Orderable groups, elementary theory, and the Kaplansky conjecture

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.