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Article

Morphisms and Order Ideals of Toric Posets

by
Matthew Macauley
Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, USA
Mathematics 2016, 4(2), 39; https://doi.org/10.3390/math4020039
Submission received: 1 January 2015 / Revised: 19 May 2016 / Accepted: 25 May 2016 / Published: 4 June 2016

Abstract

:
Toric posets are in some sense a natural “cyclic” version of finite posets in that they capture the fundamental features of a partial order but without the notion of minimal or maximal elements. They can be thought of combinatorially as equivalence classes of acyclic orientations under the equivalence relation generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper, we define toric intervals and toric order-preserving maps, which lead to toric analogues of poset morphisms and order ideals. We develop this theory, discuss some fundamental differences between the toric and ordinary cases, and outline some areas for future research. Additionally, we provide a connection to cyclic reducibility and conjugacy in Coxeter groups.
MSC:
06A06, 52C35

1. 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 [1], Coxeter groups [2,3,4], graph polynomials [5], lattices [6,7], and quiver representations [8]. These equivalence classes were first formalized as toric posets in [9], 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 [10] and later extended to signed graphs by Zaslavsky [11].
Combinatorially, a poset over a graph G is determined by an acyclic orientationω 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 [9] 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 $ω ′ ∈ [ ω ]$. (Proposition 5.3)
The edge ${ i , j }$ is in the toric transitive closure of $P ( G , [ ω ] )$ iff ${ i , j }$ is in the transitive closure of $P ( G , ω ′ )$ for all $ω ′ ∈ [ ω ]$. (Proposition 5.15)
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 $( x , y ) ∈ R$, and say that $x < R y$ if $x ≤ R y$ and $y ≰ R x$. Much of the basics on preposets can be found in [12].
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 a directed edge (which we denote by $i → j$) 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 if $i ≠ k$. However, for any directed graph Γ, there is a unique minimal preorder $P Γ$ containing the edge set of Γ; $ω ( P Γ )$ is the transitive closure of Γ. 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 $R o p$ denote the opposite binary relation, meaning that $( x , y ) ∈ R o p$ 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 $Q = P ∪ R o p ¯$. Intuitively, each added edge $( x , y ) ∈ R o p$ 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

Every finite poset corresponds to a chamber of at least one graphic hyperplane arrangement. The interested reader is encouraged to consult the excellent texts of Zaslavsky [13] on the combinatorial geometry of arrangements, or of Wachs [14] on poset topology. A brief correspondence between posets and chambers of graphic hyperplane arrangements will be descsribed 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 $R V$:
$c = c ( P ) : = { x ∈ R V : x i < x j if i < P j }$
The (topological) closure of the cone $c ( P )$ is $c ( P ) ¯ = { x ∈ R V : x i ≤ x j if i < P j }$.
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 $H i j$ is the set
$H i j : = { x ∈ R V : x i = x j }$
The graphic arrangement of G is the set $A ( G )$ of all hyperplanes $H i j$ in $R V$ where ${ i , j }$ is in E. 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 graph $G ^ Hasse ( P )$ called the Hasse diagram of P and a unique maximal graph $G ¯ ( P )$, where $( G , ω ) ⟼ ( G ¯ ( P ) , ω ¯ )$ 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 some 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 $B i ≤ P B j$ whenever $x ≤ P y$ 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 fundamentally different.
Compatibility of partitions with a poset can be characterized by a closure operator on $Π V$. If $P π = ( π , ≤ P )$ is a preposet, 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 simply $π ¯ : = cl P ( π )$. A partition π is closed (with respect to P) if $π ¯ = π$, which is equivalent to being compatible with P. Geometrically, it means that for any $i ≠ j$, there is some $x ∈ c ( P ) ¯ ∩ D π$ such that $x b i ≠ x b j$ for some $b i ∈ B i$ and $b j ∈ B j$. If π is not closed, then $c ( P ) ¯ ∩ D π$ has strictly lower dimension than $D π$. In this case, $π ¯$ is the unique coarsening that is closed with respect to P and satisfies $c ( P ) ¯ ∩ D π ¯ = c ( P ) ¯ ∩ D π$.
Still assuming that P is a poset over $G = ( V , E )$, and π is a partition of V, define
$F ¯ π ( P ) : = c ( P ) ¯ ∩ D π$
If $D π$ is a flat of $A ( G ¯ ( P ) )$, then $F ¯ π ( P )$ is a face of the (topologically) closed polyhedral cone $c ( P ) ¯$. In the latter case, we say that π is a face partition of P. Since it is almost always clear what P is, we will usually write $F ¯ π$ instead of $F ¯ π ( P )$. 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 [15], and also done in 1 of Stanley [16] in the characterization of the face structure of the order polytope of a poset, defined by
$O ( P ) = { x ∈ [ 0 , 1 ] V : x i ≤ x j if i ≤ P j } = c ( P ) ¯ ∩ [ 0 , 1 ] V$
Our notation is a little different from [16]; for example, we do not adjoin a minimal or maximal element ($0 ^$ or $1 ^$) to P. Also, our definition of “closed faced partitions” is in terms of $c ( P ) ¯$ rather than $O ( P )$, but since $O ( P ) = c ( P ) ¯ ∩ [ 0 , 1 ] V$, our “closed face partitions” correspond to those of $P ^ = P ∪ { 0 ^ , 1 ^ }$ with $0 ^$ and $1 ^$ as singleton blocks, yielding the faces of $O ( P )$ that don’t lie in any hyperplane $x i = 0 , 1$. These correspond to the faces of $c ( P ) ¯$, and each closed face intersects at least the diagonal $x 1 = ⋯ = x n$. 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.
Theorem 2.1.
([16], Theorem 1.2) Let P be a poset over $G = ( V , E )$. A partition π of V is a closed face partition of P if and only if it is connected and compatible with 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 the flat $D π$ might cut through the interior of $O ( P )$ (and hence of $c ( P )$); see Example 2.2, or it might lie in a proper face of $O ( P )$; see Example 2.5. In either 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: $cl P ( π ) = cl P ( π ′ ) = 1234$.
If $π ¯ = π = { B 1 , ⋯ , B r }$ is a closed face partition of P, then $D π$ is an r-dimensional flat of $A ( G )$, and the closed face $F ¯ π$ is an r-dimensional subset of $D π ⊆ R V$. The interior of $F ¯ π$ 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 of $F ¯ π$ 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:
$Face A ( G ) = ⋃ ω ∈ Acyc ( G ) Face ( P ( G , ω ) ) , Face ¯ A ( G ) = ⋃ ω ∈ Acyc ( G ) Face ¯ ( P ( G , ω ) )$
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 0-dimensional faces of $A ( G )$. The open n-dimensional faces of $A ( G )$ are its chambers. Additionally, $R V$ is a disjoint union of open faces of $A ( G )$:
$R V = ⋃ F ∈ Face A ( G ) ˙ F .$
Example 2.5.
Let P be the poset over $V = { 1 , 2 , 3 , 4 }$, where the only covering relation is $1 < P 2$, and consider the partition $π = 12 / 34$, which is closed but not connected with respect to P. Explicitly, $c ( P )$ and $D π$ are the following sets:
$c ( P ) = { x ∈ R 4 : x 1 < x 2 } , D π = { x ∈ R 4 : x 1 = x 2 , x 3 = x 4 }$
Since $D π ⊆ c ( P ) ¯ = { x ∈ R 4 : x 1 ≤ x 2 }$, the intersection is $F ¯ π = D π ∩ c ( P ) ¯ = D π$, which is not a face of the polyhedral cone $c ( P ) ¯$; rather it is a proper subset of the face
$F ¯ 12 / 3 / 4 = D 12 / 3 / 4 ∩ c ( P ) ¯ = D 12 / 3 / 4 = { x ∈ R 4 : x 1 = x 2 }$
Finally, note that $F ¯ π$ is a face of an extension $P ′$ of P, namely the one formed by adding the covering relation $3 < P ′ 4$. Therefore, π is not a face partition of P, but it is a face partition of $P ′$.
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, where one allows bi-directed edges. These were called Type B fourientations in a recent paper by Backman and Hopkins [17], though we will simply call them “orientations” since we don’t refer to them often, and when we do, it should always be clear from the context that bi-directed edges are allowed. The notation $ω π$ 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,
$Pre ( G ) = { ω π ∣ ω ∈ Acyc ( G ) , π closed face partition of P ( G , ω ) }$
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 $( π , ≤ )$.
The mapping $R V - A ( G ) ⟶ α G Acyc ( G )$ in Equation (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 $F π ( P ) = F π ¯ ( P )$ for some poset P and closed face partition $π ¯ = cl P ( π )$, then the preposet $P π = ( π , ≤ P )$ has vertex set $π = { B 1 , ⋯ , B r }$; these are the strongly connected components of the orientation $α G ( x )$.

