Morphisms and order ideals of toric posets

Toric posets are cyclic analogues of finite posets. They can be viewed combinatorially as equivalence classes of acyclic orientations generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper we study toric intervals, morphisms, and order ideals, and we provide a connection to cyclic reducibility and conjugacy in Coxeter groups.


Introduction
A finite poset can be described by at least one directed acyclic graph where the elements are vertices and directed edges encode relations. We say "at least one" because edges implied by transitivity may be present or absent. The operation of converting a source vertex into a sink generates an equivalence relation on finite posets over a fixed graph. Equivalence classes are called toric posets. These objects have arisen in a variety of contexts in the literature, including but not limited to chip-firing games [Eri94], Coxeter groups [EE09,Shi01,Spe09], graph polynomials [Che10], lattices [ES09,Pro93], and quiver representations [MRZ03]. These equivalence classes were first formalized as toric posets in [DMR15], where the effort was made to develop a theory of these objects in conjunction with the existing theory of ordinary posets. The name "toric poset" is motivated by a bijection between toric posets over a fixed (undirected) graph and chambers of the toric graphic hyperplane arrangement of that graph. This is an analogue to the well-known bijection between ordinary posets over a fixed graph G and chambers of the graphic hyperplane arrangement of G, first observed by Greene [Gre77] and later extended to signed graphs by Zaslavsky [Zas91].
Combinatorially, a poset over a graph G is determined by an acyclic orientation ω of G. We denote the resulting poset by P (G, ω). A toric poset over G is determined by an acyclic orientation, up to the equivalence generated by converting sources into sinks. We denote this by P (G, [ω]). Though most standard features of posets have elegant geometric interpretations, this viewpoint is usually unnecessary. In contrast, for most features of toric posets, i.e., the toric analogues of standard posets features, the geometric viewpoint is needed to see the natural proper definitions and to prove structure theorems. Once this is done, the definitions and characterizations frequently have simple combinatorial (non-geometric) interpretations.
To motivate our affinity for the geometric approach, consider one of the fundamental hallmarks of an ordinary poset P : its binary relation, < P . Most of the classic features of posets (chains, transitivity, morphisms, order ideals, etc.) are defined in terms of this relation. Toric posets have no such binary relation, and so this is why we need to go to the geometric setting to define the basic features. Perhaps surprisingly, much of the theory of posets carries over to the toric setting despite the absence of a relation, and current toric poset research strives to understand just how much and what does carry over. As an analogy from a different area of mathematics, topology can be thought of as "analysis without the metric." A fundamental hallmark of a metric space is its distance function. Many of the classic features of metric spaces, such as open, closed, and compact sets, and continuous functions, are defined using the distance function. However, once one establishes an equivalent characterization of continuity in terms of the inverse image of open sets, many results can be proven in two distinct ways: via epsilon-delta proofs, or topologically. When one moves from metric spaces to topological spaces, one loses the distance function and all of the tools associated with it, so this first approach goes out the window. Remarkably, much of the theory of real analysis carries over from metric spaces to topological spaces. Back to the poset world, one can prove theorems of ordinary posets using either the binary relation or the geometric definitions. However, upon passing to toric posets, the binary relation and all of the tools associated with it are lost, so one is forced to go to the geometric setting. Remarkably, much of the theory of ordinary posets still carries over to toric posets. This analogy is not perfect, because toric posets are not a generalization of ordinary posets like how topological spaces extend metric spaces. However, it should motivate the reliance on geometric methods throughout this paper.
An emerging theme of toric poset structure theorems, from both the original paper [DMR15] and this one, is that characterizations of toric analogues, when they exist, usually have one of two forms. In one, the feature of a toric poset P (G, [ω]) is characterized by it being the analogous feature of the ordinary poset P (G, ω ′ ), for all ω ′ ∈ [ω]. In the other, the feature of P (G, [ω]) is characterized by it being the analogous feature of P (G, ω ′ ) for some ω ′ ∈ [ω]. Several examples of this are given below. It is not obvious why this should happen or which type of characterization a given toric analogue should have a priori. Most of these are results from this paper, and so this list provides a good overview for what is to come. The "For All " structure theorems: A set C ⊆ V is a toric chain of P (G, [ω]) iff C is a chain of P (G, ω ′ ) for all ω ′ ∈ [ω]. The "For Some" structure theorems: A partition π ∈ Π V is a closed toric face partition of P (G, [ω]) iff π is a closed face partition of P (G, ω ′ ) for some ω ′ ∈ [ω]. (Theorem 4.7) A set A ⊆ V is a (geometric) toric antichain of P (G, [ω]) iff A is an antichain of P (G, ω ′ ) for some ω ′ ∈ [ω]. (Proposition 5.17) If a set I ⊆ V is a toric interval of P (G, [ω]), then I is an interval of P (G, ω ′ ) for some ω ′ ∈ [ω]. (Proposition 5.14) A set J ⊆ V is a toric order ideal of P (G, [ω]) iff J is an order ideal of P (G, ω ′ ) for some ω ′ ∈ [ω]. (Proposition 7.3) Collapsing P (G, [ω]) by a partition π ∈ Π V is a morphism of toric posets iff collapsing P (G, ω ′ ) by π is a poset morphism for some ω ′ ∈ [ω]. (Corollary 6.2) If an edge {i, j} is in the Hasse diagram of P (G, ω ′ ) for some ω ′ ∈ [ω], then it is in the toric Hasse diagram of P (G, [ω]). (Proposition 5.15) This paper is organized as follows. In the next section, we formally define posets and preposets and review how to view them geometrically in terms of faces of chambers of graphic hyperplane arrangements. In Section 3, we translate well-known properties of poset morphisms to this geometric setting. In Section 4, we define toric posets and preposets geometrically in terms of faces of chambers of toric hyperplane arrangements, and we study the corresponding "toric face partitions" and the bijection between toric preposets and lower-dimensional faces. In Section 5, we define the notion of a toric interval and review some features of toric posets needed for toric order-preserving maps, or morphisms, which are finally presented in Section 6. In Section 7, we introduce toric order ideals and filters, which are essentially the preimage of one element upon mapping into a two-element toric poset. The toric order ideals and filters of a toric poset turn out to coincide. They form a graded poset J tor (P ), but unlike the ordinary case, this need not be a lattice. In Section 8, we provide a connection of this theory to Coxeter groups, and then we conclude with a summary and discussion of current and future research in Section 9.
2. Posets geometrically 2.1. Posets and preposets. A binary relation R on a set V is a subset R ⊆ V × V . A preorder or preposet is a binary relation that is reflexive and transitive. This means that (x, x) ∈ R for all x ∈ V , and if (x, y), (y, z) ∈ R, then (x, z) ∈ R. We will use the notation x R y instead of (x, y), and say that x ≺ R y if x R y and y R x. Much of the basics on preposets can be found in [PRW08].
An equivalence relation is a preposet whose binary relation is symmetric. For any preposet P , we can define an equivalence relation ∼ P on P by saying x ∼ P y if and only if x P y and y P x both hold. A partially ordered set, or poset, is a preposet P such that every ∼ P -class has size 1. We say that a preposet is acyclic if it is also a poset.
Every preorder P over V determines a directed graph ω(P ) over V that contains an edge i → j if and only if i P j and i = j. Not every directed graph arises from such a preorder, since edge transitivity is required. That is, if i → j and j → k are edges, then i → k must also be an edge. However, any directed graph can be completed to a transitive graph via transitive closure, which adds in all "missing" edges. Note that the graph ω(P ) is acyclic if and only if P is a poset. The strongly connected components are the ∼ P -classes, and so the quotient ω(P )/∼ P is acyclic and P/∼ P inherits a natural poset structure from P .
If R 1 and R 2 are preorders on V , then we can define their union R 1 ∪ R 2 as the union of the subsets R 1 and R 2 of V × V . This need not be a preorder, but its transitive closure R 1 ∪ R 2 will be.
Another way we can create a new preorder from an old one is by an operation called contraction. Given a binary relation R ⊆ V × V , let R op denote the opposite binary relation, meaning that (x, y) ∈ R op if and only if (y, x) ∈ R. If P and Q are preposets on V , then Q is a contraction of P if there is a binary relation R ⊆ P such that Q = P ∪ R op . Intuitively, each added edge (x, y) ∈ R op forces x ∼ Q y because (y, x) ∈ P ⊆ Q by construction. Note that in this context, contraction is a different concept than what it often means in graph theory -modding out by a subset of vertices, or "collapsing" a set of vertices into a single vertex.
2.2. Chambers of hyperplane arrangements. It is well known that every finite poset corresponds to a chamber of a graphic hyperplane arrangement [Wac07]. This correspondence will be reviewed here. Let P be a poset over a finite set V = [n] := {1, . . . , n}. This poset can be identified with the following open polyhedral cone in R V : It is easy to see how the cone c determines the poset P = P (c): one has i < P j if and only if x i < x j for all x in c. Each such cone c is a connected component of the complement of the graphic hyperplane arrangement for at least one graph G = (V, E). In this case, we say that P is a poset over G (or "on G"; both are used interchangeably). Given distinct vertices i, j of a simple graph G, the hyperplane H ij is the set Under a slight abuse of notation, at times it is convenient to refer to A(G) as the set of points in R V on the hyperplanes, as opposed to the actual finite set of hyperplanes themselves. It should always be clear from the context which is which.
