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The weak order on integer posets
Chatel, Grégory1; Pilaud, Vincent2; Pons, Viviane3
1 LIGM, Univ. Paris-Est Marne-la-Vallée, 5 Boulevard Descartes, 77420 Champs-sur-Marne, France
2 CNRS & LIX, École Polytechnique, 1 Rue Honoré d’Estienne d’Orves, 91120 Palaiseau, France
3 LRI, Univ. Paris-Sud, Bâtiment 650, Rue Noetzlin, 91190 Gif-sur-Yvette, France
Algebraic Combinatorics, Volume 2 (2019) no. 1, pp. 1-48.

We explore lattice structures on integer binary relations (i.e. binary relations on the set {1,2,⋯,n} for a fixed integer n) and on integer posets (i.e. partial orders on the set {1,2,⋯,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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.36
Classification: 03G10, 06A07, 06B99, 52B12
Keywords: Integer binary relations, Weak order, Lattices

Chatel, Grégory 1; Pilaud, Vincent 2; Pons, Viviane 3

1 LIGM, Univ. Paris-Est Marne-la-Vallée, 5 Boulevard Descartes, 77420 Champs-sur-Marne, France
2 CNRS & LIX, École Polytechnique, 1 Rue Honoré d’Estienne d’Orves, 91120 Palaiseau, France
3 LRI, Univ. Paris-Sud, Bâtiment 650, Rue Noetzlin, 91190 Gif-sur-Yvette, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Chatel, Grégory; Pilaud, Vincent; Pons, Viviane. The weak order on integer posets. Algebraic Combinatorics, Volume 2 (2019) no. 1, pp. 1-48. doi : 10.5802/alco.36. https://alco.centre-mersenne.org/articles/10.5802/alco.36/

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