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\title{\bf On the Topology of the Cambrian Semilattices}
\author{Myrto Kallipoliti\thanks{Supported by FWF research grant no. Z130-N13.} 
\qquad Henri M{\"u}hle\footnotemark[1]\\
\small Fakult{\"a}t f{\"u}r Mathematik\\[-0.8ex]
\small Universit{\"a}t Wien\\[-0.8ex] 
\small 1090 Vienna, Austria\\
\small\tt myrto.kallipoliti@univie.ac.at \qquad henri.muehle@univie.ac.at\\
}
\date{\dateline{Nov 21, 2012}{May 13, 2013}\\
\small Mathematics Subject Classifications: 20F55, 06A07, 05E15}

\begin{document}

\maketitle

\begin{abstract}
	For an arbitrary Coxeter group $W$, Reading and Speyer defined Cambrian semilattices 
	$\mathcal{C}_{\gamma}$ as sub-semilattices of the weak order on $W$ induced by so-called $\gamma$-sortable elements.
	In this article, we define an edge-labeling of $\mathcal{C}_{\gamma}$, and show that this is an EL-labeling for
	every closed interval of $\mathcal{C}_{\gamma}$. In addition, we use our labeling to show that every finite open
	interval in a Cambrian semilattice is either contractible or spherical, and we characterize 
	the spherical intervals, generalizing a result by Reading.
\end{abstract}

\section{Introduction}
  \label{sec:introduction}
In \cite{bjorner97shellable}*{Theorem~9.6}, Bj\"orner and Wachs observed that the Tamari 
lattice $\mathcal{T}_{n}$, introduced in \cite{tamari62algebra}, can be regarded as the subposet of the weak-order
lattice on the symmetric group $\mathfrak{S}_{n}$, consisting of 312-avoiding permutations. More precisely,
there exists an order-preserving surjection $\sigma:\mathfrak{S}_{n}\to\mathcal{T}_{n}$ such that $\mathcal{T}_{n}$ is
isomorphic to the subposet of the weak-order lattice on $\mathfrak{S}_{n}$ consisting of the bottom elements in the
fibers of $\sigma$. In \cite{reading06cambrian}, the map $\sigma$ was realized as a map from
$\mathfrak{S}_{n}$ to the triangulations of an $(n+2)$-gon, where the partial order on the latter is given
by diagonal flips. It was shown that the fibers of $\sigma$ induce a congruence relation on the weak-order
lattice on $\mathfrak{S}_{n}$, and that the Tamari lattice is isomorphic to the lattice quotient induced by
this congruence. Moreover, it was observed that different embeddings of the $(n+2)$-gon in the plane yield
different lattice quotients of the weak-order lattice on $\mathfrak{S}_{n}$. The realization of
$\mathfrak{S}_{n}$ as the Coxeter group $A_{n-1}$ was then used to connect the embedding of the $(n+2)$-gon
in the plane with a Coxeter element of $A_{n-1}$. This connection eventually led to the definition of
Cambrian lattices, which can analogously be defined for an arbitrary finite Coxeter group $W$ as lattice
quotients of the weak-order lattice on $W$ with respect to certain lattice congruences induced by
orientations of the Coxeter diagram of $W$, see \cite{reading07sortable}. 

As suggested in \cite{stasheff97from}*{Appendix~B}, and later in \cite{loday04realization}*{Theorem~1},
the Hasse diagram of the Tamari lattice corresponds to the $1$-skeleton of the classical associa\-hedron.
(Due to the connection to the symmetric group, which was elaborated in \cite{loday04realization}, the
classical associahedron is also referred to as \emph{type $A$-associahedron}.) In 
\cites{bott94on, simion03type, fomin03systems, chapoton02polytopal}, generalized associahedra 
were defined for all crystallographic Coxeter groups which generalize the type $A$-associahedron. The 
Cambrian lattices provide another viewpoint for the generalized associahedra, namely that the fan
associated to a Cambrian lattice of crystallographic type is the normal fan of the generalized
associahedron of the same type, see \cite{reading09cambrian} for the details of this construction.
Moreover, since the Cambrian lattices are defined for all finite Coxeter groups, this connection defines a
generalized associahedron for the non-crystallographic types as well, see
\cite{reading09cambrian}*{Corollary~8.1}.

In \cite{reading11sortable}, Reading and Speyer generalized the construction of Cambrian 
lattices to infinite Coxeter groups. Since there exists no longest element in an infinite Coxeter group, the weak 
order constitutes only a meet-semilattice. Using the realization of the Cambrian 
lattices in terms of Coxeter-sortable elements, which was first described in \cite{reading07sortable} and 
later extended in \cite{reading11sortable}, the analogous construction as in the finite case yields a 
sub-semilattice of the weak-order semilattice, the so-called \emph{Cambrian semilattice}.

This article is dedicated to the investigation of the topological properties of the order complex of 
the proper part of closed intervals in a Cambrian semilattice. One (order-theoretic) tool to 
investigate these properties is EL-shellability, which was introduced in \cite{bjorner80shellable}, and
further developed in \cites{bjorner83lexicographically,bjorner96shellable,bjorner97shellable}. The fact
that a poset is EL-shellable implies a number of properties of the associated order complex: this order
complex is Cohen-Macaulay, it is homotopy equivalent to a wedge of spheres and the dimensions of its 
homology groups can be computed from the labeling. The first main result of the present article is the
following.
\begin{theorem}
  \label{thm:infinite_shellability}
	Every closed interval in $\mathcal{C}_{\gamma}$ is EL-shellable for every (possibly infinite) Coxeter group
	$W$ and every Coxeter element $\gamma\in W$.
\end{theorem}
We prove this result uniformly using the realization of $\mathcal{C}_{\gamma}$ in terms of Coxeter-sortable elements, 
and thus our proof does not require $W$ to be finite or even crystallographic. 
For finite crystallographic Coxeter groups, Theorem~\ref{thm:infinite_shellability}
is implied by \cite{ingalls09noncrossing}*{Theorem~4.17}. Ingalls and Thomas considered in
\cite{ingalls09noncrossing} the category of finite dimensional representations of an orientation of the
Coxeter diagram of a finite crystallographic Coxeter group $W$, and considered the corresponding Cambrian 
lattices as a poset of torsion classes of this category. However, their approach cannot be 
applied directly to non-crystallographic or to infinite Coxeter groups. Moreover, we remark that the labeling from
Ingalls' and Thomas' approach is different from our labeling.

Finally, using the fact that every closed interval of $\mathcal{C}_{\gamma}$ is EL-shellable, we are able to
determine the homotopy type of the proper parts of these intervals by counting the number of falling chains
with respect to our labeling. It turns out that every open interval is either contractible or spherical,
\emph{i.e.} homotopy equivalent to a sphere. We can further characterize which intervals of $\mathcal{C}_{\gamma}$
are contractible and which are spherical, as our second main result shows. Recall that a closed interval
$[x,y]$ in a lattice is called \emph{nuclear} if $y$ is the join of atoms of $[x,y]$.
\begin{theorem}
  \label{thm:cambrian_topology}
	Let $W$ be a (possibly infinite) Coxeter group and let $\gamma\in W$ be a Coxeter element. Every 
	finite open interval in the Cambrian semilattice $\mathcal{C}_{\gamma}$ is either contractible or spherical. 
	Furthermore, a finite open interval $(x,y)_{\gamma}$ is spherical if and only if the corresponding 
	closed interval $[x,y]_{\gamma}$ is nuclear. 
\end{theorem}
For finite Coxeter groups, Theorem~\ref{thm:cambrian_topology} is implied by concatenating
\cite{reading05lattice}*{Theorem~1.1} and \cite{reading05lattice}*{Propositions~5.6~and~5.7}. Reading's approach in the 
cited article was to investigate fan posets of central hyperplane arrangements. He showed that for a finite Coxeter 
group $W$ the Cambrian lattices can be viewed as fan posets of a fan induced by regions of the Coxeter arrangement of 
$W$ which are determined by orientations of the Coxeter diagram of $W$. The tools Reading developed in 
\cite{reading05lattice} apply to a much larger class of fan posets, but cannot be applied directly to infinite Coxeter 
groups. 

The proofs of Theorems~\ref{thm:infinite_shellability} and \ref{thm:cambrian_topology} are obtained 
completely within the framework of Coxeter-sortable elements and thus have the advantage that 
they are uniform and direct. 

\smallskip

This article is organized as follows. In Section~\ref{sec:preliminaries}, we recall the necessary
order-theoretic concepts, as well as the definition of EL-shellability. Furthermore, we recall the 
definition of Coxeter groups, and the construction of the Cambrian semilattices. In
Section~\ref{sec:shellability_cambrian}, we define a labeling of the Hasse diagram of a Cambrian
semilattice and give a case-free proof that this labeling is indeed an EL-labeling for every closed
interval of this semilattice, thus proving Theorem~\ref{thm:infinite_shellability}. In 
Section~\ref{sec:applications}, we prove Theorem~\ref{thm:cambrian_topology}, by counting the falling
maximal chains with respect to our labeling and by applying \cite{bjorner96shellable}*{Theorem~5.9} which
relates the number of falling maximal chains in a poset to the homotopy type of the corresponding order
complex. The characterization of the spherical intervals of $\mathcal{C}_{\gamma}$ follows from 
Theorem~\ref{thm:mobius_nuclear}. 
 
