Linearization of analytic order relations

Abstract

We prove that if $\leq$ is an analytic partial order then either $\leq$ can be extended to a (boldface) $Δ^1_2$ linear order similar to an antichain in $2^{<ω_1}$ ordered lexicographically or a certain Borel partial order $\leq_0$ embeds in $\leq.$ Some corollaries for analytic equivalence relations are given, for instance, if $E$ is a $Σ^1_1[z]$ equivalence relation such that $E_0$ does not embed in $E$ then $E$ is determined by intersections with E-invariand Borel sets coded in $L[z]$.