Notre Dame Journal of Formal Logic

Reverse Mathematics and Fully Ordered Groups

Reed Solomon

Abstract

We study theorems of ordered groups from the perspective of reverse mathematics. We show that $\mathit{RCA}_0$ suffices to prove Hölder's Theorem and give equivalences of both $\mathit{WKL}_0$ (the orderability of torsion free nilpotent groups and direct products, the classical semigroup conditions for orderability) and $\mathit{ACA}_0$ (the existence of induced partial orders in quotient groups, the existence of the center, and the existence of the strong divisible closure).

Article information

Source
Notre Dame J. Formal Logic, Volume 39, Number 2 (1998), 157-189.

Dates
First available in Project Euclid: 7 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1039293061

Digital Object Identifier
doi:10.1305/ndjfl/1039293061

Mathematical Reviews number (MathSciNet)
MR1714964

Zentralblatt MATH identifier
0973.03076

Subjects
Primary: 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35]
Secondary: 03F35: Second- and higher-order arithmetic and fragments [See also 03B30]

Citation

Solomon, Reed. Reverse Mathematics and Fully Ordered Groups. Notre Dame J. Formal Logic 39 (1998), no. 2, 157--189. doi:10.1305/ndjfl/1039293061. https://projecteuclid.org/euclid.ndjfl/1039293061


Export citation

References

  • Baumslag, G., F. Cannonito, D. Robinson, and D. Segal, “The algorithmic theory of polycyclic-by-finite groups,” Journal of Algebra, vol. 141 (1991), pp. 118–49. Zbl 0774.20019 MR 92i:20036
  • Buttsworth, R. N., “A family of groups with a countable infinity of full orders,” Bulletin of the Australian Mathematical Society, vol. 4 (1971), pp. 97–104. Zbl 0223.06008 MR 43:4739
  • Downey, R. G., and S. A. Kurtz, “Recursion theory and ordered groups,” Annals of Pure and Applied Logic, vol. 32 (1986), pp. 137–51. Zbl 0629.03020 MR 87m:03062
  • Friedman, H. M., S. G. Simpson, and R. L. Smith, “Countable algebra and set existence axioms,” Annals of Pure and Applied Logic, vol. 25 (1983), pp. 141–81. Zbl 0575.03038 MR 85i:03157
  • Fuchs, L., “Note on ordered groups and rings,” Fundamenta Mathematicae, vol. 46 (1958), pp. 167–74. Zbl 0100.26701 MR 20:7069
  • Fuchs, L., Partially Ordered Algebraic Systems, Pergamon Press, New York, 1963. Zbl 0137.02001 MR 30:2090
  • Hatzikiriakou, K. and S. G. Simpson, “$\mbox{\emph{WKL}}_{0}$ and orderings of countable abelian groups," Contemporary Mathematics, vol. 106 (1990), pp. 177–80. Zbl 0703.03038 MR 91i:03111
  • Jockusch, C. G., Jr., and R. I. Soare, “$\Pi_{1}^{0}$ classes and degrees of theories,” Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56. Zbl 0262.02041 MR 47:4775
  • Kargapolov, M. I., A. Kokorin, and V. M. Kopytov, “On the theory of orderable groups,” Algebra i Logika, vol. 4 (1965), pp. 21–27. MR 33:4162
  • Kokorin, A., and V. M. Kopytov, Fully Ordered Groups, translated by D. Louvish, John Wiley and Sons, New York, 1974. MR 51:306
  • Lorenzen, P., “Über halbgeordnete gruppen,” Archiv der Mathematik, vol. 2 (1949), pp. 66–70. Zbl 0038.15901
  • Łos, J., “On the existence of linear order in a group,” Bulletin de L'Académie des Polonaise des Sciences Cl. III, vol. 2 (1954), pp. 21–23. Zbl 0057.25302 MR 16,564c
  • Mura, R B., and A. Rhemtulla, Orderable Groups, vol. 27, Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 1977. Zbl 0358.06038 MR 58:10652
  • Ohnishi, M., “Linear order on a group,” Osaka Mathematics Journal, vol. 4 (1952), pp. 17–18. Zbl 0047.02206 MR 14,241f
  • Simpson, S. G., Subsystems of Second Order Arithmetic, Springer-Verlag, New York, 1998. Zbl 0909.03048 MR 2001i:03126
  • Smith, R. L., “Two theorems on autostability in p-groups,” pp. 302–11 in Logic Year 1979–80, vol. 859, Lecture Notes In Mathematics, edited by A. Dold and B. Eckmann, Springer-Verlag, New York, 1981. Zbl 0488.03024 MR 83h:03064
  • Soare, R. I., Recursively Enumerable Sets and Degrees, Perspectives in Mathematical Logic, Springer–Verlag, New York, 1987. Zbl 0667.03030 Zbl 0623.03042 MR 88m:03003
  • Teh, H. H., “Construction of orders in abelian groups,” Cambridge Philosophical Society Proceedings, vol. 57 (1960), pp. 476–82. Zbl 0104.24603 MR 23:A950