12]Hjorth, G., The isomorphism relation on countable torsion free abelian groups. Fundamenta Mathematicae, vol. 175 (2002), no. 3, pp. 241257.
[13]Hyttinen, T. and Kulikov, V., On ${\rm{\Sigma }}_1^1$-complete equivalence relations on the generalized Baire space. Mathematical Logic Quarterly, vol. 61 (2015), no. 1–2, pp. 6681.
[14]Hyttinen, T. and Moreno, M., On the reducibility of isomorphism relations. Mathematical Logic Quarterly, vol. 63 (2017), no. 3–4, pp. 175192.
[15]Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.
[16]Louveau, A. and Rosendal, C., Complete analytic equivalence relations. Transactions of the American Mathematical Society, vol. 357 (2005), no. 12, pp. 48394866.
[17]Ros, L. M., The descriptive set-theoretical complexity of the embeddability relation on models of large size. Annals of Pure and Applied Logic, vol. 164 (2013), no. 12, pp. 14541492.
[18]Przeździecki, A. J., An almost full embedding of the category of graphs into the category of abelian groups. Advances in Mathematics, vol. 257 (2014), pp. 527545.
[19]Rotman, J. J., An Introduction to Homological Algebra, second ed., Universitext, Springer, New York, 2009.
[20]Thomas, S., The classification problem for torsion-free abelian groups of finite rank. Journal of the American Mathematical Society, vol. 16 (2003), no. 1, pp. 233258.
[21]Williams, J., Universal countable Borel quasi-orders, this Journal, vol. 79 (2014), no. 3, pp. 928–954.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed