MUTUALLY COMPLEMENTARY PARTIAL ORDERS

Two partial orders P = (X, less-than-or-equal-to) and Q = (X, less-than-or-equal-to') are complementary if P and Q = {(X, X): x is-an-element-of x) and the transitive closure of P or Q is {(X, Y): X, Y is-an-element-of X). We investigate here the size omega(n) of the largest set of pairwise complementary partial orders on a set of size n. In particular, for large n we construct OMEGA(n/log n) mutually complementary partial orders of order n, and show on the other hand that omega(n) < 0.486n for all sufficiently large n. This provides an estimate of the maximum number of mutually complementary T0 topologies on a set of size n.