Each point x = (x 1 , . . . , x n ) in the complement R V −A(G) determines an acyclic orientation ω(x) of the edge set E: direct the edge {i, j} in E as i → j if and only if x i < x j . Clearly, the fibers of the mapping α G : x −→ ω(x) are the chambers of the hyperplane arrangement A(G). Thus, α G induces a bijection between the set Acyc(G) of acyclic orientations of G and the set Cham A(G) of chambers of A(G): We denote the poset arising from an acyclic orientation ω ∈ Acyc(G) by P = P (G, ω). An open cone c = c(P ) may be a chamber in several graphic arrangements, because adding or removing edges implied by transitivity does not change the poset. Geometrically, the hyperplanes corresponding to these edges do not cut c, though they intersect its boundary. Thus, there are, in general, many pairs (G, ω) of a graph G and acyclic orientation ω that lead to the same poset P = P (G, ω) = P (c). Fortunately, this ambiguity is not too bad, in that with respect to inclusion of edge sets, there is a unique minimal graphĜ Hasse (P ) called the Hasse diagram of P and a unique maximal graphḠ(P ), where (G, ω) −→ (Ḡ(P ),ω) is transitive closure.
Given two posets P, P ′ on a set V , one says that P ′ is an extension of P when i < P j implies i < P ′ j. Geometrically, P ′ is an extension of P if and only if c(P ′ ) ⊆ c(P ). Moreover, P ′ is a linear extension if c(P ′ ) is a chamber of A(K V ), where K V is the complete graph.
2.3. Face structure of chambers. Let π = {B 1 , . . . , B r } be a partition of V into nonempty blocks. The set Π V of all such partitions has a natural poset structure: π ≤ V π ′ if every block in π is contained in some block in π ′ . When this happens, we say that π is finer than π ′ , or that π ′ is coarser than π.
Intersections of hyperplanes in A(G) are called flats, and the set of flats is a lattice, denoted L(A(G)). Flats are partially ordered by reverse inclusion: If X 1 ⊆ X 2 , then X 2 ≤ L X 1 . Every flat of A(G) has the form D π := {x ∈ R V : x i = x j for every pair i, j in the same block B k of π} , for at least one partition π of V . Note that D π ≤ L D π ′ if and only if π ≤ V π ′ ; this should motivate the convention of partially ordering L(A(G)) by reverse inclusion.
Given a poset P = P (G, ω) over G = (V, E), a partition π of V defines a preposet P π := (π, ≤ P ) on the blocks, where B i ≤ P B j for x ≤ P y for some x ∈ B i and y ∈ B j (and taking the transitive closure). This defines a directed graph ω/∼ π , formed by collapsing out each block B i into a single vertex. Depending on the context, we may use P π or ω/∼ π interchangeably. If this preposet is acyclic (i.e., if P π is a poset, or equivalently, the directed graph ω/∼ π is acyclic), then we say that π is compatible with P . In this case, there is a canonical surjective poset morphism P → P π . We call such a morphism a quotient, as to distinguish it from inclusions and extensions which are inherently different.
Compatibility of partitions with respect to a poset can be characterized by a closure operator on Π V . If P π = (π, ≤ P ) is a preposet that is not acyclic, then there is a unique minimal coarsening cl P (π) of π such that the contraction (cl P (π), ≤ P ) is acyclic. This is the partition achieved by merging all pairs of blocks B i and B j such that B i ∼ P B j , and we call it the closure of π with respect to P . If P is understood, then we may write this as simplyπ := cl P (π). A partition π is closed (with respect to P ) ifπ = π, which is equivalent to being compatible with respect to P . Geometrically, it means that for any i = j, there is some x ∈ c(P ) ∩ D π such that x bi = x bj for some b i ∈ B i and b j ∈ B j . If π is not closed, then the polyhedral face c(P ) ∩ D π has strictly lower dimension than D π . In this case,π is the unique coarsening that is closed with respect to P and satisfies c(P ) ∩ Dπ = c(P ) ∩ D π .
Still assuming that P is a poset over G = (V, E), and π is a partition of V , define (3)F π (P ) := c(P ) ∩ D π .
If D π is a flat of A(Ḡ(P )), thenF π (P ) is a face of the (topologically) closed polyhedral cone c(P ).
In this case, we say that π is a face partition of P . Since it is almost always clear what P is, we will usually writeF π instead ofF π (P ). If D π is not a flat of A(G), then the subspace D π still intersects c(P ) in at least the line x 1 = · · · = x n . Though this may intersect in the interior of c(P ′ ), it is a face of c(P ′ ) ⊆ c(P ), for at least one extension P ′ of P .
To characterize the facial structure of the cone c(P ), it suffices to characterize the closed face partitions. This is well known -it was first described by Geissinger [Gei81], and also done by Stanley [Sta86] in the characterization of the face structure of the order polytope of a poset, defined by Clearly, if π is a closed face partition of P , then the subposets induced by the individual blocks are connected (that is, their Hasse diagrams are connected). We call such a partition connected, with respect to P . To summarize Theorem 2.1, characterizing the faces of O(P ) amounts to characterizing which face partitions π are closed. If π is not compatible with P , then it is not closed. On the other hand, if π not connected, then it is either not closed, or the flat D π cuts through the interior of O(P ) (and hence of c(P )), in which case π is not a face partition.
Example 2.2. Let P be the poset shown at left in Figure 1; its lattice of closed face partitions is shown at right. In this and in later examples, we denote the blocks of a partition using dividers rather than set braces, e.g., π = B 1 /B 2 / · · · /B r .
The partition σ = 1/23/4 is closed but not connected; it is not a face partition because D σ = H 23 intersects the interior of c(P ). The partition π = 124/3 is connected but not closed. Finally, the partition π ′ = 14/23 is neither connected nor closed. However, both π and π ′ are face partitions because the subspaces D π and D π ′ intersect c(P ) in the line x 1 = x 2 = x 3 = x 4 , which is the flat D V . Therefore, both of these partitions have the same closure: cl P (π) = cl P (π ′ ) = 1234. Ifπ = π = {B 1 , . . . , B r } is a closed face partition of P , then D π is an r-dimensional flat of A(G), and the closed faceF π is an r-dimensional subset of D π ⊆ R V . The interior ofF π with respect to the subspace topology of D π will be called an open face. So as to avoid confusion between open and closed faces, and open and closed chambers, we will speak of faces as being features of the actual poset, not of the chambers. It should be easy to relate these definitions back to the chambers if one so desires.
Definition 2.3. A set F ⊆ R V is a closed face of the poset P if F =F π = c(P ) ∩ D π for some closed face partition π = cl P (π) of V . The interior ofF π with respect to the subspace topology of D π is called an open face of P , and denoted F π . Let Face(P ) and Face(P ) denote the set of open and closed faces of P , respectively. Finally, define the faces of the graphic arrangement A(G) to be the faces of the posets over G: Face(P (G, ω)) .
Faces of co-dimension 1 are called facets.
Remark 2.4. The dimension of the face F π (P (G, ω)) is the number of strongly connected components of ω/∼ π . As long as G is connected, there is a unique 1-dimensional face of A(G), which is the line x 1 = · · · = x n and is contained in the closure of every chamber. There are no 0dimensional faces of A(G). The n-dimensional faces of A(G) are its chambers. Additionally, R V is a disjoint union of open faces of A(G): If P is a fixed poset over G, then there is a canonical isomorphism between the lattice of closed face partitions and the lattice of faces of P , given by the mapping π → F π . Recall that since π is closed, P π = (π, ≤ P ) is an acyclic preposet (i.e., poset) of size |π| = r. This induces an additional preposet over V (i.e., of size |V | = n), which is ≤ P with the additional relations that x ∼ π y for all x, y ∈ B i . We will say that this is a preposet over G, because it can be described by an (not necessarily acyclic) orientation ω π of G. The notation reflects the fact that this orientation can be constructed by starting with some ω ∈ Acyc(G) and then making each edge bidirected if both endpoints are contained in the same block of π. Specifically, ω π orients edge {i, j} as i → j if i ≤ P j and as i ↔ j if additionally i ∼ π j. Let Pre(G) be the set of all such orientations of G that arise in this manner. That is, When working with preposets over G, sometimes it is more convenient to quotient out by the strongly connected components and get an acyclic graph ω π /∼ π . Note that this quotient is the same as ω ′ /∼ π for at least one ω ′ ∈ Acyc(G). In particular, ω ′ = ω will always do. In summary, a preposet over G can be expressed several ways: (i) as a unique orientation ω π of G, where π is the partition into the strongly connected components; (ii) as a unique acyclic quotient ω/∼ π of an acyclic orientation ω ∈ Acyc(G). Note that while the orientation ω π and acyclic quotient ω/∼ π are both unique to the preposet, the choice of representative ω is not. Regardless of how an element in Pre(G) is written, it induces a canonical partial order (π, ≤) on the blocks of π. However, information is lost by writing it this way; in particular, the original graph G cannot necessarily be determined from just (π, ≤).