\section{Preliminaries}
  \label{sec:preliminaries}
In this section, we recall the necessary definitions, which are used throughout the article. For 
further background on posets, we refer to \cite{davey02introduction} or to 
\cite{stanley97enumerative}, where in addition some background on lattices and lattice congruences 
is provided. An introduction to poset topology can be found in either 
\cite{bjorner96topological} or \cite{wachs07poset}. For more background on Coxeter groups, we refer to
\cite{bjorner05combinatorics} and \cite{humphreys90reflection}.
 
\subsection{Posets and EL-Shellability}
  \label{sec:shellability}
Let $(P,\leq_{P})$ be a finite partially ordered set (\emph{poset} for short). We say that $P$ is 
\emph{bounded} if it has a unique minimal and a unique maximal element, which we usually denote by
$\hat{0}$ and $\hat{1}$, respectively. For $x,y\in P$, we say that $y$ \emph{covers} $x$ (and write 
$x\lessdot_{P} y$) if $x\leq_{P} y$ and there is no $z\in P$ such that $x<_{P} z<_{P} y$. We denote the
set of all covering relations of $P$ by $\mathcal{E}(P)$.  

For $x,y\in P$ with $x\leq_{P} y$, we define the closed interval $[x,y]$ to be the set 
$\{z\in P\mid x\leq_{P} z\leq_{P} y\}$. Similarly, we define the open interval 
$(x,y)$ to be the set $\{z\in P\mid x<_{P}z<_{P}y\}$. A chain $c:x=p_0\leq_{P} p_1\leq_{P}\cdots\leq_{P} p_s=y$ is 
called \emph{maximal} if $(p_i,p_{i+1})\in \mathcal{E}(P)$ for every $0\leq i\leq s-1$. 

Let $(P,\leq_{P})$ be a bounded poset and let
$c:\hat{0}=p_0\lessdot_{P} p_1\lessdot_{P}\cdots\lessdot_{P} p_s=\hat{1}$ be a maximal chain of $P$. Given 
another poset $(\Lambda,\leq_{\Lambda})$, a map $\lambda:\mathcal{E}(P)\to\Lambda$ is called an
\emph{edge-labeling of $P$}. We denote the sequence 
$\bigl(\lambda(p_0,p_1),\lambda(p_1,p_2),\ldots,\lambda(p_{s-1},p_s)\bigr)$ of edge-labels of $c$ by 
$\lambda(c)$. The chain $c$ is called \emph{rising} (respectively \emph{falling}) if
$\lambda(c)$ is a strictly increasing (respectively weakly decreasing) sequence. 
For two words $(p_1,p_2,\ldots,p_s)$ and $(q_1,q_2,\ldots ,q_t)$ in the alphabet $\Lambda$, we write 
$(p_1,p_2,\dots,p_s)\leq_{\Lambda^*}(q_1,q_2,\dots,q_t)$ if and only if either 
\begin{align*}
	& p_i=q_i, && \hspace*{-3cm}\mbox{for}\;1\leq i\leq s\;\mbox{and}\;s\leq t,\quad\mbox{or}\\
	& p_i<_{\Lambda}q_i, && \hspace*{-3cm}\mbox{for the least}\;i\;\mbox{such that}\;p_i\neq q_i.
\end{align*}
A maximal chain $c$ of $P$ is called \emph{lexicographically first} among the maximal chains of $P$ 
if for every other maximal chain $c'$ of $P$ we have $\lambda(c)\leq_{\Lambda^*} \lambda(c')$. 
An edge-labeling of $P$ is called \emph{EL-labeling} if for every closed interval $[x,y]$ in $P$ there
exists a unique rising maximal chain which is lexicographically first among all maximal chains in
$[x,y]$. A bounded poset that admits an EL-labeling is called \emph{EL-shellable}.

Let us further recall that the M\"obius function $\mu$ of $P$ is the map $\mu:P\times P\to\mathbb{Z}$ defined
recursively by
\begin{displaymath}
	\mu(x,y)=\begin{cases}
		1, & x=y\\
		-\sum_{x\leq_{P}z<_{P}y}{\mu(x,z)}, & x<_{P}y\\
		0, & \mbox{otherwise}.
	\end{cases}
\end{displaymath}

A remarkable property of EL-shellable posets is that we can compute the value of the M\"obius function
for every closed interval of $P$ from the labeling, as is stated in the following 
proposition\footnote{Actually, Proposition~5.7 in \cite{bjorner96shellable} is stated for posets admitting
a so-called \emph{CR-labeling}. EL-shellable posets are a particular instance of this class of posets, and 
for the scope of this article it is sufficient to restrict our attention to these.}.
\begin{proposition}[\cite{bjorner96shellable}*{Proposition~5.7}]
	\label{prop:mobius}
	Let $(P,\leq_{P})$ be an EL-shellable poset, and let $x,y\in P$ with $x\leq_{P} y$. Then,
	\begin{multline*}
		\mu(x,y)=\mbox{number of even length falling maximal chains in}\;[x,y]\\
			-\;\mbox{number of odd length falling maximal chains in}\;[x,y].
	\end{multline*}
\end{proposition}

\subsection{Coxeter Groups and Weak Order}
  \label{sec:weak_order}
Let $W$ be a (possibly infinite) group being generated by the finite set $S=\{s_{1},s_{2},\ldots,s_{n}\}$. Let 
$m=(m_{i,j})_{1\leq i,j\leq n}$ be a symmetric $(n\times n)$-matrix, where the entries are either 
positive integers or the formal symbol $\infty$, and which satisfies $m_{i,i}=1$ for all $1\leq i\leq n$, 
and $m_{i,j}\geq 2$ otherwise. (We use the convention that $\infty$ is formally larger than any natural 
number.) We call $W$ a \emph{Coxeter group} if it has the presentation 
\begin{displaymath}
	W = \bigl\langle S\mid (s_{i}s_{j})^{m_{i,j}}=\varepsilon\;\mbox{for}\;1\leq i,j\leq n\bigr\rangle,
\end{displaymath}
where $\varepsilon\in W$ denotes the identity, and we call $S$ the set of \emph{simple generators of $W$}. 
We interpret the case $m_{i,j}=\infty$ as stating that there is no relation between the generators $s_{i}$ 
and $s_{j}$, and call the matrix $m$ the \emph{Coxeter matrix of $W$}. 

Since $S$ is a generating set of $W$, we can write every element $w\in W$ as a product of the elements
in $S$, and we call such a word a \emph{reduced word for $w$} if it has minimal length. More
precisely, define the \emph{word length} on $W$ (with respect to $S$) as
\begin{displaymath}
	\ell_{S}:W\to\mathbb{N},\quad 
		w\mapsto\min\{k\mid w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{k}}\;
		\mbox{and}\;s_{i_{j}}\in S\;\mbox{for all}\;1\leq j\leq k\}.
\end{displaymath}
If $\ell_{S}(w)=k$, then every product of $k$ generators which yields $w$ is a reduced word for $w$. 
Define the \emph{(right) weak order of $W$} by 
\begin{displaymath}
	u\leq_{S} v\quad\mbox{if and only if}\quad \ell_{S}(v)=\ell_{S}(u)+\ell_{S}(u^{-1}v).
\end{displaymath}
The poset $(W,\leq_{S})$ is a ranked meet-semilattice, the so-called \emph{weak-order semilattice of $W$}, 
and $\ell_{S}$ is its rank function. Moreover, $(W,\leq_{S})$ is \emph{finitary} meaning that every principal order 
ideal of $(W,\leq_{S})$ is finite. In the case where $W$ is finite, there exists a unique longest word 
$w_{o}$ of $W$, and $(W,\leq_{S})$ is thus a lattice. 

It is often convenient to have the following alternative characterization of the (right) weak order on $W$.
Let $T=\{wsw^{-1}\mid w\in W, s\in S\}$, and define for $w\in W$, the \emph{(left) inversion set of $w$} as 
\begin{displaymath}
	\mbox{inv}(w)=\{t\in T\mid\ell_{S}(tw)\leq\ell_{S}(w)\}.
\end{displaymath}
It is the statement of \cite{bjorner05combinatorics}*{Proposition~3.1.3} that $u\leq_{S}v$ if and only if 
$\mbox{inv}(u)\subseteq \mbox{inv}(v)$, and \cite{bjorner05combinatorics}*{Corollary~1.4.5} states that
$\ell_{S}(w)=\lvert\mbox{inv}(w)\rvert$ for all $w\in W$. Thus, every $w\in W$ is uniquely determined by its (left)
inversion set. 

