Document Type


Article Version

Publisher's PDF

Publication Date



We show that each of the classes of left-orderable groups and orderable groups is a quasivariety with undecidable theory. In the case of orderable groups, we find an explicit set of universal axioms. We then consider the relationship with the Kaplansky group rings conjecture and show that K, the class of groups which satisfy the conjecture, is the model class of a set of universal sentences in the language of group theory. We also give a characterization of when two groups in K or more generally two torsion-free groups are universally equivalent.


© 2018 Walter de Gruyter GmbH, Berlin/Boston

The final publisher PDF has been archived here with permission from the copyright holder.

Publication Title

Groups Complexity Cryptology

Published Citation

Fine, B., Gaglione, A., Rosenberger, G., & Spellman, D. (2018). Orderable groups, elementary theory, and the Kaplansky conjecture. Groups Complexity Cryptology, 10(1), 43-52.



Peer Reviewed

Available for download on Friday, April 26, 2019