3. 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 $ϕ : V → V ′$ characterized
• combinatorially by the condition that $i < P j$ is equivalent to $ϕ ( i ) < P ′ ϕ ( j )$ for all $i , j ∈ V$;
• geometrically by the equivalent condition that the induced isomorphism $Φ : R V → R V ′$ 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:
$( x 1 , x 2 , ⋯ , x n ) ⟼ Φ ( x ϕ - 1 ( 1 ) , x ϕ - 1 ( 2 ) , ⋯ , x ϕ - 1 ( n ) )$
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 $ϕ : V → V ′$ such that $i ≤ P j$ implies $ϕ ( i ) ≤ P ′ ϕ ( j )$ for all $i , j ∈ V$. 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 to $F ¯ π ¯ : = c ( P ) ¯ ∩ D π ¯$, where $π ¯ = cl P ( π )$. This is the mapping
$d π : c ( P ) ⟶ D π , d π ( x ) = ( x + D π ¯ ⊥ ) ∩ D π ¯$
By construction, the image of this map is $F π ¯$, 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 at least two (closed) chambers along facets, and intersects the boundary of every chamber (because each hyperplane and each closed chamber contains the line $x 1 = ⋯ = x n$).
Remark 3.1.
The map $d π$ simply replaces every coordinate of x by the average of the coordinates from the same π-block. This is a basic fact from statistics about the usual estimator for the mean; see a book on linear models, e.g., Chapter 2 (Estimation) of [18] for details.
Example 3.2.
Let $G = K 3$, the complete graph on three 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, $O ( P i )$.
Contracting $ω 1$ and $ω 2$ by the partition $π = { B 1 = { 1 } , B 2 = { 2 , 3 } }$ yields a poset over ${ B 1 , B 2 }$; these are shown directly below the orientations in Figure 2. Therefore, π is closed with respect to $P 1$ and $P 2$. Geometrically, the flat $D π$ intersects the closed chambers $c ( P i ) ¯$ for $i = 1 , 2$ in 2-dimensional faces.
In contrast, contracting $P 3$ by π yields a preposet that is not a poset. Therefore, π is not closed with respect to $P 3$. Indeed, $π ¯ : = cl P ( π ) = 123 : = { { 1 , 2 , 3 } }$, and the flat $D π ¯$ intersects the closed chamber $c ( P 3 ) ¯$ in a line. Modding out the preposet $P π : = ( π , ≤ P 3 )$ by its strongly connected components yields a one-element poset. Geometrically, the chamber $c ( P 3 )$ projects onto the one-dimensional face $F ¯ π ¯$.
To see why the map $d π$ from Equation (7) does not extend to the closure of the chambers in a well-defined 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 Equation (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 two-dimensional closed face $F ¯ π ( P 4 ) = c ( P 4 ) ¯ ∩ D π$. The point y is projected orthogonally onto the plane $D π$, and does not end up on the line $x 1 = x 2 = x 3$.
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 coarsening $π ¯ : = cl P ( π )$ of π such that $( π ¯ , ≤ P )$ is acyclic. The map $d π$ can now be extended to all of $R V$, as
$d π : R V ⟶ D π , d π ( x ) = ( x + D π ¯ ⊥ ) ∩ D π ¯ , where π ¯ = π ( x ) ¯ : = cl α G ( x ) ( π )$
where $α G$ is the map from Equation (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 with $π ¯ = cl P ( π )$, the subset $F ¯ π : = c ( P ) ¯ ∩ D π ¯$ 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 set $F ¯ π ¯ = F ¯ π = c ( P ) ¯ ∩ D π$, for $π ¯ = cl P ( π )$, 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 Equation (8) projects a chamber $c ( P )$ onto a flat $D π ¯$ of $A ( G π ′ )$, where $π ¯ = cl P ( π )$. 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. Each hyperplane in $A ( G )$ either gets mapped to a hyperplane in $A ( G / ∼ π )$ or to all of $R W$, and every hyperplane in $A ( G / ∼ π )$ is the $r π$-image of at least one hyperplane in $A ( G )$. In other words, $r π$-preimage of $A ( G / ∼ π )$ is $A ( G )$. The following ensures that $r π$ is a homeomorphism, and that the choice of W does not matter. We omit the elementary proof.
Lemma 3.3.
Let $π = { B 1 , ⋯ , B r }$ be a partition of V, and $W = { b 1 , ⋯ , b r } ⊆ V$ with $b i ∈ B i$. The restriction $r π | D π : D π ⟶ R W$ is a homeomorphism.
Moreover, all such projection maps for a fixed π are topologically conjugate in the following sense: If $W ′ = { b 1 ′ , ⋯ , b r ′ } ⊆ V$ with $b i ′ ∈ B i$ and projection map $r π ′ | D π : D π ⟶ R W ′$, and σ is the permutation of V that transposes each $b i$ with $b i ′$, then the following diagram commutes:
Here, $σ ′ : W → W ′$ is the map $b i ↦ b i ′$, with Σ and $Σ ′$ being the induced linear maps as defined in Equation (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 map $δ ¯ π$ on closed faces. Let $F ¯ = c ( P ) ¯ ∩ D σ$ be a closed face of $A ( G )$, for some closed face partition $σ ∈ Π V$. Then the map $δ ¯ π$ is defined by
$δ ¯ π : Face ¯ A ( G ) ⟶ Face ¯ A ( G π ′ ) , δ ¯ π : c ( P ) ¯ ∩ D σ ⟼ c ( P ) ¯ ∩ D σ ∩ D π$
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 $r π | D π : D π ⟶ R W$. [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
$Pre ( G ) ⟶ q π Pre ( G π ′ ) ⟶ p π Pre ( G / ∼ π ) , ω σ = ω ⟼ q π ω π ′ ⟼ p π ω π ′ / ∼ π ¯ = ω / ∼ π ¯$
Here, $π ¯ = cl P σ ( π )$, 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 $D π ¯$ of $A ( G π ′ )$ for some partition $π ¯ = cl P ( π )$, and then homeomorphically mapping this down to a chamber of a lower-dimensional graphic arrangement $A ( G / ∼ π )$. Equivalently, contracting $P ( G , ω )$ by $π ¯$ yields an acyclic preposet $P π ¯ = ( π ¯ , ≤ P )$. 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 3.4.
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
$π S = { B 1 = S , B 2 , ⋯ , B r } , where | B i | = 1 for i = 2 , ⋯ , r$
Contracting an interval $I ⊆ V$ in a poset P yields the poset $P π I = ( π I , ≤ P )$. 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 $P π A = ( π A , ≤ P )$. Note that $D π I$ is a flat of $A ( G )$ and lies on the boundary of $c ( P )$, but the $( | V | - | A | + 1$)-dimensional subspace $D π A$ cuts through the interior of $c ( P )$. For both of these cases, $S = I$ and $S = A$, the subspace $D π S$ is trivially a flat of $A ( G π S ′ )$.

