Poset of Regions in Hyperplane Arrangements

• Poset of Regions is a partial order defined on the chambers of a hyperplane arrangement by comparing the hyperplanes that separate them from a fixed base region.
• It structures arrangement theory by yielding graded orders, establishing canonical links to acyclic orientations and embedding into complete ortholattices via Dedekind–MacNeille completions.
• Applications extend to graphic, Shi, and toric arrangements, providing concrete models that connect combinatorial properties with geometric and lattice-theoretic frameworks.

Searching arXiv for papers on posets of regions, hyperplane arrangements, and related completions. The poset of regions is the partial order obtained from a hyperplane arrangement after choosing a base region and comparing regions by inclusion of the hyperplanes that separate them from that base. In arrangement theory this construction organizes chambers of real arrangements, yields canonical graded orders such as the weak order for the braid arrangement, and, for central arrangements, admits a Dedekind–MacNeille completion described by lattices of regular closed sets. Related work also uses specialized posets to encode regions of Shi arrangements inside fixed Coxeter cones, and develops toric analogues in which regions correspond not to single acyclic orientations but to source-to-sink flip classes (Develin et al., 2012, Santocanale et al., 2013, Meszaros, 2011).

1. Base-region order in ordinary hyperplane arrangements

For a finite hyperplane arrangement $\mathcal{H}\subset \mathbb{R}^n$, a region is a connected component of

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$

If a base region $R_0$ is fixed, the separation set $\mathrm{Sep}(R,R_0)$ consists of the hyperplanes of $\mathcal{H}$ that separate $R$ from $R_0$. The poset of regions $PR(\mathcal{A},R_0)$ is then defined by

$$R\le R' \quad \text{if} \quad \mathrm{Sep}(R,R_0)\subseteq \mathrm{Sep}(R',R_0),$$

with cover relations $R\mathbin{\cdot\!\!\!<}R'$ when $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$0 and $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$1 are adjacent and $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$2 for a single hyperplane $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$3. This order is graded by

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$4

In the central case considered for arrangements $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$5 in $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$6, one writes $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$7 for the set of regions and, after choosing a base region $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$8, defines

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$9

and

$R_0$0

The resulting ordered set is denoted $R_0$1 (Develin et al., 2012, Santocanale et al., 2013).

For graphic arrangements this order has a standard combinatorial model. If $R_0$2 is a finite simple graph, the graphic hyperplane arrangement is

$R_0$3

A point $R_0$4 in the complement determines an acyclic orientation $R_0$5 by directing $R_0$6 as $R_0$7 if $R_0$8. This gives a bijection between chambers and acyclic orientations, and Greene–Zaslavsky and Stanley showed that

$R_0$9

where $\mathrm{Sep}(R,R_0)$0 is the Tutte polynomial. For the complete graph $\mathrm{Sep}(R,R_0)$1, $\mathrm{Sep}(R,R_0)$2 is the braid arrangement; its regions are indexed by permutations $\mathrm{Sep}(R,R_0)$3 via

$\mathrm{Sep}(R,R_0)$4

and $\mathrm{Sep}(R,R_0)$5 is isomorphic to the right weak order on $\mathrm{Sep}(R,R_0)$6 (Develin et al., 2012).

2. Central arrangements and Dedekind–MacNeille completion

For a central hyperplane arrangement $\mathrm{Sep}(R,R_0)$7 in $\mathrm{Sep}(R,R_0)$8, the poset $\mathrm{Sep}(R,R_0)$9 admits a completion described in terms of closure spaces. A closure operator on a set $\mathcal{H}$0 is an extensive, idempotent, isotone map $\mathcal{H}$1 with $\mathcal{H}$2. Its associated kernel operator is

$\mathcal{H}$3

A subset $\mathcal{H}$4 is closed if $\mathcal{H}$5, open if $\mathcal{H}$6, regular closed if $\mathcal{H}$7, regular open if $\mathcal{H}$8, and clopen if it is simultaneously closed and open. The regular closed subsets form the complete lattice $\mathcal{H}$9, and a subset is regular closed iff it is the closure of some open set. The orthogonal operation

$R$0

defines an orthocomplementation of $R$1, so $R$2 is an ortholattice (Santocanale et al., 2013).

In the arrangement-theoretic application, one chooses, for each $R$3, a vector $R$4 on the same side of $R$5 as the base region $R$6, normalized by $R$7 for a fixed $R$8. With

$R$9

and for a region $R_0$0,

$R_0$1

the assignment $R_0$2 is an order-isomorphism from $R_0$3 onto the set $R_0$4 of strongly bi-convex subsets of $R_0$5, namely those $R_0$6 such that $R_0$7. Here $R_0$8 is the relative convex hull operator on $R_0$9, and $PR(\mathcal{A},R_0)$0 is an atomistic convex geometry.