\subsection{Coxeter-Sortable Elements}
  \label{sec:coxeter_sortable}
Let $\gamma=s_{1}s_{2}\cdots s_{n}\in W$ be a Coxeter element, and define the half-infinite word 
\begin{displaymath}
	\gamma^{\infty}=s_{1}s_{2}\cdots s_{n}\vert s_{1}s_{2}\cdots s_{n}\vert\cdots. 
\end{displaymath}
The vertical bars in the representation of $\gamma^{\infty}$ are ``dividers'', which have no influence
on the structure of the word, but shall serve for a better readability. Clearly, every reduced word
for $w\in W$ can be considered as a subword of $\gamma^{\infty}$. Among all reduced words for $w$,
there is a unique reduced word, which is lexicographically first considered as a subword of 
$\gamma^{\infty}$. This reduced word is called the \emph{$\gamma$-sorting word of $w$}. 
\begin{example}
	Consider the Coxeter group $W=\mathfrak{S}_{5}$, generated by $S=\{s_{1},s_{2},s_{3},s_{4}\}$, where 
	$s_{i}$ corresponds to the transposition $(i,i+1)$ for all $i\in\{1,2,3,4\}$ and let
	$\gamma=s_1s_2s_3s_4$. Clearly, $s_{1}$ and $s_{4}$ commute. Hence, 
	$w_{1}=s_{1}s_{2}\vert s_{1}s_{4}$ and $w_{2}=s_{1}s_{2}s_{4}\vert s_{1}$ are reduced words
	for the same element $w\in W$. Considering $w_{1}$ and $w_{2}$ as subwords of $\gamma^{\infty}$, we 
	find that $w_{2}$ is a lexicographically smaller subword of $\gamma^{\infty}$ than $w_{1}$ is. There 
	are six other reduced words for $w$, namely
	\begin{align*}
		w_{3}=s_{1}s_{4}\vert s_{2}\vert s_{1}, && w_{4} = s_{4}\vert s_{1}s_{2}\vert s_{1}, 
		  && w_{5} = s_{4}\vert s_{2}\vert s_{1}s_{2},\\
		w_{6}=s_{2}s_{4}\vert s_{1}s_{2}, && w_{7} = s_{2}\vert s_{1}s_{4}\vert s_{2}, 
		  && w_{8} = s_{2}\vert s_{1}s_{2}s_{4}.
	\end{align*}
	It is easy to see that among these $w_{2}$ is the lexicographically first subword of 
	$\gamma^{\infty}$, and hence $w_{2}$ is the $\gamma$-sorting word of $w$.
\end{example}

In the following, we consider only $\gamma$-sorting words, and we write
\begin{equation}
  \label{eq:presentation}
	w=s_1^{\delta_{1,1}}s_2^{\delta_{1,2}}\cdots s_n^{\delta_{1,n}}\,\vert\,
	s_1^{\delta_{2,1}}s_2^{\delta_{2,2}}\cdots s_n^{\delta_{2,n}}\,\vert\,
	\cdots\,\vert\,s_1^{\delta_{l,1}}s_2^{\delta_{l,2}}\cdots s_n^{\delta_{l,n}},
\end{equation}
where $\delta_{i,j}\in\{0,1\}$ for $1\leq i\leq l$ and $1\leq j\leq n$. For each 
$i\in\{1,2,\ldots,l\}$, we say that 
\begin{displaymath}
	b_{i}=\{s_{j}\mid \delta_{i,j}=1\}\subseteq S
\end{displaymath}
is the \emph{$i$-th block of $w$}. We consider the blocks of $w$ sometimes as sets and sometimes as
subwords of $\gamma$, depending on how much structure we need. We say that $w$ is 
\emph{$\gamma$-sortable} if and only if $b_{1}\supseteq b_{2}\supseteq\cdots\supseteq b_{l}$, and we 
denote the set of $\gamma$-sortable elements of $W$ by $C_{\gamma}$.
\begin{example}
	Let us continue the previous example. We have seen that $w_{2}=s_{1}s_{2}s_{4}\vert s_{1}$ is a 
	$\gamma$-sorting word in $W$, and $b_{1}=\{s_{1},s_{2},s_{4}\}$, and $b_{2}=\{s_{1}\}$. Since 
	$b_{2}\subseteq b_{1}$, we see that $w_{2}$ is indeed $\gamma$-sortable.
\end{example}

The $\gamma$-sortable elements of $W$ are characterized by a recursive property which we will describe next. A
generator $s\in S$ is called \emph{initial in $\gamma$} if it is the first letter in some reduced word for
$\gamma$. For some subset $J\subseteq S$, let $W_{J}$ denote the parabolic subgroup of $W$ generated by
the set $J$, and let $W^{J}=\{w\in W\mid w<_{S}ws\;\mbox{for all}\;s\in J\}$. It is the statement of 
\cite{bjorner05combinatorics}*{Proposition~2.4.4\;(i)} that $w$ can be written uniquely as $w=w^{J}\cdot w_{J}$ for 
$w^{J}\in W^{J}$ and $w_{J}\in W_{J}$. It is straightforward to verify that the map $w\mapsto w_{J}$ is defined by the 
property that $\mbox{inv}(w_{J})=\mbox{inv}(w)\cap W_{J}$, see for instance \cite{reading11sortable}*{p.\;10} for a 
proof. For $s\in S$ we abbreviate $\langle s\rangle=S\setminus\{s\}$. 

\begin{proposition}[\cite{reading11sortable}*{Proposition~2.29}]
  \label{prop:recursion}
	Let $W$ be a Coxeter group, let $\gamma\in W$ be a Coxeter element and let $s$ be initial in $\gamma$. Then, an 
	element $w\in W$ is $\gamma$-sortable if and only if
	\begin{enumerate}
		\item[(i)] $s\leq_{S}w$ and $sw$ is $s\gamma s$-sortable,\quad or
		\item[(ii)] $s\not\leq_{S}w$ and $w$ is an $s\gamma$-sortable element of $W_{\langle s\rangle}$.
	\end{enumerate}
\end{proposition}

\begin{remark}
  \label{rem:dependence}
	The property of being $\gamma$-sortable does not depend on the choice of a reduced word for $\gamma$,
	see \cite{reading11sortable}*{Section~2.7}. 
	For $w\in W$, let $w_{1}$ and $w_{2}$ be the $\gamma$-sorting words of $w$
	with respect to two different reduced words $\gamma_{1}$ and $\gamma_{2}$ for $\gamma$. 
	Since $\gamma_{1}$ and $\gamma_{2}$ differ only in commutations of letters, it is clear 
	that $w_{1}$ and $w_{2}$ differ also only in commutations of letters, with no commutations across 
	dividers. Hence, the $i$-th block of $w_{1}$, considered as a subset of $S$, is equal to the $i$-th 
	block of $w_{2}$, considered as a subset of $S$. However, the $i$-th block of $w_{1}$, considered as 
	a subword of $\gamma_{1}$, is different from the $i$-th block of $w_{2}$, considered as a subword 
	of $\gamma_{2}$. 
\end{remark}

\subsection{Cambrian Semilattices}
  \label{sec:cambrian}
In \cite{reading11sortable}*{Section~7} the \emph{Cambrian semilattice $\mathcal{C}_{\gamma}$} is defined as the 
sub-semilattice of the weak order on $W$ consisting of all $\gamma$-sortable elements. That $\mathcal{C}_{\gamma}$ is
well-defined follows from the following theorem.

\begin{theorem}[\cite{reading11sortable}*{Theorem~7.1}]
  \label{thm:joins_meets}
	Let $A$ be a collection of $\gamma$-sortable elements of $W$. If $A$ is nonempty, then $\bigwedge_{S} A$
	is $\gamma$-sortable. If $A$ has an upper bound, then $\bigvee_{S} A$ is $\gamma$-sortable. 
\end{theorem}

It turns out that $\mathcal{C}_{\gamma}$ is not only a sub-semilattice of the weak order, but also a quotient 
semilattice. The key role in the proof of this property is played by the projection $\pi_{\downarrow}^{\gamma}$ which
maps every word $w\in W$ to the unique largest $\gamma$-sortable element below $w$. More precisely if $s$
is initial in $\gamma$, then define
\begin{equation}
  \label{eq:projection}
	\pi_{\downarrow}^{\gamma}(w)=
	  \begin{cases}
	    s\pi_{\downarrow}^{s\gamma s}(sw), & \mbox{if}\;s\leq_{S}w\\
	    \pi_{\downarrow}^{s\gamma}(w_{\langle s\rangle}), & \mbox{if}\;s\not\leq_{S}w,
	  \end{cases}
\end{equation}
and set $\pi_{\downarrow}^{\gamma}(\varepsilon)=\varepsilon$, see \cite{reading11sortable}*{Section~6}. 
The most important properties of this map are stated in the following theorems.

\begin{theorem}[\cite{reading11sortable}*{Theorem~6.1}]
  \label{thm:projection_homomorphism}
	The map $\pi_{\downarrow}^{\gamma}$ is order-preserving. 
\end{theorem}

\begin{theorem}[\cite{reading11sortable}*{Theorem~7.3}]
	For some subset $A\subseteq W$, if $A$ is nonempty, then 
	$\bigwedge_{\gamma}\pi_{\downarrow}^{\gamma}(A)=\pi_{\downarrow}^{\gamma}\bigl(\bigwedge_{S} A\bigr)$ and 
	if $A$ has an upper bound, then 
	$\bigvee_{\gamma}\pi_{\downarrow}^{\gamma}(A)=\pi_{\downarrow}^{\gamma}\bigl(\bigvee_{S} A\bigr)$.
\end{theorem}

