Hostname: page-component-54dcc4c588-wlffp Total loading time: 0 Render date: 2025-10-12T20:59:49.429Z Has data issue: false hasContentIssue false
  • English
  • Français

TORSION-FREE ABELIAN GROUPS OF FINITE RANK AND FIELDS OF FINITE TRANSCENDENCE DEGREE

Part of: Model theory Computability and recursion theory

Published online by Cambridge University Press:  13 February 2025

MENG-CHE TURBO HO Open the ORCID record for MENG-CHE TURBO HO [Opens in a new window] ,
JULIA FRANDSEN KNIGHT  and
RUSSELL GEDDES MILLER Open the ORCID record for RUSSELL GEDDES MILLER [Opens in a new window]
MENG-CHE TURBO HO*
Affiliation:
DEPARTMENT OF MATHEMATICS CALIFORNIA STATE UNIVERSITY, NORTHRIDGE 18111 NORDHOFF STREET, NORTHRIDGE CA 91330, USA
JULIA FRANDSEN KNIGHT
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME 255 HURLEY HALL, NOTRE DAME IN 46556-5641, USA E-mail: julia.f.knight.1@nd.edu
RUSSELL GEDDES MILLER
Affiliation:
DEPARTMENT OF MATHEMATICS QUEENS COLLEGE 65-30 KISSENA BLVD. QUEENS, NY 11367 USA AND PH.D. PROGRAMS IN MATHEMATICS AND COMPUTER SCIENCE CUNY GRADUATE CENTER 365 FIFTH AVENUE NEW YORK, NY 10016 USA E-mail: russell.miller@qc.cuny.edu
*
E-mail: turboho@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $\operatorname {TFAb}_r$ be the class of torsion-free abelian groups of rank r, and let $\operatorname {FD}_r$ be the class of fields of characteristic $0$ and transcendence degree r. We compare these classes using various notions. Considering the Scott complexity of the structures in the classes and the complexity of the isomorphism relations on the classes, the classes seem very similar. Hjorth and Thomas showed that the $\operatorname {TFAb}_r$ are strictly increasing under Borel reducibility. This is not so for the classes $\operatorname {FD}_r$. Thomas and Velickovic showed that for sufficiently large r, the classes $\operatorname {FD}_r$ are equivalent under Borel reducibility. We try to compare the groups with the fields, using Borel reducibility, and also using some effective variants. We give functorial Turing computable embeddings of $\operatorname {TFAb}_r$ in $\operatorname {FD}_r$, and of $\operatorname {FD}_r$ in $\operatorname {FD}_{r+1}$. We show that under computable countable reducibility, $\operatorname {TFAb}_1$ lies on top among the classes we are considering. In fact, under computable countable reducibility, isomorphism on $\operatorname {TFAb}_1$ lies on top among equivalence relations that are effective $\Sigma _3$.

Keywords

isomorphism problemtorsion-free abelian groupsfields with finite transcendence degree

MSC classification

Primary: 03C57: Effective and recursion-theoretic model theory 03D45: Theory of numerations, effectively presented structures 03D55: Hierarchies

