Lattice structure of Grassmann-Tamari orders

International audienceThe Tamari order is a central object in algebraic combinatorics and many other areas. Defined as the transitive closure of an associativity law, the Tamari order possesses a surprisingly rich structure: it is a congruence-uniform lattice. In this work, we consider a larger class of posets, the Grassmann-Tamari orders, which arise as an ordering on the facets of the non-crossing complex introduced by Pylyavskyy, Petersen, and Speyer. We prove that the Grassmann-Tamari orders are congruence-uniform lattices, which resolves a conjecture of Santos, Stump, and Welker. Towards this goal, we define a closure operator on sets of paths inside a rectangle, and prove that the biclosed sets of paths, ordered by inclusion, form a congruence-uniform lattice. We then prove that the Grassmann-Tamari order is a quotient lattice of the corresponding lattice of biclosed sets.L’ordre Tamari est un objet central dans la combinatoire algébrique et de nombreux autres domaines. Définie comme la fermeture transitive d’une loi d’associativité, l’ordre Tamari possède une structure étonnamment riche: il est un treillis congruence uniforme. Dans ce travail, nous considérons une classe plus large de posets, les ordres Grassmann-Tamari, qui découlent comme un ordre sur les facettes du complexe non-croisement introduit par Pylyavskyy, Petersen, et Speyer. Nous démontrons que les ordres Grassmann-Tamari sont treillis congruence uniformes, ce qui résout une conjecture de Santos, Stump, et Welker. Pour atteindre cet objectif, nous définissons un opérateur de fermeture sur des ensembles de chemins à l’intérieur d’un rectangle, et prouver que les ensembles bifermé de chemins, ordonné par inclusion, forment un réseau de congruence uniforme. Nous démontrons ensuite que l’ordre Grassmann-Tamari est un treillis quotient du treillis correspondant d’ensembles bifermés

The extended permutohedron on a transitive binary relation

For a given transitive binary relation e on a set E, the transitive closures of open (i.e., co-transitive in e) sets, called the regular closed subsets, form an ortholattice Reg(e), the extended permutohedron on e. This construction, which contains the poset Clop(e) of all clopen sets, is a common generalization of known notions such as the generalized permutohedron on a partially ordered set on the one hand, and the bipartition lattice on a set on the other hand. We obtain a precise description of the completely join-irreducible (resp., completely meet-irreducible) elements of Reg(e) and the arrow relations between them. In particular, we prove that (1) Reg(e) is the Dedekind-MacNeille completion of the poset Clop(e); (2) Every open subset of e is a set-theoretic union of completely join-irreducible clopen subsets of e; (3) Clop(e) is a lattice iiff every regular closed subset of e is clopen, iff e contains no "square" configuration, iff Reg(e)=Clop(e); (4) If e is finite, then Reg(e) is pseudocomplemented iff it is semidistributive, iff it is a bounded homomorphic image of a free lattice, iff e is a disjoint sum of antisymmetric transitive relations and two-element full relations. We illustrate the strength of our results by proving that, for n greater than or equal to 3, the congruence lattice of the lattice Bip(n) of all bipartitions of an n-element set is obtained by adding a new top element to a Boolean lattice with n2^{n-1} atoms. We also determine the factors of the minimal subdirect decomposition of Bip(n).Comment: 25 page

Optimum basis of finite convex geometry

Convex geometries form a subclass of closure systems with unique criticals, or UC-systems. We show that the F-basis introduced in [6] for UC- systems, becomes optimum in convex geometries, in two essential parts of the basis: right sides (conclusions) of binary implications and left sides (premises) of non-binary ones. The right sides of non-binary implications can also be optimized, when the convex geometry either satis es the Carousel property, or does not have D-cycles. The latter generalizes a result of P.L. Hammer and A. Kogan for acyclic Horn Boolean functions. Convex geometries of order convex subsets in a poset also have tractable optimum basis. The problem of tractability of optimum basis in convex geometries in general remains to be ope

Optimum basis of finite convex geometry

Convex geometries form a subclass of closure systems with unique criticals, or UC-systems. We show that the F-basis introduced in [6] for UC- systems, becomes optimum in convex geometries, in two essential parts of the basis: right sides (conclusions) of binary implications and left sides (premises) of non-binary ones. The right sides of non-binary implications can also be optimized, when the convex geometry either satis es the Carousel property, or does not have D-cycles. The latter generalizes a result of P.L. Hammer and A. Kogan for acyclic Horn Boolean functions. Convex geometries of order convex subsets in a poset also have tractable optimum basis. The problem of tractability of optimum basis in convex geometries in general remains to be ope

Lattice structure of Grassmann-Tamari orders

The Tamari order is a central object in algebraic combinatorics and many other areas. Defined as the transitive closure of an associativity law, the Tamari order possesses a surprisingly rich structure: it is a congruence-uniform lattice. In this work, we consider a larger class of posets, the Grassmann-Tamari orders, which arise as an ordering on the facets of the non-crossing complex introduced by Pylyavskyy, Petersen, and Speyer. We prove that the Grassmann-Tamari orders are congruence-uniform lattices, which resolves a conjecture of Santos, Stump, and Welker. Towards this goal, we define a closure operator on sets of paths inside a rectangle, and prove that the biclosed sets of paths, ordered by inclusion, form a congruence-uniform lattice. We then prove that the Grassmann-Tamari order is a quotient lattice of the corresponding lattice of biclosed sets

LATTICES OF REGULAR CLOSED SUBSETS OF CLOSURE SPACES

Abstract. For a closure space (P,ϕ) with ϕ(∅) = ∅, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg(P,ϕ), extending the poset Clop(P,ϕ) of all clopen subsets. If (P,ϕ) is a finite convex geometry, then Reg(P,ϕ) is pseudocomplemented. The Dedekind-MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained inthisway, hence itispseudocomplemented. The lattice Reg(P,ϕ) carries a particularly interesting structure for special types of convex geometries, that we call closure spaces of semilattice type. For finite such closure spaces, — Reg(P,ϕ) satisfies an infinite collection of stronger and stronger quasiidentities, weaker than both meet- and join-semidistributivity. Nevertheless it may fail semidistributivity. — If Reg(P,ϕ) is semidistributive, then it is a bounded homomorphic image of a free lattice. — Clop(P,ϕ) is a lattice iff every regular closed set is clopen. The extended permutohedron R(G) on a graph G, and the extended permutohedron RegS on a join-semilattice S, are both defined as lattices of regular closed sets of suitable closure spaces. While the lattice of regular closed sets is, in the semilattice context, always the Dedekind Mac-Neille completion of the poset of clopen sets, this does not always hold in the graph context, although it always does so for finite block graphs and for cycles. Furthermore, both R(G) and RegS are bounded homomorphic images of free lattices