The lattice of closure relations on a poset

Abstract

In this paper we show that the set of closure relations on a finite posetP forms a supersolvable lattice, as suggested by Rota. Furthermore this lattice is dually isomorphic to the lattice of closed sets in a convex geometry (in the sense of Edelman and Jamison [EJ]). We also characterize the modular elements of this lattice (whenP has a greatest element) and compute its characteristic polynomial.