Self complementary topologies and preorders

Abstract

A topology on a set X is self complementary if there is a homeomorphic copy on the same set that is a complement in the lattice of topologies on X. The problem of characterizing finite self complementary topologies leads us to redefine the problem in terms of preorders (i.e. reflexive, transitive relations). A preorder P on a set X is self complementary if there is an isomorphic copy P′ of P on X that is arc disjoint to P (except for loops) and with the property that P∪P′ is strongly connected. We characterize here self complementary finite partial orders and self complementary finite equivalence relations.