Title: The weak order on integer posets,
Abstract: We explore lattice structures on integer binary relations (i.e. binary
relations on the set $\{1, 2, \dots, n\}$ for a fixed integer $n$) and on
integer posets (i.e. partial orders on the set $\{1, 2, \dots, n\}$ for a fixed
integer $n$). We first observe that the weak order on the symmetric group
naturally extends to a lattice structure on all integer binary relations. We
then show that the subposet of this weak order induced by integer posets
defines as well a lattice. We finally study the subposets of this weak order
induced by specific families of integer posets corresponding to the elements,
the intervals, and the faces of the permutahedron, the associahedron, and some
recent generalizations of those.