Groups with prescribed finite numbers of lattice orders

We prove that the possible finite numbers of lattice group orders on an ℓ-orderable group are exactly 1 and the positive even integers. We also construct, for every n ≥ 1, a countable nonabelian simple group with exactly 2n bi-invariant total orders. These results answer the finite-spectrum questions in the 2009 problem list of Bludov, Glass, Kopytov, and Medvedev. The lattice-order construction rests on a classification of all lattice orders on certain groups of homeomorphisms of the line. For the complementary examples, we show that every lattice order on the affine family used here, due to Kopytov, is total. Repository with the manuscript source, compiled PDF, and supporting materials: https://github.com/eulogica/finite-lattice-order-spectrum