The weak order on integer posets

Vincent Pilaud (1) , Viviane Pons (2) , Grégory Chatel (3)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 GALaC - LRI - Graphes, Algorithmes et Combinatoire (LRI)
3 LIGM - Laboratoire d'Informatique Gaspard-Monge

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.

Mots clés: Weak order; Lattices; Integer binary relations

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Vincent Pilaud, Viviane Pons, Grégory Chatel. The weak order on integer posets. Algebraic Combinatorics, 2019, 2 (1), pp.1-48. ⟨10.5802/alco.36⟩. ⟨hal-02017630⟩