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Orderable groups, elementary theory, and the Kaplansky conjecture

Orderable groups, elementary theory, and the Kaplansky conjecture AbstractWe show that each of the classes of left-orderable groups and orderablegroups is a quasivariety with undecidable theory. In the case of orderablegroups, we find an explicit set of universal axioms. We then consider the relationship with the Kaplansky group rings conjecture and show that 𝒦{{\mathcal{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{{\mathcal{K}}}or more generally two torsion-free groups are universally equivalent. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups Complexity Cryptology de Gruyter

Orderable groups, elementary theory, and the Kaplansky conjecture

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Publisher
de Gruyter
Copyright
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1869-6104
eISSN
1869-6104
DOI
10.1515/gcc-2018-0005
Publisher site
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Abstract

AbstractWe show that each of the classes of left-orderable groups and orderablegroups is a quasivariety with undecidable theory. In the case of orderablegroups, we find an explicit set of universal axioms. We then consider the relationship with the Kaplansky group rings conjecture and show that 𝒦{{\mathcal{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{{\mathcal{K}}}or more generally two torsion-free groups are universally equivalent.

Journal

Groups Complexity Cryptology – de Gruyter

Published: May 1, 2018

References

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  • Elementary theory of free non-abelian groups
  • Polycyclic right-ordered groups