When a partial Borel order is linearizable

We prove a classification theorem of the “Glimm–Effros” type for Borel order relations: a Borel partial order on the reals either is Borel linearizable or includes a copy of a certain Borel partial order ≤0 which is not Borel linearizable. Notation. A binary relation 4 on a set X is a partial quasi-order , or p.q.-o. in brief, on X, iff x 4 y∧y 4 z ⇒ x 4 z, and x 4 x for any x ∈ X. In this case, ≈ is the associated equivalence relation, i.e. x ≈ y iff x 4 y∧y 4 x. If in addition x ≈ x ⇒ x = x for any x then 4 is a partial order , or p.o., so that, say, forcing relations are p.q.-o.’s, but, generally speaking, not p.o.’s in this terminology. A p.o. is linear (l.o.) iff we have x 4 y ∨ y 4 x for all x, y ∈ X. Let 4 and 4′ be p.q.-o.’s on resp. X and X ′. A map h : X → X ′ will be called half order preserving , or h.o.p., iff x 4 y ⇒ h(x) 4′ h(y). Definition 1. A Borel p.q.-o. 〈X; 4〉 is Borel linearizable iff there is a Borel l.o. 〈X ′; 4′〉 and a Borel h.o.p. map h : X → X ′ (called a linearization map) satisfying x ≈ y ⇔ h(x) = h(y) (1). Introduction. Harrington, Marker, and Shelah [2] proved several theorems on Borel partial order relarions, mainly concerning thin p.q.-o.’s, i.e. those which do not admit uncountable pairwise incomparable subsets. In particular, they demonstrated that any such Borel p.q.-o. is Borel linearizable, and moreover the corresponding l.o. 〈X ′; 4′〉 can be chosen as a suborder of 〈2;≤lex〉 for some α < ω1, where ≤lex is the lexicographical order.