(2) can be extended to all of R V by adding both edges i → j and j → i if x i = x j . This induces a bijection between the set Pre(G) of all preposets on G and the set of faces of the graphic hyperplane arrangement: Consequently, for any preposet ω π over G, we can let c(ω π ) denote the open face of A(G) containing any (equivalently, all) x ∈ R V such that α G (x) = ω π . Moreover, if we restrict to the preposets on exactly r strongly connected components, then the α G -fibers are the r-dimensional open faces of A(G). If x lies on a face F π (P ) = Fπ(P ) for some poset P and closed face partitionπ = cl P (π), then the preposet P π = (π, ≤ P ) has vertex set π = {B 1 , . . . B r }; these are the strongly connected components of the orientation α G (x).

Morphisms of ordinary posets
Poset isomorphisms are easy to describe both combinatorially and geometrically. An isomorphism between two finite posets P and P ′ on vertex sets V and V ′ is a bijection φ : • geometrically by the equivalent condition that the induced isomorphism Φ on R V → R V ′ maps c(P ) to c(P ′ ) bijectively. By "induced isomorphism," we mean that Φ permutes the coordinates of R V in the same way that φ permutes the vertices of V : Morphisms of ordinary posets are also well understood. The "combinatorial" definition is easiest to modify. If P and P ′ are as above, then a morphism, or order-preserving map, is a function The geometric characterization is trickier because quotients, injections, and extensions are inherently different. These three types of order-preserving maps generate all poset morphisms, up to isomorphism. Below we will review this and give a geometric interpretation of each, which will motivate their toric analogues.
3.1.1. Contracting partitions. Roughly speaking, a quotient morphism of a poset P (G, ω) is described combinatorially by contracting ω by the blocks of a partition π = {B 1 , . . . , B r } while preserving acyclicity. Geometrically, the chamber c(P ) is orthogonally projected toFπ := c(P ) ∩ Dπ, whereπ = cl P (π). This is the mapping By construction, the image of this map is Fπ, which is a face of P if π is a face partition. Though the map d π extends to the closure c(P ), it does not do so in a well-defined manner; the image d π (x) for some x on a hyperplane depends on the choice of P , and each hyperplane intersects two (closed) chambers along facets, and intersects the boundary of every chamber.
Example 3.1. Let G = K 3 , the complete graph on 3 vertices. There are six acyclic orientations of G, and three of them are shown in Figure 2. The curved arrows point to the chamber c(P i ) of A(G) for each P i := P (G, ω i ), i = 1, 2, 3. The intersection of each closed chamber c(P i ) with [0, 1] 3 is the order polytope, O(P i ).
To see why the map d π from Eq. (7) does not extend to the closure of the chambers in a welldefined manner, consider the point y shown in Figure 2 that lies on the hyperplane x 1 = x 2 , and the same partition, π = 1/23. The orthogonal projection d π : c(P 3 ) → D π as defined in Eq. (7) and extended continuously to the closed chamber maps c(P 3 ) onto the line x 1 = x 2 = x 3 . However, if c(P 4 ) is the other closed chamber containing y (that is, the one for which x 3 ≤ x 2 ≤ x 1 ), then d π : c(P 4 ) → D π extended to the closure maps c(P 4 ) onto a 2-dimensional closed faceF π (P 4 ) = c(P 4 ) ∩ D π . The point y is projected orthogonally onto the plane D π , and does not end up on the line Despite this, there is a natural way to extend d π to all of R V , though not continuously. To do this, we first have to extend the notion of the closure of a partition π with respect to a poset, to a preposet P over G. This is easy, since the original definition did not specifically require P to actually be a poset. Specifically, the closure of π with respect to a preposet P is the unique minimal coarseningπ := cl P (π) of π such that (π, ≤ P ) is acyclic. The map d π can now be extended to all of R V , as Figure 2. The hyperplane arrangement A(G) for G = K 3 . Three orientations in Acyc(G) are shown, along with the corresponding chambers of A(G), and the preposet that results when contracting where α G is the map from Eq. (5) sending a point to the unique open face (i.e., preposet over G) containing it. Let us return to the case where P is a poset over G, and examine the case when π is not a face partition of P . Indeed, for an arbitrary partition π of V withπ = cl P (π), the subset F π := c(P ) ∩ Dπ need not be a face of P ; it could cut through the interior of the chamber. In this case, it is the face of at least one extension of P . Specifically, let G ′ π be the graph formed by making each block B i a clique, and let G/∼ π be the graph formed by contracting these cliques into vertices, with loops and multiedges removed. Clearly, D π is a flat of the graphic arrangement A(G ′ π ) (this choice is not unique, but it is a canonical one that works). Thus, the setFπ =F π = c(P ) ∩ D π , forπ = cl P (π), is a closed face of A(G ′ π ), and hence a face of some poset P ′ over G ′ π for which π = cl P ′ (π).
Whether or not π is a face partition of a particular poset P over G, the map d π in Eq. (8) projects a chamber c(P ) onto a flat Dπ of A(G ′ π ), whereπ = cl P (π). From here, we need to project it homeomorphically onto a coordinate subspace of R V so it is a chamber of a lower-dimensional arrangement. Specifically, for a partition π = {B 1 , . . . , B r }, let W ⊆ V be any subset formed by removing all but 1 coordinate from each B i , and let r π : R V −→ R W be the induced projection. The r π -image of A(G) is the graphic arrangement of G/∼ π . The following ensures that r π is a homeomorphism, and that the choice of W does not matter. We omit the elementary proof.
Moreover, all such projection maps for a fixed π are topologically conjugate in the following sense: i , then the following diagram commutes: with Σ and Σ ′ being the induced linear maps as defined in Eq. (6).
By convexity, d π induces a well-defined map δ π : Face A(G) −→ Face A(G ′ π ) making the following diagram commute: The map δ π is best understood by looking at a related mapδ π on closed faces. LetF = c(P )∩D σ be a closed face of A(G), for some closed face partition σ ∈ Π V . Then the mapδ π is defined bȳ The map δ π is between the corresponding open faces. These faces are then mapped to faces of the arrangement A(G/∼ π ) under the projection r π | Dπ : D π −→ R W . [Alternatively, we could simply identity the quotient space R V /D ⊥ π with R W .] To summarize, the open faces of A(G) arise from preposets ω = ω σ in Pre(G), where without loss of generality, the blocks for σ ∈ Π V are the strongly connected components. The contraction of this preposet formed by adding all relations (edges) of the form (v, w) for v, w ∈ B i yields a preposet ω ′ π over G ′ π . Then, modding out by the strongly connected components yields an acyclic preposet, i.e., a poset. This two-step process is a composition of maps , the closure of π with respect to the preposet P σ , which we have been denoting by ω σ under a slight abuse of notation.
Putting this all together gives a commutative diagram that illustrates the relationship between the points in R V , the open faces of the graphic arrangement A(G), and the preposets over G. The left column depicts the acyclic preposets -those that are also posets.
3.1.2. Intervals and antichains. Poset morphisms that are quotients are characterized geometrically by projecting the chamber c(P ) onto a flat Dπ of A(G ′ π ) for some partitionπ = cl P (π), and then homeomorphically mapping this down to a chamber of a lower-dimensional graphic arrangement A(G/∼ π ). Equivalently, contracting P (G, ω) byπ yields an acyclic preposet Pπ = (π, ≤ P ). It is well known that contracting a poset by an interval or an antichain yields an acyclic preposet. Verification of this is elementary, but first recall how these are defined.
Definition 3.3. Let P be a poset over V . An interval of P is a subset I ⊆ V , sometimes denoted [i, j], such that I = {x ∈ P : i ≤ P x ≤ P j}, for some fixed i, j ∈ P . An antichain of P is a subset A ⊆ V such that any two elements are incomparable.
We will take a moment to understand how contracting an interval or antichain fits in the partition framework described above, which will help us understand the toric analogue. Given a nonempty subset S ⊆ V , define the partition π S of V by where |B i | = 1 for i = 2, . . . , r.
