Order extension in Chehata's group

In the real case, Chehata's group consists of the increasing piecewise-linear homeomorphisms of the real line with finitely many pieces and bounded support. We prove that every bi-invariant partial order on this group extends to a bi-invariant total order, answering the real-field form of the question recorded in the 2009 survey of Bludov, Glass, Kopytov, and Medvedev. More precisely, a set of nonidentity elements generates a normal semigroup containing the identity if and only if the logarithms of its initial slopes and those of its final slopes each admit a nonempty positive integer relation. Every partial order therefore extends to an endpoint-slope order induced by an additive total order on the abstract group of real numbers. These orders form two disjoint families and exhaust all bi-orders; their number is 2^{2^{aleph_0}}. The proof creates an arbitrary endpoint germ after cancellation of a positive germ relation and then applies an equal-germ positive factorization argument. Over an Archimedean ordered field F, the analogous criterion and classification use additive orders on the logarithmic slope group and give 2^{|F|} bi-orders. Author manuscript and full paper archive: https://github.com/eulogica/order-extension-chehata