Mutually complementary partial orders

Two partial orders P = (X, S) and Q = (X, s’) are complementary if P fl Q = {(x, x): x E x} and the transitive closure of P U Q is {(x. y): x, y E X}. We investigate here the size w(n) of the largest set of pairwise complementary par!iai orders on a set of size n. In particular, for large n we construct L?(n/iogrt) mutually complementary partial orders of order n, and show on the other hand that o(n) <0.486n for ail sufficiently large II. This provides an estimate of the maximum number of mutually complementary ?;, topologies on a set of size n.