Hence, $\pi_{\downarrow}^{\gamma}$ is a semilattice homomorphism from the weak order on $W$ to $\mathcal{C}_{\gamma}$, 
and $\mathcal{C}_{\gamma}$ can be considered as the quotient semilattice of the weak order modulo the semilattice 
congruence $\theta_{\gamma}$ induced by the fibers of $\pi_{\downarrow}^{\gamma}$. This semilattice 
congruence is called \emph{Cambrian congruence}. Since the lack of a maximal element is the only obstruction
for the weak order to be a lattice, it follows immediately that the restriction of 
$\pi_{\downarrow}^{\gamma}$ (and hence $\theta_{\gamma}$) to closed intervals of the weak order yields a 
lattice homomorphism (and hence a lattice congruence). Figure~\ref{fig:cambrian_congruence} shows the
Hasse diagram of the weak order on the Coxeter group $A_{3}$ and the congruence classes of 
$\theta_{\gamma}$ for $\gamma=s_{1}s_{2}s_{3}$. 

\begin{figure}
	\centering
	\begin{tikzpicture}\scriptsize
		\def\x{1.75};
		\def\y{1.5};
		%%
		\draw(3.5*\x,1.25*\y) node(n0){$\varepsilon$};
		\draw(2.5*\x,2*\y) node(n1){$s_{1}$};
		\draw(3.5*\x,2*\y) node(n2){$s_{2}$};
		\draw(4.5*\x,2*\y) node(n3){$s_{3}$};
		\draw(1.5*\x,3*\y) node(n4){$s_{1}s_{2}$};
		\draw(2.5*\x,3*\y) node(n5){$s_{2}\vert s_{1}$};
		\draw(3.5*\x,3*\y) node(n6){$s_{1}s_{3}$};
		\draw(4.5*\x,3*\y) node(n7){$s_{2}s_{3}$};
		\draw(5.5*\x,3*\y) node(n8){$s_{3}\vert s_{2}$};
		\draw(1*\x,4*\y) node(n9){$s_{1}s_{2}\vert s_{1}$};
		\draw(2*\x,4*\y) node(n10){$s_{1}s_{2}s_{3}$};
		\draw(3*\x,4*\y) node(n11){$s_{1}s_{3}\vert s_{2}$};
		\draw(4*\x,4*\y) node(n12){$s_{2}s_{3}\vert s_{1}$};
		\draw(5*\x,4*\y) node(n13){$s_{3}\vert s_{2}\vert s_{1}$};
		\draw(6*\x,4*\y) node(n14){$s_{2}s_{3}\vert s_{2}$};
		\draw(1.5*\x,5*\y) node(n15){$s_{1}s_{2}s_{3}\vert s_{1}$};
		\draw(2.5*\x,5*\y) node(n16){$s_{1}s_{2}s_{3}\vert s_{2}$};
		\draw(3.5*\x,5*\y) node(n17){$s_{1}s_{3}\vert s_{2}\vert s_{1}$};
		\draw(4.5*\x,5*\y) node(n18){$s_{2}s_{3}\vert s_{1}s_{2}$};
		\draw(5.5*\x,5*\y) node(n19){$s_{2}s_{3}\vert s_{2}\vert s_{1}$};
		\draw(2.5*\x,6*\y) node(n20){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}$};
		\draw(3.5*\x,6*\y) node(n21){$s_{1}s_{2}s_{3}\vert s_{2}\vert s_{1}$};
		\draw(4.5*\x,6*\y) node(n22){$s_{2}s_{3}\vert s_{1}s_{2}\vert s_{1}$};
		\draw(3.5*\x,6.75*\y) node(n23){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}\vert s_{1}$};
		%%
		\draw(n0) -- (n1);
		\draw(n0) -- (n2);
		\draw(n0) -- (n3);
		\draw(n1) -- (n4);
		\draw(n1) -- (n6);
		\draw(n2) -- (n5);
		\draw(n2) -- (n7);
		\draw(n3) -- (n6);
		\draw(n3) -- (n8);
		\draw(n4) -- (n9);
		\draw(n4) -- (n10);
		\draw(n5) -- (n9);
		\draw(n5) -- (n12);
		\draw(n6) -- (n11);
		\draw(n7) -- (n12);
		\draw(n7) -- (n14);
		\draw(n8) -- (n13);
		\draw(n8) -- (n14);
		\draw(n9) -- (n15);
		\draw(n10) -- (n15);
		\draw(n10) -- (n16);
		\draw(n11) -- (n16);
		\draw(n11) -- (n17);
		\draw(n12) -- (n18);
		\draw(n13) -- (n17);
		\draw(n13) -- (n19);
		\draw(n14) -- (n19);
		\draw(n15) -- (n20);
		\draw(n16) -- (n21);
		\draw(n17) -- (n21);
		\draw(n18) -- (n20);
		\draw(n18) -- (n22);
		\draw(n19) -- (n22);
		\draw(n20) -- (n23);
		\draw(n21) -- (n23);
		\draw(n22) -- (n23);
		%%
		\begin{pgfonlayer}{background}
		  \draw[line width=15pt,line cap=round,rounded corners,blue!40!white](3.5*\x,2*\y) -- 
		    (2.5*\x,3*\y);
		  \draw[line width=15pt,line cap=round,rounded corners,blue!40!white](4.5*\x,3*\y) -- 
		    (4*\x,4*\y) -- (4.5*\x,5*\y);
		  \draw[line width=15pt,line cap=round,rounded corners,blue!40!white](6*\x,4*\y) -- 
		    (5.5*\x,5*\y) -- (4.5*\x,6*\y);
		  \draw[line width=15pt,line cap=round,rounded corners,blue!40!white](4.5*\x,2*\y) -- 
		    (5.5*\x,3*\y) -- (5*\x,4*\y);
		  \draw[line width=15pt,line cap=round,rounded corners,blue!40!white](3.5*\x,3*\y) -- 
		    (3*\x,4*\y) -- (3.5*\x,5*\y);
		  \draw[line width=15pt,line cap=round,rounded corners,blue!40!white](2.5*\x,5*\y) -- 
		    (3.5*\x,6*\y);
		\end{pgfonlayer}
	\end{tikzpicture}
	\caption{The Cambrian congruence on the weak-order lattice on $A_{3}$ induced by the Coxeter
	element $s_{1}s_{2}s_{3}$. The non-singleton congruence classes are highlighted.}
	\label{fig:cambrian_congruence}
\end{figure}

In the remainder of this article, we switch frequently between the weak-order semilattice on $W$ and the
Cambrian semilattice $\mathcal{C}_{\gamma}$. In order to point out properly which semilattice we consider, we denote
the order relation of the weak-order semilattice by $\leq_{S}$, and the order relation of $\mathcal{C}_{\gamma}$ by
$\leq_{\gamma}$. Analogously, we denote a closed (respectively open) interval in the weak-order semilattice 
by $[u,v]_{S}$ (respectively $(u,v)_S$), and a closed (respectively open) interval in $\mathcal{C}_{\gamma}$
by $[u,v]_{\gamma}$ (respectively $(u,v)_{\gamma}$). Moreover, we denote the meet (respectively join) in the weak-order
semilattice by $\wedge_{S}$ (respectively $\vee_{S}$), and the meet (respectively join) in $\mathcal{C}_{\gamma}$ by 
$\wedge_{\gamma}$ (respectively $\vee_{\gamma}$).

\section{EL-Shellability of the Closed Intervals in $\mathcal{C}_{\gamma}$}
  \label{sec:shellability_cambrian}
In this section, we define an edge-labeling of $\mathcal{C}_{\gamma}$, discuss some of its properties and eventually
prove Theorem~\ref{thm:infinite_shellability}. 

\subsection{The Labeling}
  \label{sec:labeling}
In the remainder of this article, we focus on a fixed reduced word for $\gamma$, namely 
$\gamma=s_{1}s_{2}\cdots s_{n}$. Define for every $w\in W$ the set of positions of the $\gamma$-sorting word of $w$ as 
\begin{displaymath}
	\alpha_{\gamma}(w)=\bigl\{(i-1)\cdot n+j\mid\delta_{i,j}=1\bigr\}\subseteq\mathbb{N},
\end{displaymath}
where the $\delta_{i,j}$'s are the exponents from \eqref{eq:presentation}.

\begin{example}
	Let $W=\mathfrak{S}_{4}$, $\gamma=s_1s_2s_3$  and consider 
	$u=s_{1}s_{2}s_{3}\vert s_{2}$, and $v=s_{2}s_{3}\vert s_{2}\vert s_{1}$. 
	Then, $\alpha_{\gamma}(u)=\{1,2,3,5\}$, and $\alpha_{\gamma}(v)=\{2,3,5,7\}$, where $u\in \mathcal{C}_{\gamma}$,
	while $v\notin\mathcal{C}_{\gamma}$. 
\end{example}
It is not hard to see that an element $w\in W$ lies in $C_{\gamma}$ if and only if the following holds: if 
$i\in\alpha_{\gamma}(w)$ and $i>n$, then $i-n\in\alpha_{\gamma}(w)$. In the previous example, we see that
$\alpha_{\gamma}(u)$ contains both $5$ and $2$, while $\alpha_{\gamma}(v)$ does not contain $7-3=4$. 