Information

Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 30
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Alvir, R., Scott analysis of countable structures , Ph.D. thesis, University of Notre Dame, 2022.Google Scholar
Beauville, A., Colliot-Thélène, J., Sansuc, J., and Swinnerton-Dyer, P., Variétés stablement rationnelles non rationnelles . Annals of Mathematics , vol. 121 (1985), no. 2, pp. 283–318.10.2307/1971174CrossRefGoogle Scholar
Calvert, W., Cummins, D., Knight, J. F., and Miller, S., Comparing classes of finite structures . Algebra and Logic , vol. 43 (2004), pp. 374–392.10.1023/B:ALLO.0000048827.30718.2cCrossRefGoogle Scholar
Coles, R. J., Downey, R. G., and Slaman, T. A., Every set has a least jump enumeration . Journal of the London Mathematical Society , vol. 62 (2000), no. 2, pp. 641–649.10.1112/S0024610700001459CrossRefGoogle Scholar
Engler, A. J. and Prestel, A., Valued Fields , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.Google Scholar
Friedman, H. and Stanley, L., A Borel reducibility theory for classes of countable structures . Journal of Symbolic Logic , vol. 54 (1989), pp. 894–914.10.2307/2274750CrossRefGoogle Scholar
Frolov, A., Kalimullin, I., and Miller, R., Spectra of algebraic fields and subfields , Mathematical Theory and Computational Practice: Fifth Conference on Computability in Europe, CiE 2009 (Ambos-Spies, K., Löwe, B., and Merkle, W., editors), Lecture Notes in Computer Science, 5635, Springer-Verlag, Berlin, 2009, pp. 232–241.10.1007/978-3-642-03073-4_24CrossRefGoogle Scholar
Hall, C., Knight, J. F., and Lange, K., Complexity of well-ordered sets in an ordered Abelian group. to appear in Monatshefte für Mathematik.Google Scholar
Harrison-Trainor, M., Computable valued fields . Archive for Mathematical Logic , vol. 57 (2018), nos. 5–6, pp. 473–495.10.1007/s00153-017-0589-9CrossRefGoogle Scholar
Harrison-Trainor, M., Melnikov, A., Miller, R., and Montalbán, A., Computable functors and effective interpretability . Journal of Symbolic Logic , vol. 82 (2017), pp. 77–97.10.1017/jsl.2016.12CrossRefGoogle Scholar
Harrison-Trainor, M., Miller, R., and Montalbán, A., Borel functors and infinitary interpretations . Journal of Symbolic Logic , vol. 83 (2018), pp. 1434–1456.10.1017/jsl.2017.81CrossRefGoogle Scholar
Hirschfeldt, D., Kramer, K., Miller, R., and Shlapentokh, A., Categoricity properties for computable algebraic fields . Transactions of the American Mathematical Society , vol. 367 (2015), no. 6, pp. 3955–3980.Google Scholar
Hjorth, G., Around nonclassifiability for countable torsion free abelian groups , Abelian Groups and Modules (Eklof, P. C. and Göbel, R., editors), Trends in Mathematics, Birkhäuser, Basel, 1999, pp. 269–292.10.1007/978-3-0348-7591-2_22CrossRefGoogle Scholar
Knight, J. F., Lange, K., and McCoy, C., Computable ${\varPi}_2$ Scott sentences, pre-print.Google Scholar
Knight, J. F., Miller, S., and Vanden Boom, M., Turing computable embeddings . Journal of Symbolic Logic , vol. 72 (2007), no. 3, pp. 901–918.10.2178/jsl/1191333847CrossRefGoogle Scholar
Knight, J. F. and Saraph, V., Scott sentences for certain groups . Archive for Mathematical Logic , vol. 57 (2018), pp. 453–472.10.1007/s00153-017-0578-zCrossRefGoogle Scholar
Laskowski, M. and Ulrich, D., A proof of the Borel completeness of torsion free abelian groups , preprint, 2022, arXiv preprint arXiv:2202.07452.Google Scholar
Laskowski, M. and Ulrich, D., Borel complexity of modules , preprint, 2022, arXiv preprint arXiv:2209.06898.Google Scholar
Lopez-Escobar, E. G. K., An interpolation theorem for denumerably long formulas . Fundamenta Mathematicae , vol. 57 (1965), pp. 253–272.10.4064/fm-57-3-253-272CrossRefGoogle Scholar
Miller, R., Computable reducibility for Cantor space , Structure and Randomness in Computability and Set Theory (Cenzer, D., Porter, C., and Zapletal, J., editors), World Sci. Publ., Hackensack, NJ, 2020, pp. 155–196.10.1142/9789813228238_0005CrossRefGoogle Scholar
Miller, R. and Ng, K. M., Finitary reducibility on equivalence relations . Journal of Symbolic Logic , vol. 81 (2016), no. 4, pp. 1225–1254.10.1017/jsl.2016.23CrossRefGoogle Scholar
Miller, R., Poonen, B., Schoutens, H., and Shlapentokh, A., A computable functor from graphs to fields . Journal of Symbolic Logic , vol. 83 (2018), pp. 326–348.10.1017/jsl.2017.50CrossRefGoogle Scholar
Montalbán, A., A robuster Scott rank . PAMS , vol. 143 (2015), pp. 5427–5436.10.1090/proc/12669CrossRefGoogle Scholar
Montalbán, A., Computability theoretic classifications for classes of structures . Proccedings of the ICM , vol. 2014 (2014), no. 2, pp. 79–101.Google Scholar
Paolini, G. and Shelah, S., Torsion-free abelian groups are Borel complete . Ann. of Math. , (2) 199 (2024), no. 3, pp. 1177–1224.10.4007/annals.2024.199.3.4CrossRefGoogle Scholar
Schinzel, A., On the number of terms of a power of a polynomial . Acta Arithmetica , vol. 49 (1987), no. 1, pp. 55–70.10.4064/aa-49-1-55-70CrossRefGoogle Scholar
Thomas, S., The classification problem for torsion-free abelian groups of finite rank . Journal of the American Mathematical Society , vol. 16 (2003), pp. 233–258.10.1090/S0894-0347-02-00409-5CrossRefGoogle Scholar
Thomas, S. and Velickovic, B., On the complexity of the isomorphism relation for fields of finite transcendence degree . Journal of Pure and Applied Algebra , vol. 159 (2001), pp. 347–363.10.1016/S0022-4049(00)00050-5CrossRefGoogle Scholar
Vanden Boom, M., The effective Borel hierarchy . Fundamenta Mathematicae , vol. 195 (2007), pp. 269–289.10.4064/fm195-3-4CrossRefGoogle Scholar
Vaught, R., Invariant sets in topology and logic . Fundamenta Mathematicae , vol. 82 (1974/75/1975), pp. 269–294.10.4064/fm-82-3-269-294CrossRefGoogle Scholar