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 $H i < j : = { x ∈ R V : x i < x j }$.

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 of $V - V ′$ to P but no extra relations, and then the map $P ′ ′ → P ′$ is an extension. This gives an inclusion of polyhedral cones:
$c ( P ′ ) : = { x ∈ R V ′ : x i < x j for i < P ′ j } ⊆ c ( P ′ ′ ) = { x ∈ R V ′ : x i < x j for i < P j } ≅ c ( P ) × R V ′ - V$

3.4. Summary

Up to isomorphism, every morphism from 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)
(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.

4. Toric Posets and Preposets

4.1. Toric Chambers and Posets

Toric posets, introduced in [9], 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 map $R V ⟶ q R V / Z V$. Elements of $A tor ( G )$ are toric hyperplanes
$H i j tor : = { x ∈ R V / Z V : x i mod 1 = x j mod 1 } = q ( H i j )$
Though a point $x ∈ R V / Z V$ is a residue class, we will at times want to compare its coordinates, modulo 1. The convenient way to do this is to identify x with its unique representative in $[ 0 , 1 ) V$, and under this slight abuse of notation we can unambiguously make statements such as $x i mod 1 ≤ x j mod 1$.
Definition 4.1.
A connected component c of the complement $R V / Z V - A tor ( G )$ 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 map $α ¯ G : x ↦ ω ( x )$. 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 mapping $α ¯ G$ is essentially the same as $α G$ from Equation (5) except done modulo 1, so many of its properties are predictably analogous. For example, the undirected version of $ω ( x )$ is G. The edge ${ i , j }$ is bidirected in $ω ( x )$ if and only if $x i mod 1 = x j mod 1$. Therefore, $ω ( x )$ is acyclic if and only if x lies in $R V / Z V - A tor ( G )$; 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 [9]. 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 4.3.
([9], Theorem 1.4) The map $α ¯ G$ induces a bijection between $Cham A tor ( G )$ and $Acyc ( G ) / ≡$ as follows:
In other words, two points $x , x ′$ in $R V / Z V - A tor ( G )$ have $α ¯ G ( x ) ≡ α ¯ G ( x ′ )$ if and only if $x , x ′$ lie in the same toric chamber of $A tor ( G )$.
By Theorem 4.3, every pair $( G , [ ω ] )$ of a graph G and $ω ∈ Acyc ( G )$ determines a toric poset, and we denote this by $P ( G , [ ω ] )$. Specifically, $P ( G , [ ω ] )$ is the toric poset $P ( c )$ such that $α ¯ G ( c ) = [ ω ]$. If the graph G is understood, then we may denote the corresponding toric chamber by $c [ ω ] : = c ( P ( G , [ ω ] ) )$.
If $G = ( V , E )$ is fixed, then the unit cube $[ 0 , 1 ] V$ in $R V$ is the union of order polytopes $O ( P ( G , ω ) )$, any two of which only intersect in a subset of a flat of $A ( G )$:
$[ 0 , 1 ] V = ⋃ ω ∈ Acyc ( G ) O ( P ( G , ω ) )$
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 chamber $c ¯ [ ω ]$. Moreover, by Theorem 4.3, the closed chambers of $A tor ( G )$ are unions of q-images of torically equivalent order polytopes.
Corollary 4.4.
Let $P = P ( G , [ ω ] )$ be a toric poset, and $q : [ 0 , 1 ] V → R V / Z V$ the natural quotient. The closure of the chamber $c ( P )$ is
$c ( P ) ¯ = ⋃ ω ′ ∈ [ ω ] q ( O ( P ( G , ω ′ ) ) )$