The decisive structural statement is Theorem 6.7: $PR(\mathcal{A},R_0)$1 is the Dedekind–MacNeille completion of $PR(\mathcal{A},R_0)$2 via the embedding $PR(\mathcal{A},R_0)$3. Because finite convex geometries yield pseudocomplemented lattices of regular closed sets, the Dedekind–MacNeille completion of $PR(\mathcal{A},R_0)$4 is pseudocomplemented. Thus the ordinary poset of regions of a central arrangement sits canonically inside a complete ortholattice with an explicitly defined orthocomplementation and with pseudocomplementation guaranteed by convex-geometric finiteness (Santocanale et al., 2013).

3. Shi arrangements and local posets encoding regions

In the study of Shi arrangements, the phrase “poset of regions” must be distinguished from a different construction that encodes regions inside a fixed Coxeter cone by antichains. For type $PR(\mathcal{A},R_0)$5, the Coxeter arrangement is

$PR(\mathcal{A},R_0)$6

with dominant cone

$PR(\mathcal{A},R_0)$7

and cones $PR(\mathcal{A},R_0)$8 indexed by permutations $PR(\mathcal{A},R_0)$9. The Shi arrangement used there is

$$R\le R' \quad \text{if} \quad \mathrm{Sep}(R,R_0)\subseteq \mathrm{Sep}(R',R_0),$$0

whose number of regions is

$$R\le R' \quad \text{if} \quad \mathrm{Sep}(R,R_0)\subseteq \mathrm{Sep}(R',R_0),$$1

For $$R\le R' \quad \text{if} \quad \mathrm{Sep}(R,R_0)\subseteq \mathrm{Sep}(R',R_0),$$2, the relevant poset is

$$R\le R' \quad \text{if} \quad \mathrm{Sep}(R,R_0)\subseteq \mathrm{Sep}(R',R_0),$$3

ordered by

$$R\le R' \quad \text{if} \quad \mathrm{Sep}(R,R_0)\subseteq \mathrm{Sep}(R',R_0),$$4

Antichains in $$R\le R' \quad \text{if} \quad \mathrm{Sep}(R,R_0)\subseteq \mathrm{Sep}(R',R_0),$$5 are exactly the sets of ceilings, equivalently floors, of regions of $$R\le R' \quad \text{if} \quad \mathrm{Sep}(R,R_0)\subseteq \mathrm{Sep}(R',R_0),$$6 contained in $$R\le R' \quad \text{if} \quad \mathrm{Sep}(R,R_0)\subseteq \mathrm{Sep}(R',R_0),$$7. Consequently,

$$R\le R' \quad \text{if} \quad \mathrm{Sep}(R,R_0)\subseteq \mathrm{Sep}(R',R_0),$$8

where $$R\le R' \quad \text{if} \quad \mathrm{Sep}(R,R_0)\subseteq \mathrm{Sep}(R',R_0),$$9 is the number of antichains of $R\mathbin{\cdot\!\!\!<}R'$0 (Meszaros, 2011).

The same pattern persists in type $R\mathbin{\cdot\!\!\!<}R'$1. The type $R\mathbin{\cdot\!\!\!<}R'$2 Shi arrangement is

$R\mathbin{\cdot\!\!\!<}R'$3

and its number of regions is

$R\mathbin{\cdot\!\!\!<}R'$4

For a signed permutation $R\mathbin{\cdot\!\!\!<}R'$5, one defines

$R\mathbin{\cdot\!\!\!<}R'$6

again with interval order

$R\mathbin{\cdot\!\!\!<}R'$7

The regions of $R\mathbin{\cdot\!\!\!<}R'$8 lying in $R\mathbin{\cdot\!\!\!<}R'$9 are in bijection with the antichains of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$00, so

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$01

These local posets support refined bijections. In type $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$02, regions of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$03 correspond to parking functions; in type $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$04, regions of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$05 correspond to sequences $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$06 with $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$07. The ceiling and floor statistics satisfy

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$08

and

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$09

A standard misconception is that the posets $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$10 and $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$11 are global posets of regions. They are not: they are specialized local encoding devices whose antichains recover the sets of ceilings or floors for regions inside a fixed Coxeter cone (Meszaros, 2011).

4. Graphic arrangements, chamber posets, and toric analogues

In the ordinary graphic setting, every chamber determines an acyclic orientation and hence an ordinary poset. For a chamber $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$12, the associated orientation $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$13 defines a poset $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$14 by taking the transitive closure of the relation “$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$15 precedes $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$16 along a directed path in $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$17.” Chains in $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$18 correspond to coordinate inequalities $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$19 valid on $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$20, the Hasse diagram is obtained by deleting redundant transitive edges, and linear extensions of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$21 index braid chambers whose union closure equals the closure of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$22. This is the standard combinatorial content carried by regions of graphic arrangements (Develin et al., 2012).

The toric theory modifies this picture by passing from $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$23 to the torus $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$24. For a graph $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$25, the toric graphic arrangement is

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$26

A toric chamber is a connected component of the complement. Directing $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$27 by $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$28 when the fractional parts satisfy $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$29 defines a map to $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$30, but unlike the classical case, points in the same toric chamber need not determine the same orientation: moving one coordinate across $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$31 can turn a source into a sink without crossing a toric hyperplane.