\begin{lemma}
  \label{lem:alpha}
  	Let $u,v\in W$ with $u\leq_S v$. Then $\alpha_{\gamma}(u)$ is a subset of $\alpha_{\gamma}(v)$.
\end{lemma}
\begin{proof}
    The $\gamma$-sorting word of an element $w\in W$ is a reduced word for $w$. 
	Thus, it follows immediately from the definition of the weak order that any letter appearing in 
	the $\gamma$-sorting word of $u$ has to appear also in the $\gamma$-sorting word of every element
	that is greater than $w$ in the weak order. 
	Thus, if $u,v\in C_{\gamma}$ with $u\leq_{\gamma}v$, then $\alpha_{\gamma}(u)\subseteq\alpha_{\gamma}(v)$. 
\end{proof}

Denote by $\mathcal{E}(\mathcal{C}_{\gamma})$ the set of covering relations of $\mathcal{C}_{\gamma}$, and define 
an edge-labeling of $\mathcal{C}_{\gamma}$ by
\begin{equation}
  \label{eq:labeling}
	\lambda_{\gamma}:\mathcal{E}(\mathcal{C}_{\gamma})\to\mathbb{N},
	  \quad (u,v)\mapsto\min\bigl\{i\mid i\in\alpha_{\gamma}(v)\smallsetminus\alpha_{\gamma}(u)\bigr\}.
\end{equation}
Figures~\ref{fig:camb_a3} and \ref{fig:camb_b3} show the Hasse diagrams of a Cambrian lattice 
$\mathcal{C}_{\gamma}$ of the Coxeter groups $A_{3}$ and $B_{3}$, respectively, together with the labels defined by
the map $\lambda_{\gamma}$. 

\begin{figure}
	\centering
	\begin{tikzpicture}\scriptsize
		\def\x{1.75};
		\def\y{1.5};
		%%
		\draw(3*\x,1*\y) node(v0){$\varepsilon$};	
		\draw(2*\x,2*\y) node(v1){$s_{1}$};
		\draw(3*\x,2*\y) node(v2){$s_{2}$};
		\draw(4*\x,2*\y) node(v3){$s_{3}$};
		\draw(1*\x,3*\y) node(v4){$s_{1}s_{2}$};
		\draw(3*\x,3*\y) node(v5){$s_{1}s_{3}$};
		\draw(4*\x,3*\y) node(v6){$s_{2}s_{3}$};
		\draw(.5*\x,4*\y) node(v7){$s_{1}s_{2}\vert s_{1}$};
		\draw(1.5*\x,4*\y) node(v8){$s_{1}s_{2}s_{3}$};
		\draw(5*\x,4*\y) node(v9){$s_{2}s_{3}\vert s_{2}$};
		\draw(1*\x,5*\y) node(v10){$s_{1}s_{2}s_{3}\vert s_{1}$};
		\draw(2*\x,5*\y) node(v11){$s_{1}s_{2}s_{3}\vert s_{2}$};
		\draw(2*\x,6*\y) node(v12){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}$};
		\draw(3*\x,7*\y) node(v13){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}\vert s_{1}$};
		%%
		\draw[very thick](v0) -- (v1) node[fill=white] at(2.5*\x,1.5*\y){\tiny $1$};
		\draw(v0) -- (v2) node[fill=white] at(3*\x,1.5*\y){\tiny $2$};
		\draw(v0) -- (v3) node[fill=white] at(3.5*\x,1.5*\y){\tiny $3$};
		\draw[very thick](v1) -- (v4) node[fill=white] at(1.5*\x,2.5*\y){\tiny $2$};
		\draw(v1) -- (v5) node[fill=white] at(2.67*\x,2.67*\y){\tiny $3$};
		\draw(v2) -- (v6) node[fill=white] at(3.67*\x,2.67*\y){\tiny $3$};
		\draw(v2) -- (v7) node[fill=white] at(1.75*\x,3*\y){\tiny $1$};
		\draw(v3) -- (v5) node[fill=white] at(3.33*\x,2.67*\y){\tiny $1$};
		\draw(v3) -- (v9) node[fill=white] at(4.5*\x,3*\y){\tiny $2$};
		\draw(v4) -- (v7) node[fill=white] at(.75*\x,3.5*\y){\tiny $4$};
		\draw[very thick](v4) -- (v8) node[fill=white] at(1.33*\x,3.67*\y){\tiny $3$};
		\draw(v5) -- (v11) node[fill=white] at(2.5*\x,4*\y){\tiny $2$};
		\draw(v6) -- (v9) node[fill=white] at(4.5*\x,3.5*\y){\tiny $5$};
		\draw(v6) -- (v12) node[fill=white] at(3*\x,4.5*\y){\tiny $1$};
		\draw(v7) -- (v10) node[fill=white] at(.75*\x,4.5*\y){\tiny $3$};
		\draw[very thick](v8) -- (v10) node[fill=white] at(1.25*\x,4.5*\y){\tiny $4$};
		\draw(v8) -- (v11) node[fill=white] at(1.75*\x,4.5*\y){\tiny $5$};
		\draw(v9) -- (v13) node[fill=white] at(4*\x,5.5*\y){\tiny $1$};
		\draw[very thick](v10) -- (v12) node[fill=white] at(1.5*\x,5.5*\y){\tiny $5$};
		\draw(v11) -- (v13) node[fill=white] at(2.62*\x,6.25*\y){\tiny $4$};
		\draw[very thick](v12) -- (v13) node[fill=white] at(2.5*\x,6.5*\y){\tiny $7$};
	\end{tikzpicture}
	\caption{An $A_{3}$-Cambrian lattice with the labeling as defined in \eqref{eq:labeling}.}
	\label{fig:camb_a3}
\end{figure}

\begin{figure}
	\centering
	\begin{tikzpicture}\scriptsize
		\def\x{1.25};
		\def\y{1.5};
		%%
		\draw(4*\x,1*\y) node(v0){$\varepsilon$};
		\draw(2*\x,2*\y) node(v1){$s_{3}$};
		\draw(4*\x,2*\y) node(v2){$s_{2}$};
		\draw(6*\x,2*\y) node(v3){$s_{1}$};
		\draw(3*\x,3*\y) node(v4){$s_{2}s_{3}$};
		\draw(4*\x,3*\y) node(v5){$s_{1}s_{3}$};
		\draw(7*\x,3*\y) node(v6){$s_{1}s_{2}$};
		\draw(2*\x,4*\y) node(v7){$s_{2}s_{3}\vert s_{2}$};
		\draw(5.67*\x,3.67*\y) node(v8){$s_{1}s_{2}\vert s_{1}$};
		\draw(8*\x,4*\y) node(v9){$s_{1}s_{2}s_{3}$};
		\draw(1*\x,5*\y) node(v10){$s_{2}s_{3}\vert s_{2}s_{3}$};
		\draw(6.67*\x,4.67*\y) node(v11){$s_{1}s_{2}s_{3}\vert s_{1}$};
		\draw(8.67*\x,4.67*\y) node(v12){$s_{1}s_{2}s_{3}\vert s_{2}$};
		\draw(6.7*\x,5.75*\y) node(v13){$s_{1}s_{2}s_{3}\vert s_{2}s_{3}$};
		\draw(8*\x,6*\y) node(v14){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}$};
		\draw(7*\x,7*\y) node(v15){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}s_{3}$};
		\draw(9*\x,7*\y) node(v16){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}\vert s_{1}$};
		\draw(8*\x,8*\y) node(v17){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}s_{3}\vert s_{1}$};
		\draw(7*\x,9*\y) node(v18){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}s_{3}\vert s_{1}s_{2}$};
		\draw(4*\x,10*\y) node(v19){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}s_{3}\vert s_{1}s_{2}s_{3}$};
		%%
		\draw(v0) -- (v1) node[fill=white] at (3*\x,1.5*\y){\tiny $3$};
		\draw(v0) -- (v2) node[fill=white] at (4*\x,1.5*\y){\tiny $2$};
		\draw[very thick](v0) -- (v3) node[fill=white] at (5*\x,1.5*\y){\tiny $1$};
		\draw(v1) -- (v5) node[fill=white] at (3*\x,2.5*\y){\tiny $1$};
		\draw(v1) -- (v10) node[fill=white] at (1.5*\x,3.5*\y){\tiny $2$};
		\draw(v2) -- (v4) node[fill=white] at (3.5*\x,2.5*\y){\tiny $3$};
		\draw(v2) -- (v8) node[fill=white] at (5.17*\x,3.17*\y){\tiny $1$};
		\draw(v3) -- (v5) node[fill=white] at (5.33*\x,2.33*\y){\tiny $3$};
		\draw[very thick](v3) -- (v6) node[fill=white] at (6.5*\x,2.5*\y){\tiny $2$};
		\draw(v4) -- (v7) node[fill=white] at (2.5*\x,3.5*\y){\tiny $5$};
		\draw(v4) -- (v15) node[fill=white] at (5*\x,5*\y){\tiny $1$};
		\draw(v5) -- (v13) node[fill=white] at (5.5*\x,4.5*\y){\tiny $2$};
		\draw(v6) -- (v8) node[fill=white] at (6.33*\x,3.33*\y){\tiny $4$};
		\draw[very thick](v6) -- (v9) node[fill=white] at (7.5*\x,3.5*\y){\tiny $3$};
		\draw(v7) -- (v10) node[fill=white] at (1.5*\x,4.5*\y){\tiny $6$};
		\draw(v7) -- (v18) node[fill=white] at (4.5*\x,6.5*\y){\tiny $1$};
		\draw(v8) -- (v11) node[fill=white] at (6.17*\x,4.17*\y){\tiny $3$};
		\draw[very thick](v9) -- (v11) node[fill=white] at (7.33*\x,4.33*\y){\tiny $4$};
		\draw(v9) -- (v12) node[fill=white] at (8.33*\x,4.33*\y){\tiny $5$};
		\draw(v10) -- (v19) node[fill=white] at (2.5*\x,7.5*\y){\tiny $1$};
		\draw[very thick](v11) -- (v14) node[fill=white] at (7.17*\x,5.17*\y){\tiny $5$};
		\draw(v12) -- (v13) node[fill=white] at (8*\x,5.05*\y){\tiny $6$};
		\draw(v12) -- (v16) node[fill=white] at (8.8*\x,5.75*\y){\tiny $4$};
		\draw(v13) -- (v19) node[fill=white] at (5.25*\x,8*\y){\tiny $4$};
		\draw[very thick](v14) -- (v15) node[fill=white] at (7.5*\x,6.5*\y){\tiny $6$};
		\draw(v14) -- (v16) node[fill=white] at (8.5*\x,6.5*\y){\tiny $7$};
		\draw[very thick](v15) -- (v17) node[fill=white] at (7.5*\x,7.5*\y){\tiny $7$};
		\draw(v16) -- (v17) node[fill=white] at (8.5*\x,7.5*\y){\tiny $6$};
		\draw[very thick](v17) -- (v18) node[fill=white] at (7.5*\x,8.5*\y){\tiny $8$};
		\draw[very thick](v18) -- (v19) node[fill=white] at (5.5*\x,9.5*\y){\tiny $9$};
	\end{tikzpicture}
	\caption{A $B_{3}$-Cambrian lattice, with the labeling as defined in \eqref{eq:labeling}.}
	\label{fig:camb_b3}
\end{figure}

