Linearization of definable order relations Original Research Article

We prove that if ≼ is an analytic partial order then either ≼ can be extended to a Δ21 linear order similar to an antichain in 2<ω1, ordered lexicographically, or a certain Borel partial order ⩽0 embeds in ≼. Similar linearization results are presented, for κ-bi-Souslin partial orders and real-ordinal definable orders in the Solovay model. A corollary for analytic equivalence relations says that any (lightface) Σ11 equivalence relation View the MathML source such that View the MathML source does not embed in View the MathML source is fully determined by intersections with View the MathML source-invariant Borel sets coded in L.