Linearization of analytic order relations

Abstract

We prove that if $\le$ is an analytic partial order then either $\le$ can be extended to a (boldface) ${\Delta }_2^1$ linear order similar to an antichain in $2^{<{\omega }_1}$ ordered lexicographically or a certain Borel partial order ${\le }_0$ embeds in $\le .$ Some corollaries for analytic equivalence relations are given, for instance, if $E$ is a ${\Sigma }_1^1[z]$ equivalence relation such that $E_0$ does not embed in $E$ then $E$ is determined by intersections with E-invariand Borel sets coded in $L[z]$.

References in corpus (3)

• On a Glimm -- Effros dichotomy and an Ulm--type classification in Solovay model
• When a partial Borel order is linearizable
• On a Glimm -- Effros dichotomy theorem for Souslin relations in generic universes

Cited by in corpus (2)

• Linear ROD subsets of Borel partial orders are countably cofinal in Solovay's model
• Linear ROD subsets of Borel partial orders are countably cofinal in the Solovay model