Acyclic Sourcing Poset in Hypergraphs

• Acyclic sourcing poset is a subposet of the sourcing poset of a hypergraph, defined by selecting source elements that avoid hypercycles via coordinatewise comparisons.
• The hypergraphic polytope, formed as a Minkowski sum of simplices, provides a geometric realization where its oriented 1-skeleton reveals acyclic orientations and increasing flips.
• Related constructions in valued-digraphs and k-fold acyclic simplicial complexes illustrate how acyclicity drives combinatorial decompositions and lattice properties.

Searching arXiv for relevant papers on acyclic sourcing posets and related hypergraphic posets. An acyclic sourcing poset is, in the hypergraphic setting, the induced subposet

$$\mathrm{ASour}(\mathcal H)\subseteq \mathrm{Sour}(\mathcal H)$$

on the acyclic sourcings of a hypergraph $\mathcal H$ on $[n]$. A sourcing is a map $S:\mathcal H\to [n]$ such that $S(H)\in H$ for every hyperedge $H\in\mathcal H$, and $\mathrm{Sour}(\mathcal H)$ is the product of chains $\prod_{H\in\mathcal H} H$, ordered coordinatewise. Acyclicity excludes hypercycles of the form $H_0\to H_1\to \cdots \to H_k=H_0$ with $S(H_{i-1})\in H_i\setminus\{S(H_i)\}$ for all $\mathcal H$0. Through the hypergraphic polytope $\mathcal H$1, this poset is realized as the transitive closure of an oriented $\mathcal H$2-skeleton, and recent work shows that its order is determined by coordinatewise comparison of source choices for arbitrary hypergraphs (Gélinas, 21 Aug 2025, Abram et al., 3 Aug 2025). Closely related usages also appear in Viard’s valued-digraph construction and in the face-poset theory of $\mathcal H$3-fold acyclic simplicial complexes (Viard, 2015, Doolittle et al., 2018).

1. Hypergraphic definition and basic order structure

Fix $\mathcal H$4 and let $\mathcal H$5 be a hypergraph, always assumed to contain all singletons $\mathcal H$6. For each hyperedge $\mathcal H$7, a sourcing chooses one distinguished element $\mathcal H$8. The ambient sourcing poset is

$\mathcal H$9

with order

$[n]$0

A sourcing is acyclic if there is no hypercycle $[n]$1 with $[n]$2 for every $[n]$3. The acyclic sourcing poset is the induced subposet on these acyclic sourcings: $[n]$4

An equivalent orientation language is standard. An orientation of $[n]$5 is an assignment $[n]$6 with $[n]$7, where $[n]$8 is the source of the hyperedge $[n]$9. From $S:\mathcal H\to [n]$0, one draws a directed graph on $S:\mathcal H\to [n]$1 having, for each hyperedge $S:\mathcal H\to [n]$2, all arcs

$S:\mathcal H\to [n]$3

Then $S:\mathcal H\to [n]$4 is acyclic exactly when this directed graph has no directed cycle. In this form, the acyclic sourcing poset is also described as the hypergraphic poset (Gélinas, 21 Aug 2025, Abram et al., 3 Aug 2025).

A first structural point is that $S:\mathcal H\to [n]$5 is an induced subposet of a product of chains, not a priori a lattice. The ambient coordinatewise order is simple, but the acyclicity constraint removes source-vectors that create directed cycles. This distinction is central in later lattice-theoretic results.

2. Hypergraphic polytopes and source characterization

For any hyperedge $S:\mathcal H\to [n]$6, write

$S:\mathcal H\to [n]$7

where $S:\mathcal H\to [n]$8 is the $S:\mathcal H\to [n]$9th standard basis vector. The hypergraphic polytope of $S(H)\in H$0 is the Minkowski sum

$S(H)\in H$1

Since each singleton edge $S(H)\in H$2 contributes only a translate of $S(H)\in H$3, singletons do not affect the combinatorial type of $S(H)\in H$4 and are usually ignored in the polyhedral discussion.

