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\title{Biclosed Sets in Affine Root Systems}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[G.~T.~Barkley and D.~E~Speyer]{Grant T. Barkley\thanks{\href{mailto:barkleygrant@gmail.com}{barkleygrant@gmail.com}. Supported by NSF grant DMS-1854512 and as an REU student by NSF grant DMS-1600223.}\addressmark{1} and David E. Speyer\thanks{Supported by NSF grants, DMS-1600223, DMS-1854225 and DMS-1855135.}\addressmark{2}}

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\address{\addressmark{1}Dept. of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138 \\ \addressmark{2}Dept. of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109}

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%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{The extended weak order is a partial order associated to a Coxeter system $(W,S)$. It is the containment order on ``biclosed'' sets of positive roots in the (real) root system associated to $W$. When $W$ is finite, this order coincides with the weak order on $W$, and is a lattice;  when $W$ is infinite, the weak order on $W$ is a proper order ideal in the extended weak order. It is a longstanding conjecture of Matthew Dyer that the extended weak order is a lattice for any $W$. We prove this conjecture for the affine Coxeter groups. Furthermore, we give a parametrization of the extended weak order for these groups in terms of the face lattice of the Coxeter arrangement for the associated finite group. }

%% put your French abstract here, or comment this out if you don't have one
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\keywords{Coxeter groups, root systems, affine Coxeter groups, lattice theory}


\begin{document}


%\author{Grant Barkley and David E Speyer}
\maketitle


\section{Key examples: The finite and affine symmetric groups}
This is a paper about Coxeter groups, root systems, and the weak order.
Discussing these topics in general requires a lot of notation, so we begin with two examples: The symmetric group $S_n$ and the affine symmetric group $\tS_n$.
%Everything we say in $S_n$ is extremely classical and well understood but, even in $\tS_n$, we will already be able to see new ideas.
Some of the notation in this section will be replaced with more general notation in the following sections.

The symmetric group $S_n$ is the group of permutations of $[n]:=\{ 1, 2, \ldots, n \}$.
%We write $s_i$ for the transposition $(i\ i+1)$; then $S_n$ is generated by $s_1$, $s_2$, \ldots, $s_{n-1}$ modulo the relations 
%\[ s_i^2 =1 \qquad s_i s_j = s_j s_i\ \text{for}\ |i-j| \geq 2 \qquad s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1} . \]
Put $T = \{ (i, j) : 1 \leq i < j  \leq n \}$. For $\sigma \in S_n$, the set of \newword{inversions} of $\sigma$ is
\[ I(\sigma) : = \{ (i,j) \in T : \sigma^{-1}(i) > \sigma^{-1}(j) \} . \]


The (right) weak order on $S_n$ is the partial order on $S_n$ defined by $\sigma \leq \tau$ if and only if $I(\sigma) \subseteq I(\tau)$. 
Surprisingly, weak order is a lattice, meaning that every subset of $S_n$ has a unique least upper bound and a unique greatest lower bound.

The group $\tS_n$ is the group of bijections $\widetilde{\sigma} \colon \ZZ \to \ZZ$, obeying the conditions:
\[ \widetilde{\sigma}(i+n) = \widetilde{\sigma}(i) + n, \qquad \sum_{k=1}^n (f(k)-k) = 0. \]
Put $\tT = \{ (i,j) \in \ZZ^2/{\big(} \ZZ \cdot (n,n) {\big)} : i < j,\ i \not\equiv j \bmod n \}$. To be clear, $\ZZ^2/{\big(} \ZZ \cdot (n,n) {\big)}$ is the quotient of $\ZZ^2$ by the subgroup  $\ZZ \cdot (n,n)$.
For $\widetilde{\sigma} \in \tS_n$, the set of \newword{inversions} of $\widetilde{\sigma}$ is
\[ I(\widetilde{\sigma}) : = \{ (i,j) \in \tT : \widetilde{\sigma}^{-1}(i) > \widetilde{\sigma}^{-1}(j) \} . \]
We once again define weak order on $\tS_n$ by $\widetilde{\sigma} \leq \widetilde{\tau}$ if and only if $I(\widetilde{\sigma}) \subseteq I(\widetilde{\tau})$. 

This time, weak order is only a semilattice. This means that, if a subset $\cX$ of $\tS_n$ has an upper bound, then it has a least upper bound, but no upper bound need exist at all. 
(Every non-empty subset of $\tS_n$ continues to have a greatest lower bound.) 

Weak order has many applications, for example in the representation theory of quivers \cite{iyama_reading_reiten_thomas_2018} and in the structure of cluster algebras \cite{Reading2016}. These applications are well-understood for $S_n$. When applied to $\tS_n$, however, the weak order fails to capture all of the structure we would like. For instance, in using Cambrian (semi-)lattices to model cluster algebra exchange graphs, the framework in \cite{Reading2016} gives only ``half'' of the exchange graph for $\tS_n$. Since quotients of the $S_n$ weak order as a lattice are important for these constructions, we are motivated to find an extension of the weak order on $\tS_n$ to some larger poset which is a (complete) lattice. 

It is natural to imagine doing this by defining a collection of subsets of $\tT$ that would play the role of ``generalized inversion sets''. 
% First mention of conjecture seems to be Remark 2.14 in \cite{Dyer1994a}
Such a definition was proposed and studied by Matthew Dyer~(see, \textit{e.g.}, \cite{Dyer2019}); we will describe it here for the case of $S_n$ and $\tS_n$, and for any Coxeter group in the next section.
\textbf{The main result of this paper will be that Dyer's construction (the \newword{extended weak order}) is a lattice for all affine Coxeter groups.}

We define a subset $J$ of $T$ to be \newword{closed} if, whenever $1 \leq i < j < k \leq n$ and $(i,j)$ and $(j,k) \in J$, then $(i,k) \in J$.

We define a subset $J$ of $\tT$ to be \newword{closed} if it obeys the three conditions
\begin{enumerate}
\vspace{-.25cm}
\item If $i<j<k$ with $(i,j)$ and $(j,k) \in J$, and $i\not\equiv k \bmod n$, then $(i,k) \in J$.
%\item If $i < j < i+n$ with $(i,j)$ and $(j,i+n) \in J$, then $(a,b) \in J$ for any $a < b$ with either $(a,b) \equiv (i,j)$ or $(a,b) \equiv (j,i) \bmod n$.
\vspace{-1em}
\item If $i<j<k$ with $(i,j)$ and $(j,k) \in J$, and $i\equiv k \bmod n$, then $(i,j+\ell n)$ and $(j,k+\ell n) \in J$ for all $\ell\geq 0$.
\vspace{-1em}
\item If $i<j<k$ with $(i,j)$ and $(i,k) \in J$, and $j\equiv k \bmod n$, then $(i,j+\ell n) \in J$ for all $\ell\geq 0$ such that $j+\ell n \leq k$.
\vspace{-.1cm}
\end{enumerate}
\vspace{-.15cm}

We define $J$ to be \newword{coclosed} if the complement of $J$ is closed, and we define $J$ to be \newword{biclosed} if $J$ is both closed and coclosed. 
We note that the intersection of any collection of closed sets is closed, and, for an arbitrary subset $K$ of $T$ or $\tT$, we define the \newword{closure} $\overline{K}$, to be the intersection of all closed sets containing $K$. 
Similarly, the union of any collection of coclosed sets is coclosed; we define the \newword{interior} $K^{\circ}$ to be the union of all coclosed sets contained in $K$. The following is a special case of a result for all Coxeter groups:
\begin{proposition}
The finite biclosed subsets of $T$ (respectively $\tT$) are exactly the sets of inversions of the permutations in $S_n$ (respectively $\tS_n$).
\end{proposition}

The set $T$ is finite, so all subsets of $T$ are finite. But $\tT$ is infinite, so it is natural to consider all biclosed subsets of $\tT$, without the assumption of finiteness. 
The following is a special case of Dyer's conjectures and Dyer tells us that he already checked this case; we will provide the first published proof here:
\begin{theorem} \label{AtildeLattice}
The collection of all biclosed subsets of $\tT$, ordered by containment, is a complete lattice. For any collection of biclosed sets $\cX$, we have $\bigvee \cX = \overline{\bigcup_{J \in \cX} J}$ and $\bigwedge \cX = \left( \bigcap_{J \in \cX} J \right)^{\circ}$. 
\end{theorem}

Furthermore, we give a combinatorial model for the biclosed subsets of $\tT$.
Define a \newword{translationally invariant total order} on $\ZZ$ to be a total order $\prec$ obeying the condition that $i \prec j$ if and only if $i+n \prec j+n$.
Each translationally invariant total order $\prec$ defines a biclosed subset $I(\prec)$ of $\tT$ by $I(\prec)  = \{ (i,j) \in \tT : i \succ j \}$. 
%\begin{proposition}
%A subset $J$ of $\tT$ is biclosed if and only if there is a translationally invariant total order $\prec$ on $\ZZ$ for which $J = I(\prec)$.
%\end{proposition}

It turns out that any biclosed subset of $\tT$ is of this form, but not in a unique way. 
To see the issue, take $n=2$, and consider the following total orders on $\ZZ$:
\begin{gather*}
 \cdots\prec_1 -5 \prec_1 -3 \prec_1 -1 \prec_1 1 \prec_1 3 \prec_1 5 \prec_1 \cdots \prec_1 -4 \prec_1 -2 \prec_1 0 \prec_1 2 \prec_1 4 \prec_1 \cdots,
\\
\cdots\prec_2 -5 \prec_2 -3 \prec_2 -1 \prec_2 1 \prec_2 3 \prec_2 5 \prec_2 \cdots \prec_2 4 \prec_2 2 \prec_2 0 \prec_2 -2 \prec_2 -4 \prec_2 \cdots .
\end{gather*}
In other words, both $\prec_1$ and $\prec_2$ put all the odd numbers before all the even numbers, but $\prec_1$ preserves the standard ordering within each parity class whereas $\prec_2$ reverses the even numbers. 
Then $I(\prec_1) = I(\prec_2) = \{ (i,j) : i<j,\ i \equiv 0 \bmod 2,\ j \equiv 1 \bmod 2 \}$. 

%In general, if $\prec$ is a translationally invariant total order on $\ZZ$ and $k$ is a congruence class in $\ZZ/n \ZZ$, then we say $k$ is \newword{isolated} if there do \textbf{not} exist $k_1$, $k_2$ and $\ell$, with $k_1 \equiv k_2 \equiv k \bmod n$ and $\ell \not \equiv k \bmod n$, such that $k_1 \prec \ell \prec k_2$. The orientation of an isolated congruence class will not be detected by $I(\prec)$.

\begin{theorem} \label{AtildeModel}
Biclosed subsets of $\tT$ are in bijection with translationally invariant total orders, modulo reversing the order on intervals of the form
\[ \cdots \prec k-3n \prec k-2n \prec k-n \prec k \prec k+n \prec k+2n \prec k+3n \prec \cdots.  \]
\end{theorem}

The main results of this paper are not simply \Cref{AtildeLattice,AtildeModel}, but generalizations of these results to all affine Coxeter types, and a reduction of Dyer's conjectures in general to a more precise conjecture for rank three Coxeter groups. 
In the next section, we will introduce the vocabulary necessary to state our results. 
%We close with an example of a join in $\tS_4$:
%\begin{eg}
%Put $n=4$. Define the permutations
%\[ u(i) = \begin{cases} 
%i+2 & i \equiv 0 \bmod 4 \\
%i-1 & i  \equiv 1 \bmod 4 \\
%i-1 & i \equiv i \equiv 2 \bmod 4 \\
%i & i \equiv 3 \bmod 4 \\
%\end{cases} \qquad 
%v(i) = \begin{cases} 
%i+2 & i \equiv 2 \bmod 4 \\
%i-1 & i  \equiv 3 \bmod 4 \\
%i-1 & i \equiv 4 \equiv 2 \bmod 4 \\
%i & i \equiv 5 \bmod 4 \\
%\end{cases}.\]
%In other words, $u$ takes $(\cdots, 0,1,2,3,4,5,6,7, \cdots)$ to $(\cdots, 2,0,1,3,6,4,5,7,\cdots)$ and $v$ takes $(\cdots, 0,1,2,3,4,5,6,7, \cdots)$ to $(\cdots, -1,1,4,2,3,5,8,6, \cdots)$.
%The inversion sets are $I(u)  = \{ (0,2), (1,2) \}$ and $I(v) = \{ (2,4), (3,4) \}$. 
%The corresponding total orders are just to insert inequalities into this one line notation:
%\[ \cdots \prec 2 \prec 0 \prec 1 \prec 3 \prec 6 \prec 4 \prec 3 \prec 7 \cdots \ \text{and}\  \cdots \prec -1 \prec 1 \prec 4 \prec 2 \prec 3 \prec 5 \prec 8 \prec 6 \cdots . \]
%To compute the join of $I(u)$ and $I(v)$, we must compute $\overline{I(u) \cup I(v)}$. After a bit of experimentation, one sees that
%\[ \overline{I(u) \cup I(v)} = \{ (a,b)  : a \leq b,\ (a,b) \equiv (0,2), (2,4), (1,2), (1,4), (3,4)\ \text{or} \ (3,6) \bmod 4 \} . \]
%These are exactly the inversions of the total order
%\[ \cdots \prec 4 \prec 2 \prec 0 \prec -2 \prec -4 \prec \cdots \prec -3 \prec -1 \prec 1 \prec 3 \prec \cdots .\]
%In other words, we take the even numbers in reverse order, followed by the odd numbers in order.
%\end{eg}
%

%We can think of $S_n$ geometrically as a group of symmetries of $\RR^n$, with $S_n$ permuting the coordinates in the obvious way.
%Then the transposition $(i\ j)$ is the reflection over the hyperplane $(e_i - e_j)^{\perp}$, where $e_1$, $e_2$, \dots, $e_n$ is the standard basis for $\RR^n$.
%The complement of these $\binom{n}{2}$ hyperplanes gives $n!$ cones, and $S_n$ acts freely and transitively on the set of these cones. 
%
%For each $\sigma \in S_n$, we keep track of which side of $(e_i - e_j)^{\perp}$ it is on. To this end, let 

\section{Background}\label{Background}

\subsection{Coxeter groups and root systems}
% properties of decompositions into irreducible coxeter groups/root systems
% reflection subgroups have canonical coxeter group structure
% parabolic subsystems of reflections/positive roots/ signed roots
% the span of a set S of roots is a root subsystem, which has a base contained in S? no
Let $(m_{ij})_{i,j=1}^n$ be a symmetric matrix of nonnegative integers such that $m_{ii}=1$ for all $i$, and such that $m_{ij} \geq 2$ for $i\neq j$. We further allow $\infty$ to appear in the matrix. Such a matrix is a \newword{Coxeter matrix}. A \newword{Coxeter group} is a group with a presentation of the form
\[ W = \langle s_1,\ldots,s_n : (s_is_j)^{m_{ij}} = 1, \ 1\leq i,j\leq n \rangle \]
for some Coxeter matrix $(m_{ij})$. In this case we say $(W,S)$ is a \newword{rank $n$ Coxeter system}, where $S=\{s_1,\ldots,s_n\}$ is the set of \newword{simple reflections}. The \newword{length} $\ell(w)$ of an element $w$ is the minimal number $k$ of simple reflections in any word $s_1\cdots s_k = w$. We further write
\[T=\{wsw^{-1} : w\in W, s\in S \}\]
for the set of \newword{reflections} in $W$.