Contracting an interval I ⊆ V in a poset P yields the poset P πI = (π I , ≤ P ). In this case, π I = cl P (π I ) is a face partition and F π is a (|V | − |I| + 1)-dimensional face of P . Similarly, collapsing an antichain A ⊆ V yields the poset P πA = (π A , ≤ P ). Note that D πI is a flat of A(G) and lies on the boundary of c(P ), but the (|V | − |A| + 1)-dimensional subspace D πA cuts through the interior of c(P ). For both of these cases, S = I and S = A, the subspace D πS is trivially a flat of A(G ′ πS ). 3.2. Extension. Given two posets P, P ′ on a set V , one says that P ′ is an extension of P when i < P j implies i < P ′ j. In this case, the identity map P −→ P ′ is a poset morphism. Geometrically, P ′ is an extension of P if and only if one has an inclusion of their open polyhedral cones c(P ′ ) ⊆ c(P ). Each added relation i < P ′ j amounts to intersecting c(P ) with the half-space 3.3. Inclusion. The last operation that yields a poset morphism is an injection φ : P ֒→ P ′ . This induces a canonical inclusion Φ : R V ֒→ R V ′ . Note that i < P j implies i < P ′ j, but not necessarily vice-versa. Thus, up to isomorphism, an inclusion can be decomposed into the composition P ֒→ P ′′ → P ′ , where the first map adds the elements {n + 1, . . . , m} to P but no extra relations, and then the map P ′′ → P ′ is an extension. This gives an inclusion of polyhedral cones: 3.4. Summary. Up to isomorphism, every morphism of a poset P = P (G, ω) can be decomposed into a sequence of three steps: (i) quotient : Collapsing G by a partition π that preserves acyclicity of ω (projecting c(P ) to a flat D π of A(G ′ π ) for some closed partition π = cl P (π)). (ii) inclusion: Adding vertices (adding dimensions). (iii) extension: Adding relations (intersecting with half-spaces). In the special case of the morphism P −→ P ′ being surjective, the inclusion step is eliminated and the entire process can be described geometrically by projecting c(P ) to a flat D π and then intersecting with a collection of half-spaces.

Toric posets and preposets
4.1. Toric chambers and posets. Toric posets, introduced in [DMR15], arise from ordinary (finite) posets defined by acyclic orientations under the equivalence relation generated by converting maximal elements into minimal elements, or sources into sinks. Whereas an ordinary poset corresponds to a chamber of a graphic arrangement A(G), a toric poset corresponds to a chamber of a toric graphic arrangement A tor (G) = q(A(G)), which is the image of A(G) under the quotient is called a toric chamber for G, or simply a chamber of A tor (G). Let Cham A tor (G) denote the set of all chambers of A tor (G). A toric poset P is a set c that arises as a toric chamber for at least one graph G. We may write P = P (c) or c = c(P ), depending upon the context.
If we fix a graph G = (V, E) and consider the arrangement A tor (G), then each point in R V /Z V naturally determines a preposet on G via a mapᾱ G : x → ω(x). Explicitly, for x = (x 1 mod 1, . . . , x n mod 1) in R V /Z V , the directed graph ω(x) is constructed by doing the following for each edge {i, j} in E: If x i mod 1 ≤ x j mod 1, then include edge i → j; If x j mod 1 ≤ x i mod 1, then include edge j → i. The mappingᾱ G is essentially the same as α G from Eq. (5) except done modulo 1, so many of its properties are predictably analogous. For example, the undirected version of ; in this case ω(x) describes a poset. Otherwise it describes a preposet (that is not a poset). Modding out by the strongly connected components yields an acyclic graph ω(x)/∼ x that describes a poset.
Definition 4.2. When two preposets ω(x) and ω(y) are such that the directed graphs ω(x)/∼ x and ω(y)/∼ y differ only by converting a source vertex (equivalence class) into a sink, or vice-versa, we say they differ by a flip. The transitive closure of the flip operation generates an equivalence relation on Pre(G), denoted by ≡.
In the special case of restricting to preposets that are acyclic, we get Acyc(G) ⊆ Pre(G) and a bijective correspondence between toric posets and chambers of toric graphic arrangements. This is Theorem 1.4 in [DMR15]. A generalization of this to a bijection between toric preposets and faces of the toric graphic arrangement appears later in this section (Proposition 4.11). Theorem 1.4) The mapᾱ G induces a bijection between Cham A tor (G) and Acyc(G)/≡ as follows: When G is understood, we will say that the order polytopes O(P (G, ω)) and O(P (G, ω ′ )) are torically equivalent whenever ω ′ ∈ [ω]. Under the natural quotient q : [0, 1] V → R V /Z V , each order polytope O(P (G, ω)) is mapped into the closed toric chamberc [ω] . Moreover, by Theorem 4.3, the closed chambers of A tor (G) are unions of q-images of torically equivalent order polytopes.
) be a toric poset, and q : The closure of the chamber c(P ) is

4.2.
Toric faces and preposets. Let P = P (c) be a toric poset over G = (V, E). To define objects like a face of P or its dimension, it helps to first lift c up to a chamber of the affine graphic arrangement which lies in R V : The affine chambers are open unbounded convex polyhedral regions in R V , the universal cover of R V /Z V . The path lifting property guarantees that two points x and y in R V /Z V −A tor (G) are in the same toric chamber if and only if they have liftsx andŷ that are in the same affine chamber. Moreover, since Corollary 4.4 characterizes the closed toric chamber c(P ) as a union of torically equivalent order polytopes under a universal covering map, each closed affine chamber is a union of translated copies of torically equivalent order polytopes in R V . We usually denote an affine chamber byĉ or c aff . Each hyperplane H tor ij has a unique preimage containing the origin in R V called its central preimage; this is the ordinary hyperplane Thus, the set of central preimages of A tor (G) is precisely the graphic arrangement A(G) in R V . Each closed affine chamberc aff contains at most one order polytope O(P (G, ω)) for ω ∈ Acyc(G). Affine chambers whose closures contain precisely one order polytope O(P (G, ω)) are central affine chambers.
We will call nonempty sets that arise as intersections of hyperplanes in A aff (G) affine flats and nonempty sets that are intersections of hyperplanes in A tor (G) toric flats. Since the toric flats have a nonempty intersection, they form a lattice that is denoted L(A tor (G)), and partially ordered by reverse inclusion.
Since a toric flat of A tor (G) is the image of a flat of A(G), it too is determined by a partition π of V , and so it is of the form Since R V q −→ R V /Z V is a covering map, it is well-founded to declare the dimension of a toric flat D tor π in R V /Z V to be the same as the dimension of its central preimage D π in R V .
Recall that a partition π = {B 1 , . . . , B r } is compatible with an ordinary poset P if contracting the blocks of π yields a preposet P π = (π, ≤ P ) that is acyclic (also a poset). The notion of compatible partitions does not carry over well to toric posets, because compatibility is not preserved by toric equivalence. Figure 2 shows an example of this: the preposets (π, ≤ P1 ) and (π, ≤ P2 ) are acyclic but (π, ≤ P3 ) is not. Despite this, every set D tor π , whether or not it is a toric flat of A tor (G), intersects the closed toric chamber c(P ) in at least the line x 1 = · · · = x n . We denote this intersection by (15)F tor π (P ) := c(P ) ∩ D tor π . If D tor π does not intersect c(P ), then we say that π is a toric face partition, since it intersects the closed toric chamber along its boundary. Compare this to the definition of face partitions of an ordinary poset P (G, ω), which are those π ∈ Π V characterized by D π being a flat of the graphic arrangement of the transitive closure, or equivalently, by D π ∩ c(P (G, ω)) = ∅. The transitive closureḠ(P (G, ω)) is formed from G by adding all additional edges {i, j} such that H i,j ∩ c(P (G, ω)) = ∅. Similarly, we can define the toric transitive closure of P (G, [ω]) as the graph G along with the extra edges {i, j} such that H tor i,j ∩ c(P ) = ∅. This was done in [DMR15], and we will return to it in the Section 5.3 when we discuss toric Hasse diagrams. Now, let π ∈ Π V be an arbitrary partition. Since flats of A tor (K V ) are closed under intersections, there is a unique maximal toric subspace D tor π (that is, of minimal dimension) for which F tor π = c(P ) ∩ D tor π . The partitionπ is the unique minimal coarsening of π for whichF tor π =F tor π , and it is the lattice-join of all such partitions. We call it the closure of π with respect to the toric poset P , denoted cl tor P (π), and we define dim(F tor π ) := dim(D tor π ). A partition π is closed with respect to the toric poset P if π = cl tor P (π). Note that the closure is defined for all partitions, not just toric face partitions. Toric faces of co-dimension 1 are called facets.
The following remark is the toric analogue of Remark 2.4.
Remark 4.6. Let P = P (G, [ω]) be a toric poset. The dimension ofF tor π (P ) := c(P ) ∩ D tor π is simply the maximum dimension of c aff (P ) ∩ D π taken over all affine chambers that descend down to c(P ). Since closed affine chambers are unions of translations of order polytopes, this is the maximum dimension of c(P (G, ω ′ )) ∩ D π taken over all ω ′ ∈ [ω]. In other words, dim F π (P (G, ω ′ )) .
On the level of graphs, this is the maximum number of strongly connected components that ω ′ /∼ π can have for some ω ′ ∈ [ω]. In particular, a partition π is closed with respect to P (G, [ω]) if and only if ω ′ /∼ π is acyclic for some ω ′ ∈ [ω].
