Mutually complementary partial orders

AbstractTwo partial orders P=(X,⩽) and Q=(X, ⩽′) are complementary ifP ∩ Q={(x, x): x ε x} and the transitive closure of P ∩ Q is {(x, y : x, y ε x}. We investigate here the size ω(n) of the largest set of pairwise complementary partial orders on a set of size n. In particular, for large n we constructπ(n/log n) mutually complementary partial orders of order n, and show on the other hand that ω(n)