Associated to each Coxeter group is a faithful $n$-dimensional representation $V$ called the \newword{geometric} or \newword{reflection representation}. Fix a \newword{Cartan matrix} $(A_{ij})_{i,j=1}^n$ such that $A_{ii}=2$, such that $A_{ij}=0$ if $m_{ij}=2$, such that $A_{ij}$, $A_{ji} <0$ with $A_{ij} A_{ji} = 4 \cos^2 \tfrac{\pi}{m_{ij}}$ if $3 \leq m_{ij} < \infty$, and such that  $A_{ij}$, $A_{ji} <0$ with $A_{ij} A_{ji} \geq 4$ if $m_{ij}=\infty$.

The geometric representation has a basis $\a_1,\ldots,\a_n$. We will also want to work with the dual representation $V^{\vee}$, which has dual basis $\omega_1$, $\omega_2$, \ldots, $\omega_n$. We put $\alpha_j^{\vee} = \sum_i a_{ij} \omega_i$, so $\langle \alpha_i, \alpha_j^{\vee} \rangle = a_{ij}$.  (Note that $\alpha_i^{\vee}$ need not be a basis of $V^{\vee}$.)
Then $W$ acts on $V$ by $s_i(v) = v - \langle \alpha_i^{\vee}, v \rangle \alpha_i$ and acts on $V^{\vee}$ by the dual formula $s_i(f) = f - \langle f, \alpha_i \rangle \alpha_i^{\vee}$. 
It is a standard result that $V$ and $V^{\vee}$ are faithful representations of $W$.

The (real) \newword{roots} are the vectors of the form $w \alpha_i$ for $w \in W$, $1 \leq i \leq n$. Each root is either \newword{positive}, meaning in the positive linear span of the $\alpha_i$, or \newword{negative}, meaning the negation of a positive root. We write $\Phi$ and $\Phi_+$ for the sets of roots and positive roots respectively; these are called the \newword{root system} and the \newword{positive root system}. Similarly, we define the co-root system $\Phi^{\vee} = \{ w \alpha_i^{\vee} : w \in W,\ i \in I \}$. 

The reflections in $T$ are precisely those elements of $W$ which act on $V$ and $V^{\vee}$ by involutions fixing a codimension $1$ subspace. 
Specifically, for each $t \in T$, there is a unique positive root $\beta_t$ and co-root $\beta^{\vee}_t$ such that $t(v) = v - \langle \beta_t^{\vee}, v \rangle \beta_t$. 

We are interested in the weak order on $W$. To this end, define the set of (left) \newword{inversions} of a group element $w\in W$ to be
\[N(w) = \{t\in T : \ell(tw) < \ell(w) \}. \]
The inversion set also admits a geometric description: Take a point $f$ in $V^{\vee}$ such that $\langle f, \alpha_i \rangle >0$ for each $i$, then $t$ is an inversion of $w$ if and only if $\langle w f, \beta_t \rangle < 0$. We say $w\leq v$ in \newword{weak order} if $N(w)\subseteq N(v)$. 

%\begin{definition}
%A set of roots $B$ is \newword{rank two biconvex} (or \newword{biclosed} for brevity) if and only if for all $\a,\b,\g$ such that $\g \in \cone\{\a,\b\}$, 
%\end{definition}

%\begin{proposition}\label{thm:finite}
%Let $B \subseteq \pos$ be a set of positive roots. Then $B = N(w)$ for some $w\in W$ if and only if $B$ is finite and biclosed.
%\end{proposition}
%
%%When dealing with biclosed sets, it is useful to keep an image in mind of what they ``look like''. 
%
%% Biclosed sets in \oDel look like semicircles, circles, or empty
%% The set of positive roots in such a biclosed set is a biclosed set of positive roots
%
%%\begin{lemma}\label{thm:irrclass}%probably dont need this
%%	Let $\Delta$ be any root system. Define an equivalence relation on $\Delta$ generated by the relations $\a \sim \b$ whenever there is some $\g$ in the span of $\a,\b$, distinct from $\{\pm\a, \pm\b\}$, and the relations $\a\sim -\a$. Then the equivalence classes of $\sim$ are the irreducible root subsystems of $\Delta$.
%%\end{lemma}
%
%
%\begin{definition}
%	Let $S\subseteq \re$ be a set of real roots. The \newword{dihedral span} (or \newword{2-span}) of $S$ is the minimal set of real roots $S'$ containing $S$ which is closed under adding elements $\gamma \in \spn\{\alpha,\beta\}\cap\re$ for any $\alpha,\beta\in S'$.
%\end{definition}
%
%The following lemma is new and is essential to our work in Section 5. 
%
A subset $\Phi'$ of $\Phi$ is called a \newword{subsystem} of $\Phi$ if $\Phi'$ is preserved by reflections over any $\beta\in \Phi'$. We define $\Phi'_+ = \Phi'\cap \Phi_+$ and $\Phi'_-=\Phi'\cap \Phi_-$. The following is a collection of results from \cite{Dyer1990} and \cite{Dyer1991}.

\begin{proposition}
	Let $\Phi'$ be a subsystem of $\Phi$. There is a unique minimal set $\Pi'\subseteq \Phi'_+$ such that 
	\[ \RR_{\geq 0}\Pi'\cap \Phi' = \Phi'_+ = -\Phi'_-. \]
	Let $W'$ be the reflection subgroup of $W$ generated by reflections over the roots in $\Phi'$, and let $S'$ be the reflections over elements of $\Pi'$. Then $(W',S')$ is a Coxeter system, and the bijection between $T$ and $\Phi_+$ restricts to a bijection between $\Phi'_+$ and reflections in $W'$. Furthermore,
	\[ S' = \{t\in T : N(t)\cap W' = \{t\} \}. \]
	If additionally the span of $\Phi'$ is 2-dimensional, then $|\Pi'|=2$. 
	%^ this is the only part from the 1991 paper
\end{proposition}
The elements of $\Pi'$ are called the \newword{simple roots} of $\Phi'$. The cardinality of $\Pi'$ is called the \newword{rank} of $\Phi'$.
The group $W'$ is called the \newword{reflection subgroup} generated by $\Phi'$. 
We say a subsystem $\Phi'$ is \newword{full} if whenever $\alpha$ and $\beta\in \Phi'$, it follows that any $\gamma$ in the span $\RR\alpha+\RR\beta$ is also in $\Phi'$. The \newword{type} of $\Phi'$ is a description of the Coxeter matrix of $(W',S')$. If $X_n$ is a Coxeter type, then a \emph{full} subsystem of $\Phi$ with type $X_n$ is called an \newword{$X_n$-subsystem}. (\textit{E.g.}, an $A_2$-subsystem consists of roots $\alpha,\beta,\alpha+\beta$ and their negations, such that no other elements of $\Phi$ are in their span.)

A root system is \newword{indecomposable} when it cannot be written as the disjoint union of two nonempty full subsystems. A Coxeter group with an indecomposable root system is called \newword{irreducible}. Every root system can be written uniquely as a disjoint union of indecomposable subsystems, and each Coxeter group can be written uniquely as a product of irreducible factors.

%\begin{lemma} \label{FullSubsystem}
%	Let $\alpha_1,\ldots,\alpha_n\in \Phi$. Then the minimal full subsystem $\Phi'$ which contains $\alpha_1,\ldots,\alpha_n$ has rank at most $n$.
%\end{lemma}
%\begin{proof}
%	Assume $\Phi'$ has rank $m>n$. Construct an ($m$-dimensional) geometric representation $\Delta$ for the Coxeter system $(W',S')$. By the Proposition, we can identify the elements of $\Delta$ with the elements of $\Phi'$. Let $\Delta'$ be the set of roots in $\Delta$ contained in the span of $\alpha_1,\ldots,\alpha_n$ (as roots in $\Delta$). Then $\Delta'$ contains the minimal full subsystem of $\Delta$ containing $\alpha_1,\ldots,\alpha_n$. But since $\Delta'$ is contained in $n$-dimensional space, it must not contain one of the $m$ linearly independent simple roots in $\Delta$. So $\Delta'$ is a proper subset of $\Phi'$. But, using the bijection of $T$ with $\Phi_+$, the minimal full subsystem containing a set of roots can be described intrinsically to $W$ using the reflections $t_1,\ldots,t_n$ associated to $\alpha_1,\ldots,\alpha_n$. Specifically, $W'$ is the minimal subgroup containing $t_1,\ldots,t_n$ such that if $s,t\in T$ and the subgroup $H$ generated by $s$ and $t$ satisfies $|H\cap W'| \geq 2$, then $s$ and $t\in W'$. Thus, $\Delta'$ must equal $\Delta$, a contradiction.
%\end{proof}


\subsection{Finite and affine Coxeter groups} \label{FiniteAffineGroups}
Let $(A_{ij})$ be a Cartan matrix with integer entries such that the associated root system $\Phi_0$ is finite and indecomposable. 
Such matrices are classified by \newword{Dynkin diagrams}. There is a unique root $\theta\in (\Phi_0)_+$ such that $\theta-\alpha$ is in the positive span of $(\Phi_0)_+$ for any $\alpha\in (\Phi_0)_+$. The root $\theta$ is called the \newword{highest root} of $\Phi_0$.

We now construct a new root system $\Phi$. If $\Phi_0$ is of type $X_n$, then we say $\Phi$ is of type $\widetilde X_n$, and call it the \newword{affine root system} associated to $\Phi_0$. If $V_0$ is the ambient $n$-dimensional vector space of $\Phi_0$, then let $V := V_0 \oplus \RR\delta$ to be an $(n+1)$-dimensional vector space, with new basis vector $\delta$. We define $\langle \alpha_i^\vee,\delta\rangle=0$ for all simple coroots $\alpha_i$. Then $\Phi$ is the set
\[ \Phi = \{ \alpha+k\delta : \alpha\in\Phi_0,\ k\in\ZZ \}. \]
We endow $\Phi$ with the set of simple roots $\alpha_1,\ldots,\alpha_n,\alpha_0$, where $\alpha_1,\ldots,\alpha_n$ are the simple roots of $\Phi_0$ and $\alpha_0=\delta-\theta$. The corresponding new coroot is $\alpha_0^\vee:=-\theta^\vee$. Then $\Phi$ is a root system \cite{Kac1983}. The associated Coxeter group is called the \newword{affine Coxeter group} of type $\widetilde X$. Up to isomorphism, the irreducible affine Coxeter groups are classified into the families $\widetilde A_n, \widetilde B_n, \widetilde C_n, \widetilde D_n,$ or one of the exceptional types $\widetilde E_6, \widetilde E_7, \widetilde E_8,\widetilde F_4,\widetilde G_2$. The affine symmetric group $\tS_n$ from the introduction is the affine Coxeter group of type $\tA_{n-1}$. 

For each pair $\pm \beta$ of roots in $\Phi_0$, there is a type $\tA_1$ root subsystem of $\Phi$, whose elements are $\{ \pm \beta + k \delta : k \in \ZZ \}$. 
Each root of $\Phi$ lies in exactly one such $\tA_1$ subsystem.

To give a concrete model for biclosed sets in affine type in \Cref{AffineModels}, we recall the definition of the \newword{Coxeter fan} of a finite Coxeter group $W_0$. This is a complete fan living in the coroot representation $V^\vee_0$ of $W_0$. The set $F_0$ of $f\in V^\vee_0$ such that $\langle f,\alpha\rangle\geq 0$ for all $\alpha\in (\Phi_0)_+$ is called the \newword{dominant chamber}. A \newword{chamber} is any cone of the form $wF_0$ for $w\in W_0$. The Coxeter fan consists of all cones formed by intersections of chambers. These cones are called the \newword{faces} of the fan. Let $\faces(W_0)$ denote the set of faces of the Coxeter fan of $W_0$. The action of $W_0$ on $V^\vee_0$ permutes the elements of $\faces(W_0)$. 

There is also an action of the affine Coxeter group $W$ associated to $W_0$ on $\faces(W_0)$, described as follows. If $V^\vee$ is the coroot representation for $W$, then notice that $W$ preserves the subspace consisting of functionals which annihilate the vector $\delta\in V$. This subspace has a basis given by $\alpha_1^\vee,\ldots,\alpha_n^\vee$ and thus can be canonically identified with $V_0^\vee$; we do so from here on. This identification respects the action of $W_0\subseteq W$. Given a face $F\in \faces(W_0)$, let $W_F$ denote the stabilizer of $F$ under the $W$ action, and let $\Phi_F$ be the set of roots in $\Phi$ which are annihilated by all functionals in $F$. Then $W_F$ is the reflection subgroup corresponding to the root subsystem $\Phi_F$, called a \newword{parahoric subgroup} of $W$.


\subsection{Closed and biclosed sets} \label{ClosedBiclosedSection}
A subset $J$ of $\Phi_+$ is called \newword{closed} if, for any three roots $\alpha$, $\beta$ and $\gamma$ with $\gamma \in \RR_{>0} \alpha + \RR_{>0} \beta$, if $\alpha$ and $\beta \in J$ then $\gamma \in J$.
A subset $J$ of $\Phi_+$ is called \newword{coclosed} if $\Phi_+ \setminus J$ is closed. 
The set $J$ is \newword{biclosed} if it is both closed and coclosed.
As in the introduction, for a subset $A$ of $\Phi_+$, we define the \newword{closure} $\overline{A}$ to be the intersection of all closed sets containing $A$ and we define the \newword{interior} $A^{\circ}$ to be the union of all coclosed sets contained in $A$. 