As long as G is connected, there is a unique 1-dimensional face of A tor (G), which is the line x 1 = · · · = x n and is contained in the closure of every chamber. There are no 0-dimensional faces of A tor (G). The n-dimensional faces of A tor (G) are its chambers. Additionally, R V /Z V is a disjoint union of open faces of A tor (G): As in the case of ordinary posets, there is a canonical bijection between the closed toric face partitions of P and open faces (or closed faces) of P , via π → F tor π . To classify the faces of a toric poset, it suffices to classify the closed toric face partitions. The proof of Theorem 4.7 will be done later in this section, after the following lemma, which establishes that cl tor P is a closure operator [War42] on the partition lattice Π V and compares it with cl P .
Conversely, suppose that π is connected and compatible with respect to P (G, ω ′ ) for some ω ′ ∈ [ω]. By Theorem 2.1, π is a closed face partition of P (G, ω ′ ). Since π is connected, D π is a flat of A(G). Therefore, D tor π is a toric flat of A tor (G), and soF tor π := c(P ) ∩ D tor π is a face of the toric poset P (G, [ω ′ ]) = P (G, [ω]) = P . Therefore, π is a toric face partition. Closure of π with respect to P = P (G, [ω]) = P (G, [ω ′ ]) follows immediately from Lemma 4.8(a) applied to the fact that π is closed with respect to P (G, ω ′ ).
Unlike the ordinary case, where faces of posets are literally faces of a convex polyhedral cone, it is not quite so "geometrically obvious" what subsets can be toric faces. The following example illustrates this. The subtlety in Example 4.10 does not arise for ordinary posets, because distinct ordinary posets never have chambers with the same topological closure. In contrast, if G = (V, E) and G ′ = (V, E ′ ) are both forests, then A tor (G) and A tor (G ′ ) both have a single toric chamber. This is because the number of chambers is counted by the Tutte polynomial evaluation T G (1, 0), which is always 1 for a forest; see [DMR15]. In this case, the closures of both chambers will be all of R V /Z V . A more complicated example involving a toric poset over a graph with three vertices, will appear soon in Example 4.13.
Recall the mapᾱ G from Eq. (11) that sends a point x in R V /Z V to a preposet ω(x). By Theorem 4.3, when restricted to the points in R V /Z V −A tor (G), this map induces a bijection between toric posets and toric chambers. Toric faces F tor π that are open in D tor π are chambers in lower-dimensional arrangements that are contractions of A tor (G), namely by the subspace D tor π . Thus, the bijection between toric equivalence classes of Acyc(G) (n-element preposets) and toric chambers (n-dimensional faces) extends naturally to a bijection between toric preposets over G and open faces of A tor (G).
Proposition 4.11. The mapᾱ G induces a bijection between Face A tor (G) and Pre(G)/ ≡ as follows: In other words, given two points x,  If x lies on a toric face F tor π of P , where (without loss of generality) π = cl tor P (π), then the strongly connected components of the preposetᾱ G (x) are π = {B 1 , . . . B r }.
Example 4.13. Let G = K 3 , as in Example 3.1. The six acyclic orientations of G fall into two toric equivalence classes. The three orientations shown in Figure 3.1 comprise one class, and so the corresponding toric poset is P = P (G, [ω i ]) for any i = 1, 2, 3. Equivalently, the closed toric chamber is a union of order polytopes under the natural quotient map: This should be visually clear from Figure 3.1. The two chambers in A tor (G) are the threedimensional faces of P . Each of the three toric hyperplanes in A tor (G) = {H tor 12 , H tor 13 , H tor 23 } are two-dimensional faces of P , and these (toric preposets) correspond to the following toric equivalence classes of size-2 preposets ω π /∼ π over K 3 : The toric flat D {V } is the unique one-dimensional face of P , and this corresponds to the unique size-1 preposet over K 3 ; when x 1 = x 2 = x 3 , which is trivially in its own toric equivalence class.

Toric intervals and antichains
Collapsing an interval or antichain of an ordinary poset defines a poset morphism. This remains true in the toric case, as will be shown in Section 6, though the toric analogues of these concepts are trickier to define. Toric antichains were introduced in [DMR15], but toric intervals are new to this paper. First, we need to review some terminology and results about toric total orders, chains, transitivity, and Hasse diagrams. This will also be needed to study toric order ideals and filters in Section 7. Much of the content in Sections 5.1-5.3 can be found in [DMR15]. Throughout, G = (V, E) is a fixed undirected graph with |V | = n, and coordinates x i of points x = (x 1 , . . . , x n ) in a toric chamber c(P ) are assumed to be reduced modulo 1, i.e., x i ∈ [0, 1). 5.1. Toric total orders. A toric poset P ′ is a total toric order if c(P ′ ) is a chamber of A tor (K V ). If P (G, [ω]) is a total toric order, then P (G, ω ′ ) is a total order for each ω ′ ∈ [ω], and thus [ω] has precisely |V | elements. Since each P (G, ω ′ ) has exactly one linear extension, total toric orders are indexed by the (n − 1)! cyclic equivalence classes of permutations of V : [w] = [(w 1 , . . . , w n )] := (w 1 , . . . , w n ), (w 2 , . . . , w n , w 1 ), . . . , (w n , w 1 , . . . , w n−1 ) .
Recall that if P and P ′ are toric posets over G, then P ′ is an extension of P if c(P ′ ) ⊆ c(P ). Moreover, P ′ is a total toric extension if P ′ is a total toric order. Analogous to how a poset is determined by its linear extensions, a toric poset P is determined by its set of total toric extensions, denoted L tor (P ).

5.2.
Toric directed paths, chains, and transitivity. A chain in a poset P (G, ω) is a totally ordered subset C ⊆ V . Equivalently, this means that the elements in C all lie on a common directed path i 1 → i 2 → · · · → i m in ω. Transitivity can be characterized in this language: if i and j lie on a common chain, then i and j are comparable in P (G, ω). Geometrically, i and j being comparable means the hyperplane H i,j does not cut (i.e., is disjoint from) the chamber c(P (G, ω)).
The toric analogue of a chain is "essentially" a totally cyclically ordered set, but care must be taken in the case when |C| = 2 because every size-two subset C ⊆ V is trivially totally cyclically ordered. Define a toric directed path in ω, to be a directed path i 1 → i 2 → · · · → i m such that the edge i 1 → i m is also present. We denote such a path by i 1 → tor i m . Toric directed paths of size 2 are simply edges, and every singleton set is a toric directed path of size 1. A fundamental property of toric directed paths is that up to cyclic shifts, they are invariants of toric-equivalence classes. That is, i 1 → tor i m is a toric directed path of ω if and only if each ω ′ ∈ [ω] has a toric directed path j 1 → tor j m , for some cyclic shift (j 1 , . . . , j m ) in [(i 1 , . . . , i m )]. This is Proposition 4.2 of [DMR15], and it leads to the notion of a toric chain, which is a totally cyclically ordered subset.
The following is a reformulation of Proposition 6.3 of [DMR15] using the language of this paper, where notation such as P (G, ω) and P (G, [ω]) is new. Having the concept of a toric chain leads to the notion of toric transitivity, which is completely analogous to ordinary transitivity when stated geometrically.
Proposition 5.5. Let i, j ∈ V be distinct. Then the hyperplane H tor i,j does not cut the chamber c(P (G, [ω])) if and only if i and j lie on a common toric chain.

Toric Hasse diagrams.
One of the major drawbacks to studying toric posets combinatorially, as equivalences of acyclic orientations (rather than geometrically, as toric chambers), is that a toric poset P or chamber c = c(P ) generally arises in multiple toric graphic arrangements A tor (G) over the same vertex set. That is, one can have P (G, [ω]) = P (G ′ , [ω ′ ]) for different graphs, leading to ambiguity in labeling a toric poset P with a pair (G, [ω]) consisting of a graph G and equivalence class [ω] in Acyc(G)/≡. Toric transitivity resolves this issue. As with ordinary posets, there is a well-defined notion for toric posets of what it means for an edge to be "implied by transitivity." The toric Hasse diagram is the graphĜ torHasse with all such edges removed. In Section 5.3, we encountered the toric transitive closure, which is the graphḠ tor with all such edges included. In other words, given any toric poset P = P (G, [ω]), there is always a unique minimal pair (Ĝ torHasse (P ), [ω torHasse (P )]) and maximal pair (Ḡ tor (P ), [ω tor (P )]) with the property that the set c(P ) is in Cham A tor where ⊆ is inclusion of edges. In this case, ω can be taken to be the restriction to G of any orientation in [ω tor (P )].