The $S(H)\in H$5-skeleton of $S(H)\in H$6 is oriented by the generic linear functional

$S(H)\in H$7

Each edge is oriented from $S(H)\in H$8 to $S(H)\in H$9 when $H\in\mathcal H$0. The resulting oriented graph is acyclic, and its transitive closure is isomorphic to the Hasse diagram of $H\in\mathcal H$1. In parallel language, one defines the hypergraphic poset $H\in\mathcal H$2 as the transitive closure of the oriented skeleton of $H\in\mathcal H$3; equivalently, the vertices of $H\in\mathcal H$4 are in bijection with certain acyclic orientations of $H\in\mathcal H$5, and the edges correspond to elementary increasing flips among those orientations (Abram et al., 3 Aug 2025, Gélinas, 21 Aug 2025).

The source characterization theorem sharpens this picture. Bergeron–Pilaud had shown, for interval hypergraphs, that two acyclic orientations $H\in\mathcal H$6 satisfy

$H\in\mathcal H$7

Gélinas extends this to arbitrary hypergraphs: if $H\in\mathcal H$8 is any hypergraph on $H\in\mathcal H$9, and $\mathrm{Sour}(\mathcal H)$0 are acyclic orientations of $\mathrm{Sour}(\mathcal H)$1, then

$\mathrm{Sour}(\mathcal H)$2

In the formulation given in the paper, this means that in every hypergraphic poset, the cover-relations can be read off by a simple coordinate-wise comparison of source-vectors (Gélinas, 21 Aug 2025).

This removes the interval hypothesis from the order-theoretic description. A common misconception was that coordinatewise comparison was a phenomenon tied to interval hypergraphs; the extension shows that it is in fact intrinsic to every hypergraphic poset.

3. Increasing flips and the proof architecture

The local moves in the hypergraphic picture are increasing flips. If $\mathrm{Sour}(\mathcal H)$3 and $\mathrm{Sour}(\mathcal H)$4 are two acyclic orientations, they differ by an increasing flip along a transposition $\mathrm{Sour}(\mathcal H)$5 precisely when, for every hyperedge $\mathrm{Sour}(\mathcal H)$6 on which they disagree, $\mathrm{Sour}(\mathcal H)$7 and $\mathrm{Sour}(\mathcal H)$8, and moreover no hyperedge containing $\mathrm{Sour}(\mathcal H)$9 “hides” one orientation inside the other. These flips are exactly the oriented edges of $\prod_{H\in\mathcal H} H$0.

The proof of the general source characterization proceeds by reducing arbitrary comparable pairs $\prod_{H\in\mathcal H} H$1 to coherent local flips. Lemma 3.3 shows that a flip $\prod_{H\in\mathcal H} H$2 preserves acyclicity exactly when, in the old orientation, there is no long “non-edge” path from $\prod_{H\in\mathcal H} H$3 to $\prod_{H\in\mathcal H} H$4. Among the hyperedges $\prod_{H\in\mathcal H} H$5 for which $\prod_{H\in\mathcal H} H$6, one selects a “small” or “minimized” hyperedge. If the corresponding flip is not coherent, the source path algorithm follows a combinatorial path $\prod_{H\in\mathcal H} H$7 in the oriented graph of $\prod_{H\in\mathcal H} H$8, climbing through successively refined hyperedges until eventually one finds a hyperedge admitting a coherent flip. Induction on the total distance

$\prod_{H\in\mathcal H} H$9

then yields a finite path from $H_0\to H_1\to \cdots \to H_k=H_0$0 to $H_0\to H_1\to \cdots \to H_k=H_0$1 (Gélinas, 21 Aug 2025).

The basic non-interval example used in the exposition is

$H_0\to H_1\to \cdots \to H_k=H_0$2

The $H_0\to H_1\to \cdots \to H_k=H_0$3-skeleton of $H_0\to H_1\to \cdots \to H_k=H_0$4 has eight acyclic orientations as vertices. In every covering relation $H_0\to H_1\to \cdots \to H_k=H_0$5, the sources differ in exactly one coordinate by increasing that entry by $H_0\to H_1\to \cdots \to H_k=H_0$6, and one checks directly that

$H_0\to H_1\to \cdots \to H_k=H_0$7

Geometrically, the argument exploits that $H_0\to H_1\to \cdots \to H_k=H_0$8 is a generalized permutahedron, so its vertices are naturally in bijection with acyclic orientations and its directed edges with increasing flips (Gélinas, 21 Aug 2025).