We pause to connect this notion  to the closed subsets from the introduction.
Let $e_1$, $e_2$, \ldots, $e_n$ be the standard basis of $\RR^n$, then $\{ e_i-e_j : 1 \leq i < j \leq n \}$ form a root system of type $A_{n-1}$, which spans an $(n-1)$-dimensional subspace of $\RR^n$.
 The rank two subsystems of $\Phi_+$ come in two forms: The type $A_2$ subsystems are $\{ e_i-e_j,\ e_i-e_k,\ e_j-e_k \}$ for $i<j<k$, and the type $A_1 \times A_1$ subsystems are $\{ e_i - e_j,\ e_k - e_{\ell} \}$ for $i < j$ and $k < \ell$ distinct.
 The closure conditions  for the $A_2$-subsystems are the conditions from the introduction; the closure conditions for the $A_1 \times A_1$-subsystems are trivial, since they only have two positive roots. 
%The symmetric group $S_n$ is the Coxeter group $A_{n-1}$, whose positive root system can be described as $\{ e_i - e_j : 1 \leq i < j \leq n \}$, where $e_1$, $e_2$, \dots, $e_n$ is the standard basis of $\RR^n$. The rank two subsystems of $\Phi_+$ come in two forms: For $1 \leq i < j < k \leq n$, we have the $A_2$-subsystem $\{ e_i-e_j,\ e_i-e_k,\ e_j-e_k \}$ and, for $i < j$, $k < \ell$ distinct, we have the $A_1 \times A_1$-subsystem $\{ e_i - e_j,\ e_k - e_{\ell} \}$. The closure condition with respect to the $A_2$-subsystems is the closure condition from the introduction; the closure condition with respect to the $A_1 \times A_1$-subsystems is trivial, since it only has two positive roots. 

Similarly, define $\widehat{V}$ to be the $n+1$-dimensional vector space spanned by a collection of vectors $\te_i$, for $i \in \ZZ$, modulo the relation $\te_{i+n} - \te_i = \te_{j+n} - \te_j$ for all $i$, $j$. Then the vectors $\{ \te_i - \te_j : i<j,\ i \not \equiv j \bmod n \}$ form a copy of the positive $\widetilde{A}_{n-1}$-system, inside an $n$-dimensional subspace of $\widehat{V}$. There are three types of full rank two subsystems: For $i<j<k$ all distinct modulo $n$, we have an $A_2$-subsystem $\{ \te_i-\te_j,\ \te_i-\te_k,\ \te_j-\te_k \}$; for $i < j < i+n$ we have an $\tA_1$-subsystem $\{ \te_{i+pn}- \te_{j+qn} :  i+pn < j+qn \}\cup \{ \te_{j+pn} - \te_{i+qn} : j+pn < i+qn \}$ and, for $i < j$, $k < \ell$ distinct modulo $n$, we have an $A_1 \times A_1$-subsystem $\{ \te_i - \te_j,\ \te_k - \te_{\ell} \}$. 
The closure conditions for the $A_2$ and $\widetilde A_1$-subsystems are the conditions from the introduction; the closure conditions  for  the 
$A_1 \times A_1$-subsystems are trivial.

Recall that the set of inversions of a Coxeter group element can be described geometrically: $N(w) = \{ t \in T : \langle wf, \beta_t \rangle < 0 \}$, where $f \in V^{\vee}$ obeys $\langle f, \alpha_i \rangle >0$. 
From this geometric description, it is clear that $\{ \beta_t : t \in N(w) \}$ is biclosed for any $w \in W$. This characterizes the finite biclosed sets:
%
%Recall that any inversion set $N(w)$ is a finite of the form $\{t\in T : \langle f, \beta_t \rangle < 0 \}$ for some $f\in V^\vee$. Note that any set of the form $\{ \beta \in \Phi_+: \langle f, \beta\rangle<0  \}$, is biclosed, by inspection. The strongest version of this observation is the following: 

\begin{proposition}[\cite{Dyer2019}]\label{FiniteBiclosed}
	A set $B$ of positive roots is finite and biclosed if and only if there exists $w\in W$ such that $B = \{\beta_t : t\in N(w)  \}$. In this case, $w$ is uniquely defined.
\end{proposition}

Define the \newword{extended weak order} to be the poset of biclosed sets ordered by inclusion. This paper is motivated by Dyer's conjecture, first discussed in \cite[Remark 2.14]{Dyer1994a}:
\begin{conj} \label{DyerMain}
The extended weak order is a complete lattice. For $X$ a collection of biclosed sets, we have $\bigvee X = \overline{\bigcup_{J \in X} J}$ and $\bigwedge X = \left( \bigcup_{J \in X} J \right)^{\circ}$. 
\end{conj}


This conjecture is known in finite type \cite[Theorem 1.5]{Dyer2019} and rank two, and was recently shown by Weijia Wang to hold for rank three affine Coxeter groups \cite{Wang2018}. In \Cref{LatticeProof}, we give the first proof of Dyer's conjecture for all affine Coxeter groups. In \Cref{AffineModels}, we give a useful parametrization of biclosed sets for these groups (which is independent of \Cref{LatticeProof}).

\begin{figure}
	\centering 
	\begin{tikzpicture}[yscale=1]
		\draw[fill,blue] (0,0) node[below] {$\varnothing$} circle (1.2pt) 
		-- ++(1.707,.307) node[right] {$\{\te_2-\te_3\}$} circle (1.2pt)
		(-1.707,.307) node[left] {$\{\te_1-\te_2\}$} circle (1.2pt) -- (0,0);
		\draw[fill,blue] (-1.707,.307) -- ++(0,.5) node[left] {$\{\te_1-\te_2,\te_1-\te_4\}$} circle (1.2pt)
		%-- ++ (0, .5) node[left] {$\{\te_1-\te_2,\te_1-\te_4,\te_1-\te_6\}$} circle (1.2pt)
		-- ++ (0, .3) circle (1.2pt) -- ++(0, .2) circle (1.2pt)
		++ (0, .1) circle (.3pt) ++ (0, .1) circle (.3pt)
		++ (0, .1) circle (.3pt);
		\draw[fill,blue] (1.707,.307) -- ++(0,.5) node[right] {$\{\te_2-\te_3,\te_2-\te_5\}$} circle (1.2pt)
		%-- ++ (0, .5) node[right] {$\{\te_2-\te_3,\te_2-\te_5,\te_2-\te_7\}$} circle (1.2pt)
		-- ++ (0, .3) circle (1.2pt) -- ++(0, .2) circle (1.2pt)
		++ (0, .1) circle (.3pt) ++ (0, .1) circle (.3pt)
		++ (0, .1) circle (.3pt);
		
		\draw[fill] (-1.707, .307+1.4) node[left] {$\{\te_1-\te_{2k}:k\geq 1\}$} circle (1.2pt)
		(1.707, .307+1.4) node[right] {$\{\te_2-\te_{2k+1}:k\geq 1\}$} circle (1.2pt);
		
		\begin{scope}[shift={(0,.307+1.4)}]
			\draw[fill] (-1.707,.1) circle (.3pt) ++ (0,.1) circle (.3pt) ++ (0,.1) circle (.3pt) ++ (0, .1) circle (1.2pt) -- ++ (0,.2) circle (1.2pt) %-- ++ (0,.3) node[left] {$ \Phi_+\setminus\{\te_2-\te_3,\te_2-\te_5,\te_2-\te_7\} $} circle (1.2pt)
			-- ++(0,.3) node[left] {$ \Phi_+\setminus\{\te_2-\te_3,\te_2-\te_5\}$} circle (1.2pt)
			-- ++(0,.5) node[left] {$ \Phi_+\setminus\{\te_2-\te_3\} $} circle (1.2pt)
			-- ++(1.707,.307) node[above] {$\Phi_+$} circle (1.2pt);
			
			\draw[fill] (1.707,.1) circle (.3pt) ++ (0,.1) circle (.3pt) ++ (0,.1) circle (.3pt) ++ (0, .1) circle (1.2pt) -- ++ (0,.2) circle (1.2pt) %-- ++ (0,.3) node[left] {$ \Phi_+\setminus\{\te_2-\te_3,\te_2-\te_5,\te_2-\te_7\} $} circle (1.2pt)
			-- ++(0,.3) node[right] {$ \Phi_+\setminus\{\te_1-\te_2,\te_1-\te_4\}$} circle (1.2pt)
			-- ++(0,.5) node[right] {$ \Phi_+\setminus\{\te_1-\te_2\} $} circle (1.2pt)
			-- ++(-1.707,.307);
		\end{scope}
	\end{tikzpicture}
	\vspace{-.2cm}
	\caption{The extended weak order for $\widetilde A_1$. The finite biclosed sets are shaded in blue. 
		The subposet consisting of the finite biclosed sets is the usual weak order for $\widetilde A_1$; note the usual weak order is not a lattice, but the extended weak order is a complete lattice.
	}
	\label{fig:infdi}
	\vspace{-.7cm}
\end{figure}

\section{A combinatorial model for biclosed sets in affine type}\label{AffineModels}

We now use the material from \Cref{FiniteAffineGroups} to give a description of the biclosed sets for affine Coxeter groups generalizing \Cref{AtildeModel}. 
We omit proofs in this section.

We define two biclosed sets, $B$ and $C$, to be \newword{commensurable} if the symmetric difference $B \oplus C := (B \setminus C) \sqcup (C \setminus B)$ is finite.
We will begin by describing biclosed sets up to commensurability.
For a biclosed subset $B$ of $\Phi_+$, we define 
\[ B_{\infty} = \{ \beta \in \Phi_0 : \beta+k \delta \in B \ \text{for}\ k \gg 0 \}. \]
It is easy to see that $B_{\infty} = C_{\infty}$ if and only if $B$ and $C$ are commensurable. 
To understand the sets which can appear as $B_\infty$, 
we will use the Coxeter fan described in \Cref{FiniteAffineGroups}.

Let $F$ be a face of the $W_0$-Coxeter fan and let $f$ lie in the relative interior of $F$.
Then $\{ \beta \in \Phi_+ : \langle f, \beta \rangle < 0 \}$ and $\{ \beta \in \Phi_+ : \langle f, \beta \rangle \leq 0 \}$ are both biclosed sets. 
More generally, let $\Phi_F = \{ \beta \in \Phi : \langle f, \beta \rangle  = 0\}$, then $\Phi_F$ is the root system associated to the parahoric subgroup $W_F$.
If $\Phi'$ is any union of indecomposable components of $\Phi_F$, then $B(F, \Phi') :=  \{ \beta \in \Phi_+ : \langle f, \beta \rangle  < 0\} \cup (\Phi' \cap \Phi_+)$ is biclosed.

\begin{prop}
Each biclosed set is commensurable to a unique biclosed set of the form $B(F, \Phi')$ for $F \in \faces(W_0)$ and $\Phi'$ a (possibly empty) union of indecomposable components of $\Phi_F$.
\end{prop}

We note that $B(\mathbf{0}, \emptyset) = \emptyset$ and that the commensurability class of $B(\mathbf{0}, \emptyset)$ consists of the finite biclosed sets which, by \Cref{FiniteBiclosed}, are in bijection with the elements of $W$.
Following~\cite[Section 4.1]{Dyer2019}, we define an action of $W$ on the set of biclosed classes by
\[ w \cdot B = (wB \cap \Phi_+) \cup {\big(} \{ \beta_t : t \in N(w) \} \setminus (-wB) {\big)} . \]
Then we can generalize our description of $B(\mathbf{0}, \emptyset)$ to 

\begin{prop}
The set of biclosed sets commensurable to $B(F, \Phi')$ is the $W_F$-orbit of $B(F, \Phi')$, and $W_F$ acts freely on this orbit.
\end{prop}

Combining our results, biclosed sets are in bijection with triples $(F, \Phi', w)$ where $F$ is a face of $\faces(W_0)$, where $\Phi'$ is a union of indecomposable components of $\Phi_F$, and where $w \in W_F$. 

\begin{Remark}
The limit weak order of Lam and Pylyavskyy~\cite{LP} corresponds to triples $(F, \emptyset, w)$.
\end{Remark}

Let us work out what this looks like for $W=\tS_n$, the affine Coxeter group of type $\tA_{n-1}$. 
We will reuse the notation $\widehat{V}$  from \Cref{ClosedBiclosedSection} and we let $V$ be the subspace of $\widehat{V}$ spanned by $\te_i - \te_j$.
Similarly, we let $V_0$ be the subspace of $\RR^n$ spanned by $e_i - e_j$. The projection $V \to V_0$ is induced by the map $\widehat{V} \to \RR^n$ sending $\te_i$ to $e_{i \bmod n}$.
So the dual vector space, $(V_0)^{\vee}$, is $\RR^n / \RR(1,1,\ldots, 1)$, and we will write elements of $(V_0)^{\vee}$ as $(f_1, f_2, \ldots, f_n)$. 

The faces of the $S_n$ Coxeter fan are in bijection with ordered set partitions (equivalently, total preorders) of $[n]$. For example, with $n=4$, the ordered set partition $(\{1,3\}, \{2,4 \})$ corresponds to the face $\{ f_1 = f_3 \leq f_2 = f_4 \}$. 
Given an ordered set partition $(B_1, B_2, \ldots, B_r)$ of $[n]$ with corresponding face $F$, a permutation $w \in S_n$ stabilizes $F$ if and only if $w$ sends each block $B_i$ to itself. For example, the stabilizer of  $\{ f_1 = f_3 \leq f_2 = f_4 \}$ is $\langle (1 3),\ (2 4) \rangle$.

The parahoric subgroup $W_F$ is the preimage of this stabilizer in $\tS_n$. 
To describe this, put $\tB_a = \{ i \in \ZZ : i \bmod n \in B_a \}$. Then $\Phi_F$ is $\bigcup_{a=1}^r \{ \te_i - \te_j : i,j \in \tB_q,\  i \not\equiv j \bmod n \}$.
The irreducible components of $\Phi_F$ correspond to the non-singleton blocks of the set partition.
So, to give a pair $(F, \Phi')$ is to give an ordered set partition $(B_1, B_2, \ldots, B_r)$ of $[n]$ and a subset $R$ of the non-singleton blocks of this partition.

