Hypergraphic Poset: Polyhedral & Lattice Theory

• Hypergraphic posets are defined as partial orders on acyclic orientations derived from hypergraphic polytopes constructed via Minkowski sums.
• They encompass classical models such as the weak Bruhat order and Tamari lattice by interpreting hypergraph intervals and cyclic arrangements.
• Lattice criteria based on sourcing and intersection properties guide when these posets become distributive, semidistributive, or full lattices.

A hypergraphic poset is the partial order $P_{\mathbb H}$ attached to a hypergraph $\mathbb H$ on $[n]=\{1,2,\dots,n\}$ by orienting the $1$-skeleton of its hypergraphic polytope $\Delta_{\mathbb H}=\sum_{H\in\mathbb H}\Delta_H$, where $\Delta_H=\operatorname{conv}\{e_h:h\in H\}$, with a generic linear functional $\omega$. Its elements can be described as acyclic orientations, or source assignments, on the hyperedges, and its order is generated by $\omega$-increasing flips. The construction subsumes several classical posets: the weak Bruhat order on the permutahedron, the Tamari lattice on the associahedron, and, in the cyclic-interval setting, the cyclohedron poset of maximal tubings on the cycle (Bergeron et al., 2024, Gélinas et al., 5 May 2026).

1. Polyhedral definition

Fix a hypergraph $\mathcal H\subseteq 2^{[n]}$, usually with all singletons $\{i\}\in\mathcal H$. For each hyperedge $\mathbb H$0, the simplex

$\mathbb H$1

is the standard simplex on the corresponding coordinate basis vectors, and the hypergraphic polytope is the Minkowski sum

$\mathbb H$2

This polytope is a deformed permutahedron, and singletons affect only translation, not the combinatorics (Gélinas, 21 Aug 2025).

A standard choice of linear functional is

$\mathbb H$3

although the construction is formulated for a generic $\mathbb H$4. Every edge of the $\mathbb H$5-skeleton of $\mathbb H$6 is oriented in the direction of increasing $\mathbb H$7-value, producing an acyclic digraph on the vertices of $\mathbb H$8. The hypergraphic poset $\mathbb H$9 is then the transitive closure of this oriented $[n]=\{1,2,\dots,n\}$0-skeleton; equivalently, its Hasse diagram is the transitive reduction of the same orientation (Gélinas, 21 Aug 2025).

This polyhedral definition is fundamental because it places order-theoretic questions inside the geometry of generalized permutahedra. In particular, the central structural problem is not whether the orientation exists—it always does for generic $[n]=\{1,2,\dots,n\}$1—but when the resulting acyclic digraph defines a lattice.

2. Acyclic orientations, sourcings, and coordinate order

An orientation of a hypergraph $[n]=\{1,2,\dots,n\}$2 is a choice of a source in each hyperedge,

$[n]=\{1,2,\dots,n\}$3

Equivalently, for every $[n]=\{1,2,\dots,n\}$4, one draws an arrow $[n]=\{1,2,\dots,n\}$5. The orientation is acyclic if the resulting directed graph on $[n]=\{1,2,\dots,n\}$6 has no directed cycle. Vertices of $[n]=\{1,2,\dots,n\}$7 are indexed by these acyclic orientations, and $[n]=\{1,2,\dots,n\}$8-increasing edges in the polytope correspond to increasing flips, in which the source of a collection of covering hyperedges changes from $[n]=\{1,2,\dots,n\}$9 to $1$0 with $1$1 while preserving acyclicity (Gélinas, 21 Aug 2025).

The same structure can be phrased as a sourcing problem. A sourcing $1$2 of a hypergraph chooses $1$3 for each hyperedge $1$4, and the set of all sourcings is ordered componentwise: $1$5 All sourcings therefore form the product of chains $1$6. The acyclic sourcing poset $1$7 is the induced subposet on acyclic sourcings; in general, $1$8 need not be a lattice (Abram et al., 3 Aug 2025).

A major structural result is the source characterization for arbitrary hypergraphs: for any two acyclic orientations $1$9 of $\Delta_{\mathbb H}=\sum_{H\in\mathbb H}\Delta_H$0,

$\Delta_{\mathbb H}=\sum_{H\in\mathbb H}\Delta_H$1

Thus $\Delta_{\mathbb H}=\sum_{H\in\mathbb H}\Delta_H$2 is the coordinatewise order on the set of acyclic source-sequences, not merely a reachability order defined abstractly by flips (Gélinas, 21 Aug 2025). A plausible implication is that many lattice-theoretic questions about $\Delta_{\mathbb H}=\sum_{H\in\mathbb H}\Delta_H$3 reduce to asking when the set of acyclic source-sequences forms a sublattice of the ambient product of chains.

3. Classical models and standard examples

The hypergraphic-poset framework recovers several canonical combinatorial orders. These examples are central because they show that hypergraphic posets are not a niche generalization, but a common language for familiar polyhedral orders (Bergeron et al., 2024, Gélinas et al., 5 May 2026).