\begin{remark}
	By definition, the Cambrian semilattice $\mathcal{C}_{\gamma}$ does not depend on a reduced word for $\gamma$,
	see Remark~\ref{rem:dependence}. In contrast to that, the labeling defined in \eqref{eq:labeling} \emph{does} depend 
	on the reduced word for $\gamma$. That means, if $\gamma_{1}$ and $\gamma_{2}$ are two different reduced words
	for $\gamma$, the Cambrian semilattices $\mathcal{C}_{\gamma_{1}}$ and $\mathcal{C}_{\gamma_{2}}$ are isomorphic,
	but the labeling of $\mathcal{C}_{\gamma_{1}}$ is structurally different from the labeling of 
	$\mathcal{C}_{\gamma_{2}}$. However, we show that the property that $\lambda$ is an EL-labeling does not depend 
	on the reduced word for $\gamma$.
\end{remark}

\subsection{Properties of the Labeling}
  \label{sec:properties}

\begin{lemma}
  \label{lem:i_0}
	Let $\mathcal{C}_{\gamma}$ be a Cambrian semilattice, and let $u,v\in C_{\gamma}$ such that $u\leq_{\gamma}v$. 
	Let $i_0=\min\{i\mid i\in\alpha_{\gamma}(v)\smallsetminus\alpha_{\gamma}(u)\}$. Then, 
	the following hold.
	\begin{enumerate}
		\item[(i)] The label $i_0$ appears in every maximal chain of the interval $[u,v]_{\gamma}$. 
		\item[(ii)] The labels of a maximal chain in $[u,v]_{\gamma}$ are distinct.
	\end{enumerate}
\end{lemma}
\begin{proof}
	(i) Suppose that this is not the case. Then there exists a maximal chain 
	$u=x_0\lessdot_{\gamma}x_1\lessdot_{\gamma}\cdots\lessdot_{\gamma}x_{t-1}\lessdot_{\gamma}x_t=v$
	with $\lambda_{\gamma}(x_i,x_{i+1})\neq i_0$ for every $i\in\{0,1,\ldots,t-1\}$. Hence, 
	$i_0\in\alpha_{\gamma}(u)$ if and only if $i_0\in\alpha_{\gamma}(v)$, which contradicts the definition of 
	$i_0$. 

	(ii) Let $u=x_{0}\lessdot_{\gamma}x_{1}\lessdot_{\gamma}\cdots\lessdot_{\gamma}x_{t}=v$ be a
	maximal chain in $[u,v]_{\gamma}$. Assume that there are $i,j\in\{0,1,\ldots,t-1\}$ with $i<j$
	such that $\lambda_{\gamma}(x_{i},x_{i+1})=k=\lambda_{\gamma}(x_{j},x_{j+1})$. By definition, 
	$k\in\alpha_{\gamma}(x_{i+1})$, and $k\notin\alpha_{\gamma}(x_{j})$. Since $x_{i+1}\leq_{S} x_{j}$, we can 
	conclude from Lemma~\ref{lem:alpha} that $\alpha_{\gamma}(x_{i+1})\subseteq\alpha_{\gamma}(x_{j})$, which 
	yields a contradiction.
\end{proof}

The $\gamma$-sortable elements of $W$ are defined recursively as described in
Proposition~\ref{prop:recursion}. Before we investigate how our labeling behaves with respect to this recursion, 
we need to recall one more result. For $s\in S$, let $W_{\geq s}=\{w\in W\mid s\leq_{S}w\}$ and let 
$W_{\not\geq s}=\{w\in W\mid s\not\leq_{S}w\}$.
\begin{proposition}[\cite{reading11sortable}*{Proposition~2.18}]
  \label{prop:left_multiplication}
	Let $w\in W$ and $s\in S$. Then $\ell_{S}(sw)<\ell_{S}(w)$ if and only if $s\leq_{S}w$ if and only if 
	$s\in\mbox{inv}(w)$. Left multiplication by $s$ is a poset isomorphism from 
	$\bigl(W_{\not\geq s},\leq_{S}\bigr)$ to $\bigl(W_{\geq s},\leq_{S}\bigr)$. If $w\lessdot_{S} w',s\leq_{S}w'$ 
	and $s\not\leq_{S}w$, then $w'=sw$.
\end{proposition}

\begin{lemma}
   \label{lem:labeling_recursion}
	Let $W$ be a Coxeter group and let $\gamma=s_{1}s_{2}\cdots s_{n}$ be a Coxeter element of $W$. 
	For $u,v\in C_{\gamma}$ with $u\lessdot_{\gamma}v$, we have
	\begin{displaymath}
		\lambda_{\gamma}(u,v)=
		  \begin{cases}
		  	  1, & \mbox{if}\;s_{1}\not\leq_{S}u\;\mbox{and}\;s_{1}\leq_{S}v,\\
		  	  \lambda_{s_{1}\gamma s_{1}}(s_{1}u,s_{1}v)+1, & \mbox{if}\;s_{1}\leq_{S}u,\\
		  	  \lambda_{s_{1}\gamma}\bigl(u_{\langle s_{1}\rangle},v_{\langle s_{1}\rangle}\bigr)+k, 
			    & \mbox{if}\;s_{1}\not\leq_{S}v\;\mbox{and the first position where}\;u\;\mbox{and}\;v\\
		  	    & \mbox{differ is in their}\;k\mbox{-th block}.
		  \end{cases}
	\end{displaymath}
\end{lemma}
\begin{proof}
	Let first $s_{1}\not\leq_{S}u$ and $s_{1}\leq_{S}v$. By definition of the weak order, $s_{1}$ does not occur 
	in the first position of any reduced word for $u$, in particular it does not occur in the first position 
	of the $\gamma$-sorting word of $u$. Hence, $1\notin\alpha_{\gamma}(u)$. On the other hand, $s_{1}$ does occur 
	in the first position of the $\gamma$-sorting word of $v$, and hence $1\in\alpha_{\gamma}(v)$. By
	definition this implies $\lambda_{\gamma}(u,v)=1$.
	
	Let now $s_{1}\leq_{S}u$. Then, $s_{1}\leq_{S}v$, and with Proposition~\ref{prop:recursion}, we find that
	$s_{1}u$ and $s_{1}v$ are $s_{1}\gamma s_{1}$-sortable. It follows from 
	Propositions~\ref{prop:recursion} and \ref{prop:left_multiplication} and from the definition of the weak 
	order that $s_{1}u\lessdot_{s_{1}\gamma s_{1}}s_{1}v$. Say $\lambda_{s_{1}\gamma s_{1}}(s_{1}u,s_{1}v)=k$. By 
	construction, the $s_{1}\gamma s_{1}$-sorting word of $s_{1}u$ is precisely the subword of $u$ starting at the 
	second position. Thus, the $s_{1}\gamma s_{1}$-sorting word of $s_{1}u$ is the leftmost subword of 
	$\gamma^{\infty}$ where the first position is empty, and likewise for $s_{1}v$. If the first position of 
	$(s_{1}\gamma s_{1})^{\infty}$ where $s_{1}u$ and $s_{1}v$ differ is $k$, then the first position of 
	$\gamma^{\infty}$ where $u$ and $v$ differ is $k+1$. Hence, 
	$\lambda_{\gamma}(u,v)=\lambda_{s_{1}\gamma s_{1}}(s_{1}u,s_{1}v)+1$.
	
