Positive common multiples and order extension in abelian-by-nilpotent groups

Let N be a nilpotent group. We prove that any nonzero p, q ∈ ℤ≥0[N] have nonzero cofactors r, s ∈ ℤ≥0[N] satisfying rp = sq. Thus common left multiples can be formed with nonnegative coefficients throughout. As a consequence, for abelian-by-nilpotent groups, absence of nontrivial generalized torsion, bi-orderability, and the property that every partial bi-invariant order extends to a total one are equivalent. These results give affirmative answers to the orderability and order-extension questions recorded by Longobardi, Maj and Rhemtulla in 2003. The proof of the common-multiple theorem separates the construction of a signed relation with nonnegative boundary coefficients from a simultaneous positive-multiplier argument, and joins them by induction on Hirsch rank. GitHub repository: positive-common-multiples-order-extension