4. Path hypergraphs, ornamentations, and lattice criteria

Let $H_0\to H_1\to \cdots \to H_k=H_0$9 be a directed graph on vertex set $S(H_{i-1})\in H_i\setminus\{S(H_i)\}$0, and let $S(H_{i-1})\in H_i\setminus\{S(H_i)\}$1 be its path-hypergraph, namely the collection of vertex-sets of all directed paths in $S(H_{i-1})\in H_i\setminus\{S(H_i)\}$2. When $S(H_{i-1})\in H_i\setminus\{S(H_i)\}$3, one writes $S(H_{i-1})\in H_i\setminus\{S(H_i)\}$4. In general, $S(H_{i-1})\in H_i\setminus\{S(H_i)\}$5 need not be a lattice.

The special case of increasing trees is substantially better behaved. A directed tree $S(H_{i-1})\in H_i\setminus\{S(H_i)\}$6 on $S(H_{i-1})\in H_i\setminus\{S(H_i)\}$7 is called unstarred if there do not exist two vertices $S(H_{i-1})\in H_i\setminus\{S(H_i)\}$8 with $S(H_{i-1})\in H_i\setminus\{S(H_i)\}$9 having at least $\mathcal H$00 incoming edges, $\mathcal H$01 having at least $\mathcal H$02 outgoing edges, and a directed path $\mathcal H$03 in $\mathcal H$04. If $\mathcal H$05 is an unstarred increasing tree on $\mathcal H$06, then the acyclic reorientation poset $\mathcal H$07, the acyclic sourcing poset $\mathcal H$08, and the acyclic ornamentation poset $\mathcal H$09 are all lattices. Moreover, every ornamentation of $\mathcal H$10 is automatically acyclic, so

$\mathcal H$11

and the natural map of oriented graphs

$\mathcal H$12

is a surjective lattice homomorphism (Abram et al., 3 Aug 2025).

An ornamentation of a directed graph $\mathcal H$13 on $\mathcal H$14 is an assignment

$\mathcal H$15

such that $\mathcal H$16 is a rooted subset containing $\mathcal H$17 and closed under paths in $\mathcal H$18, and if $\mathcal H$19 then $\mathcal H$20. The map $\mathcal H$21 from sourcings of $\mathcal H$22 to ornamentations of $\mathcal H$23 is an order-preserving surjection

$\mathcal H$24

and it restricts to a bijection

$\mathcal H$25

For any increasing tree $\mathcal H$26, the ornamentation lattice $\mathcal H$27 is the MacNeille completion of $\mathcal H$28. More generally, if $\mathcal H$29 is an intreeval hypergraph, then $\mathcal H$30 is a lattice if and only if $\mathcal H$31 is path-intersection-closed and star-sparse. In particular, the lattice property is not automatic; it is governed by explicit combinatorial closure and sparsity conditions (Abram et al., 3 Aug 2025).

5. The valued-digraph “acyclic-sourcing” construction

A distinct use of the phrase “acyclic-sourcing” appears in Viard’s construction of a poset $\mathcal H$32 from a simple acyclic directed graph together with a valuation on its vertices. Let $\mathcal H$33 be a simple acyclic directed graph, and let $\mathcal H$34. An out-degree-compatible valuation is a map $\mathcal H$35 satisfying

$\mathcal H$36

A vertex $\mathcal H$37 is erasable in $\mathcal H$38 if $\mathcal H$39 and, for every $\mathcal H$40 with $\mathcal H$41, one has $\mathcal H$42. Starting from $\mathcal H$43, one repeatedly chooses an erasable vertex, removes it and all incident arcs, and decrements $\mathcal H$44 by $\mathcal H$45 for each arc $\mathcal H$46 deleted. This produces a peeling sequence

$\mathcal H$47

The collection of all initial sections of all peeling sequences,

$\mathcal H$48

ordered by inclusion, is the poset

$\mathcal H$49

Several structural properties are explicit. An $\mathcal H$50 lies in $\mathcal H$51 if and only if, for every $\mathcal H$52,

$\mathcal H$53

and for every $\mathcal H$54,

$\mathcal H$55

The poset has unique minimum $\mathcal H$56 and rank $\mathcal H$57. Its Möbius function satisfies, for intervals $\mathcal H$58,

$\mathcal H$59

where

$\mathcal H$60

This construction recovers several weak orders. For type $\mathcal H$61, one obtains the right weak order on $\mathcal H$62; for affine type $\mathcal H$63, one obtains the right weak order on the affine Coxeter group of type $\mathcal H$64; and for the wreath product $\mathcal H$65, one obtains the flag weak order of Adin–Brenti–Roichman. Associated quasi-symmetric generating functions recover Stanley’s symmetric function in type $\mathcal H$66 and Lam’s affine Stanley symmetric function in affine type $\mathcal H$67 (Viard, 2015).