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 $R V$:
$A aff ( G ) : = q - 1 ( A tor ( G ) ) = q - 1 ( q ( A ( G ) ) ) = ⋃ { i , j } ∈ E k ∈ Z { x ∈ R V : x i = x j + k }$
where $q : R V → R V / Z V$ is the natural quotient. 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 lifts $x ^$ and $y ^$ 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 by $c ^$ or $c aff$. Each hyperplane $H i j tor$ has a unique preimage containing the origin in $R V$ called its central preimage; this is the ordinary hyperplane $H i j = { x ∈ R V : x i = x j }$. Thus, the set of central preimages of $A tor ( G )$ is precisely the graphic arrangement $A ( G )$ in $R V$. Each closed affine chamber $c ¯ aff$ 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
$D π tor = { x ∈ R V / Z V : x i = x j for every pair i , j in the same block B k of π } = q ( D π )$
Since $R V ⟶ q R V / Z V$ is a covering map, it is well-founded to declare the dimension of a toric flat $D π tor$ 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 $( π , ≤ P 1 )$ and $( π , ≤ P 2 )$ are acyclic but $( π , ≤ P 3 )$ is not. Despite this, every set $D π tor$, 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
$F ¯ π tor ( P ) : = c ( P ) ¯ ∩ D π tor$
If $D π tor$ 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 closure $G ¯ ( P ( G , ω ) )$ 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 $H i , j tor ∩ c ( P ) = ⌀$. This was done in [9], 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 $A tor ( G )$ are closed under intersections, there is a unique maximal toric subspace $D π ¯ tor$ (that is, of minimal dimension) for which $F ¯ π tor = c ( P ) ¯ ∩ D π ¯ tor$. The partition $π ¯$ is the unique minimal coarsening of π for which $F ¯ π tor = F ¯ π ¯ tor$, 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 $cl P tor ( π )$, and we define $dim ( F ¯ π tor ) : = dim ( D π ¯ tor )$. A partition π is closed with respect to the toric poset P if $π = cl P tor ( π )$. Note that the closure is defined for all partitions, not just toric face partitions.
Definition 4.5.
A set $F ⊆ R V / Z V$ is a closed face of the toric poset P if $F = F ¯ π tor = c ( P ) ¯ ∩ D π tor$ for some closed toric face partition $π = cl P tor ( π )$. The interior of $F ¯ π tor$ with respect to the subspace topology of $D π tor$ is called an open face of P, and denoted $F π tor$. Let $Face ( P )$ and $Face ¯ ( P )$ denote the set of open and closed faces of P, respectively. Finally, define the faces of the toric graphic arrangement $A tor ( G )$ to be the faces of the toric posets over G:
$Face A tor ( G ) = ⋃ ω ∈ Acyc ( G ) Face ( P ( G , [ ω ] ) ) , Face ¯ A tor ( G ) = ⋃ ω ∈ Acyc ( G ) Face ¯ ( P ( G , [ ω ] ) )$
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 of $F ¯ π tor ( P ) : = c ( P ) ¯ ∩ D π tor$ 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,
$dim F π tor ( P ( G , [ ω ] ) ) = max ω ′ ∈ [ ω ] dim F π ( P ( G , ω ′ ) )$
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 $A tor ( G )$:
$R V / Z V = ⋃ F ∈ Face A tor ( G ) ˙ F$
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 $π ↦ F π tor$. To classify the faces of a toric poset, it suffices to classify the closed toric face partitions.
Theorem 4.7.
Let $P = P ( G , [ ω ] )$ be a toric poset over $G = ( V , E )$. A partition π of V is a closed toric face partition of P if and only if it is connected and compatible with $P ( G , ω ′ )$, for some $ω ′ ∈ [ ω ]$.
The proof of Theorem 4.7 will be done later in this section, after the following lemma, which establishes that $cl P tor$ is a closure operator [19] on the partition lattice $Π V$ and compares it with $cl P$.
Lemma 4.8.
Let ω be an acyclic orientation of a graph $G = ( V , E )$, and π a partition of V.
(a)
If π is closed with respect to $P ( G , ω )$, then π is closed with respect to $P ( G , [ ω ] )$.
(b)
Closure is monotone: if $π ≤ V π ′$, then $cl P tor ( π ) ≤ V cl P tor ( π ′ )$.
(c)
If $π ≤ V π ′ ≤ V cl P tor ( π )$, then $cl P tor ( π ′ ) = cl P tor ( π )$.
(d)
$cl P ( G , [ ω ] ) tor ( π ) ≤ V cl P ( G , ω ) ( π )$.
Proof.
If π is closed with respect to $P ( G , ω )$, then the preposet $ω / ∼ π$ is acyclic. By Remark 4.6, this means that π is closed with respect to $P ( G , [ ω ] )$, which establishes (a).
Part (b) is obvious. Part (c) follows from taking the closure of each term in the chain of inequalities $π ≤ V π ′ ≤ V cl P tor ( π )$:
$cl P tor ( π ) ≤ V cl P tor ( π ′ ) ≤ V cl P tor cl P tor ( π ) = cl P tor ( π )$
To prove (d), let $π ¯ = cl P ( G , ω ) ( π )$, which is closed with respect to $P ( G , ω )$. By (a), $π ¯$ is closed with respect to $P ( G , [ ω ] )$. Using this, along with (b) applied to $π ≤ V π ¯$, yields
$cl P ( G , [ ω ] ) tor ( π ) ≤ V cl P ( G , [ ω ] ) tor ( π ¯ ) = π ¯ = cl P ( G , ω ) ( π )$
whence the theorem. ☐
Example 4.9.
For an example both of where the converse to Lemma 4.8 (a) fails, and where $cl P ( G , [ ω ] ) tor ( π ) ⪇ V cl P ( G , ω ) ( π )$, consider $G = K 3$, and the partition $π = 1 / 23$. Using the same notation as in Example 3.2 and Figure 2, we see that $cl P ( G , ω 3 ) ( π ) = 123$, because the intersection of $c ( P ( G , ω 3 ) ) ¯$ and $D π$ is one-dimensional. Equivalently, the preposet $ω 3 / ∼ π$ has one strongly connected component. In contrast, the intersection of $c ( P ( G , [ ω 3 ] ) ¯$ with $D π tor$ is two-dimensional. Indeed, the preposets $ω 1 / ∼ π$ and $ω 2 / ∼ π$ both have two strongly connected components, and $ω 1 ≡ ω 2 ≡ ω 3$. Therefore,
$1 / 23 = π = cl P ( G , [ ω 3 ] ) tor ( π ) ⪇ V cl P ( G , ω 3 ) ( π ) = 123$
and so π is closed with respect to $P ( G , [ ω 3 ] )$ but not with respect to $P ( G , ω 3 )$.
Proof of Theorem 4.7.
Suppose π is a closed toric face partition of $P = P ( G , [ ω ] )$, and that $F ¯ π tor : = c ( P ) ¯ ∩ D π tor$ has dimension k. Then for some $ω ′ ∈ [ ω ]$, the hyperplane $D π$ must intersect the order polytope $O ( P ( G , ω ′ ) )$ in a k-dimensional face. Thus, $F ¯ π : = c ( P ( G , ω ′ ) ) ¯ ∩ D π$ has dimension k for some $ω ′ ∈ [ ω ]$, and so π is a face partition of $P ( G , ω ′ )$. To see why π is closed with respect to $P ( G , ω ′ )$, suppose there were a coarsening $π ′ ≥ V π$ such that
$F ¯ π ′ = c ( P ( G , ω ′ ) ) ¯ ∩ D π ′ = c ( P ( G , ω ′ ) ) ¯ ∩ D π = F ¯ π$
It suffices to show that $π ′ = π$. Descending down to the torus, the intersection $F ¯ π ′ tor : = c ( P ) ¯ ∩ D π ′ tor$ must have dimension at least k by Equation (17), but no more than k because $F ¯ π ′ tor ⊆ F ¯ π tor$. Thus, we have equality $F ¯ π ′ tor = F ¯ π tor$, which means $π ′ ≤ V cl P tor ( π ) = π$, whence $π ′ = π$. Since π is a closed face partition with respect to $P ( G , ω ′ )$, it is connected and compatible with $P ( G , ω ′ )$ by Theorem 2.1.
Conversely, suppose that π is connected and compatible with $P ( G , ω ′ )$ 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, $D π tor$ is a toric flat of $A tor ( G )$, and so $F ¯ π tor : = c ( P ) ¯ ∩ D π tor$ 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.
Example 4.10.
There are only two simple graphs $G = ( V , E )$ over $V = { 1 , 2 }$: The edgeless graph $G 0 = ( V , ⌀ )$, and the complete graph $K 2 = ( V , { { 1 , 2 } } )$. For both graphs, the complement $R 2 / Z 2 - A ( G )$ is connected. The respective chambers are
$c 0 : = R 2 / Z 2 , and c : = R 2 / Z 2 - H 12 tor$
and so they represent different toric posets, $P ( c 0 )$ and $P ( c )$. Despite this, these chambers have the same topological closures: $c 0 ¯ = c ¯ = R 2 / Z 2$. The lattice of flats of $G 0$ contains one element: $A ( G 0 ) = { D 1 / 2 tor }$, and this flat arises from the permutation $π = 1 / 2$. The lattice of flats of $K 2$ has two elements: $A ( K 2 ) = { D 1 / 2 tor , D 12 tor }$, where $D 1 / 2 tor = R V / Z V$, and $D 12 tor = H 12 tor = { x 1 = x 2 }$. Thus, the closed faces of the corresponding toric posets are
$Face ¯ ( P ( c 0 ) ) = { R V / Z V } , and Face ¯ ( P ( c ) ) = { R V / Z V , H 12 tor }$
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 [9]. 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 map $α ¯ G$ from Equation (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 $F π tor$ that are open in $D π tor$ are chambers in lower-dimensional arrangements that are contractions of $A tor ( G )$, namely by the subspace $D π tor$. 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 map $α ¯ G$ induces a bijection between $Face A tor ( G )$ and $Pre ( G ) / ≡$ as follows:
In other words, given two points $x , x ′$ in $R V / Z V$ the equivalence $α ¯ G ( x ) ≡ α ¯ G ( x ′ )$ holds if and only if $x , x ′$ lie on the same open face of $A tor ( G )$.
Definition 4.12.
A toric preposet is a set that arises as an open face of a toric poset $P = P ( c )$ for at least one graph G.
If x lies on a toric face $F π tor$ of P, where (without loss of generality) $π = cl P tor ( π )$, then the strongly connected components of the preposet $α ¯ G ( x )$ are $π = { B 1 , ⋯ , B r }$.
Example 4.13.
Let $G = K 3$, as in Example 3.2. The six acyclic orientations of G fall into two toric equivalence classes. The three orientations shown in Figure 2 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:
$c ( P ) ¯ = c ¯ [ ω i ] = ⋃ i = 1 3 q ( O ( P ( G , ω i ) ) )$
This should be visually clear from Figure 2. The two chambers in $A tor ( G )$ are the three-dimensional faces of P. Each of the three toric hyperplanes in $A tor ( G ) = { H 12 tor , H 13 tor , H 23 tor }$ 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 $K 3$:
${ { 3 } → { 1 , 2 } , { { 2 } → { 1 , 3 } , { { 1 } → { 2 , 3 } , { 3 } → { 1 , 2 } } { 2 } → { 1 , 3 } } { 1 } → { 2 , 3 } }$
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.