This leads to flip equivalence. Two acyclic orientations are equivalent if they differ by a sequence of source-to-sink flips. The main correspondence theorem states that

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$32

and therefore

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$33

For $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$34, the toric chambers are indexed by cyclic classes $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$35 of permutations, called toric total orders, and their number is $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$36.

The toric framework develops analogues of chains, transitivity, Hasse diagrams, and total extensions. A toric chain is a subset $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$37 for which there exists a cyclic order $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$38 such that every $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$39 in the toric chamber has some cyclic shift $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$40 with

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$41

Toric transitive closure adds edges implied by toric directed paths, and this closure operator is convex. The toric Hasse diagram is obtained by removing chord edges from toric directed cycles. The paper does not define a full toric poset-of-regions structure analogous to $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$42; instead it treats chambers, flip classes, toric transitivity, and toric Hasse diagrams as the appropriate toric replacement (Develin et al., 2012).

5. Regular closed lattices, clopen subposets, and extended permutohedra

The lattice-theoretic completion of a poset of regions belongs to a broader theory of closure spaces and regular closed sets. For any closure space $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$43, $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$44 is always a complete lattice. If $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$45 is a family of regular closed sets, then

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$46

For any closure space $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$47 and any subset $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$48, $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$49 is the Dedekind–MacNeille completion of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$50 iff every regular closed set is a join of members of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$51; this occurs, in particular, if every regular open set is a union of members of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$52 (Santocanale et al., 2013).

A particularly important class is that of closure spaces of semilattice type, where $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$53 is a poset and every minimal covering $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$54 of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$55 satisfies $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$56. Such closure spaces are atomistic convex geometries. For well-founded closure spaces of semilattice type, $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$57 is tight in $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$58, meaning that the inclusion preserves all existing joins and meets. In the same context, $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$59 is a lattice iff $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$60. The paper also establishes a hierarchy of quasi-identities $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$61 weaker than both meet- and join-semidistributivity, gives a finite semilattice-type example where $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$62 is not semidistributive, and proves that for finite semilattice-type closure spaces,

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$63

This framework includes two families closely connected with region posets. For a graph $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$64, if $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$65 is the set of all nonempty connected subsets of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$66 and $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$67 closes under disjoint unions, then

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$68

is the extended permutohedron on $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$69, while

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$70

is the permutohedron on $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$71. For a join-semilattice $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$72, the canonical closure operator $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$73 yields $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$74 and $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$75. In the semilattice case, every open subset is a union of clopen sets, $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$76 is generated as a complete ortholattice by the ideals of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$77, $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$78 is the Dedekind–MacNeille completion of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$79, and every completely join-irreducible element of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$80 is clopen. For finite join-semilattices, $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$81 is a bounded homomorphic image of a free lattice (Santocanale et al., 2013).

6. Special cases, failures, and conceptual boundaries

Several boundary phenomena clarify what the poset-of-regions perspective does and does not capture. In the graph setting, Theorem 14.1 gives an exact criterion: $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$82 More generally, if $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$83 is a finite graph and either a block graph or a cycle, then $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$84 is the Dedekind–MacNeille completion of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$85. This need not hold in general: for $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$86, the extended permutohedron $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$87 is not the Dedekind–MacNeille completion of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$88. The failure is tied to the existence of completely join-irreducible regular closed sets that are not clopen. The paper also exhibits a minimal regular open neighborhood $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$89 of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$90 containing no clopen neighborhood of $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$91, showing that for complete graphs $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$92 with $\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$93, not every regular open subset is a union of clopen sets (Santocanale et al., 2013).

The toric theory has a different limitation. Although one can formally mimic a base-region order by separation sets of toric hyper-subtori, the torus is not simply connected, orientations vary within a chamber by flips, and the resulting global order need not be graded or a lattice. The toric framework therefore shifts emphasis from a classical poset of regions to the correspondence

$\mathbb{R}^n\setminus\bigcup_{H\in\mathcal{H}}H.$94

together with toric chains, toric total extensions, toric transitive closure, and toric Hasse diagrams. In this sense toric partial orders are an analogue of region posets rather than a direct transplantation of the classical construction (Develin et al., 2012).

Taken together, these results place the poset of regions in a broader structural landscape. In ordinary real arrangements it is a base-region order on chambers; in central arrangements it embeds into a pseudocomplemented ortholattice of regular closed sets; in Shi arrangements it coexists with local interval-order encodings tailored to ceilings and floors; and in the toric setting it is replaced by a chamber theory organized by flip equivalence and cyclic order. This suggests that the enduring role of the poset of regions is not only to order chambers, but also to act as the interface between arrangement geometry and the combinatorics of lattices, orientations, antichains, and completion procedures (Santocanale et al., 2013, Meszaros, 2011).