The biclosed set $B(F, \Phi')$ corresponds to the inversions of a translation invariant total order defined as follows: If $a<b$, $i \in \tB_a$ and $j \in \tB_b$, then we put $i \prec j$.
If $B_a$ is a non-singleton block, and $i$ and $j \in \tB_a$ with $i<j$, then we put $i \prec j$ if $B_a \not\in R$ and $i \succ j$ if $B_a \in R$.
If $B_a$ is a singleton block, the order on $\tB_a$ is undetermined, this will give the indeterminacy in \Cref{AtildeModel}.

More generally, an element $w$ of $W_F$ is a permutation in $\tS_n$ which takes each $\tB_a$ to itself. 
Applying such permutations to the translation invariant total orders described above gives the whole of \Cref{AtildeModel}.

%
%
%%
%%
%%\[ F_{\{1,3\}\prec \{2,4\}} = \{ f\in \o V^{\vee} : f(\alpha_{13}) = f(\alpha_{24})=0,~ f(\alpha_{12})\geq 0  \}. \]
%%In general, if $\preceq$ is a preorder on $[n]$, then define $F_{\preceq} = \{ f\in \o V^{\vee} : f(\alpha_{ij})\leq 0 ~\text{if } i\preceq j \}$.
%%The face $F_{\preceq}$ is fixed by $w\in \tS_n$ if and only if $i\preceq j$ whenever $w(i) \preceq w(j)$ (where we reduce $w(i),w(j)$ modulo $n$ before comparing). In effect, this means that $w$ just permutes elements within parts in the partition. For our partition $\{1,3\}\prec \{2,4\}$, the parahoric subgroup $W_F$ is all affine permutations which send integers which are 1 or 3 mod $4$ to integers that are 1 or 3 mod $4$, and similarly with integers that are 2 or 4 mod $4$. The root system of this parahoric is 
%%\[\Phi_F = \{ \alpha_{ij} : i\not\equiv j\mod 4,~i,j\equiv 1,3 \} \sqcup \{ \alpha_{ij} : i\not\equiv j\mod 4,~i,j\equiv 2,4 \}. \]
%%In this case, $\Phi_F$ is decomposable; the above disjoint union is a decomposition into indecomposables. So there are $4$ choices of $\Phi'\subseteq \Phi_F$ we can use to build a valid triple (we have a choice of whether to include the $\{1,3\}$ subsystem and whether to include the $\{2,4\}$ subsystem). For a general preorder $\preceq$, the root system $\Phi_F$ will have as many (nonempty) indecomposable subsystems as there are parts of size $\geq 2$ in the partition induced by $\preceq$. For instance, if $F=F_{1\prec 2 \prec 3 \prec 4}$, then $\Phi_F$ is empty (and there is just one choice for $\Phi'$), and if $F=F_{\{1,2,3,4\}}$, then $\Phi_F=\Phi$ is indecomposable (and there are two choices for $\Phi'$).
%
%Using \Cref{AffineClassification}, we thus have a parametrization of the path components for $\tS_n$. These are in bijection with pairs $(F,\Phi')$ as above; and thus, in bijection with tuples $(\preceq, X)$ consisting of a preorder on $[n]$ and a union $X$ of non-singleton $\preceq$-equivalence classes. It remains to parametrize the elements of the path component. \Cref{AffineClassification} gives a canonical biclosed set $B(F,\Phi')$ in each path component. Using the preorder, this can be written:
%\[ B(\preceq, X) = \{ \alpha_{ij}\in \Phi_+ : i<j,~ i\succ j  \} \cup \{\alpha_{ij} : i<j,~ i,j\in X\}.  \]
%%To turn this biclosed set into a translationally invariant total order on the integers, as in \Cref{AtildeModel}, for any integers $i<j$,
%%\[ i \succ_B j \iff \alpha_{ij}\in B.   \]
%%This leaves the relative order of $i$ and $i+n$ ambiguous if $i$ covers $i+n$ or vice versa.
%Using $X$, we can always extend the preorder $\preceq$ on $[n]$ into a total order $\prec'$ on $\mathbb Z$ as follows. For each $i<j$ which are $\preceq$-equivalent, we say $i\succ' j$ if $i,j\in X$, and we say $i\prec' j$ otherwise. The end result is a total order, where the $\preceq$-equivalent pairs $i,j$ are exactly the pairs with finitely many elements between them in the total order.
%
%For our running example $F_{\{1,3\}\preceq \{2,4\}}$, there are 4 options for $X$:
%\begin{align*} B(\preceq, \varnothing): &\cdots \prec' -3 \prec' -1 \prec' 1 \prec' 3 \prec' \cdots \prec' -4\prec' -2\prec' 0\prec' 2 \prec' 4\prec'\cdots   \\
% B(\preceq, \{1,3\}): &\cdots \prec'3\prec' 1 \prec' -1 \prec' -3 \prec' \cdots \prec' -4\prec' -2\prec' 0\prec' 2 \prec' 4\prec'\cdots   \\
% B(\preceq, \{2,4\}): &\cdots \prec' -3 \prec' -1 \prec' 1 \prec' 3 \prec' \cdots \prec'4 \prec' 2\prec' 0\prec' -2\prec' -4 \prec' \cdots   \\
% B(\preceq, \{1,2,3,4\}): &\cdots \prec'3\prec' 1 \prec' -1 \prec' -3  \prec' \cdots \prec'4 \prec' 2\prec' 0\prec' -2\prec' -4 \prec' \cdots.  
%\end{align*}
%The usual order on the integers corresponds to $B(F_{\{1,2,3,4\}}, \varnothing)=\varnothing$, whereas the biclosed set $B(F_{\{1,2,3,4\}},\Phi_+)$ corresponds to the reverse order on the integers.
%
%Note that if $i$ is alone in its $\preceq$-equivalence class, then this procedure for creating a total order does not determine whether $i\prec i+n$ or $i \succ i+n$. This is exactly the ambiguity discussed in the introduction. For instance, there is a unique biclosed set associated to the preorder $\{1\}\prec \{2\} \prec \{3\} \prec \{4\}$, given by
%\[ B(F_{\{1\}\preceq \{2\} \preceq \{3\} \preceq \{4\}},\varnothing) = \{ \alpha_{14}+k\delta,\alpha_{25}+k\delta, \alpha_{36}+k\delta, \alpha_{47}+k\delta : k\geq 0  \}. \]
%But there are several translationally invariant total orders on $\mathbb Z$ which describe this biclosed set: for instance, 
%\begin{align*}
%\cdots \prec -3 \prec 1 \prec 5 \prec \cdots \prec -2 \prec 2 \prec 6 \prec \cdots -1 \prec 3 \prec 7 \prec \cdots \prec -4 \prec 0 \prec 4 \prec\cdots \\
%\cdots \prec -3 \prec 1 \prec 5 \prec \cdots \prec 6 \prec 2 \prec -2 \prec \cdots -1 \prec 3 \prec 7 \prec \cdots \prec -4 \prec 0 \prec 4 \prec\cdots
%\end{align*}
%both have the same underlying biclosed set, since the only difference is the orientation of $2\prec 2+4$, which has no (real) root to detect it.
%
%Now that we have seen that path components can be parametrized by pairs $(\preceq, X)$ and realized as total orders which group together $\preceq$-equivalence classes and reverse all pairs in $X$, it remains to understand the parametrization of the elements of the path component. Indeed, \Cref{AffineClassification} indicates that such elements are of the form $w\cdot B(F,\Phi')$ for $w\in W_F$. We can easily see what this action should look like on a total order: it is just the permutation action of $\tS_n$ on $\mathbb Z$! For instance, the permutation with cycle decomposition $\cdots(-3,-2)(1,2)(5,6)\cdots$ acts on \(\prec -2 \prec -1 \prec 0 \prec 1 \prec 2 \prec \cdots\) by sending it to \[\prec -3 \prec -1 \prec 0 \prec 2 \prec 1 \prec \cdots. \]
%The condition that $W_F$ stabilizes $F$ is exactly the condition that this action moves each $i\in \mathbb Z$ across only finitely many integers. As a result, we see that $B(\{1,2,3,4\},\varnothing,w)$ corresponds to the total order describing the one-line notation for $w$ as a permutation of $\mathbb Z$. Since any translationally invariant total order can be permuted into one of the standard form $B(\preceq, X)$, it follows that \Cref{AffineClassification} implies \Cref{AtildeModel}, the description of biclosed sets given in the introduction.
%\begin{remark}
%	The boundary of the Tits cone is the hyperplane dual to the imaginary root $\delta$. The restriction of the Coxeter arrangement of $W$ to this subspace is the Coxeter arrangement of $\o W$. 
%\end{remark}


