## 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

https://projecteuclid.org/euclid.ndjfl/1039293061

Digital Object Identifier
doi:10.1305/ndjfl/1039293061

Mathematical Reviews number (MathSciNet)
MR1714964

Zentralblatt MATH identifier
0973.03076

#### 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

#### 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