Mutually complementary partial orders

Abstract

Two partial orders P=( X,⩽) and Q=( X, ⩽′) are complementary if P ∩ 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)<0.486 n for all sufficiently large n. This provides an estimate of the maximum number of mutually complementary T 0 topologies on a set of size n.