%
%\section{A combinatorial model for biclosed sets in affine type}\label{AffineModels}
%
%Using the constructions in \Cref{Background}, we may give a description of the biclosed sets for affine Coxeter groups generalizing \Cref{AtildeModel}. It follows from \Cref{FiniteBiclosed} that, in a finite Coxeter group, biclosed sets are parametrized by the elements of $W$. Thus in that case the classification problem is equivalent to describing the group elements, which is well-understood. In the infinite case we will wish to reduce to the finite. 
%%To do so, we group the sets into equivalence classes called \newword{path components}. Biclosed sets $B,B'$ are in the same path component iff there is a finite sequence $B=B_1,\ldots,B_k=B'$ of biclosed sets such that $B_i$ and $B_{i+1}$ differ by the addition or removal of one root. We say that two biclosed sets $B,B'$ are \newword{commensurable} if their symmetric difference 
%To do so, we consider an equivalence relation on biclosed sets called commensurability. Biclosed sets $B,B'$ are \newword{commensurable} if their symmetric difference $B \oplus B' := (B\setminus B') \ \cup\  (B'\setminus B)$
%is a finite set. It turns out equivalence classes under this relation are exactly the path-connected components of the Hasse diagram for extended weak order; we call these classes \newword{path components}.
%%The following fact is nontrivial.
%%
%%\begin{lemma}
%%	Path components are the maximal path-connected subsets of the Hasse diagram for extended weak order. Furthermore, biclosed sets $B,B'$ are in the same path component if and only if they are commensurable.
%%\end{lemma}
%
%
%
%
%%We shall show in the course of proving \Cref{thm:affineparam} that two biclosed sets are commensurable if and only if they are connected via a finite sequence of covering relations in the extended weak order; thus a path component of $W$ is equivalently a path component of the Hasse diagram of its extended weak order. 
%%
%%The next two lemmas show that the type $\tA_1$-subsystems govern the path components.
%%
%%\begin{lemma}\label{thm:affrank2}
%%	Let $W$ be an affine Coxeter group. Then every reflection is contained in exactly one full root subsystem of type $\tA_1$. Additionally, the number of type $\tA_1$ full subsystems is finite.
%%\end{lemma}
%%\begin{proof}
%%	Let $W_0$ be the finite Coxeter group associated to $W$ and let $\Phi_0$ be the corresponding root system. The $\tA_1$-subsystems are the subsystems of the form $\{ \pm \beta + k \delta : k \in \ZZ \}$ for $\beta \in \Phi_0$; since $\Phi_0$ is finite, there are finitely many of these.
%%	%	 Then the type $\tA$ rank two systems are exactly the set of reflections in $W$ which reduce to a given reflection in $\o W$, proving the first claim. The second claim follows since there are a finite number of reflections in $\o W$. 
%%%	% make sure ``lying over'' is defined somewhere, or explain this better
%%\end{proof}
%
%%Thus the type $\tA$ rank two systems partition the set $T$ of reflections into finitely many parts. As a result, commensurability can be checked independently on each of these parts.
%%
%%\begin{lemma}\label{lem:fincofspl}
%%	Let $W$ be an affine Coxeter group. Then two biclosed sets $B_1$ and $B_2$ in $W$ are commensurable if and only if, for any full $\tA_1$-subsystem $\cA$, exactly one of the following holds:
%%	\begin{itemize}
%%		\item $\cA \cap B_1$ and $\cA \cap B$ are both finite.
%%		\item  $\cA \setminus B_1$ and $\cA \setminus B_2$ are both finite.
%%		\item  $\cA \cap B_1 = \cA \cap B_2$ and both $\cA \cap B_1$ and $\cA \setminus B_1$ are infinite.
%%	\end{itemize}
%%\end{lemma}
%%\begin{proof}
%%	Since each reflection in $W$ is contained in exactly one of finitely $\tA_1$-subsystems, two biclosed sets $B_1$, $B_2$ are commensurable if and only if $|(B_1\cap  \cA) \oplus (B_2\cap  \cA)|$ is finite for all such $\cA$. We then consider the possible biclosed subsets of $\tA_1$, where this is an immediate verification.
%%\end{proof}
%
%%The previous lemma shows 
%
%
%
%%It turns out that path components are characterized by their \emph{asymptotic behavior}, meaning that two biclosed sets $B$ and $B'$ are commensurable if and only if $\alpha +r\delta \in B$ exactly when $\alpha+r\delta \in B'$, for large enough $r$. 
%%For a biclosed set $B$ in $\Phi_+$, put \[ B_{\infty} =  \{ \beta \in \Phi_0 :  \beta+r\delta \in B \ \text{for} \ r \gg 0 \} \]
%%so $B_{\infty}$ encodes the asymptotic behavior of $B$. Even though $B_\infty$ can contain both positive and negative roots of $\Phi_0$, it turns out to be biclosed in an appropriate sense. This is an observation that was independently discovered by Dyer in an unreleased work \cite{Dyerpreprint}. Here we will describe how the possible choices for $B_\infty$ can be described using the geometry of the finite Coxeter group associated to $W$.
%
%
%%
%%The following shows that the allowed behavior at infinity is constrained by the structure of the finite root system $\Phi_0$ associated to $W$. 
%%
%%\begin{lemma}\label{thm:inj}
%%With the above notation, $B_{\infty}$ is a biclosed set in $\Phi_0$. Two biclosed sets $B_1$ and $B_2$ are commensurable if and only if $(B_1)_{\infty} = (B_2)_{\infty}$.
%%\end{lemma}
%%\begin{proof}
%%	First we check that if $B$ is in $\bic(W)$, then $B_{\infty}$ is biclosed. Let $\a,\b,\g\in \Phi_0$ satisfy $\a+\b=\g$. First assume $\a, \b$ are both elements of $B_0$, so that for some $r_0$ and all $r\geq r_0$, we have $\a+r\d$ and $\b+r \d \in B$. 
%%	%(We may take $r_0\geq 1$ so that these roots are in fact positive.) 
%%	Then 
%%	\[ (\a + r\d) + (\b+r_0\d) = \a+\b +(r+r_0)\d  \]
%%	necessarily forces $\a+\b+(r+r_0)\d$ to be an element of $B$, by closure. In particular, if $\ell \geq 2r_0$, then $\a+\b+\ell\d$ is in $B$. So $\a+\b \in B_{\infty}$. Thus, $\o B$ is a closed set. Coclosure is less obvious. Assume now that $\g$ is in $\o B$, so that $\g+r\d\in B$ for all $r\geq r_0$. For each such $r$, consider $(\a+r\d) + (\b+ r\d) = \a + \b +2r\d$. Coclosure of $B$ implies that for each such $r$, we have at least one of $t_{\a+r\d},t_{\b+r\d}$ in $B$. Since there are infinitely many values of $r\geq r_0$, one of those two options must appear in $B$ infinitely many times. Say (without loss of generality) that it is $t_{\a+r\d}$ which is in $B$ for infinitely many values of $r$; then in fact $t_{\a+r\d}\in B$ for every choice of $r$ greater than the minimal such value, since $B$ is closed. So $\a \in \o B$. Thus $\o B$ is coclosed.
%%	
%%	Now we verify that $B_1,B_2\in \bic(W)$ are commensurable if and only if $\o B_1=\o B_2$. Indeed, the criterion set out in \Cref{lem:fincofspl} can be reinterpreted using the following. First, note that each $\mathcal A$ of type $\tA_1$ in $W$ corresponds to a unique pair $\pm \a$ in $\oDel$ (these are the two roots for which $t_{\a+r\d}$ is in $\mathcal{A}$). Then
%%	\begin{itemize}
%%		\item $B_1$ is finite in $\mathcal A$ iff $\{\pm \a\}\cap \o B_1 = \varnothing$,
%%		
%%		\item $B_1$ is cofinite in $\mathcal A$ iff $\{\pm \a\}\cap \o B_1 = \{\pm \a\}$, and
%%		
%%		\item the two split cases correspond to when $\{\pm \a\}\cap \o B_1$ is a singleton.
%%	\end{itemize}
%%	From this viewpoint, \Cref{lem:fincofspl} says that $B_1,B_2$ are commensurable iff $\o B_1= \o B_2$, as desired.
%%\end{proof}
%%\begin{remark}
%%	The boundary of the Tits cone for an (irreducible) affine Coxeter group $W$ is a linear space of codimension 1 which is subdivided by reflection hyperplanes of $W$ into an arrangement equivalent to the reflection arrangement for $\o W$. When $B$ is separable (its set of positive roots is determined by a linear functional on $V$), it corresponds to a region of the $W$ arrangement. The operation $B\mapsto \o B$ can be thought of as intersecting this region with the boundary of the Tits cone. The resulting $\face(\o B)\in \faces(\o W)$ is exactly this intersection.
%%	%TODO put a picture of this
%%\end{remark}
%%
%%Thus we may label each path component by an element of $\bic(\oDel)$, which determines the status of each $\tA_1$ rank two system (finite, cofinite, split one way, split the other way). The parametrization problem for path components reduces to determining which labels produce valid path components. The following lemma shows that all such labels are valid, so that path components are in bijection with elements of $\bic(\oDel)$. 
%%
%%\begin{lemma}\label{thm:sur}
%%	Let $\o B \in \bic(\oDel)$. Then the set
%%	\[ B = \{ t_{\alpha+r\delta} : \alpha\in \o B, ~ r\geq 0, ~ \a+r\d \in \rep \} \]
%%	is biclosed in $W$.
%%\end{lemma}
%%\begin{proof}
%%	First note that if $\mathcal A$ is any type $\tA$ rank two systems, then $B\cap \mathcal A$ is either empty, is all of $\mathcal A$, or is split. In all cases $B\cap\mathcal A$ is biclosed. To check that $B$ is biclosed in the finite type rank twos, it is sufficient (by \Cref{thm:finitesystem}(ii)) to check that $B$ is biclosed for all triples of the form
%%	\[ t_{\alpha + r\delta}\rightarrow t_{\gamma + (r+r')\delta } \rightarrow t_{\beta+r'\delta} \]
%%	with $\alpha,\beta,\gamma \in \oDel$ such that $\alpha+\beta=\gamma$. By definition of $B$, this is equivalent to $\o B$ being biclosed for the triple 
%%	\[ \alpha \rightarrow \gamma \rightarrow \beta, \]
%%	which is tautological.
%%\end{proof}
%%
%%For each path component $\block$, let $B_\block$ denote the canonical element of $\block$ described in the lemma. Once we have a valid nonempty path component, it remains to be seen how to parametrize its contents. It will turn out that the path component $\block$ is exactly the orbit of $B_\block$ under a certain reflection subgroup $W_\block$ of $W$, called a \emph{parahoric subgroup}\footnotemark. 
%%To define it, consider the type $\tA$ rank two systems for which elements of $\block$ are finite or cofinite (\textit{i.e.}, not split). Define $W_\block$ to be the subgroup of $W$ generated by all reflections in such rank twos. Recall that for a face $F$ in $\faces(\oDel)$, $\o W_F$ denotes the subgroup of $\o W$ generated by reflections across hyperplanes containing $F$. Then $W_\block$ is equivalently the group generated by reflections laying over those in $W_{\face(\o B)}$, for any representative $B$ in $\block$.
%%\footnotetext{In the theory of affine buildings associated to a reductive group $G$ over a $p$-adic ring $A$, a parahoric subgroup is the subgroup of $G(A)$ which fixes a given face in the spherical building at infinity. For us the definition is (almost) the same; given an element $\o B \in \bic(\oDel)$ (a ``face'' in the spherical Coxeter complex at infinity), \Cref{thm:para} shows that its stabilizing reflection subgroup of $W$ is exactly $W_{\block}$.}
%%%TODO make this definition clearer
%%
%%\begin{lemma}\label{thm:para}
%%	If $t \in T_W$ is a reflection, then set $t\cdot B \doteq N(t) \oplus tBt^{-1}$. If $t=t_{\a}$ for a root $\alpha\in \re$, then 
%%	\[ \o{(t\cdot B)} = t_{\o\alpha}\o B \]
%%	for any biclosed set $B\in \bic(W)$. 
%%	In particular, if $\face(\o B) = F$, then $t\cdot B$ is in the same path component as $B$ if and only if $t_{\o\alpha} \in \o W_F$ if and only if $t\in W_\block$.
%%\end{lemma}
%%\begin{proof}
%%	We first note that $N(t)\oplus tBt^{-1}$ and $tBt^{-1}$ have the same asymptotic behavior, since $N(t)$ is finite. Thus we just need to check that $\o\beta+r\delta \in tBt^{-1}$ for large enough $r$ exactly when $\o\beta \in t_{\o\alpha}\o B$. The first statement is equivalent to $t(\o\beta+r\delta) \in B$. We have
%%	\[ t(\o\beta + r\delta) = \o\beta + r\delta - 2\frac{(\beta:\alpha)}{(\alpha:\alpha)} \alpha 
%%	= \o\beta + r\delta - 2\frac{(\o\beta:\o\alpha)}{(\o\alpha:\o\alpha)} (\o\alpha+k\delta) 
%%	= (r-2\frac{(\o\beta:\o\alpha)}{(\o\alpha:\o\alpha)}k)\delta + t_{\o \alpha}\beta \]
%%	for some $k\geq 0$, where we use that $(\delta: \gamma)=0$ for all $\gamma$. Thus $\o\beta+r\delta \in tBt^{-1}$ for $r\gg 0$ exactly when $t_{\o\alpha}\beta + r\delta \in B$ for $r\gg 0$, which by definition of $\o B$ occurs if and only if $t_{\o\alpha}\beta \in \o B$. This is equivalent to $\beta \in t_{\o\alpha} \o B$, and the lemma follows.
%%\end{proof}
%
%%\begin{lemma}
%%	Let $\mathcal A$ be a type $A$ rank two system of an affine Coxeter group $W$. Then there is a unique irreducible rank three reflection subgroup $W'$ of $W$ which contains $\mathcal A$. This group is of type $\tA_2$. 
%%\end{lemma}
%%
%%Now we can describe the full parametrization of biclosed sets in affine type.
%%
%%\begin{theorem}\label{thm:affineparam}
%%	Let $W$ be an affine Coxeter group and $\oDel$ the associated finite root system. Then the map $B_\block \mapsto \o B_\block$ induces a bijection between path components of $W$ and elements of $\bic(\oDel)$. 
%%	Furthermore, if $W_\block$ is the parahoric subgroup stabilizing $\block$, then the map
%%	\begin{align*}
%%		W_\block &\rightarrow \block \\
%%		w &\mapsto w\cdot B_\block 
%%	\end{align*}
%%	is a bijection, and the equality
%%	\[ w\cdot B_\block = N_{W_\block}(w) \oplus B_\block  \]
%%	holds.
%%\end{theorem}
%%\begin{proof}
%%	The first assertion is a combination of Lemmas \ref{thm:inj} and \ref{thm:sur}. For the rest, we first check that 
%%	\[wB_\block w^{-1} \cap W_\block = B_\block \cap W_\block \tag{$*$}\]
%%	for $w\in W_\block$. Then the stated equality
%%	\[ w\cdot B_\block = N_W(w) \oplus wB_\block w^{-1} 
%%	%= N_{W_\block}(w) \oplus N(w)\setminus W_\block \oplus wB_\block w^{-1}\cap W_\block \oplus wB_\block w^{-1}\setminus W_\block 
%%	= N_{W_\block}(w) \oplus B_\block \]
%%	follows from the following observations:
%%	\begin{itemize}
%%		\item[(1)] $(w\cdot B_\block) \setminus W_\block = B_\block\setminus W_\block,$ since each reflection in $W\setminus W_\block$ is contained in an infinite rank two system for which all elements of $\block$ are split in the same direction.
%%		
%%		\item[(2)] $N_W(w)\cap W_\block = N_{W_\block}(w)$.
%%		
%%		\item[(3)] $(X\oplus Y)\cap Z = (X\cap Z)\oplus (Y\cap Z)$ for any sets $X,Y,Z$.
%%	\end{itemize}
%%	($*$), (2), and (3) together imply that 
%%	\[(w\cdot B_\block)\cap W_\block = (N_{W_\block}\oplus B_\block)\cap W_\block.\]
%%	Combined with (1), we find that $N_W(w) \oplus wB_\block w^{-1}	= N_{W_\block}(w) \oplus B_\block$, as desired.
%%	
%%	Now we show ($*$). It is sufficient to show that $tB_\block t^{-1} \cap W_\block = B_\block \cap W_\block$ for reflections $t$ in $W_\block$. Observe that conjugating by a reflection takes rank two systems of $W$ to rank two systems of $W$ of the same type (since a reflection takes two dimensional subspaces of $V$ to two dimensional subspaces of $V$). Furthermore, this action fixes a given rank two system $\mathcal A$ if and only if $t\in\mathcal A$. If $\mathcal A$ is a type $\tA$ rank two system for which $B$ is finite, then
%%	\[ |tB_\block t^{-1} \cap \mathcal A| = |B_\block \cap t^{-1}\mathcal A t| = 0 \]
%%	since the intersection of $B_\block$ with a type $\tA$ is either empty or not finite, and the second case would imply that $tB_\block t^{-1}$ is in a different path component from $B_\block$, which can't happen by the previous lemma. A similar argument shows that $tB_\block t^{-1} \cap \mathcal A = \mathcal A$ whenever elements of $\block$ are cofinite in $\mathcal A$. 
%%	%Since a set of roots is determined by its intersections with infinite rank two systems, 
%%	It follows that $tB_\block t^{-1} \cap W_\block = B_\block \cap W_\block$, so ($*$) is shown.
%%	% We don't really need to restrict to reflections in this
%%	
%%	Since $B_\block \oplus w\cdot B_\block= N_{W_\block}(w)$ determines uniquely the element $w\in W_\block$, each $w\cdot B_\block$ is a distinct element of $\block$. To show surjectivity, let $B$ be any element of $\block$. Then $B\oplus B_\block$ is finite. We claim $B\oplus B_\block$ contains only reflections from $W_\block$, and that it is in fact an element of $\bic(W_\block)$. \Cref{thm:finite} will then imply that $B\oplus B_\block$ is an inversion set of an element of $W_\block$, as desired. Elements of $B\oplus B_\block$ must come from infinite rank twos in which $B$ and $B_\block$ are both finite or both cofinite (cf. \Cref{lem:fincofspl}). So by definition of $W_\block$, we have that $B\oplus B_\block \subseteq T_{W_\block}$. 
%%	
%%	We check that $B\oplus B_\block$ is biclosed in $W_\block$. It follows from the definition of $B_\block$ that $B\oplus B_\block$ is biclosed for type $\tA$ rank twos. Indeed, for $\mathcal A$ such a rank two, $(B\oplus B_\block)\cap \mathcal A$ is either empty or equal to the intersection of $\mathcal A$ with $B$ or with its complement. Thus we just need to check nontrivial $\mathcal A$ of finite type. Each such $\mathcal A$ lies over $\o{\mathcal A}$, a rank two in $\oDel$ of the same type, and we have the identity $|B_\block\cap \mathcal A| = |\o B_\block \cap \mathcal A|$. If $\mathcal A \subseteq W_\block$, then it follows (cf. \Cref{thm:undirectedtypes} and the definition of $W_\block$) that $\o B_\block \cap \o{\mathcal A}$ is either empty or all of $\o{\mathcal A}$. In one case $(B\oplus B_\block)\cap \mathcal A = B\cap \mathcal A$, in the other case $(B\oplus B_\block)\cap \mathcal A = B^c\cap \mathcal A$, and in both cases $B\oplus B_\block$ is biclosed in $\mathcal A$. Thus $B\oplus B_\block$ is biclosed in every rank two in $W_\block$, and we are done.
%%\end{proof}
%
%
%There is an action of $W$ on biclosed sets which extends the natural action $v\cdot N(w):=N(vw)$ on inversion sets. We define for a set $B\subseteq \Phi_+$ and an element $w\in W$:
%\[ w\cdot B := \Phi_+\cap \{ w\alpha : \alpha \in \Phi_-\cup B \}. \]
%Now we can state the full classification of biclosed sets in an affine Coxeter group.
%
%\begin{theorem}\label{AffineClassification}
%	Let $W$ be an affine Coxeter group and $W_0$ the associated finite Coxeter group. Biclosed sets for $W$ are in bijection with triples $(F, \Phi', w)$, where: 
%	\begin{itemize} 
%	\item $F$ is a face of the Coxeter fan for $W_0$
%	
%	\item $\Phi'$ is a (possibly empty) union of indecomposable components of the root system $\Phi_F$
%	
%	\item $w\in W$ stabilizes $F$.
%	\end{itemize} 
%	If $f$ is a functional in the interior of $F$, define
%	\[ B(F,\Phi') := \{ \alpha \in \Phi_+ : \langle f,\alpha\rangle>0 \} \cup \Phi'_+. \]
%	Then the biclosed set associated to the triple $(F,\Phi',w)$ is
%	%equivalent: \[ B(F, \Phi', w) = \{ \alpha_t \in \Phi_F : t\in N(w)\cap W_F \} \oplus (\{ \alpha \in \Phi_+ : f(\alpha)>0 \} \cup \Phi') . \]
%	\begin{align*} 
%	B(F, \Phi', w) &:= w\cdot B(F,\Phi') \\
%	&\phantom{:}= \{ \alpha_t \in \Phi_F : t\in N(w)\cap W_F \} \oplus B(F,\Phi'). 
%	\end{align*}
%	The sets $B(F_1, \Phi'_1, w_1)$, $B(F_2,\Phi'_2,w_2)$ are commensurable if and only if $F_1=F_2$ and $\Phi'_1=\Phi'_2$.
%\end{theorem}
%
%Let us work out what this looks like for $W=\tS_n$, the affine Coxeter group of type $\tA_{n-1}$. In this case, $\o W = S_n$, the usual symmetric group on $n$ letters. The Coxeter fan for $S_n$ is often called the \newword{braid fan} (the fan associated to the \newword{braid arrangement}). The set $\faces(S_n)$ is in bijection with ordered set partitions (equivalently, preorders) of $[n]$. For example, in $\tS_4$, the ordered partition $\{1,3\}\prec \{2,4\}$ corresponds to the face
%\[ F_{\{1,3\}\prec \{2,4\}} = \{ f\in \o V^{\vee} : f(\alpha_{13}) = f(\alpha_{24})=0,~ f(\alpha_{12})\geq 0  \}. \]
%In general, if $\preceq$ is a preorder on $[n]$, then define $F_{\preceq} = \{ f\in \o V^{\vee} : f(\alpha_{ij})\leq 0 ~\text{if } i\preceq j \}$.
%The face $F_{\preceq}$ is fixed by $w\in \tS_n$ if and only if $i\preceq j$ whenever $w(i) \preceq w(j)$ (where we reduce $w(i),w(j)$ modulo $n$ before comparing). In effect, this means that $w$ just permutes elements within parts in the partition. For our partition $\{1,3\}\prec \{2,4\}$, the parahoric subgroup $W_F$ is all affine permutations which send integers which are 1 or 3 mod $4$ to integers that are 1 or 3 mod $4$, and similarly with integers that are 2 or 4 mod $4$. The root system of this parahoric is 
%\[\Phi_F = \{ \alpha_{ij} : i\not\equiv j\mod 4,~i,j\equiv 1,3 \} \sqcup \{ \alpha_{ij} : i\not\equiv j\mod 4,~i,j\equiv 2,4 \}. \]
%In this case, $\Phi_F$ is decomposable; the above disjoint union is a decomposition into indecomposables. So there are $4$ choices of $\Phi'\subseteq \Phi_F$ we can use to build a valid triple (we have a choice of whether to include the $\{1,3\}$ subsystem and whether to include the $\{2,4\}$ subsystem). For a general preorder $\preceq$, the root system $\Phi_F$ will have as many (nonempty) indecomposable subsystems as there are parts of size $\geq 2$ in the partition induced by $\preceq$. For instance, if $F=F_{1\prec 2 \prec 3 \prec 4}$, then $\Phi_F$ is empty (and there is just one choice for $\Phi'$), and if $F=F_{\{1,2,3,4\}}$, then $\Phi_F=\Phi$ is indecomposable (and there are two choices for $\Phi'$).
%
%Using \Cref{AffineClassification}, we thus have a parametrization of the path components for $\tS_n$. These are in bijection with pairs $(F,\Phi')$ as above; and thus, in bijection with tuples $(\preceq, X)$ consisting of a preorder on $[n]$ and a union $X$ of non-singleton $\preceq$-equivalence classes. It remains to parametrize the elements of the path component. \Cref{AffineClassification} gives a canonical biclosed set $B(F,\Phi')$ in each path component. Using the preorder, this can be written:
%\[ B(\preceq, X) = \{ \alpha_{ij}\in \Phi_+ : i<j,~ i\succ j  \} \cup \{\alpha_{ij} : i<j,~ i,j\in X\}.  \]
%%To turn this biclosed set into a translationally invariant total order on the integers, as in \Cref{AtildeModel}, for any integers $i<j$,
%%\[ i \succ_B j \iff \alpha_{ij}\in B.   \]
%%This leaves the relative order of $i$ and $i+n$ ambiguous if $i$ covers $i+n$ or vice versa.
%Using $X$, we can always extend the preorder $\preceq$ on $[n]$ into a total order $\prec'$ on $\mathbb Z$ as follows. For each $i<j$ which are $\preceq$-equivalent, we say $i\succ' j$ if $i,j\in X$, and we say $i\prec' j$ otherwise. The end result is a total order, where the $\preceq$-equivalent pairs $i,j$ are exactly the pairs with finitely many elements between them in the total order.
%
%For our running example $F_{\{1,3\}\preceq \{2,4\}}$, there are 4 options for $X$:
%\begin{align*} B(\preceq, \varnothing): &\cdots \prec' -3 \prec' -1 \prec' 1 \prec' 3 \prec' \cdots \prec' -4\prec' -2\prec' 0\prec' 2 \prec' 4\prec'\cdots   \\
% B(\preceq, \{1,3\}): &\cdots \prec'3\prec' 1 \prec' -1 \prec' -3 \prec' \cdots \prec' -4\prec' -2\prec' 0\prec' 2 \prec' 4\prec'\cdots   \\
% B(\preceq, \{2,4\}): &\cdots \prec' -3 \prec' -1 \prec' 1 \prec' 3 \prec' \cdots \prec'4 \prec' 2\prec' 0\prec' -2\prec' -4 \prec' \cdots   \\
% B(\preceq, \{1,2,3,4\}): &\cdots \prec'3\prec' 1 \prec' -1 \prec' -3  \prec' \cdots \prec'4 \prec' 2\prec' 0\prec' -2\prec' -4 \prec' \cdots.  
%\end{align*}
%The usual order on the integers corresponds to $B(F_{\{1,2,3,4\}}, \varnothing)=\varnothing$, whereas the biclosed set $B(F_{\{1,2,3,4\}},\Phi_+)$ corresponds to the reverse order on the integers.
%
%Note that if $i$ is alone in its $\preceq$-equivalence class, then this procedure for creating a total order does not determine whether $i\prec i+n$ or $i \succ i+n$. This is exactly the ambiguity discussed in the introduction. For instance, there is a unique biclosed set associated to the preorder $\{1\}\prec \{2\} \prec \{3\} \prec \{4\}$, given by
%\[ B(F_{\{1\}\preceq \{2\} \preceq \{3\} \preceq \{4\}},\varnothing) = \{ \alpha_{14}+k\delta,\alpha_{25}+k\delta, \alpha_{36}+k\delta, \alpha_{47}+k\delta : k\geq 0  \}. \]
%But there are several translationally invariant total orders on $\mathbb Z$ which describe this biclosed set: for instance, 
%\begin{align*}
%\cdots \prec -3 \prec 1 \prec 5 \prec \cdots \prec -2 \prec 2 \prec 6 \prec \cdots -1 \prec 3 \prec 7 \prec \cdots \prec -4 \prec 0 \prec 4 \prec\cdots \\
%\cdots \prec -3 \prec 1 \prec 5 \prec \cdots \prec 6 \prec 2 \prec -2 \prec \cdots -1 \prec 3 \prec 7 \prec \cdots \prec -4 \prec 0 \prec 4 \prec\cdots
%\end{align*}
%both have the same underlying biclosed set, since the only difference is the orientation of $2\prec 2+4$, which has no (real) root to detect it.
%
%Now that we have seen that path components can be parametrized by pairs $(\preceq, X)$ and realized as total orders which group together $\preceq$-equivalence classes and reverse all pairs in $X$, it remains to understand the parametrization of the elements of the path component. Indeed, \Cref{AffineClassification} indicates that such elements are of the form $w\cdot B(F,\Phi')$ for $w\in W_F$. We can easily see what this action should look like on a total order: it is just the permutation action of $\tS_n$ on $\mathbb Z$! For instance, the permutation with cycle decomposition $\cdots(-3,-2)(1,2)(5,6)\cdots$ acts on \(\prec -2 \prec -1 \prec 0 \prec 1 \prec 2 \prec \cdots\) by sending it to \[\prec -3 \prec -1 \prec 0 \prec 2 \prec 1 \prec \cdots. \]
%The condition that $W_F$ stabilizes $F$ is exactly the condition that this action moves each $i\in \mathbb Z$ across only finitely many integers. As a result, we see that $B(\{1,2,3,4\},\varnothing,w)$ corresponds to the total order describing the one-line notation for $w$ as a permutation of $\mathbb Z$. Since any translationally invariant total order can be permuted into one of the standard form $B(\preceq, X)$, it follows that \Cref{AffineClassification} implies \Cref{AtildeModel}, the description of biclosed sets given in the introduction.
%%\begin{remark}
%%	The boundary of the Tits cone is the hyperplane dual to the imaginary root $\delta$. The restriction of the Coxeter arrangement of $W$ to this subspace is the Coxeter arrangement of $\o W$. 
%%\end{remark}