Geometrically, the existence of a unique toric Hasse diagram is intuitive; it corresponds to the minimal set of toric hyperplanes that bound the chamber c(P ), and the edges implied by transitivity correspond to the additional hyperplanes that do not cut c(P ). The technical combinatorial reason for the existence of a unique Hasse diagram (respectively, toric Hasse diagram) follows because the transitive closure (respectively, toric transitive closure) A −→Ā is a convex closure, meaning it satisfies the following anti-exchange condition; also see [EJ85]: Edges {i, j} in the Hasse diagram (respectively, toric Hasse diagram) are precisely those whose removal "change" the poset (respectively, toric poset), and the geometric definitions make this precise. Though the ordinary and toric cases are analogous, there are a few subtle differences. For example, consider the following "folk theorem." Since toric posets are defined geometrically as subsets of R V /Z V that are chambers of a graphic hyperplane arrangement, the equivalence (i)⇔(ii) is immediate for toric posets. Condition (iii) says that the edges {i, j} of the Hasse diagram are precisely the size-2 intervals, and Condition (iv) says these are the closed face partitions having two blocks of the form π = {{i, j}, V −{i, j}}. Finally, note that the implication (i)⇒(v) in Proposition 5.6 trivially fails for toric posets. For a simple counterexample, take any tree G with at least one edge. Since A tor (G) has only one chamber, removing any H tor i,j will never increase the number of chambers. Since adding or removing edges implied by toric transitivity does not change the toric poset, it does not change which sets are toric chains. Thus, to characterize the toric chains of P (G, [ω]), it suffices to characterize the toric chains of P (Ḡ tor (P ), [ω tor (P )]). The following is immediate.
Remark 5.7. Let P be a toric poset. A size-2 subset C = {i, j} of V is a toric chain of P if and only if {i, j} is an edge ofḠ tor (P ). In particular, if C is a maximal toric chain, then {i, j} is an edge inĜ torHasse (P ).

Toric intervals.
To motivate the definition of a toric interval, it helps to first interpret the classical definition in several different ways. We will define toric intervals geometrically, motivated by Condition (i), and show how it is equivalent to the toric version of Condition (ii). In contrast, Condition (iii) has a small wrinkle - the property of lying on a directed path from i to j does not depend on the choice of (G, ω) for P . Specifically, if P (G, ω) = P (G ′ , ω ′ ) and k lies on an ω-directed path from i to j, then k lies on an ω ′ -directed path from i to j. This is not the case for toric directed paths in toric posets, as the following example illustrates. As a result, we will formulate and prove a modified version of Condition (iii) for toric intervals.
Example 5.9. Consider the circular graph G = C 4 , and ω ∈ Acyc(C 4 ) as shown at left in Figure 3. Let P = P (C 4 , [ω]), which is a total toric order. Therefore, the toric transitive closure of (C 4 , [ω]) is the pair (K 4 , [ω ′ ]), where ω ′ is shown in Figure 3 on the right. Therefore, Now, let i = 1 and j = 3. The set {i, j} lies on a toric directed path from 1 to 3 in ω ′ (which also contains 2). However, none of the 4 representatives in [ω] contain a toric directed path from 1 to 3.
Another obstacle to formulating the correct toric analogue of an interval is how to characterize which size-2 subsets should be toric intervals. This ambiguity arises from the aforementioned "size-2 chain problem" of all size-2 subsets being totally cyclically ordered. Since ordinary intervals are unions of chains, we will require this to be a feature of toric intervals. Remark 5.11. If i, j, k are distinct elements of the toric interval I = [i, j] tor of P = P (G, [ω]), then for each x in c(P ), exactly one of the following must hold: By Theorem 5.1, we can rephrase Remark 5.11 as the toric analogue of Definition 5.8(ii): the toric interval [i, j] tor in P is the set of elements between i and j in the cyclic order of any total toric extension of P .  Finally, the toric analogue of Definition 5.8(iii) can be obtained by first passing to the toric transitive closure.
Proposition 5.13. Fix a toric poset P = P (G, [ω]). An element k is in [i, j] tor if and only if k lies on a toric directed path i → tor j inω ′ , for someω ′ ∈ [ω tor (P )].
Proof. Throughout, let C = {i, j, k}. Assume that |C| ≥ 3; the result is trivial otherwise. Suppose k is in [i, j] tor , which means that P | C = [(i, k, j)]. Take any ω ′ ∈ [ω] for which i is a source. By Proposition 5.3, the elements of C occur as a subsequence of a toric directed path in ω ′ , ordered (i, k, j). Since this is a toric chain, the edges {i, k}, {k, j}, and {i, j} are all implied by toric transitivity. Thus, k lies on a toric directed path i → tor j inω ′ , the unique orientation of [ω tor (P )] whose restriction to G is ω ′ .
Conversely, suppose that k lies on a toric directed path i → tor j inω ′ , for someω ′ ∈ [ω tor (P )]. Then C is a toric chain, ordered P | C = [(i, k, j)], hence k is in [i, j] tor .
Proposition 5.14. Let P = P (G, [ω]) be a toric poset. If a set I ⊆ V is a toric interval I = [i, j] tor , then there is some ω ′ ∈ [ω] for which the set I is the interval [i, j] of P (G, ω ′ ). The converse need not hold.
Proof. Without loss of generality, assume that G =Ḡ tor (P ). The statement is trivial if |I| < 2. We need to consider the cases |I| = 2 and |I| ≥ 3 separately. In both cases, we will show that one can take ω ′ to be any orientation that has i as a source.
First, suppose |I| = 2, which means that I = {i, j} is an edge of G. Take any ω ′ ∈ [ω] for which i is a source. Since |I| = 2, there is no other k ∈ {i, j} on a directed path from i to j in ω ′ , as this would form a toric directed path. Therefore, the interval [i, j] in P (G, ω ′ ) is simply {i, j}.
Next, suppose |I| ≥ 3. As before, take any ω ′ ∈ [ω] such that i is a source in ω ′ . Since G =Ḡ tor (P ), the directed edge i → j is present, and so by Proposition 5.13, [i, j] tor consists of all k ∈ V that lie on a directed path from i to j. This is precisely the definition of the interval [i, j] in P (G, ω ′ ).
Proof. Given the toric Hasse diagram of P (G, [ω]), the ordinary Hasse diagram of P (G, ω) is obtained by removing the edge {i 1 , i m } for each toric directed path i 1 → tor i m in P (G, ω) of size at least m ≥ 3. This establishes the first inequality in Eq. (21). The second inequality is obvious. Loosely speaking, the final equality holds because edges in the toric transitive closure are precisely the size-2 toric chains, which are precisely the subsets that are size-2 chains in every representative poset. We will prove each containment explicitly. For "⊆", take an edge {i, j} ofḠ tor (P ), which is a size-2 toric chain. By Proposition 5.3, {i, j} is a toric chain of P (G, ω ′ ) for all ω ′ ∈ [ω], which means that it is an edge of the transitive closurē G(P (G, ω ′ )). The "⊇" containment is analogous: suppose {i, j} is an edge ofḠ(P (G, ω ′ )) for each ω ′ ∈ [ω]. Then by Proposition 5.3, it is a toric chain of P , and hence an edge ofḠ tor (P ).

Toric antichains. An antichain of an ordinary poset P is a subset A ⊆ V characterized
• combinatorially by the condition that no pair {i, j} ⊆ A with i = j are comparable, that is, they lie on no common chain of P , or • geometrically by the equivalent condition that the (|V | − |A| + 1)-dimensional subspace D πA intersects the open polyhedral cone c(P ) in R V . As shown in [DMR15], these two conditions in the toric setting lead to different notions of toric antichains which are both easy to formulate. Unlike the case of ordinary posets, these two definitions are non-equivalent; leading to two distinct versions of a toric antichains, combinatorial and geometric. The following is the geometric one which we will use in this paper. Its appearance in Proposition 5.17, which is one of the "For Some" structure theorems listed in the Introduction, suggests that it is the more natural toric analogue of the two. Proposition 5.17. Let P = P (G, [ω]) be a toric poset. Then a set A ⊆ V is a geometric toric antichain of P if and only if A is an antichain of P (G, ω ′ ) for some ω ′ ∈ [ω].

Morphisms of toric posets
Morphisms of ordinary posets have equivalent combinatorial and geometric characterizations. In contrast, while there seems to be no simple or obvious combinatorial description for morphisms of toric posets, the geometric version has a natural toric analogue.
Firstly, it is clear how to define a toric isomorphism between two toric posets P and P ′ on vertex sets V and V ′ : a bijection φ : V → V ′ such that the induced isomorphism on R V /Z V → R V ′ /Z V ′ maps c(P ) to c(P ′ ) bijectively. The other types of ordinary poset morphisms have the following toric analogues: quotients that correspond to projecting the toric chamber onto a flat of A tor (G ′ π ) for some closed toric face partition π = cl tor P (π); inclusions that correspond to embedding a toric chamber into a higher-dimensional chamber; extensions that add relations (toric hyperplanes). Since every poset morphism can be expressed as the composition of a quotient, an inclusion, and an extension, it is well-founded to define a toric poset morphism to be the composition of the toric analogues of these maps. In the remainder of this section, we will describe toric morphisms in detail. Most of the difficulties have already been done in Section 3, when interpreting the wellknown concept of an ordinary poset morphism geometrically. In contrast, this section is simply an adaptation of this geometric framework from R V to R V /Z V , though there are some noticeable differences. For example, there is no toric analogue of intersecting a chamber with a half-space, because the torus minus a hyperplane is connected.
