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Reverse mathematics and fully ordered groups. (English) Zbl 0973.03076
This article is a detailed presentation of some of the author’s work in the reverse mathematics of ordered groups, including proofs of many of the results listed in his survey article [Bull. Symb. Log. 5, No. 1, 45-58 (1999; Zbl 0922.03078)]. Theorems pertaining to quotient groups, product groups, and semigroup conditions for orderability are analyzed, and a proof of Hölder’s theorem is carried out in RCA\(_0\). Related results can also be found in the author’s paper [J. Symb. Log. 66, No. 1, 192-206 (2001; Zbl 0981.03060)].

MSC:
03F35 Second- and higher-order arithmetic and fragments
06F15 Ordered groups
03B30 Foundations of classical theories (including reverse mathematics)
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[1] 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 · doi:10.1016/0021-8693(91)90221-S
[2] 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 · doi:10.1017/S000497270004630X
[3] 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 · doi:10.1016/0168-0072(86)90049-7
[4] 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 · doi:10.1016/0168-0072(83)90012-X
[5] Fuchs, L., “Note on ordered groups and rings,” Fundamenta Mathematicae , vol. 46 (1958), pp. 167–74. · Zbl 0100.26701 · eudml:213501
[6] Fuchs, L., Partially Ordered Algebraic Systems , Pergamon Press, New York, 1963. · Zbl 0137.02001
[7] Hatzikiriakou, K. and S. G. Simpson, “\(\mbox\emphWKL_0\) and orderings of countable abelian groups,” Contemporary Mathematics , vol. 106 (1990), pp. 177–80. · Zbl 0703.03038
[8] 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. JSTOR: · Zbl 0262.02041 · doi:10.2307/1996261 · links.jstor.org
[9] Kargapolov, M. I., A. Kokorin, and V. M. Kopytov, “On the theory of orderable groups,” Algebra i Logika , vol. 4 (1965), pp. 21–27.
[10] Kokorin, A., and V. M. Kopytov, Fully Ordered Groups , translated by D. Louvish, John Wiley and Sons, New York, 1974. · Zbl 0192.36401
[11] Lorenzen, P., “Über halbgeordnete gruppen,” Archiv der Mathematik , vol. 2 (1949), pp. 66–70. · Zbl 0038.15901 · doi:10.1007/BF02036756
[12] 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
[13] Mura, R B., and A. Rhemtulla, Orderable Groups , vol. 27, Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 1977. · Zbl 0358.06038
[14] Ohnishi, M., “Linear order on a group,” Osaka Mathematics Journal , vol. 4 (1952), pp. 17–18. · Zbl 0047.02206
[15] Simpson, S. G., Subsystems of Second Order Arithmetic , Springer-Verlag, New York, 1998. · Zbl 0909.03048
[16] 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
[17] Soare, R. I., Recursively Enumerable Sets and Degrees , Perspectives in Mathematical Logic, Springer–Verlag, New York, 1987. Zentralblatt MATH: · Zbl 0667.03030 · www.zentralblatt-math.org
[18] Teh, H. H., “Construction of orders in abelian groups,” Cambridge Philosophical Society Proceedings , vol. 57 (1960), pp. 476–82. · Zbl 0104.24603 · doi:10.1017/S0305004100035520
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