\section{Dyer's conjecture and the $CU$-property} \label{LatticeProof}
In this section we prove \Cref{DyerMain} for affine Coxeter groups. \Cref{DyerMain} would follow from the following conjecture (also due to Dyer, see \cite[Conjecture 2.6]{Dyer2019}). %this reference is to ``closure of a unipodal set is unipodal''
%TODO should we change this to unipodal? It would lose the symmetry (and also is no longer preserved by folding)
\begin{conj} \label{GaloisConnection}
	Let $U$ be a coclosed subset of $\Phi_+$, then $\overline{U}$ is also coclosed. Dually, let $K$ be a closed subset of $\Phi_+$, then $K^{\circ}$ is also closed. 
\end{conj}
The statements about $\overline{U}$ and about $K^{\circ}$ are equivalent, by putting $K = \Phi_+ \setminus U$.

\begin{proof}[Proof that \Cref{GaloisConnection} implies \Cref{DyerMain}]
	Let $\cX$ be any collection of biclosed subsets of $\Phi_+$ and put $V:=\overline{\bigcup_{J \in \cX} J}$. We will show that $V$ is biclosed and is the join of $\cX$.
	Indeed, $V$ is closed since it is defined as a closure. The set $\bigcup_{J \in \cX} J$ is coclosed, since it is a union of coclosed sets, so \Cref{GaloisConnection} implies that $V$ is coclosed; we have now checked that $Y$ is biclosed.
	
	For any $J_0 \in \cX$, we have $J_0 \subseteq \bigcup_{J \in \cX} J \subseteq V$, so $V$ is an upper bound for $\cX$. If $Y$ is any other biclosed upper bound for $\cX$ then, by the definition of an upper bound, we have $\bigcup_{J \in \cX} J \subseteq Y$; since $Y$ is closed, we then have $V \subseteq Y$ as well. We have thus shown that $Y$ is the unique least upper bound of $\cX$. 	The case of lower bounds is analogous.
\end{proof}
To prove \Cref{GaloisConnection}, we will need the notion of a \newword{unipodal} set of roots. A set $U\subseteq \Phi_+$ is unipodal if, for any full rank two subsystem $\Phi'$ of $\Phi$, whenever $\Phi'\cap U$ is nonempty it follows that some simple root $\alpha$ of $\Phi'$ is in $U$. All coclosed sets are unipodal, and unipodal sets are preserved by arbitrary unions. Furthermore, any closed set which is unipodal is in fact biclosed.

We now introduce a new property, which is designed to prove \Cref{GaloisConnection}. 
Let $A$ be a subset of $\Phi_+$. We will say that a positive root $\gamma$ is \newword{nocked} with respect to $A$ there are positive roots $\alpha$ and $\beta$, with $\gamma$ a positive linear combination of $\alpha$ and $\beta$, such that $\alpha,\beta \in A$ and $\gamma \not \in A$. 
We write $\Nock(A)$ for the set of $\gamma$ which are nocked with respect to $A$. 
To motivate the definition of $\Nock(A)$, note that the closure of $A$ is the ascending union of the chain $A \subseteq A \cup \Nock(A) \subseteq A \cup \Nock(A) \cup \Nock(A \cup \Nock(A)) \subseteq \cdots$, so the nocked\footnote{The word ``nocked'' comes from archery and describes an arrow ready to be fired, as the nocked roots are ready to be added.} roots are the ones which get added to the closure ``first''. 