	Finally, let $s_{1}\not\leq_{S}v$. Then, $s_{1}\not\leq_{S}u$, and with Proposition~\ref{prop:recursion}, we
	find that $u=u_{\langle s_{1}\rangle}$ and $v=v_{\langle s_{1}\rangle}$ are $s_{1}\gamma$-sortable elements 
	of the parabolic subgroup $W_{\langle s_{1}\rangle}$ of $W$, and the Cambrian lattice $\mathcal{C}_{s_{1}\gamma}$ 
	is an order ideal in $\mathcal{C}_{\gamma}$. Say that the first position filled in $v_{\langle s_{1}\rangle}$ but
	not in $u_{\langle s_{1}\rangle}$ is in the $k$-th block of $v_{\langle s_{1}\rangle}$. Considering 
	$u_{\langle s_{1}\rangle}$ and $v_{\langle s_{1}\rangle}$ as subwords of $\gamma^{\infty}$ adds the letter 
	$s_{1}$ with exponent $0$ to each block of $u_{\langle s_{1}\rangle}$ and $v_{\langle s_{1}\rangle}$. Since 
	the first difference of $u_{\langle s_{1}\rangle}$ and $v_{\langle s_{1}\rangle}$ is in the $k$-th block, the 
	first difference of $u$ and $v$ is still in the $k$-th block, but each block has an additional first 
	letter. Hence 
	$\lambda_{\gamma}(u,v)=\lambda_{s_{1}\gamma}\bigl(u_{\langle s_{1}\rangle},v_{\langle s_{1}\rangle}\bigr)+k$.
\end{proof}

\begin{example}
	Let $W=B_{3}$ generated by $S=\{s_{1},s_{2},s_{3}\}$ satisfying
	$(s_{1}s_{2})^{3}=(s_{2}s_{3})^{4}=(s_{1}s_{3})^{2}=\varepsilon$ and 
	$s_{1}^{2}=s_{2}^{2}=s_{3}^{2}=\varepsilon$, and let $\gamma=s_{1}s_{2}s_{3}$ be a Coxeter element of 
	$B_{3}$. Thus, $\mathcal{C}_{\gamma}$ is the lattice depicted in Figure~\ref{fig:camb_b3}.
	
	Consider $u_{1}=s_{2}s_{3}\vert s_{2}s_{3}$ and 
	$v_{1}=s_{1}s_{2}s_{3}\vert s_{1}s_{2}s_{3}\vert s_{1}s_{2}s_{3}$. With the definition of our 
	labeling follows $\lambda_{\gamma}(u_{1},v_{1})=1$ immediately. 
	
	Let now $u_{2}=s_{1}s_{2}s_{3}\vert s_{1}s_{2}$ and 
	$v_{2}=s_{1}s_{2}s_{3}\vert s_{1}s_{2}s_{3}$. Then, $s_{1}u_{2}=s_{2}s_{3}s_{1}\vert s_{2}$ and
	$s_{1}v_{2}=s_{2}s_{3}s_{1}\vert s_{2}s_{3}$ considered as $s_{1}\gamma s_{1}$-sorting words. We have
	\begin{displaymath}
		\lambda_{s_{1}\gamma s_{1}}(s_{1}u_{2},s_{1}v_{2})=5,\quad\mbox{and}\quad
		\lambda_{\gamma}(u_{2},v_{2})=6.
	\end{displaymath}
	
	Finally, let $u_{3}=s_{2}s_{3}\vert s_{2}$ and $v_{3}=s_{2}s_{3}\vert s_{2}s_{3}$. The
	$(s_{1}\gamma)^{\infty}$-sorting words of $(u_{3})_{\langle s_{1}\rangle}$ and 
	$(v_{3})_{\langle s_{1}\rangle}$ written as in
	\eqref{eq:presentation} are 
	\begin{displaymath}
		(u_{3})_{\langle s_{1}\rangle}=s_{2}^{1}s_{3}^{1}\vert s_{2}^{1}s_{3}^{0},
		  \quad\mbox{and}\quad 
		(v_{3})_{\langle s_{1}\rangle}=s_{2}^{1}s_{3}^{1}\vert s_{2}^{1}s_{3}^{1}.
	\end{displaymath}
	The corresponding $\gamma$-sorting words of $u_{3}$ and $v_{3}$ are
	\begin{displaymath}
		u_{3}=s_{1}^{0}s_{2}^{1}s_{3}^{1}\vert s_{1}^{0}s_{2}^{1}s_{3}^{0},\quad\mbox{and}\quad 
		v_{3}=s_{1}^{0}s_{2}^{1}s_{3}^{1}\vert s_{1}^{0}s_{2}^{1}s_{3}^{1}.
	\end{displaymath}
	Hence, 
	$\lambda_{s_{1}\gamma}\bigl((u_{3})_{\langle s_{1}\rangle},(v_{3})_{\langle s_{1}\rangle}\bigr)=4$
	and $\lambda_{\gamma}(u_{3},v_{3})=6$.
\end{example}

\subsection{Proof of Theorem~\ref{thm:infinite_shellability}}
  \label{sec:proof}
We will prove Theorem~\ref{thm:infinite_shellability} by showing that the map $\lambda_{\gamma}$ defined in 
\eqref{eq:labeling} is an EL-labeling for every closed interval in $\mathcal{C}_{\gamma}$. In particular we show the 
following.

\begin{theorem}
  \label{thm:el_labeling}
	Let $u,v\in C_{\gamma}$ with $u\leq_{\gamma}v$. Then the map $\lambda_{\gamma}$ defined in 
	\eqref{eq:labeling} is an EL-labeling for $[u,v]_{\gamma}$.
\end{theorem}

For the proof of Theorem~\ref{thm:el_labeling}, we need two more technical results. The first one is 
\cite{reading11sortable}*{Proposition~2.20} which was first proved in 
\cite{jedlicka05combinatorial}*{Lemmas~4.2~(iii)~and~4.5}. The second result uses many of the
deep results on Cambrian semilattices developed in \cite{reading11sortable}. For $w\in W$, we 
say that $t\in\mbox{inv}(w)$ is called a \emph{cover reflection of $w$} if there exists some $s\in S$ with $tw=ws$. 
We denote by $\mbox{cov}(w)$ the set of all cover reflections of $w$.

\begin{proposition}[\cite{reading11sortable}*{Proposition~2.20}]
  \label{prop:commuting_operations}
	Let $J\subseteq S$ and $A\subseteq W$ and define $A_{J}=\{w_{J}\mid w\in A\}$. If $A$ is nonempty, then 
	$\bigwedge_{S}\bigl(A_{J}\bigr)=\bigl(\bigwedge_{S}A\bigr)_{J}$, and if $A$ has an upper bound, then 
	$\bigvee_{S}\bigl(A_{J}\bigr)=\bigl(\bigvee_{S}A\bigr)_{J}$. 
\end{proposition}

\begin{lemma}
  \label{lem:joincov}
	Let $\gamma=s_{1}s_{2}\cdots s_{n}$, and let $u,v\in C_{\gamma}$ with $u\leq_{\gamma}v$. If
	$s_{1}\not\leq_{\gamma}u$ and $s_{1}\leq_{\gamma}v$, then the join $s_{1}\vee_{\gamma}u$ covers $u$ in 
	$\mathcal{C}_{\gamma}$.
\end{lemma}
\begin{proof}
	First of all, since $s_{1}\leq_{\gamma}v$ and $u\leq_{\gamma}v$, we conclude from 
	Theorem~\ref{thm:joins_meets} that $s_{1}\vee_{\gamma} u$ exists, and we set $z=s_{1}\vee_{\gamma}u$. 
	Now, we observe that if $w,w'\in C_{\gamma}$, then it follows from 
	Theorem~\ref{thm:projection_homomorphism} that
	\begin{equation}
	  \label{eq:order}
		w\leq_{S}w'\quad\mbox{implies}\quad
		  \pi_{\downarrow}^{\gamma}(w)\leq_{\gamma}\pi_{\downarrow}^{\gamma}(w')\quad\mbox{implies}\quad 
		  w\leq_{\gamma}w'.
	\end{equation}
	By assumption, we have $s_{1}\not\leq_{\gamma}u$, and by contraposition follows from \eqref{eq:order} that 
	$s_{1}\not\leq_{S}u$. Hence, with \eqref{eq:projection} follows that $u=\pi_{\downarrow}^{\gamma}(u)=
	\pi_{\downarrow}^{s_{1}\gamma}(u_{\langle s_{1}\rangle})\in W_{\langle s_{1}\rangle}$, and 
	Proposition~\ref{prop:recursion} implies $u=u_{\langle s_{1}\rangle}\in W_{\langle s_{1}\rangle}$.
	