6. Face posets of $\mathcal H$68-fold acyclic simplicial complexes

In a different acyclicity-driven setting, an acyclic sourcing poset is identified with the face poset of a simplicial complex whose links satisfy higher-order acyclicity conditions. A simplicial complex $\mathcal H$69 on a finite ground set $\mathcal H$70 is a family of subsets of $\mathcal H$71 closed under inclusion, and its face poset $\mathcal H$72 is the set of all faces $\mathcal H$73 ordered by inclusion, with rank

$\mathcal H$74

If $\mathcal H$75 is a field, $\mathcal H$76 is $\mathcal H$77-fold acyclic if $\mathcal H$78 for all $\mathcal H$79, and $\mathcal H$80 is $\mathcal H$81-fold acyclic if for every face $\mathcal H$82 with $\mathcal H$83, the link

$\mathcal H$84

is acyclic over $\mathcal H$85. Equivalently, every induced subcomplex of $\mathcal H$86 on fewer than $\mathcal H$87 vertices is acyclic.

Stanley proved in 1993 that if $\mathcal H$88 is acyclic, then $\mathcal H$89 can be decomposed into disjoint rank-$\mathcal H$90 Boolean intervals whose minimal faces form a subcomplex, and conjectured the higher-order analogue: if $\mathcal H$91 is $\mathcal H$92-fold acyclic, then

$\mathcal H$93

with each interval $\mathcal H$94 and $\mathcal H$95 itself a subcomplex of $\mathcal H$96. The conjecture is false in general. The construction begins with a relative complex $\mathcal H$97,

$\mathcal H$98

for which $\mathcal H$99 are both $[n]$00-fold acyclic but $[n]$01 cannot be partitioned into rank-$[n]$02 Boolean intervals. By thickening a $[n]$03-ball $[n]$04 and gluing in six copies of $[n]$05 along six copies of $[n]$06, and then applying the gluing theorem, one obtains a $[n]$07-fold acyclic complex $[n]$08 with

$[n]$09

yet with no decomposition into rank-$[n]$10 Boolean intervals (Doolittle et al., 2018).

The conjecture can be repaired by replacing Boolean intervals with Boolean trees. A Boolean tree of rank $[n]$11 is defined recursively: a single element is a Boolean tree of rank $[n]$12, and if $[n]$13 and $[n]$14 are two disjoint Boolean trees of rank $[n]$15 with unique minima $[n]$16 and $[n]$17 covers $[n]$18 in $[n]$19, then $[n]$20 is a Boolean tree of rank $[n]$21. The revised theorem states that if $[n]$22 is $[n]$23-fold acyclic, then

$[n]$24

where each $[n]$25 is a Boolean tree of rank $[n]$26, and the set of minima forms a subcomplex $[n]$27. The proof uses exterior algebraic shifting, the Duval–Zhang iterated-homology decomposition, and successive mergers of lower-rank trees.

At maximal acyclicity the original interval statement returns. If $[n]$28 is $[n]$29-dimensional and $[n]$30-fold acyclic, then $[n]$31 is a stacked complex; in particular, $[n]$32 admits a decomposition into disjoint Boolean intervals all of rank $[n]$33, whose minima form a subcomplex. The argument passes through purity, connectedness of the facet–ridge graph, and a stacked shelling. These results give a precise picture of how homological acyclicity controls the combinatorial structure of the face poset: in full generality, $[n]$34-fold acyclicity guarantees a tree-like decomposition into rank-$[n]$35 Boolean trees, while Boolean-interval decompositions can fail in moderate dimension and reappear when $[n]$36 (Doolittle et al., 2018).

A plausible implication of these parallel developments is that “acyclic sourcing poset” names a broader family of acyclicity-controlled partial orders rather than a single uniform object. In the hypergraphic case, acyclicity is encoded by source assignments and generalized permutahedra; in the valued-digraph case, by erasability under peeling; and in the simplicial-complex case, by face-poset decompositions governed by link acyclicity. Across all three settings, the recurring theme is that homological or combinatorial acyclicity imposes strong order-theoretic structure, but does not in general force lattice or Boolean-interval behavior without additional hypotheses.