5. 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 [9], 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 Section 5.1, Section 5.2 and Section 5.3 can be found in [9]. 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 $L tor ( P )$.
Theorem 5.1.
([9], Proposition 1.7) Any toric poset P is completely determined by its total toric extensions in the following sense:
$c ( P ) ¯ = ⋃ P ′ ∈ L tor ( P ) c ( P ′ ) ¯$

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 [9], and it leads to the notion of a toric chain, which is a totally cyclically ordered subset.
Definition 5.2.
Let $P = P ( G , [ ω ] )$ be a toric poset. A subset $C = { i 1 , … , i m } ⊆ V$ is a toric chain of P if there exists a cyclic equivalence class $[ ( i 1 , … , i m ) ]$ of linear orderings of C with the following property: for every $x ∈ c ( P )$ there exists some $( j 1 , ⋯ , j m )$ in $[ ( i 1 , … , i m ) ]$ for which
$0 ≤ x j 1 < x j 2 < ⋯ < x j m < 1$
In this situation, we will say that $P | C = [ ( i 1 , … , i m ) ]$.
The following is a reformulation of Proposition 6.3 of [9] using the language of this paper, where notation such as $P ( G , ω )$ and $P ( G , [ ω ] )$ is new.
Proposition 5.3.
Fix a toric poset $P = P ( G , [ ω ] )$, and $C = { i 1 , … , i m } ⊆ V$. The first three of the following four conditions are equivalent, and when $m = | C | ≠ 2$, they are also equivalent to the fourth.
(a)
C is a toric chain in P, with $P | C = [ ( i 1 , … , i m ) ]$.
(b)
For every $ω ′ ∈ [ ω ]$, the set C is a chain of $P ( G , ω ′ )$, ordered in some cyclic shift of $( i 1 , ⋯ , i m )$.
(c)
For every $ω ′ ∈ [ ω ]$, the set C occurs as a subsequence of a toric directed path in $ω ′$, in some cyclic shift of the order $( i 1 , ⋯ , i m )$.
(d)
Every total toric extension $[ w ]$ in $L tor ( P )$ has the same restriction $[ w | C ] = [ ( i 1 , … , i m ) ]$.
For ordinary posets, all subsets of chains are chains. The same holds in the toric case.
Proposition 5.4.
Subsets of toric chains are toric chains.
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 $H i , j tor$ does not cut the chamber $c ( P ( G , [ ω ] ) )$ if and only if i and j lie on a common toric chain.