6.1. Quotient. In the ordinary poset case, a quotient is performed by contracting P (G, ω) by a partition π = {B 1 , . . . , B r }. Each B i gets collapsed into a single vertex, and the resulting acyclic graph is denoted by ω/∼ π , which is an element of Acyc(G/∼ π ). This does not carry over to the toric case, because in general, contracting a partition will make some representatives acyclic and others not. However, the geometric definition has a natural analogue. Now, let P = P (G, [ω]) be a toric poset, and π be any partition of V closed with respect to P , i.e., π = cl tor P (π). By construction, D tor π is a flat of A tor (G ′ π ), and so the subset F tor π (P ) is a face of A tor (G ′ π ). First, we need a map that projects a point x in c(P ) onto this face, which is relatively open in the subspace topology of D tor π . This can be extended to the entire torus, by taking the unique mapd π that makes the following diagram commute, where d π is the mapping from Eq. (9): Explicitly, the mapd π takes a point x ∈ R V /Z V , lifts it to a pointx in an order polytope in R V , projects it onto the flat D π as in Eq. (9), and then maps that point down to the toric flat D tor π . In light of this, we will say that the mapd π is a projection onto the toric flat D tor π . After projecting a chamber c(P ) onto a flat D tor π of A(G ′ π ), we need to project it homeomorphically onto a coordinate subspace of R V /Z V so it is a chamber of a lower-dimensional toric arrangement. As in the ordinary case, let W ⊆ V be any subset formed by removing all but 1 coordinate from each B i , and letr π : R V /Z V −→ R W /Z W be the induced projection. Ther π -image of A tor (G) will be the toric arrangement A tor (G/∼ π ). As before, the following easily verifiable lemma ensures that our choice of W ⊆ V does not matter.
Moreover, all such projection maps for a fixed π are topologically conjugate in the following sense: and projection mapr ′ π | Dπ : D tor π −→ R W ′ /Z W ′ , and σ is the permutation of V that transposes each b i with b ′ i , then the following diagram commutes: withΣ andΣ ′ being the induced linear maps as defined in Eq. (6), but done modulo 1.
By convexity (in the fundamental affine chambers), two points in the same face of A tor (G) get mapped to the same face in A tor (G/∼ π ). In other words,d π induces a well-defined mapδ π from Face A tor (G) to Face A tor (G ′ π ) making the following diagram commute: Explicitly, the mapδ π is easiest to defined by the analogous map on closed faces: The open faces of A tor (G ′ π ) are then mapped to faces of the arrangement A tor (G/ ∼ π ) under the projectionr π | Dπ : D tor Combinatorially, the open faces of A tor (G) are toric preposets [ω] over G (i.e., in Pre(G)/≡). These are mapped to toric preposets over G/∼ π via the composition The following commutative diagram illustrates the relationship between the points in R V /Z V , the faces of the toric graphic arrangement A tor (G), and the toric preposets over G. The left column depicts the toric preposets over G that are also toric posets.
To summarize, toric poset morphisms that are quotients are characterized geometrically by projecting the toric chamber c(P ) onto a flat of A tor (G ′ π ), for some closed toric face partition π = cl tor P (π). Applying Theorem 4.7 gives a combinatorial interpretation of this, which was not a priori obvious.
The following is now immediate from Propositions 5.14 and 5.17. Corollary 6.3. Let P be a toric poset over V . Then contracting a toric interval I ⊆ V or a geometric toric antichain A ⊆ V defines a toric morphism.
6.2. Inclusion. Just like for ordinary posets, a toric poset can be included in larger one. Let P be a poset over V and let V V ′ . The simplest injection adds vertices (dimension) but no edges (extra relations). In this case, the inclusion φ : P ֒→ P ′ defines a canonical inclusion Φ : This sends the arrangement A tor (G) in R V , where G = (V, E), to the same higher-dimensional arrangement: The toric chamber c = c(P ) is sent to the chamber More generally, an injection P → P ′ can have added relations in P ′ either among the vertices in P or those in V ′ −V . Such a map is simply the composition of an inclusion described above and a toric extension, described below.
6.3. Extension. Extensions of ordinary posets were discussed in Section 3.2. A poset P ′ is an extension of P (both assumed to be over the same set V ) if any of the three equivalent conditions holds: The first of these conditions does not carry over nicely to the toric setting, but the second two do. A toric poset P ′ is a toric extension of P if and only one has an inclusion of their open polyhedral cones c(P ′ ) ⊆ c(P ) in R V /Z V , which is equivalent toĜ torHasse (P ) ⊆Ĝ torHasse (P ′ ).
Note that in the special case of the morphism P −→ P ′ being surjective, the inclusion step is eliminated and the entire process can be described geometrically by projecting c(P ) to a toric flat D tor π and a then adding toric hyperplanes.

Toric order ideals and filters
Let P be a poset over a set V of size at least 2, and suppose φ : P → P ′ is a morphism to a poset over a size-2 subset V ′ ⊆ V . This is achieved by projecting c(P ) onto a flat D π of A(G ′ π ) such that π = cl P (π) has at most two blocks, and henceF π = c(P ) ∩ D π is at most 2-dimensional. A point x = (x 1 , . . . , x n ) on F π has at most two distinct entries. Thus, the partition π = {I, J} of V satisfies The set I is called an order ideal or just an ideal of P and J is called a filter.
Ideal/filter pairs are thus characterized by closed partitions π of V such that D π intersects c(P ) in at most two dimensions. The set of ideals has a natural poset structure by subset inclusion. Allowing I or J to be empty, this poset has a unique maximal element I = V (corresponding to J = ∅) and minimal element I = ∅ (corresponding to J = V ). Moreover, the order ideal poset is a lattice; this is well-known [Sta01]. Similarly, the set of filters is a lattice as well.
Toric order ideals and filters can be defined similarly.
Definition 7.1. Let P be a toric poset over V , and suppose φ : P → P ′ is a morphism to a toric poset over a size-2 subset V ′ ⊆ V . This projects c(P ) onto a toric flat D tor π of A tor (G ′ π ) for some π = cl tor P (π) such thatF tor π = c(P ) ∩ D tor π is at most 2-dimensional. For the partition π = {I, J} of V , each point x = (x 1 , . . . , x n ) on F tor π satisfies x i k mod 1 = x i ℓ mod 1 for all i k , i ℓ in I; x j k mod 1 = x j ℓ mod 1 for all j k , j ℓ in J. The set I is called a toric order ideal of P .
Remark 7.2. By symmetry, if I is a toric order ideal, then so is J := V −I. A toric filter can be defined analogously, and it is clear that these two concepts are identical. Henceforth, we will stick with the term "toric filter" to avoid ambiguity with the well-established but unrelated notion of a toric ideal from commutative algebra and algebraic geometry [Stu96].
By construction, toric filters are characterized by closed toric partitions π of V such that D tor π intersects c(P ) in at most two dimensions -either a two-dimensional face of P or of an extension P ′ over G ′ π . Proposition 7.3. Let P (G, [ω]) be a toric poset. The following are equivalent for a subset I ⊆ V .
Proof. The result is obvious if I = ∅ or I = V , so assume that ∅ I V , and π = {I, V −I}. This forces D tor π to be two-dimensional (rather than one-dimensional). (i)⇒(ii): If I is a toric filter of P (G, [ω]), then D tor π intersects c(P (G, [ω])) in two-dimensions, and so D π intersects an order polytope O(P (G, ω ′ )) in two-dimensions, for some ω ′ ∈ [ω]. Therefore, D π intersects the chamber c(P (G, ω ′ )) in two-dimensions, and hence I is an ideal of P (G, ω ′ ).
(ii)⇔(iii): Immediate by Remark 7.2 upon reversing the roles of I and V −I.
(ii)⇒(iv): If I is a size-k ideal of P (G, ω ′ ), then by a well-known property of posets, there is a linear extension of the form (i 1 , . . . , i k , v k+1 , . . . , v n ), where each i j ∈ I. The cyclic equivalence class [(i 1 , . . . , i k , v k+1 , . . . , v n )] is a total toric extension of P (G, [ω ′ ]) = P (G, [ω]) in which the elements of I appear in consecutive cyclic order.
(iv)⇒(ii): Suppose [(i 1 , . . . , i k , v k+1 , . . . , v n )] is a toric total extension of P (G, [ω]). This means that for some x ∈ R V /Z V , The unique preimagex of this point in [0, 1) V under the quotient map q :  Given a toric poset, we can define the characteristic function χ I of a toric filter similarly. However, one must be careful because under the canonical quotient to the torus, the vertices of every order polytope get identified to (0, . . . , 0). Therefore, we will still identify χ I with a point in R V , not R V /Z V . Let J tor (P ) denote the set of toric filters of P . This has a natural poset structure by subset inclusion. Once again, there is a unique maximal element I = V and minimal element I = ∅.
Proposition 7.6. With respect to subset inclusion and cardinality rank function, J tor (P ) is a graded poset.