Hypergraph $\Delta_{\mathbb H}=\sum_{H\in\mathbb H}\Delta_H$4	Polytope $\Delta_{\mathbb H}=\sum_{H\in\mathbb H}\Delta_H$5	Poset $\Delta_{\mathbb H}=\sum_{H\in\mathbb H}\Delta_H$6
All $\Delta_{\mathbb H}=\sum_{H\in\mathbb H}\Delta_H$7-subsets $\Delta_{\mathbb H}=\sum_{H\in\mathbb H}\Delta_H$8	Permutahedron	Weak Bruhat order
All intervals $\Delta_{\mathbb H}=\sum_{H\in\mathbb H}\Delta_H$9	Stasheff associahedron / Loday’s associahedron	Tamari lattice
All regular and cyclic intervals	Bott–Taubes cyclohedron	Cyclo-lattice of maximal tubing on the cycle

For the complete-graph hypergraph $\Delta_H=\operatorname{conv}\{e_h:h\in H\}$0, the vertices of $\Delta_H=\operatorname{conv}\{e_h:h\in H\}$1 are permutations of $\Delta_H=\operatorname{conv}\{e_h:h\in H\}$2, the flips are adjacent transpositions that increase inversion number, and $\Delta_H=\operatorname{conv}\{e_h:h\in H\}$3 is the weak Bruhat order on $\Delta_H=\operatorname{conv}\{e_h:h\in H\}$4. In source-sequence terms, each edge $\Delta_H=\operatorname{conv}\{e_h:h\in H\}$5 is sent to $\Delta_H=\operatorname{conv}\{e_h:h\in H\}$6, which recovers the usual inversion-set description of weak order (Bergeron et al., 2024, Gélinas et al., 5 May 2026).

For the interval hypergraph $\Delta_H=\operatorname{conv}\{e_h:h\in H\}$7, $\Delta_H=\operatorname{conv}\{e_h:h\in H\}$8 is the associahedron, and acyclic orientations are in bijection with binary trees with $\Delta_H=\operatorname{conv}\{e_h:h\in H\}$9 leaves, or triangulations of an $\omega$0-gon. The resulting poset is the Tamari lattice. Concretely, each interval-edge $\omega$1 is oriented according to which among $\omega$2 or $\omega$3 appears first in the bracketed tree-walk around the polygon (Bergeron et al., 2024, Gélinas et al., 5 May 2026).

For the complete cyclic-interval hypergraph, containing all regular and cyclic intervals, $\omega$4 is the Bott–Taubes cyclohedron and $\omega$5 is the cyclo-lattice of maximal tubing on the cycle (Gélinas et al., 5 May 2026).

4. Lattice criteria in the interval and cyclic-interval regimes

The basic misconception to avoid is that a hypergraphic poset is automatically a lattice. Even in highly structured families, latticehood is a classification problem rather than a formal consequence of the definition.

For interval hypergraphs $\omega$6, Bergeron and Pilaud give a complete lattice-theoretic classification. The fundamental criterion is: $\omega$7 They further characterize when $\omega$8 is distributive, semidistributive, and a lattice quotient of weak order. In particular, distributivity requires an additional “initial/final” condition on overlaps; join-semidistributivity requires intersection closure together with a four-interval “diamond” condition; and the natural surjection $\omega$9 is a full lattice quotient exactly when $\omega$0 is closed under all subintervals, in which case $\omega$1 factors as a Cartesian product of Tamari lattices (Bergeron et al., 2024).

The cyclic-interval case extends this picture. An edge $\omega$2 is regular if $\omega$3 with $\omega$4 and $\omega$5, and cyclic if $\omega$6 for some $\omega$7. A cyclic-interval hypergraph is one whose hyperedges are all regular or cyclic, with all singletons included. For such a hypergraph $\omega$8, define for every interval $\omega$9

$\mathcal H\subseteq 2^{[n]}$0

Then $\mathcal H\subseteq 2^{[n]}$1 is a lattice if and only if, for every interval $\mathcal H\subseteq 2^{[n]}$2, two conditions hold: first, the collection of regular edges in $\mathcal H\subseteq 2^{[n]}$3 is closed under intersection; second, every hugging quadruple in $\mathcal H\subseteq 2^{[n]}$4 admits a fix. Here a hugging quadruple consists of $\mathcal H\subseteq 2^{[n]}$5 with $\mathcal H\subseteq 2^{[n]}$6 containing $\mathcal H\subseteq 2^{[n]}$7, $\mathcal H\subseteq 2^{[n]}$8 containing $\mathcal H\subseteq 2^{[n]}$9, and with $\{i\}\in\mathcal H$0 regular while $\{i\}\in\mathcal H$1 are cyclic, or vice versa. A fix is an edge $\{i\}\in\mathcal H$2 such that

$\{i\}\in\mathcal H$3

This gives a complete lattice characterization for cyclic-interval hypergraphs and extends both the interval result of Bergeron–Pilaud and the complete cyclic-interval result of Adenbaum et al. (Gélinas et al., 5 May 2026, Adenbaum et al., 10 Oct 2025).