5.3. 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 graph $G ^ torHasse$ with all such edges removed. In Section 5.3, we encountered the toric transitive closure, which is the graph $G ¯ tor$ with all such edges included. In other words, given any toric poset $P = P ( G , [ ω ] )$, there is always a unique minimal pair $( G ^ torHasse ( P ) , [ ω torHasse ( P ) ] )$ and maximal pair $( G ¯ tor ( P ) , [ ω ¯ tor ( P ) ] )$ with the property that the set $c ( P )$ is in $Cham A tor ( G )$ iff
$G ^ torHasse ( P ) ⊆ G ⊆ G ¯ tor ( P )$
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) $A ⟼ A ¯$ is a convex closure, meaning it satisfies the following anti-exchange condition; also see [20]:
$for a ≠ b with a , b ∉ A ¯ and a ∈ A ∪ { b } ¯ , one has b ∉ A ∪ { a } ¯$
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.”
Proposition 5.6.
Let P be a poset over $G = ( V , E )$ and ${ i , j } ∈ E$. Then the following are equivalent:
(i)
The edge ${ i , j }$ is in the Hasse diagram, $G ^ Hasse ( P )$.
(ii)
Removing $H i , j$ enlarges the chamber $c ( P )$.
(iii)
$c ( P ) ¯ ∩ H i , j$ is a (closed) facet of P.
(iv)
The interval $[ i , j ]$ is precisely ${ i , j }$.
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 $π = { { i , j } , V - { i , j } }$.
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 $P ( G ¯ tor ( P ) , [ ω ¯ tor ( P ) ] )$. 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 of $G ¯ tor ( P )$. In particular, if C is a maximal toric chain, then ${ i , j }$ is an edge in $G ^ torHasse ( P )$.

5.4. Toric Intervals

To motivate the definition of a toric interval, it helps to first interpret the classical definition in several different ways.
Definition 5.8.
Let i and j be elements of a poset $P = P ( G , ω )$. The interval$[ i , j ]$ is the set I characterized by one of the following equivalent conditions:
(i)
$I = { k ∈ V : x i ≤ x k ≤ x j , for all x ∈ c ( P ) }$;
(ii)
$I = { k ∈ V : k appears between i and j ( inclusive ) in any linear extension of P }$;
(iii)
$I = { k ∈ V : k lies on a directed path from i to j in ω }$.
Note that if $i = j$, then $[ i , j ] = { i }$, and if $i ≰ j$, then $[ i , j ] = ⌀$.
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, $P ( C 4 , [ ω ] ) = P ( K 4 , [ ω ′ ] )$.
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.
Definition 5.10.
Let i and j be elements of a toric poset $P = P ( G , [ ω ] )$. The toric interval $I = [ i , j ] tor$ is the empty set if $i , j$ do not lie on a common toric chain, and otherwise is the set
$I = [ i , j ] tor : = { i , j } ∪ { k ∈ V : P | { i , j , k } = [ ( i , k , j ) ] }$
If there is no $k ∉ { i , j }$ satisfying Equation (19), then $[ i , j ] tor = { i , j }$. If $i = j$, then $[ i , j ] tor = { i }$.
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:
$0 ≤ x i < x k < x j < 1 , 0 ≤ x j < x i < x k < 1 , 0 ≤ x k < x j < x i < 1$
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.
Corollary 5.12.
Suppose $I = [ i , j ] tor$ is a toric interval of $P = P ( G , [ ω ] )$ of size $| I | ≥ 3$. Then
$[ i , j ] tor = { i , j } ∪ { k : [ w | { i , j , k } ] = [ ( i , k , j ) ] , for all [ w ] ∈ L tor ( P ) } .$
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 $i → tor j$ in $ω ¯ ′$, for some $ω ¯ ′ ∈ [ ω ¯ tor ( P ) ]$.
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 $i → tor j$ in $ω ¯ ′$, the unique orientation of $[ ω ¯ tor ( P ) ]$ whose restriction to G is $ω ′$.
Conversely, suppose that k lies on a toric directed path $i → tor j$ in $ω ¯ ′$, for some $ω ¯ ′ ∈ [ ω ¯ tor ( P ) ]$. 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 $G = G ¯ tor ( P )$. 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 $G = G ¯ tor ( P )$, 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 $P ( G , ω ′ )$.
To see how the converse can fail, take G to be the line graph on 3 vertices, and ω to be the orientation $1 → 2 → 3$. In $P ( G , ω ′ )$, the interval $[ 1 , 3 ]$ is ${ 1 , 2 , 3 }$ but since 1 and 3 do not lie on a common toric chain, $[ 1 , 3 ] tor = ⌀$ in $P ( G , [ ω ] )$. ☐
Proposition 5.15.
For any toric poset $P = P ( G , [ ω ] )$,
$E ( G ^ Hasse ( P ( G , ω ) ) ) ⊆ E ( G ^ torHasse ( P ( G , [ ω ] ) ) ) ⊆ E ( G ¯ tor ( P ) ) = ⋂ ω ′ ∈ [ ω ] E ( G ¯ ( P ( G , ω ′ ) ) )$
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 Equation (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 ${ i , j }$ of $G ¯ tor ( P )$, 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 closure $G ¯ ( P ( G , ω ′ ) )$. The “⊇” containment is analogous: suppose ${ i , j }$ is an edge of $G ¯ ( P ( G , ω ′ ) )$ for each $ω ′ ∈ [ ω ]$. Then by Proposition 5.3, it is a toric chain of P, and hence an edge of $G ¯ tor ( P )$. ☐

5.5. 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 $D π A$ intersects the open polyhedral cone $c ( P )$ in $R V$.
As shown in [9], 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.
Definition 5.16. Given a toric poset P on V, say that $A ⊆ V$ is a geometric toric antichain if $D π A tor$ intersects the open toric chamber $c ( P )$ in $R V / Z V$.
The following characterization of toric antichains was established in [9]. It follows because if $D π S tor$ intersects the open toric chamber $c = c ( P )$ in $Cham A tor ( G )$, then $D π S$ intersects the open chamber upstairs in $Cham A ( G )$.
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 $ω ′ ∈ [ ω ]$.

