Finite pseudocomplemented lattices and ‘permutoedre’

Abstract

We study finite pseudocomplemented lattices and especially those that are also complemented. With regard to the classical results on arbitrary or distributive pseudocomplemented lattices (Glivenko, Stone, Birkhoff, Frink, Grätzer, Balbes, Horn, Varlet,…), the finiteness property allows one to bring significant, more precise, details on the structural properties of such lattices. These results can especially be applied to the lattices defined by the ‘weak Bruhat order’ on a Coxeter group (and, for instance, to the lattice of permutations, called, in french, ‘le treillis permutoèdre’) and to the lattice of binary bracketings.