Logic Seminar

Please note that the time is PST

We introduce a natural notion of order-isomorphism between equivalence relations on R and prove the following dichotomy theorem: if E = E_G is the orbit equivalence relation of a group G of orientation-preserving homeomorphisms of R all of whose orbits are dense in R, then either E is isomorphic to the orbit equivalence relation of a group of translations, or E embeds an isomorphic copy of the tail-equivalence relation.

We also discuss connections between this theorem and several related dichotomy theorems about linear orders due to Lindenbaum, Jullien, and Holland. In each of these theorems, the dichotomy in question distinguishes between linear orders that can in some sense be split into two copies of themselves and linear orders for which there is no such splitting.