6. 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 $π = cl P tor ( π )$;
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 well-known 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., $π = cl P tor ( π )$. By construction, $D π tor$ is a flat of $A tor ( G π ′ )$, and so the subset $F π tor ( P )$ 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 $D π tor$. This can be extended to the entire torus, by taking the unique map $d ¯ π$ that makes the following diagram commute, where $d π$ is the mapping from Equation (9):
Explicitly, the map $d ¯ π$ takes a point $x ∈ R V / Z V$, lifts it to a point $x ^$ in an order polytope in $R V$, projects it onto the flat $D π$ as in Equation (9), and then maps that point down to the toric flat $D π tor$. In light of this, we will say that the map $d ¯ π$ is a projection onto the toric flat $D π tor$.
After projecting a chamber $c ( P )$ onto a flat $D π tor$ 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 let $r ¯ π : R V / Z V ⟶ R W / Z W$ be the induced projection. The $r ¯ π$-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.
Lemma 6.1.
Let $π = { B 1 , ⋯ , B r }$ be a partition of V, and $W = { b 1 , ⋯ , b r } ⊆ V$ with $b i ∈ B i$. The restriction $r ¯ π | D π tor : D π tor ⟶ R W / Z W$ is a homeomorphism.
Moreover, all such projection maps for a fixed π are topologically conjugate in the following sense: If $W ′ = { b 1 ′ , ⋯ , b r ′ } ⊆ V$ with $b i ′ ∈ B i$, and projection map $r ¯ π ′ | D π : D π tor ⟶ R W ′ / Z W ′$, and σ is the permutation of V that transposes each $b i$ with $b i ′$, then the following diagram commutes:
here, $σ ′ : W → W ′$ is the map $b i ↦ b i ′$, with $Σ ¯$ and $Σ ¯ ′$ being the induced linear maps as defined in Equation (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 map $δ ¯ π$ from $Face A tor ( G )$ to $Face A tor ( G π ′ )$ making the following diagram commute:
Explicitly, the map $δ ¯ π$ is easiest to defined by the analogous map on closed faces:
$δ ¯ π : Face ¯ A tor ( G ) ⟶ Face ¯ A tor ( G π ′ ) , δ ¯ π : c ( P ) ¯ ∩ D σ tor ⟼ c ( P ) ¯ ∩ D σ tor ∩ D π tor$
The open faces of $A tor ( G π ′ )$ are then mapped to faces of the arrangement $A tor ( G / ∼ π )$ under the projection $r ¯ π | D π : D π tor ⟶ R W / Z W$. 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
$Pre ( G ) ⟶ q ¯ π Pre ( G π ′ ) ⟶ p ¯ π Pre ( G / ∼ π ) , [ ω ] ⟼ q ¯ π [ ω π ′ ] ⟼ p ¯ π [ ω π ′ / ∼ π ] = [ ω / ∼ π ]$
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 $π = cl P tor ( π )$. Applying Theorem 4.7 gives a combinatorial interpretation of this, which was not a priori obvious.
Corollary 6.2.
Let $P = P ( G , [ ω ] )$ be a toric poset. Contracting G by a partition $π ∈ Π V$ yields a morphism to a toric poset over $G / ∼ π$ if and only if $ω ′ / ∼ π$ is acyclic for some orientation $ω ′ ∈ [ ω ]$. □
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 $Φ : R V / Z V ↪ R V ′ / Z V ′$. This sends the arrangement $A tor ( G )$ in $R V$, where $G = ( V , E )$, to the same higher-dimensional arrangement:
$A tor ( G ∪ ( V ′ - V ) ) : = { H i j tor in R V ′ : { i , j } ∈ E }$
The toric chamber $c = c ( P )$ is sent to the chamber
$c ( P ′ ) = { x ∈ R V ′ : x i < x j for { i , j } ∈ E }$
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:
$i ≤ P j$ implies $i ≤ P ′ j$;
$G ^ Hasse ( P ) ⊆ G ^ Hasse ( P ′ )$, where ⊆ is inclusion of edge sets;
$c ( P ′ ) ⊆ c ( P )$.
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 to $G ^ torHasse ( P ) ⊆ G ^ torHasse ( P ′ )$.

6.4. Summary

Up to isomorphism, every toric poset morphism $P = P ( G , [ ω ] )$ can be decomposed into a sequence of three steps:
(i)
quotient: Collapsing G by a partition π that preserves acyclicity of some $ω ′ ∈ [ ω ]$ (projecting to a flat $D π tor$ of $A ( G π ′ )$ for some partition $π = cl P tor ( π )$).
(ii)
(iii)
extension: Adding relations (cutting the chamber with toric hyperplanes).
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 $D π tor$ and a then adding toric hyperplanes.

7. 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 hence $F ¯ π = c ( P ) ¯ ∩ D π$ 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
$x i k = x i ℓ$ for all $i k , i ℓ$ in I;
$x j k = x j ℓ$ for all $j k , j ℓ$ in J;
$x i ≤ x j$ for all $i ∈ I$ and $j ∈ J$.
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 [21]. 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 $D π tor$ of $A tor ( G π ′ )$ for some $π = cl P tor ( π )$ such that $F ¯ π tor = c ( P ) ¯ ∩ D π tor$ is at most 2-dimensional. For the partition $π = { I , J }$ of V, each point $x = ( x 1 , ⋯ , x n )$ on $F π tor$ 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 [22].
By construction, toric filters are characterized by closed toric partitions π of V such that $D π tor$ 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 $I ⊆ V$.
(i)
I is a toric filter of $P ( G , [ ω ] )$;
(ii)
I is an ideal of $P ( G , ω ′ )$ for some $ω ′ ∈ [ ω ]$;
(iii)
I is a filter of $P ( G , ω ′ ′ )$ for some $ω ′ ′ ∈ [ ω ]$;
(iv)
In at least one total toric extension of $P ( G , [ ω ] )$, the elements in I appear in consecutive cyclic order.
Proof.
The result is obvious if $I = ⌀$ or $I = V$, so assume that $⌀ ⊊ I ⊊ V$, and $π = { I , V - I }$. This forces $D π tor$ to be two-dimensional (rather than one-dimensional).
(i)⇒(ii): If I is a toric filter of $P ( G , [ ω ] )$, then $D π tor$ 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)⇒(i): Suppose that I is an ideal of $P ( G , ω ′ )$ for $ω ′ ∈ [ ω ]$. Then $F ¯ π = c ( P ( G , ω ′ ) ) ¯ ∩ D π$ is two-dimensional, and it descends to a two-dimensional face $F ¯ π tor$ of the toric poset $P ( G , [ ω ′ ] ) = P ( G , [ ω ] )$ or of some extension (if $F ¯ tor$ intersects the interior). Therefore, I is a toric filter 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 $x ∈ R V / Z V$,
$0 ≤ x i 1 < ⋯ < x i k < x v k + 1 < ⋯ < x v n < 1$
The unique preimage $x ^$ of this point in $[ 0 , 1 ) V$ under the quotient map $q : R V → R V / Z V$ is in the order polytope of some $P ( G , ω ′ )$ that maps into $c ( P ( G , [ ω ′ ] ) ) ¯ = c ( P ( G , [ ω ] ) ) ¯$. Since the coordinates of $x ^$ are totally ordered as in Equation (22), $( i 1 , ⋯ , i k , v k + 1 , ⋯ , v n )$ is a linear extension of $P ( G , ω ′ )$. Moreover, the coordinates in I form an initial segment of this linear extension, hence I is an ideal of $P ( G , ω ′ )$.
(iii)⇔(iv): Immediate by Remark 7.2 upon reversing the roles of I and $V - I$. ☐
Proposition 7.3 along with a result of Stanley gives a nice characterization of the toric filters in terms of vertices of order polytopes. Every filter I of a poset P has a characteristic function
$χ I : P ⟶ R , χ I ( k ) = 1 , k ∈ I 0 , k ∉ I$
We identify $χ I ( P )$ with the corresponding vector in ${ 0 , 1 } V ⊆ R V$.
Proposition 7.4.
([16], Corollary 1.3) Let P be a poset. The vertices of the order polytope $O ( P )$ are the characteristic functions $χ I$ of filters of P.
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 $R V / Z V$.
Corollary 7.5.
Let $P = P ( G , [ ω ] )$ be a toric poset and $I ⊆ V$. Then $χ I$ is the characteristic function of a toric filter of P if and only if $χ I$ is a vertex of $O ( P ( G , ω ′ ) )$ for some $ω ′ ∈ [ ω ]$.
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 graph $G ^ Hasse ( P ) = L 4$, and the transitive closure is $G ¯ ( P ) = K 4$. 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,
$L 4 = G ^ Hasse ( P ( G , ω ) ) ⊊ G ^ torHasse ( P ( G , [ ω ] ) ) = C 4 , G ¯ ( P ( G , ω ) ) = G ¯ tor ( P ( G , [ ω ] ) ) = K 4$
The 4 torically equivalent orientations are shown in Figure 4. The only total toric extension of $P ( G , [ ω ] )$ is
$[ ( 1 , 2 , 3 , 4 ) ] = { ( 1 , 2 , 3 , 4 ) , ( 2 , 3 , 4 , 1 ) , ( 3 , 4 , 1 , 2 ) , ( 4 , 1 , 2 , 3 ) ]$
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, $G ¯ ( P ( G , ω ′ ) ) ≠ G ¯ tor ( P ( G , [ ω ′ ] ) )$. 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.