Proof. Let P = P (G, [ω]) be a toric poset over G = (V, E). It suffices to show that every nonempty toric filter J contains a toric filter J ′ of cardinality |J ′ | = |J| − 1. By Proposition 7.3, the set J is an order ideal of P ′ = P (G, ω ′ ) for some ω ′ ∈ [ω]. Choose any minimal element v ∈ V of P ′ , which is a source of ω ′ . Let ω ′′ be the orientation obtained by flipping v into a sink. The set J ′ := J−{v} is an ideal of P (G, ω ′′ ), and so by Proposition 7.3, it is a toric filter of P (G, [ω ′′ ]) = P (G, [ω]).
Example 7.7. Let G = C 4 , the circle graph on 4 vertices, and let ω ∈ Acyc(G) be the orientation shown at left in Figure 4. The Hasse diagram of the poset P = P (G, ω) is a line graphĜ Hasse (P ) = L 4 , and the transitive closure isḠ(P ) = K 4 . Since V is a size-4 toric chain, it is totally cyclically ordered in every ω ′ ∈ [ω], and the dashed edges are additionally implied by toric transitivity. Thus, The 4 torically equivalent orientations are shown in Figure 4. The only total toric extension of (1, 2, 3, 4) , (2, 3, 4, 1) , (3, 4, 1, 2) , (4, 1, 2, 3)] , and this is shown at right in Figure 5. The toric filters are all subsets of V that appear as an initial segment in one of these four total orders. The poset J tor (P (C 4 , [ω])) is shown at left in Figure 5. Note that unlike the ordinary poset case, it is not a lattice.
Example 7.8. Let G = C 4 , as in Example 7.7, but now let ω ′ ∈ Acyc(G) be the orientation shown at left in Figure 6. The only nonempty toric chains are the four vertices (size 1) and the four edges (size 2). Since ω ′ has no toric chains of size greater than 2, the Hasse diagram and the transitive closure of the toric poset P (G, [ω ′ ]) are both C 4 . Note that the transitive closure of the (ordinary) poset P (G, ω ′ ) contains the edge {1, 3}, and so as graphs,Ḡ(P (G, ω ′ )) =Ḡ tor (P (G, [ω ′ ])). The 6 torically equivalent orientations of ω ′ are shown in Figure 6. There are four total toric extensions of P (G, [ω ′ ]) which are shown on the right in Figure 7, as cyclic words. The toric filters are all subsets of V that appear as a consecutive segment in one of these four total orders. The poset J tor (P (C 4 , [ω ′ ])) of toric filters is shown at left in Figure 7. In this particular case, the poset of toric filters is a lattice. In fact, it is isomorphic to a Boolean lattice, because every subset of {1, 2, 3, 4} appears consecutively (ignoring relative order) in one of the four cyclic words in Figure 7.

Application to Coxeter groups
A Coxeter system is a pair (W, S) consisting of a Coxeter group W generated by a finite set of involutions S = {s 1 , . . . , s n } with presentation W = S | s 2 i = 1, (s i s j ) mi,j = 1 where 2 ≤ m i,j ≤ ∞ for i = j. The corresponding Coxeter graph Γ has vertex set V = S and edges {i, j} for each m i,j ≥ 3 labeled with m i,j (label usually omitted if m i,j = 3). A Coxeter element is the product of the generators in some order, and every Coxeter element c ∈ W defines a partial ordering on S via an acyclic orientation ω(c) ∈ Acyc(Γ): Orient s i → s j iff s i precedes s j in some (equivalently, every) reduced expression for c. Conjugating a Coxeter element by an initial generator (note that s i = s −1 i ) cyclically shifts it: s x1 (s x1 s x2 · · · s xn )s x1 = s x2 · · · s xn s x1 , and the corresponding acyclic orientation differs by reversing the orientations of all edges incident to s x1 , thereby converting it from a source to a sink vertex. In 2009, H. and K. Eriksson showed [EE09] that two Coxeter elements c and c ′ are conjugate if and only if ω(c) ≡ ω(c ′ ). Thus, there are bijections between the set C(W ) of Coxeter elements and Acyc(Γ), as well as between the corresponding conjugacy classes and the toric equivalence classes: The toric equivalence class containing ω(c ′ ) has six orientations, which were shown in Figure 6. Each of these describes a unique conjugate Coxeter element: s 1 s 2 s 4 s 3 s 2 s 4 s 1 s 3 s 4 s 1 s 3 s 2 s 2 s 1 s 3 s 4 s 1 s 3 s 2 s 4 s 3 s 2 s 4 s 1 = s 1 s 4 s 2 s 3 = s 2 s 4 s 3 s 1 = s 4 s 3 s 1 s 2 = s 2 s 3 s 1 s 4 = s 1 s 3 s 4 s 2 = s 3 s 4 s 2 s 1 = s 4 s 2 s 3 s 1 = s 3 s 1 s 4 s 2 = s 4 s 2 s 1 s 3 = s 3 s 1 s 2 s 4 These are listed above so that the Coxeter element in the i th column corresponds to the i th orientation in Figure 6. The linear extensions of each orientation describe the reduced expressions of the corresponding Coxeter element, which are listed in the same column above. The toric poset P (G, [ω(c ′ )]) has four total toric extensions, and these were shown on the right in Figure 7 (replace k with s k ). The toric filters of P (G, [ω(c ′ )]) correspond to the subsets that appear consecutively in one of these cyclic words. The poset J tor (P (C 4 , [ω ′ ])) of toric filter appears on the left in Figure 7.

Concluding remarks
In this paper, we further developed the theory of toric posets by formalizing the notion of toric intervals, morphisms, and order ideals. In some regards, much of the theory is fairly analogous to that of ordinary posets, though there are some noticeable differences. Generally speaking, the one recurring theme was the characterization of the toric analogue of a feature in P (G, [ω]) by the characterization of the ordinary version of that feature in P (G, ω ′ ) either for some ω ′ ∈ [ω], or for all ω ′ ∈ [ω].
One question that arises immediately is whether there is a toric order complex. While there may exist such an object, there are some difficulties unique to the toric case. For example, a  Figure 8. Two non-torically equivalent orientations ω ≡ ω ′ in Acyc(C 5 ) for which P (C 5 , [ω]) and P (C 5 , [ω ′ ]) have the same set of toric chains. poset is completely determined by its chains, in that if one specifies which subsets of V are the chains of P , and then the toric order of the elements within each chain, the entire poset can be reconstructed. This is not the case for toric posets, as shown in Figure 8. Here, two torically non-equivalent orientations of C 5 are given, but the toric posets P (C 5 , [ω]) and P (C 5 , [ω ′ ]) have the same sets of toric chains: the 5 vertices and the 5 edges.
The fact that an ordinary poset is determined by its chains just means that once one specifies the total order between every chain of size k ≥ 2, then the entire partial order is determined. The problem for toric posets, which we encountered in this paper, is that every size-2 subset is trivially cyclically ordered, whether it lies on a toric chain or not. In other words, a total order can be defined on two elements, but a cyclic order needs three. The analogous statement for toric posets would be that specifying the total cyclic order between every toric chain of size k ≥ 3 specifies the entire toric order. Such a statement would establish the intuitive idea that knowing all total cyclic orders should determine the toric partial order "modulo the size-2 toric chains." Current works suggests that there is an analogue of the aforementioned properties for toric posets, but it requires a new generalization of the concept of a chain. The details are too preliminary and complicated to describe here, and it is not clear whether it will lead to a combinatorial object such as a toric order complex. Without this, there might not be a natural way to study toric posets topologically.
Another important feature of ordinary posets that does not seem to have any obvious toric analogue are Möbius functions, and this is vital to much of the theory of ordinary posets. Recall the analogy from the Introduction about how topology is like "analysis without the metric." Similarly, many of the basic features of ordinary posets have toric analogues, despite the fact that toric posets have no binary relation. However, much of the more advanced theory is likely to fail because one also seems to lose valuable tools such as an order complex and a Möbius function. Even the theory that does carry over has its shortcomings. For example, morphisms have a simple combinatorial characterization using the binary relation: i < P j implies φ(i) < P ′ φ(j). The geometric definition requires a patchwork of quotients, extensions, and inclusions. It would be desirable to have a more "holistic" characterization of toric poset morphisms, though it is not clear that that such a description should exist.
Finally, the connection of toric posets to Coxeter groups is the subject of a paper nearing completion on cyclic reducibility and conjugacy in Coxeter groups. Loosely speaking, reduced expressions can be formalized as labeled posets called heaps. This was formalized by Stembridge [Ste96,Ste98] in the 1990s. The fully commutative (FC) elements are those such that "long braid relations" (e.g., sts → tst) do not arise. Equivalently, they have a unique heap. The cyclic version of the FC elements are the cyclically fully commutative CFC elements, introduced by the author and collaborators in [BBE + 12]. In 2013, T. Marquis showed that two CFC elements are conjugate if and only if their heaps are torically equivalent [Mar14]. These elements were further studied by M. Pétréolle [Pét14]. In our forthcoming paper, we will formalize the notion of a toric heap, which will essentially be a labeled toric poset. This allows us to formalize objects such as cyclic words, cyclic commutativity classes, and develop a theory of cyclic reducibility in Coxeter groups using the toric heap framework.