We partially order $\Phi_+$ by \newword{cutting order}. If $\alpha$ and $\beta$ are positive roots, then $\beta$ is over $\alpha$ in cutting order if, in the full rank two system containing them, $\alpha$ is simple and $\beta$ is not. To show that (the transitive closure of) these relations in fact define a partial order, rather than a preorder, it is enough to exhibit a partial order which extends the cutting order. Indeed, Proposition 4.3.18(3) of \cite{Viard2015} shows that if $\alpha$ is below $\beta$ in cutting order, then $\alpha$ has a smaller depth than $\beta$.

\begin{definition} \label{CCProp}
	We say that $\Phi_+$ has the \newword{$CU$-property} if, whenever $U$ is a unipodal set and $\gamma$ is minimal in $\Nock(U)$, then $U \cup \{ \gamma \}$ is unipodal. 
\end{definition}
It is immediate to see that the $CU$-property holds if $\Phi_+$ has rank $\leq 2$.

\begin{prop}
	If $\Phi_+$ has the $CU$-property then \Cref{GaloisConnection}, and hence \Cref{DyerMain}, hold with respect to $\Phi_+$. 
\end{prop}

\begin{proof}
	Let $U$ be a unipodal set, let $K$ be the closure $\overline{U}$ and let $V$ be the interior $K^{\circ}$. We want to show that $K=V$.
	
	Since $V$ is the union of all coclosed sets contained in $K$, we have $U \subseteq V$ and thus $K$ is the closure of $V$. If $K \neq V$ then there must be some element of $K$ which is nocked with respect to $V$. Choose a $\gamma$ in $K$ which is nocked with respect to $V$ and is minimal with respect to this property. Then, by the $CU$-property, $V \cup \{ \gamma \}$ is unipodal. But then $\gamma$ would be in the interior of $K$, contradicting our choice of $\gamma$ not in $V$.
\end{proof}

The advantage of the $CU$-property is the next result:
\begin{prop} \label{Rank3Reduction}
	Suppose that every rank three subsystem of $\Phi_+$ has the $CU$-property. Then $\Phi_+$ has the $CU$-property.
\end{prop}
%
%Before proving this, we need the following lemma:
%\begin{lemma}\label{thm:2span}
%	Let $\Delta$ be a rank $n$-system, and $\beta_1$, \ldots, $\beta_r$ a set of linearly independent roots of $\Delta$. Then the 2-span of $\beta_1$, \ldots, $\beta_r$ is a root subsystem of rank $r$. 
%	%If $r=n$ and $\Delta$ is finite or affine, then the 2-span of $\a_1$,\ldots,\a_r$ is $\Delta$.
%\end{lemma}
%
%\begin{proof}
%If the proof is short, write it.
%\end{proof}
%
%\begin{remark}
%	The naive span operation would be to take $\spn(S)\cap \rep$. This set contains the 2-span of $S$, but as a root system it can have arbitrarily large rank, even for the span of $3$ roots in a rank $4$ root system. 
%\end{remark}


\begin{proof}
	Suppose that $\Phi_+$ does not obey the $CU$-property. Then there is some unipodal set $U$ and some root $\gamma$ which is minimal in $\Nock(U)$, such that $U \cup \{ \gamma \}$ is not unipodal. 
	
	The condition that $\gamma \in \Nock(U)$ means that there are positive roots $\alpha$ and $\beta$, with $\gamma$ in the positive span of $\alpha$ and $\beta$, such that $\alpha$, $\beta \in U$ and $\gamma \not \in U$. The condition that $U \cup \{ \gamma \}$ is not unipodal means that there must be some positive roots $\alpha'$, $\beta'$ which are simple in the rank two subsystem they span, and some root $\gamma'$ in the positive span of $\alpha$ and $\beta$. These satisfy $\alpha',\beta' \not \in U \cup \{ \gamma \}$ and $\gamma' \in  U \cup \{ \gamma \}$. But, if $\gamma' \in U$ then this would contradict that $U$ is unipodal, so we must have $\gamma = \gamma'$.
	
	Let $R$ and $R'$ be the full rank two subsystems containing $\{ \alpha, \gamma, \beta \}$ and $\{ \alpha', \gamma, \beta' \}$ respectively.
	Let $X$ be the minimal full subsystem containing $R$ and $R'$ and let $\Phi^X_+$ be its positive roots; $X$ is the full subsystem generated by $\alpha,\gamma,\alpha'$, and hence has rank three by the following lemma.  %by Lemma~\ref{FullSubsystem}.
	\begin{lemma}
		The minimal full subsystem containing a given set of $r$ roots has rank at most $r$.
	\end{lemma} 
	Then $U \cap \Phi^X_+$ is unipodal in $\Phi^X_+$, and the root system $R$ shows that $\gamma$ is still nocked with respect to $U \cap \Phi^X_+$.
	The partial order on $\Phi^X_+$ is the restriction of the one on $\Phi^+$ 
	%\textbf{is that right?} I do not know of any root ordering which descends to reflection subgroups (they could have different simple roots in a rank two, for instance), so I changed it to use the full subsystem description. Now that I think about it though, cutting order+dominance should descend for simply laced type (which is likely the only case we'll prove the CU property for)
	so $\gamma$ is still minimal in $\Nock(U \cap \Phi^X_+)$. 
	But then $R'$ shows that $(U \cap \Phi^X_+) \cup \{ \gamma \}$ is not unipodal in $\Phi_+^X$, contradicting the $CU$-property in the rank three subsystem $\Phi_X^+$.
\end{proof}

It is a short verification that the $CU$-property holds for the finite root systems of rank three with integral Cartan matrix: $A_3$, $B_3$, and $A_1\times X_2$ for $X_2$ a finite rank 2 root system. 
Thus, we immediately deduce that the $CU$-property, \Cref{GaloisConnection} and \Cref{DyerMain} hold for all finite crystallographic root systems. 
The fact that biclosed sets form a complete lattice is classical in this setting, but the formula for the join is already interesting. We note that the $CU$-property fails in type $\widetilde{C}_2$, so we cannot expect every root system to have this property. However, one of the main results of this extended abstract is:
\begin{theorem}
	%The $CU$-property, and thus \Cref{GaloisConnection} and \Cref{DyerMain}, hold for any affine root system.
	\Cref{GaloisConnection,DyerMain} hold for any affine root system.
	Furthermore, the \newword{simply-laced} affine root systems (types $\tA_{n}, \tD_n,$ and $\widetilde E_n$) have the $CU$-property.
\end{theorem}

The proof of the theorem reduces to the case of simply-laced types by the technique of \newword{folding} root systems (which for brevity we shall not discuss here). To show the $CU$-property in that case (and thus \Cref{GaloisConnection,DyerMain}), we use \Cref{Rank3Reduction} to reduce to checking the rank three subsystems, which in a simply-laced system are all of type $\tA_2$, $A_3$, or $A_1\times X_2$ for $X_2$ of rank two. 
The finite and decomposable systems are a quick check; we will take on the affine system $\tA_2$ %, $\tB_2$ and $\tG_2$, in \Cref{Rank3Verification}.
in the following.
\begin{theorem}
	Let $\Phi_+$ be the positive root system of type $\widetilde{A}_2$, and let $U \subseteq \Phi_+$ be a unipodal subset. Let $\gamma$ be a minimal element of $\Nock(U)$, with respect to cutting order. Then $U \cup \{ \gamma \}$ is also unipodal.
\end{theorem}

We will denote the simple roots of $\Phi_+$ as $\alpha_0$, $\alpha_1$ and $\alpha_2$, so the imaginary root is $\delta = \alpha_0+\alpha_1+\alpha_2$. 
We write $\beta_j$ for $\delta - \alpha_j$ and, for $t \geq 0$, we set $\alpha_j^t = \alpha_j+t \delta$ and $\beta_j^t = \beta_j + t \delta$. So each element of $\Phi_+$ is denoted as one of $\alpha_0^t$, $\alpha_1^t$, $\alpha_2^t$,  $\beta_0^t$, $\beta_1^t$, $\beta_2^t$ for $t \geq 0$. Note that we have $\alpha_0^t+\alpha_1^u = \beta_2^{t+u}$ and $\beta_0^t+\beta_1^u = \alpha_2^{t+u+1}$, and similarly for all permutations of the subscripts.

The condition that $\gamma$ is minimal in $\Nock(U)$ means that, anytime that we have $\eta'$ and $\eta'' \in U$ and some root $\eta$ is in the positive span of $\eta'$ and $\eta''$, with $\eta$ less than $\gamma$, we can deduce that $\eta$ is in $U$. 
In this case, we will say that $\eta'$ and $\eta''$ \newword{force} $\eta$.

We first outline the cases we will consider:
The condition that $\gamma$ is nocked means that $\gamma \not\in U$ but $\gamma$ is in the positive span of some other roots $\gamma'$ and $\gamma''$ which are in $U$; let $P$ be the rank two subsystem containing $\gamma'$ and $\gamma''$.
The only way that $U \cup \{ \gamma \}$ could fail to be unipodal is if $\gamma$ is contained in some rank two subsystem $Q$ whose simple roots are not in $U$. 
We break into cases according to the types of $P$ and $Q$:

\begin{lemma}
	If $P$ and $Q$ are both type $\widetilde{A}_1$, then $U \cup \{ \gamma \}$ is unipodal with respect to $Q$.
\end{lemma}

\begin{proof}
	In this case, since $\gamma \in P \cap Q$, we must have $P=Q$. But then $\gamma' \in Q \cap U$, and $U$ is unipodal, so $U$ contains one of the simple roots of $Q$, as desired.
\end{proof}

\begin{lemma}
	If $P$ is type $A_2$, and $Q$ is type $\widetilde{A}_1$, then $U \cup \{ \gamma \}$ is unipodal with respect to $Q$.
\end{lemma}

\begin{proof}
	Without loss of generality, $Q$ is the subsystem spanned by $\alpha_0$ and $\beta_0$, so either $\gamma = \alpha_0^t$ or $\gamma = \beta_0^t$ for some $t$. The statement is trivial for $t=0$, so we assume $t > 0$. 
	We also note that, as soon as we show that any $\alpha_0^u$ or $\beta_0^u$ is in $U$, we are done by the unipodality of $U$.
	
    \medskip

	\noindent\textbf{Case 1:} $\gamma = \alpha_0^t$,  $\gamma' = \beta_1^u$ and $\gamma''=\beta_2^{t-u-1}$:

    \smallskip
	
	\noindent Now, $\gamma' = \beta_1^u = \alpha_0^u + \alpha_2$ so, by unipodality, either $\alpha_0^u$ or $\alpha_2$ is in $U$. 
	If $\alpha_0^u \in U$, we are done, so we may assume $\alpha_2 \in U$.
	Similarly, $\gamma'' = \beta_2^{t-u-1} = \alpha_0^{t-u-1}+\alpha_1$ show that we may assume $\alpha_1 \in U$. 
	But then $\alpha_1$ and $\alpha_2$ force $\beta_0$ into $U$, and we are done.
	
    \medskip

	\noindent \textbf{Case 2:} We have $\gamma = \beta_0^t$, $\gamma' = \alpha_1^u$ and $\gamma''=\alpha_2^{t-u}$:
	
    \smallskip

	\noindent Since $t>0$, either $u > 0$ or $t-u > 0$; without loss of generality, we assume $u > 0$. Now, $\gamma' = \alpha_1^u = \beta_0^{u-1} + \beta_2$ so, by unipodality, either $\beta_0^{u-1}$ or $\beta_2$ is in $U$. If $\beta_0^{u-1} \in U$, we are done. If $\beta_2 \in U$, then unipodality implies that either $\alpha_0$ or $\alpha_1 \in U$. If $\alpha_0 \in U$, we are again done.  We have now shown that both $\alpha_1$ and $\alpha_2^{t-u}$ are in $U$. 
	But then $\alpha_1$ and $\alpha_2^{t-u}$ force $\beta_0^{t-u}$ are we are done again.
\end{proof}

The other cases are similar, but longer; they are available in the paper accompanying this extended abstract.