8. Application to Coxeter Groups

A Coxeter system is a pair $( W , S )$ consisting of a Coxeter groupW generated by a finite set of involutions $S = { s 1 , ⋯ , s n }$ with presentation
$W = 〈 S ∣ s i 2 = 1 , ( s i s j ) m i , j = 1 〉$
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 $s i = s i - 1$) cyclically shifts it:
$s x 1 ( s x 1 s x 2 ⋯ s x n ) s x 1 = s x 2 ⋯ s x n s x 1$
and the corresponding acyclic orientation differs by reversing the orientations of all edges incident to $s x 1$, thereby converting it from a source to a sink vertex. In 2009, H. and K. Eriksson showed [2] 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:
$C ( W ) ⟶ Acyc ( Γ ) Conj ( C ( W ) ) ⟶ Acyc ( Γ ) / ≡ c ⟼ ω ( c ) cl W ( c ) ⟼ [ ω ( c ) ]$
As an example, suppose $W = W ( A ˜ 3 )$, the affine Coxeter group of type A, and let $c = s 1 s 2 s 3 s 4$ (using $s 4$ instead of the usual $s 0$), as shown below:
The conjugate Coxeter elements to $c ∈ W$ are thus $s 1 s 2 s 3 s 4$, $s 2 s 3 s 4 s 1$, $s 3 s 4 s 1 s 2$, and $s 4 s 1 s 2 s 3$. The toric filters of $P ( G , [ ω ( c ) ] )$ describe which subwords can appear in an initial segment of some reduced expression of one of these conjugate Coxeter elements. The poset of these toric filters was shown in Figure 5 (replace k with $s k$).
Now, consider the element $c ′ = s 1 s 3 s 2 s 4$ in $W ( A ˜ 3 )$, as shown below:
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.

9. 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 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 [23,24] in the 1990s. The fully commutative (FC) elements are those such that “long braid relations” (e.g., $s t s ↦ t s t$) 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 [25]. In 2013, T. Marquis showed that two CFC elements are conjugate if and only if their heaps are torically equivalent [26]. These elements were further studied by M. Pétréolle [27]. 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.

Conflicts of Interest

The author declares no conflict of interest.

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Figure 1. A poset P and its lattice of closed face partitions.
Figure 1. A poset P and its lattice of closed face partitions.
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 $P i = P ( G , ω i )$ by the partition $π = { B 1 = { 1 } , B 2 = { 2 , 3 } }$ of V. The intersection of each (closed) chamber $c ( P i ) ¯$ with $[ 0 , 1 ] 3$ is the order polytope $O ( P i )$ of $P i$. The point y is supposed to lie on the hyperplane $x 1 = x 2$.
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 $P i = P ( G , ω i )$ by the partition $π = { B 1 = { 1 } , B 2 = { 2 , 3 } }$ of V. The intersection of each (closed) chamber $c ( P i ) ¯$ with $[ 0 , 1 ] 3$ is the order polytope $O ( P i )$ of $P i$. The point y is supposed to lie on the hyperplane $x 1 = x 2$.
Figure 3. Despite the equality $P ( C 4 , [ ω ] ) = P ( K 4 , [ ω ′ ] )$, the set $I = { 1 , 3 }$ lies on a toric directed path $1 → tor 3$ in $ω ′$ but not for any representative of $[ ω ]$.
Figure 3. Despite the equality $P ( C 4 , [ ω ] ) = P ( K 4 , [ ω ′ ] )$, the set $I = { 1 , 3 }$ lies on a toric directed path $1 → tor 3$ in $ω ′$ but not for any representative of $[ ω ]$.
Figure 4. The four torically equivalent orientations to $ω ∈ Acyc ( C 4 )$, shown at left. The edges implied by toric transitivity are dashed.
Figure 4. The four torically equivalent orientations to $ω ∈ Acyc ( C 4 )$, shown at left. The edges implied by toric transitivity are dashed.
Figure 5. The toric filters of the toric poset $P ( C 4 , [ ω ] )$ form a poset that is not a lattice. The vertices should be thought of as subsets of ${ 1 , 2 , 3 , 4 }$; order does not matter. That is, 134 represents ${ 1 , 3 , 4 }$.
Figure 5. The toric filters of the toric poset $P ( C 4 , [ ω ] )$ form a poset that is not a lattice. The vertices should be thought of as subsets of ${ 1 , 2 , 3 , 4 }$; order does not matter. That is, 134 represents ${ 1 , 3 , 4 }$.
Figure 6. The six torically equivalent orientations to $ω ′ ∈ Acyc ( C 4 )$, shown at left.
Figure 6. The six torically equivalent orientations to $ω ′ ∈ Acyc ( C 4 )$, shown at left.
Figure 7. The toric filters of the toric poset $P ( C 4 , [ ω ′ ] )$ form a poset that happens to be a lattice. Each toric filter appears as a consecutive sequence in one of the total toric extensions, shown at right.
Figure 7. The toric filters of the toric poset $P ( C 4 , [ ω ′ ] )$ form a poset that happens to be a lattice. Each toric filter appears as a consecutive sequence in one of the total toric extensions, shown at right.
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.
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.

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Macauley, M. Morphisms and Order Ideals of Toric Posets. Mathematics 2016, 4, 39. https://doi.org/10.3390/math4020039

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Macauley M. Morphisms and Order Ideals of Toric Posets. Mathematics. 2016; 4(2):39. https://doi.org/10.3390/math4020039

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Macauley, Matthew. 2016. "Morphisms and Order Ideals of Toric Posets" Mathematics 4, no. 2: 39. https://doi.org/10.3390/math4020039

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