The proof strategy in the cyclic-interval case is itself structurally informative. Necessity comes from restricting to $\{i\}\in\mathcal H$4 and analyzing minimal failures of join-existence inside hugging quadruples. Sufficiency is established by constructing, for any two acyclic orientations $\{i\}\in\mathcal H$5, a pseudo-join $\{i\}\in\mathcal H$6 by a regulated path of local “max-moves” on sources of edges, then proving that $\{i\}\in\mathcal H$7 is acyclic, dominates both $\{i\}\in\mathcal H$8 and $\{i\}\in\mathcal H$9, and is minimal with that property; meets are obtained symmetrically (Gélinas et al., 5 May 2026).

5. Related posets: path hypergraphs, ornamentations, and intreeval lattices

A distinct but closely related development studies hypergraphic posets through path hypergraphs of directed graphs. For a directed graph $\mathbb H$00 on $\mathbb H$01, the path hypergraph $\mathbb H$02 has as hyperedges the vertex-sets of all directed paths in $\mathbb H$03. When $\mathbb H$04 is acyclic and increasing, these hyperedges are intervals in the partial order induced by $\mathbb H$05 (Abram et al., 3 Aug 2025).

This perspective brings in two additional posets. The acyclic reorientation poset $\mathbb H$06 is defined for a directed acyclic graph $\mathbb H$07 by reversing or not reversing each arc, subject to acyclicity. The ornamentation poset $\mathbb H$08 consists of compatible choices of ornaments $\mathbb H$09, where an ornament at $\mathbb H$10 is a subset $\mathbb H$11 such that, in the induced subgraph on $\mathbb H$12, every $\mathbb H$13 has a directed path to $\mathbb H$14. Ornamentations are ordered componentwise by inclusion and form a lattice. The paper exhibits commutative squares of order-preserving surjections linking weak order on $\mathbb H$15, $\mathbb H$16, $\mathbb H$17, and $\mathbb H$18 (Abram et al., 3 Aug 2025).

For rooted, or more generally unstarred increasing trees, the acyclic sourcing poset of the path hypergraph is isomorphic to the ornamentation lattice: $\mathbb H$19 and this lattice is a lattice quotient of $\mathbb H$20. For any increasing tree, the ornamentation lattice is the MacNeille completion of $\mathbb H$21 (Abram et al., 3 Aug 2025). This gives polytopal realizations of ornamentation lattices and answers an open question of C. Defant and A. Sack.

The same work characterizes which subhypergraphs $\mathbb H$22 of a path hypergraph of an increasing tree $\mathbb H$23 again yield lattices. Such hypergraphs are called intreeval hypergraphs. The criterion is: $\mathbb H$24 This recovers the interval-hypergraph criterion on a path as a special case (Abram et al., 3 Aug 2025).

6. Structural significance and directions

The modern view of hypergraphic posets combines polyhedral realization, source-sequence combinatorics, and lattice theory. On the polyhedral side, hypergraphic polytopes realize acyclic sourcing posets through linear orientation, and graphical zonotopes realize the analogous reorientation posets (Abram et al., 3 Aug 2025). On the order-theoretic side, the coordinatewise source characterization shows that the difficulty lies not in defining the order, but in understanding the structure of the acyclic region inside the ambient product of chains (Gélinas, 21 Aug 2025).

This viewpoint clarifies why classical lattices recur. Many familiar lattices appear as $\mathbb H$25 when $\mathbb H$26 ranges over building-sets, nestohedra, graphical zonotopes, and hyperplane arrangement galleries (Gélinas et al., 5 May 2026). It also explains negative examples. The acyclic sourcing poset need not be a lattice in general; the freehedron hypergraph fails even to be a lattice in the interval setting, and a small starred tree produces ornamentations while the corresponding acyclic sourcing poset has a diamond and is not a lattice (Bergeron et al., 2024, Abram et al., 3 Aug 2025).

Several open directions are explicitly identified. Beyond the interval case, one may ask for a full classification of hypergraphs $\mathbb H$27 for which $\mathbb H$28 is semidistributive, congruence-uniform, or distributive. A second question is which hypergraphic posets arise as lattice quotients of weak order outside the interval case. A third concerns the geometry of $\mathbb H$29: characterizing those hypergraphic polytopes that are “nice” generalized permutahedra with lattice $\mathbb H$30-skeletons (Bergeron et al., 2024).

Taken together, these results place the hypergraphic poset at the intersection of generalized permutahedra, acyclic orientation theory, weak-order quotients, and lattice completions. The interval and cyclic-interval classifications provide precise test cases, while the source characterization for arbitrary hypergraphs suggests that the general theory is governed by the combinatorics of allowable source-sequences rather than by the geometry alone (Gélinas, 21 Aug 2025, Gélinas et al., 5 May 2026).