Linearization of analytic order relations

We prove that if 4 is an analytic partial order then either 4 can be extended to a ∆2 linear order similar to an antichain in 21 ordered lexicographically or a certain Borel partial order ≤0 embeds in 4 . Some corollaries for analytic equivalence relations are given, for instance, if E is a Σ1 1 [z] equivalence relation such that E0 does not embed in E then E is determined by intersections with E-invariand Borel sets coded in L[z] . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. Monotone Borel functions and the dichotomy . . . . . . . . . . . . . . . 8 3. The basic forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4. The product forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5. The construction of an embedding. . . . . . . . . . . . . . . . . . . . . . . . .14 6. Why embedding ≤0 is absolute . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 7. Borel and analytic order relations . . . . . . . . . . . . . . . . . . . . . . . . . 18 8. Special cases: Borel classes and generic models . . . . . . . . . . . . 21 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 ∗ Moscow Transport Engineering Institute † [email protected] and [email protected] ‡ This paper was accomplished in part during my visit to Caltech in April 1997. I thank Caltech for the support and A. S. Kechris and J. Zapletal for useful information and interesting discussions relevant to the topic of this paper during the visit.