First-order separation of pro-orderable groups from a bi-orderable matrix group over Z[F₂]

Abstract

Let \(\mathcal O\) be the class of bi-orderable groups and let \(\mathcal O^*\) consist of the groups in which every bi-invariant partial order extends to a bi-invariant total order. We construct a countable upper triangular matrix group \(H\) over \(\mathbb Z[F_2]\) and a sentence \(\sigma\) in the pure group language such that \(H\in\mathcal O\), every member of \(\mathcal O^*\) satisfies \(\sigma\), and \(H\) does not. Consequently, \(\operatorname{Mod}(\operatorname{Th}(\mathcal O^*))\subsetneq\mathcal O\), giving a negative answer to Kourovka Problem 3.20. The construction makes two disjoint normal subgroups definable by commutators. Their disjointness follows from Fox derivatives, while their interaction with conjugation obstructs extension of a partial order.