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We investigate a poset structure that extends the weak order on a finiteCoxeter group $W$ to the set of all faces of the permutahedron of $W$. We callthis order the facial weak order. We first provide two alternativecharacterizations of this poset: a first one, geometric, that generalizes thenotion of inversion sets of roots, and a second one, combinatorial, that usescomparisons of the minimal and maximal length representatives of the cosets.These characterizations are then used to show that the facial weak order is infact a lattice, generalizing a well-known result of A. Bj\"orner for theclassical weak order. Finally, we show that any lattice congruence of theclassical weak order induces a lattice congruence of the facial weak order, andwe give a geometric interpretation of their classes. As application, wedescribe the facial boolean lattice on the faces of the cube and the facialCambrian lattice on the faces of the corresponding generalized associahedron.

The facial weak order and its lattice quotients