Pi10 classes and orderable groups

It is known that the spaces of orders on orderable computable +elds can represent all 0 1 classes up to Turing degree. We show that the spaces of orders on orderable computable abelian and nilpotent groups cannot represent 0 1 classes in even a weak manner. Next, we consider presentations of ordered abelian groups, and we show that there is a computable ordered abelian group for which no computable presentation admits a computable set of representatives for its Archimedean classes. c © 2002 Elsevier Science B.V. All rights reserved. MSC: 03D80; 06F15

[1]  V. P. Dobritsa.  Some constructivizations of Abelian groups , 1983 .

[2]  Richard Lawrence Smith,et al.  Two theorems on autostability in p-Groups , 1981 .

[3]  H.-H. Teh,et al.  Construction of orders in Abelian groups , 1961, Mathematical Proceedings of the Cambridge Philosophical Society.

[4]  D. Robinson.  A Course in the Theory of Groups , 1982 .

[5]  R. Soare.  Recursively enumerable sets and degrees , 1987 .

[6]  D. Louvish,et al.  Fully Ordered Groups , 1974 .

[7]  Stuart A. Kurtz,et al.  Recursion theory and ordered groups , 1986, Ann. Pure Appl. Log..

[8]  Reed Solomon,et al.  Reverse Mathematics and Fully Ordered Groups , 1998, Notre Dame J. Formal Log..

[9]  M. Rabin.  Computable algebra, general theory and theory of computable fields. , 1960 .

[10]  Stephen G. Simpson,et al.  Subsystems of second order arithmetic , 1999, Perspectives in mathematical logic.

[11]  Lawrence Feiner,et al.  Hierarchies of Boolean algebras , 1970, Journal of Symbolic Logic.