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LEFT-ORDERABLE COMPUTABLE GROUPS

Published online by Cambridge University Press:  05 February 2018

MATTHEW HARRISON-TRAINOR
MATTHEW HARRISON-TRAINOR*
Affiliation:
GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE UNIVERSITY OF CALIFORNIA BERKELEY, CA, USAE-mail:matthew.h-t@berkeley.eduURL: www.math.berkeley.edu/∼mattht
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Abstract

Downey and Kurtz asked whether every orderable computable group is classically isomorphic to a group with a computable ordering. By an order on a group, one might mean either a left-order or a bi-order. We answer their question for left-orderable groups by showing that there is a computable left-orderable group which is not classically isomorphic to a computable group with a computable left-order. The case of bi-orderable groups is left open.

Keywords

03D4503C57computable structure theorycomputable algebraordered groups
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 1 , March 2018 , pp. 237 - 255
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

REFERENCES

Bergman, G. M., Ordering coproducts of groups and semigroups. Journal of Algebra, vol. 133 (1990), no. 2, pp. 313–339.CrossRefGoogle Scholar
Downey, R. G. and Kurtz, S. A., Recursion theory and ordered groups. Annals of Pure and Applied Logic, 32 (1986), no. 2, pp. 137–151.CrossRefGoogle Scholar
Dobritsa, V. P., Some constructivizations of abelian groups. Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 24 (1983), no. 2, pp. 18–25.Google Scholar
Downey, R. G. and Remmel, J. B., Questions in computable algebra and combinatorics, Computability Theory and its Applications (Boulder, CO, 1999) (Cholak, P. A., Lempp, S., Lerman, M., and Shore, R. A., editors), Contemporary Mathematics, vol. 257, American Mathematical Society, Providence, RI, 2000, pp. 95–125.CrossRefGoogle Scholar
Kopytov, V. M. and Medvedev, N. Y., Right-Ordered Groups, Siberian School of Algebra and Logic, Consultants Bureau, New York, 1996.Google Scholar
Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition.CrossRefGoogle Scholar
Shimbireva, H., On the theory of partially ordered groups. Matematicheskiĭ Sbornik, vol. 20 (1947), no. 62, pp. 145–178.Google Scholar
Solomon, R., ${\rm{\Pi }}_1^0 $ classes and orderable groups. Annals of Pure and Applied Logic, vol. 115 (2002), no. 1–3, pp. 279–302.CrossRefGoogle Scholar
Vinogradov, A. A., On the free product of ordered groups. Matematicheskiĭ Sbornik, vol. 25 (1949), no. 67, pp. 163–168.Google Scholar