%%%%%%%%%
%\begin{lemma}
%	If $P$ is type $\widetilde{A}_1$, and $Q$ is type $A_2$, then $U \cup \{ \gamma \}$ is unipodal with respect to $Q$.
%\end{lemma}
%
%\begin{proof}
%	Without loss of generality, $P$ is the subsystem spanned by $\alpha_0$ and $\beta_0$ so $\gamma$ is $\alpha_0^t$ or $\beta_0^t$. Since $\gamma$ is in the positive span of two other elements of $P$, we must have $t \geq 1$.
%	
%	\textbf{Case 1:} $Q = \{ \alpha_1^u, \beta_0^t, \alpha_2^{t-u} \}$.
%	Note that, by unipodality, we win as soon as we show any $\alpha_2^x$ or $\beta_2^x$ is in $U$.
%	%If ALb<AL<AL'', with the endpoints in U, and AL=BS+CS, need to show BS or CS in U. Have (wlog BS>BSb) AL-delta=(BS-delta)+CS is in U by minimality of AL in Nock. Either CS is in U and we are done, or we may take BS-delta to be in U by unipodality. We also know AL''=BS''+CS for some BS''>BS, so by unipodality BS'' is in U. Then BS is forced to be in U by BS''>BS>BSb and we are done.
%	
%	Let $\gamma'$ be $\beta_0^v$ for some $v<t$, so $\gamma''$ is either $\beta_0^w$ for some $w > t$ or else $\gamma''$ is $\alpha_0^w$ for some $w$.
%	If $v = t-1$, then $\gamma' = \beta_0^{t-1}$ is in $U$ and, if $v<t-1$, then $\gamma'$ and $\gamma''$ force $\beta_0^{t-1}$, so we see that $\beta_0^{t-1}$ is in $U$.
%	Now, $\beta_0^{t-1} = \alpha_1^{u-1} + \alpha_2^{t-u}$ so either $\alpha_1^{u-1}$ or $\alpha_2^{t-u}$ is in $U$. In the latter case, we are done, so we may assume that $\alpha_1^{u-1} \in U$.  NOT FINISHED
%	
%	
%	
%	
%	\textbf{Case 2:} $Q = \{ \beta_1^u, \alpha_0^t, \beta_2^{t-u-1} \}$. 
%	%If ASb<AS<AS'', with endpoints in U, and AS=BL+CL, need to show BL or CL in U. The same argument as before works unless BL=BLb and CL=CLb. AS''=BLb+(AS''-BLb), so we get AS''-BLb in U by unipodality. Then AS''-BLb= BSb + (AS''-BLb-BSb). If neither BSb nor BLb is in U, then we can continue subtracting BSb+BLb from AS'' until we find that AS is in U, a contradiction. So BSb must be in U. But then ASb+BSb=CLb, so CLb is in U and we are done.
%	NOT FINISHED
%\end{proof}
%
%\begin{lemma}
%	If $P$ and $Q$ are both type $A_2$, then $U \cup \{ \gamma \}$ is unipodal with respect to $Q$.
%\end{lemma}
%
%\begin{proof}
%	\textbf{Case 1:} $P = \{ \alpha_1^u, \beta_0^t, \alpha_2^{t-u} \}$ and $Q = \{ \alpha_1^v, \beta_0^t, \alpha_2^{t-v} \}$. Without loss of generality, we may assume that $u<v$.
%	%AL is minimal in Nock(U), pick some roots BS,CS in Phi satisfy AL=BS+CS. Assume there are roots BS'<BS and CS''>CS in U such that AL=BS'+CS''. Need to show that BS or CS is in U.
%	%First note that if BLb (resp. CSb) is in U, then BS (resp. CS) is in U by minimality of AL in Nock. So assume they are not in U; by unipodality then BSb and CLb are in U. Either BS'>BSb or BS'=BSb. In the first case, since (BS'-delta)+CS''<AL, we find that (BS'-delta)+CS'' is in U. Then so is (BS'-delta)+CS''+CLb = AL-CSb. But AL-CSb >= BS > BS', so BS is forced to be in U.
%	%In the second case, BS'=BSb. So CS''=(AL-delta)+BLb. By unipodality we find that AL-delta is in U. Then just as in the first case, (AL-delta)+CLb = AL-CSb is in U. And AL-CSb >= BS > BS', so BS is in U.
%	First note that, if $\beta_1 \in U$ then $\beta_1$ and $\alpha_1^u$ force $\alpha_1^v$ and we win.
%	Similarly, if $\alpha_2 \in U$ then $\alpha_2$ and $\alpha_2^{t-u}$ force $\alpha_2^{t-v}$ and we win again.
%	So we may assume that $\beta_1$ and $\alpha_2$ are not in $U$, then, by unipodality, $\alpha_1$ and $\beta_2$ are in $U$. 
%	
%	Our next goal is to show that $\beta_0^{t-1} \in U$. This requires two subcases:
%	
%	\textbf{Subcase 1a:} $u>0$. Then $\alpha_1$ and $\alpha_1^u$ force $\alpha_1^{u-1}$; then $\alpha_1^{u-1}$ and $\alpha_2^{t-u}$ force $\beta_0^{t-1}$
%	
%	\textbf{Subcase 1b:} $u=0$,  so $\alpha_2^t  = \alpha_2^{t-u}$ is in $U$.
%	We have $\alpha_2^t = \beta_0^{t-1} + \beta_1$, and we already argued that $\beta_1 \not \in U$, so unipodality shows that $\beta_0^{t-1} \in U$ in this case as well.
%	
%	Now, $\beta_0^{t-1}$ and $\beta_2$ force $\alpha_1^t$, and $\alpha_1$ and $\alpha_1^t$ force $\alpha_1^v$, as desired.
%	
%	
%	\textbf{Case 2:} $P = \{ \beta_1^u, \alpha_0^t, \beta_2^{t-u-1} \}$ and $Q = \{ \beta_1^v, \alpha_0^t, \beta_2^{t-v-1} \}$ Without loss of generality, we may assume that $u<v$.
%	%AS is minimal in Nock(U), pick some roots BL,CL in Phi satisfying AS=BL+CL. Assume there are roots BL'<BL and CL''>CL in U such that AS=BL'+CL''. We need to show that BL or CL is in U.
%	%First note that if CLb or any BS is in U, then we are done by forcing. So we can assume they are not, and unipodality implies that BLb and CSb are in U. We see CL''=ASb+(CL''-ASb), and so since CL''-ASb is in the BS column, ASb is in U by unipodality.
%	%Either BL'>BLb or BL'=BLb. In the first case, since (BL'-delta)+CL''<AL, we find that (BL'-delta)+CL'' is in U. Then so is (BL'-delta)+CL''+CSb = AS-CLb. But AS-CLb >= BL > BL', so BL is forced to be in U. In the second case, BL'=BLb. So CL''=(AS-delta)+BSb, and hence AS-delta is in U by unipodality. Thus, as in the first case, we find (AS-delta)+CSb = AL-CLb is in U. And BL is then forced to be in U.
%	First note that, if any $\alpha_1^w$ is in $U$, then $\alpha_1^w$ and $\beta_1^u$ force $\beta_1^v$ and we win.
%	So we can assume that no $\alpha_1^w$ is in $U$.
%	Also, if $\beta_2$ is in $U$, then $\beta_2^{t-u-1}$ and $\beta_2$ force $\beta_2^{t-v-1}$ and we win again, so we may assume that $\beta_2$ is not in $U$.
%	Unipodality then shows that $\beta_1$ and $\alpha_2$ are in $U$.
%	
%	Our next goal is to show that $\alpha_0^{t-1} \in U$. This requires two subcases:
%	\textbf{Subcase 2a:} $u>0$.
%	Then $\beta_1$ and $\beta_1^u$ force $\beta_1^{u-1}$; then $\beta_1^{u-1}$ and $\beta_2^{t-u-1}$ force $\alpha_0^{t-1}$
%	
%	\textbf{Subcase 2b:} $u=0$.
%	We have $\beta_2^t = \alpha_0^{t-1} + \alpha_1^1$, and we already argued that $\alpha_1^1 \not \in U$, so unipodality shows that $\alpha_0^{t-1} \in U$ in this case as well.
%	
%	Now, $\alpha_0^{t-1}$ and $\alpha_2$ force $\beta_1^{t-1}$, and $\beta_1$ and $\beta_1^{t-1}$ force $\beta_1^v$, as desired.
%\end{proof}
%%%%%%%%%

%
%There is a geometric way to see that the set of inversions of $w$ will give a biclosed set: Choose $\theta \in V^{\vee}$ such that $\langle \theta, \alpha_i \rangle > 0$ for all positive simple roots $\alpha_i$. Then $\langle \beta_t, w \theta \rangle < 0$ if and only if $t$ is an inversion of $w$. 
%For any $\psi$ in $V^{\vee}$, it is clear that $\{ \beta \in \Phi_+ : \langle \beta, \psi \rangle < 0 \}$ will be biclosed. 
%
%\subsection*{Some warnings}
%One might hope that other biclosed sets correspond to $\psi$ in other parts of $V^{\vee}$, but not all biclosed sets occur in this way. 
%We saw in Example~\ref{} that the following set, in type $\tA_3$, is biclosed: $\{ \alpha_0+k \delta,\  \alpha_0+\alpha_1+k \delta,\  \alpha_0 + \alpha_1 + \alpha_2 + k \delta,\  \alpha_2 + k \delta,\  \alpha_2+\alpha_3 + k \delta,\  \alpha_2+\alpha_3+\alpha_0 + k \delta \ \text{for}\ k \geq 0 \}$.
%If this set arose from a $\psi$ in $V^{\vee}$, then we would have $\langle \psi, \alpha_0+\alpha_1 \rangle < 0$ and $\langle \psi, \alpha_2+\alpha_3 \rangle < 0$, while $\langle \psi, \alpha_1+\alpha_2 \rangle \geq 0$ and $\langle \psi, \alpha_3+\alpha_0 \rangle \geq 0$, giving a contradiction, since $(\alpha_0+\alpha_1) + (\alpha_2+\alpha_3) = (\alpha_1+\alpha_2) + (\alpha_0 + \alpha_3)$. 
%Moreover, it is important to allow this set to be biclosed; if we omit it, the join of $I(u)$ and $I(v)$ from Example~\ref{} will not exist.
%
%Worse things can happen in more complicated Coxeter groups: We can have positive roots $\beta_1$, $\beta_2$, $\beta_3$ and $\gamma$, with $\gamma$ a positive linear combination of the $\beta_i$, and a biclosed set that contains $\{ \beta_1, \beta_2, \beta_3 \}$ but not $\gamma$ \cite{}.





%
%This paper is inspired by certain fascinating conjectures of Matthew Dyer. 
%In this introduction, we overview those conjectures and our accomplishments.
%Dyer's conjectures address a general Coxeter group, but our results address certain very specific Coxeter groups. 
%In this introduction, we will overview Dyer's conjectures, assuming the reader has some familiarity with Coxeter theory. 
%The reader has doesn't have this background may turn to Section~\ref{Background}, or may trust that we will soon turn to concrete examples.
%
%Let $W$ be a Coxeter group and let $\Phi^+$ be the set of positive real roots. 
%To each element $w$ of $W$, one associates a finite subset $\inv(w)$ of $\Phi^+$ known as the \newword{inversions} of $w$. 
%The \newword{weak order} on $W$ is the partial order where $u \leq v$ if and only if $\inv(u) \subseteq \inv(v)$. 
%
%When $W$ (and hence $\Phi^+$) is finite, weak order is a complete lattice, meaning that every subset of $W$ has a unique least upper bound and a unique greatest lower bound.
%When $W$ is infinite, weak order is a complete semi-lattice, meaning that every \emph{nonempty} subset has a unique greatest lower bound, and every subset which has an upper bound has a least upper bound. 
%This suggests the project of trying to construct some larger complete lattice, $\widehat{W}$, which contains $W$ as a lower ideal.
%
%Dyer defines a property of subsets $I$ of $\Phi^+$ known as \newword{biclosed}. (See Section~\ref{Defn Biclosed} for details.)
%The finite biclosed sets are precisely the sets of inversions of elements of $W$. 
%We then take $\widehat{W}$ to be the poset of biclosed sets, ordered by inclusion. 
%Dyer conjectures that this poset is a complete lattice.
%
%The simplest of the infinite Coxeter groups are the affine Coxeter groups.
%The nicest of the affine Coxeter groups are the classical groups, $\tA_{n-1}$, $\tB_n$, $\tC_n$ and $\tD_n$; these groups are pleasant to work with because they can be described as permutations of $\ZZ$ obeying certain regularities. 
%We will provide combinatorial models for biclosed subsets of $\Phi^+$ in these types, and verify that they are lattices. 
%
%We therefore describe the standard permutation models for the affine Coxeter groups, and then describe our models. 
%Our notation is meant to be as parallel as possible in the four types.
%
%Let $X$ be one of the symbols $\tA_{n-1}$, $\tB_n$, $\tC_n$, $\tD_n$. 
%Let $H(X)$ be a group of permutations of $\ZZ$ as follows: The group $H(\tA_{n-1})$ is generated by $x \mapsto x+n$, and the groups $H(\tB_n)$, $H(\tC_n)$, $H(\tD_n)$ are generated by $x \mapsto x+2n$ and $x \mapsto 2n+1-x$. 
%The centralizer, $\Cent(H(X))$, is the group of permutation $f : \ZZ \to \ZZ$ obeying $f \circ h = h \circ f$ for all $h \in H(X)$. 
%Each of the classical Coxeter groups is a subgroup of $\Cent(H(X))$.
%
%The group $\tA_{n-1}$ is the subgroup of $f$ in $\Cent(H(\tA_{n-1}))$ obeying the normalization $\sum_{i=1}^{n} f(i) = \sum_{i=1}^{n} i$. 
%In the other types, the condition that $f$ commutes with $i \mapsto 2n+1-i$ automatically imposes that $\sum_{i=1}^{2n} f(i) = \sum_{i=1}^{2n} i$, so we don't need to impose this condition. 
%
%The group $\tC_n$ is $\Cent(H(\tC_n))$. 
%The group $\tB_n$ is an index $2$ subgroup of $\tC_n$ given by the condition $\# \{ i :\  i \geq 1,\ f(i) \leq 0 \} \equiv 0 \bmod 2$.
%The group $\tD_n$ is an index $2$ subgroup of $\tB_n$ where we also impose the condition $\# \{ i :\  i \geq n+1,\ f(i) \leq n \} \equiv 0 \bmod 2$.
%
%Define a graph $\Gamma(X)$ as follows: The vertices of $\Gamma(X)$ are $\ZZ$. We will denote the set of edges of $\Gamma(X)$ by $E$; these edges are given by the following formulas:
%\[ \begin{array}{rcl@{\qquad}rcl}
%X &=& \tA_{n-1} & E &=& \{ (i,j) : i \not\equiv j \bmod n \} \\
%X &=& \tB_n & E &=& \{ (i,j) : i \not\equiv j \bmod 2n,\ i+j \not\equiv 1 \bmod 4n \} \\
%X &=& \tC_n & E &=& \{ (i,j) : i \not\equiv j \bmod 2n \} \\
%X &=& \tD_n & E &=& \{ (i,j) : i \not\equiv j \bmod 2n,\ i+j \not\equiv 1 \bmod 2n \} \\
%\end{array} \]
%The positive roots $\Phi^+$ in each root system are in bijection with the $H$-orbits on $E$, and we will identify $\Phi^+$ with $H \backslash E$ in the future. 
%(In Section~\ref{ADD REFERENCE}, we will lay out the relationship between $H \backslash E$ and the classical descriptions of root systems.)
%From this perspective, for a permutation $f$ in one of these Coxeter groups, the inversion set of $f$ is $H \backslash \{ (i,j)\in E :\ i<j ,\ f(i) > f(j) \}$.
%
%We identify subsets of $\Phi^+$ with orientations of $\Gamma(X)$: Let $(i,j) \in E$ with $i<j$, and let $R$ be a subset of $\Phi^+ = H\backslash E$. 
%If $(i,j) \not\in R$, we orient $i \to j$; if $(i,j) \in R$, we orient $j \to i$. 
%Then we can easily define biclosed sets in this context: Biclosed sets correspond to orientations of $\Gamma(X)$ such that 
%\begin{itemize}
%\item Every $3$-cycle in $\Gamma(X)$ is oriented acyclicly and
%\item If $h$ is an order preserving element of $H(X)$, and $(i,j)$, $(j,h(i))$, $(h(i), h(j))$, $(i, h(j))$ are edges in $E$, then this $4$-cycle is oriented acyclically in $\Gamma(X)$.
%\end{itemize}
%The order on biclosed sets is containment, in other words, given two orientations $\omega_1$ and $\omega_2$ of $\Gamma(X)$, we put $\omega_1 \leq \omega_2$ if, for every $(i,j) \in E$ with $i < j$, if $i \to j$ in $\omega_2$ then $i \to j$ in $\omega_1$. 
%
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