	Since $u<_{\gamma}z$, there exists some $u'\in C_{\gamma}$ with $u\leq_{\gamma}u'\lessdot_{\gamma}z$. 
	If $s_{1}\leq_{\gamma}u'$, then $u'$ is an upper bound for $s_{1}$ and $u$ which contradicts 
	$u'\lessdot_{\gamma}z$. Thus, we have $s_{1}\not\leq_{\gamma}u'$, which with \eqref{eq:order} implies 
	$s_{1}\not\leq_{S}u'$ again, and it follows from Proposition~\ref{prop:recursion} that
	$u'=u'_{\langle s_{1}\rangle}\in W_{\langle s_{1}\rangle}$. Since $\mathcal{C}_{\gamma}$ is a sub-semilattice 
	of the weak-order semilattice, the relation $u'\lessdot_{\gamma}z$ implies $u'<_{S}z$, and we obtain
	$u'_{\langle s_{1}\rangle}\leq_{S}z_{\langle s_{1}\rangle}=(s_{1}\vee_{\gamma}u)_{\langle s_{1}\rangle}$.
	In view of Proposition~\ref{prop:commuting_operations}, this implies 
	$u'_{\langle s_{1}\rangle}\leq_{S}(s_{1})_{\langle s_{1}\rangle}\vee_{\gamma}u_{\langle s_{1}\rangle}$. 
	However, since $(s_{1})_{\langle s_{1}\rangle}=\varepsilon$, and since $u=u_{\langle s_{1}\rangle}$ and
	$u'=u'_{\langle s_{1}\rangle}$, we obtain $u'\leq_{S}u$. This implies $u=u'$ and thus the result.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:el_labeling}]
	Let $[u,v]_{\gamma}$ be a closed interval of $\mathcal{C}_{\gamma}$. Since the weak order on $W$ is finitary,
	it follows that $[u,v]_{\gamma}$ is a finite lattice. We show that there 
	exists a unique maximal rising chain which is the lexicographically first among all maximal chains 
	in this interval.
	
	We proceed by induction on length and rank, using the recursive structure of $\gamma$-sortable elements, 
	see Proposition~\ref{prop:recursion}. We assume that $\ell_{S}(v)\geq 3$, and that $W$ is a 
	Coxeter group of rank $\geq 2$, since the result is trivial otherwise. Say that $W$ is of 
	rank $n$, and say that $\ell_{S}(v)=k$. Suppose that the induction hypothesis is true for 
	all parabolic subgroubs of $W$ having rank $<n$ and suppose that for every closed interval 
	$[u',v']_{\gamma}$ of $\mathcal{C}_{\gamma}$ with $\ell_{S}(v')<k$, there exists a unique rising maximal chain 
	from $u'$ to $v'$ which is lexicographically first among all maximal chains in $[u',v']_{\gamma}$. 
	We show that there is a unique rising maximal chain in the interval $[u,v]_{\gamma}$ wich is 
	lexicographically first among all maximal chains in $[u,v]_{\gamma}$. Let $\gamma=s_{1}s_{2}\cdots s_{n}$. 
	We distinguish two cases: (1) $s_{1}\not\leq_{\gamma} v$ and (2) $s_{1}\leq_{\gamma} v$. 

	(1) Since $s_{1}\not\leq_{\gamma} v$, it follows that no element of $[u,v]_{\gamma}$ contains the 
	letter $s_{1}$ in its $\gamma$-sorting word. We consider the parabolic Coxeter group 
	$W_{\langle s_{1}\rangle}$ (generated by $S\setminus\{s_{1}\}$) and the Coxeter element 
	$s_{1}\gamma$. It follows from Proposition~\ref{prop:recursion} that the interval 
	$[u,v]_{\gamma}$ is isomorphic to the interval 
	$[u_{\langle s_{1}\rangle},v_{\langle s_{1}\rangle}]_{s_{1}\gamma}$ in $W_{\langle s_{1}\rangle}$. 
	Since the rank of $W_{\langle s_{1}\rangle}$ is $n-1<n$, by induction there exists a unique 
	maximal rising chain $c':u_{\langle s_{1}\rangle}=(x_{0})_{\langle s_{1}\rangle}
	\lessdot_{s_{1}\gamma}(x_{1})_{\langle s_{1}\rangle}\lessdot_{s_{1}\gamma}\cdots
	\lessdot_{s_{1}\gamma}(x_{t})_{\langle s_{1}\rangle}=v_{\langle s_{1}\rangle}$ which is lexicographically 
	first among all maximal chains in $[u_{\langle s_{1}\rangle},v_{\langle s_{1}\rangle}]_{s\gamma}$. Let 
	$(x_{j_{a}})_{\langle s_{1}\rangle}\lessdot_{s_{1}\gamma}(x_{j_{a}+1})_{\langle s_{1}\rangle}$ and 
	$(x_{j_{b}})_{\langle s_{1}\rangle}\lessdot_{s_{1}\gamma}(x_{j_{b}+1})_{\langle s_{1}\rangle}$ be two 
	covering relations in $c'$ with $j_{a}+1\leq j_{b}$. Say that the first block where 
	$(x_{j_{a}})_{\langle s_{1}\rangle}$ and $(x_{j_{a}+1})_{\langle s_{1}\rangle}$ differ is the $d_{a}$-th 
	block of their $s_{1}\gamma$-sorting word and say that the first block where 
	$(x_{j_{b}})_{\langle s_{1}\rangle}$ and $(x_{j_{b}+1})_{\langle s_{1}\rangle}$ differ is the $d_{b}$-th 
	block of their $s_{1}\gamma$-sorting word. Since $c'$ is rising, we conclude that $d_{a}\leq d_{b}$,
	and Lemma~\ref{lem:labeling_recursion} implies that the corresponding maximal chain
	$c:u=x_{0}\lessdot_{\gamma}x_{1}\lessdot_{\gamma}\cdots\lessdot_{\gamma} x_{t}=v$ in 
	$[u,v]_{\gamma}$ is rising. Similarly, it follows that $c$ is the unique maximal rising chain and 
	that it is lexicographically first among all maximal chains in $[u,v]_{\gamma}$.

	(2a) Suppose first that $s_{1}\leq_{\gamma} u$ as well. Then, $s_{1}$ is the first letter in the 
	$\gamma$-sorting word of every element in $[u,v]_{\gamma}$. It follows from 
	Propositions~\ref{prop:recursion} and \ref{prop:left_multiplication} 
	that the interval $[u,v]_{\gamma}$ is isomorphic to the interval $[s_{1}u,s_{1}v]_{s_{1}\gamma s_{1}}$.
	Moreover, Lemma~\ref{lem:labeling_recursion} implies that for a covering relation 
	$x\lessdot_{\gamma}y$ in $[u,v]_{\gamma}$ we have 
	$\lambda_{\gamma}(x,y)=\lambda_{s_{1}\gamma s_{1}}(s_{1}x,s_{1}y)+1$. Say that $c':s_{1}u=s_{1}x_{0}
	\lessdot_{s_{1}\gamma s_{1}}s_{1}x_{1}\lessdot_{s_{1}\gamma s_{1}}\cdots
	\lessdot_{s_{1}\gamma s_{1}}s_{1}x_{t}=s_{1}v$ is
	the unique rising maximal chain in $[s_{1}u,s_{1}v]_{s_{1}\gamma s_{1}}$. (This chain exists by induction, since 
	$\ell_{S}(s_{1}v)<\ell_{S}(v)$.) Then, the chain 
	$c:u=x_{0}\lessdot_{\gamma}x_{1}\lessdot_{\gamma}\cdots\lessdot_{\gamma}x_{t}=v$ is a 
	maximal chain in $[u,v]_{\gamma}$ and clearly rising. Now consider some maximal chain 
	$\bar{c}:u=\bar{x}_{0}\lessdot_{\gamma}\bar{x}_{1}\lessdot_{\gamma}\cdots\lessdot_{\gamma}\bar{x}_{\bar{t}}=v$
	in $[u,v]_{\gamma}$ with $\bar{c}\neq c$, and let $\bar{c}':s_{1}u=s_{1}\bar{x}_{0}\lessdot_{s_{1}\gamma s_{1}}
	s_{1}\bar{x}_{1}\lessdot_{s_{1}\gamma s_{1}}\cdots\lessdot_{s_{1}\gamma s_{1}}s_{1}\bar{x}_{\bar{t}}=s_{1}v$ be the 
	corresponding chain in $[s_{1}u,s_{1}v]_{s_{1}\gamma s_{1}}$. Then, it follows from
	Propositions~\ref{prop:recursion} and \ref{prop:left_multiplication} that $\bar{c}'\neq c'$, and the induction
	hypothesis implies that $\bar{c}'$ is not rising and that $\bar{c}'$ is lexicographically larger than $c'$.
	Lemma~\ref{lem:labeling_recursion} implies now the same for $\bar{c}$. Thus, $c$ is the unique rising chain in
	$[u,v]_{\gamma}$ and every other maximal chain in $[u,v]_{\gamma}$ is lexicographically larger than $c$. 

	(2b) Suppose now that $s_{1}\not\leq_{\gamma} u$. Since $s_{1}\leq_{\gamma} v$ and $u\leq_{\gamma}v$ the 
	join $u_{1}=s_{1}\vee_{\gamma} u$ exists and lies in $[u,v]_{\gamma}$. Lemma~\ref{lem:joincov} implies 
	that $u\lessdot_{\gamma} u_{1}$. Consider the interval $[u_1,v]_{\gamma}$. Then 
	$s_{1}\leq_{\gamma} u_1$ and analogously to (2a) we can find a unique maximal rising chain 
	$c':u_1=x_1\lessdot_{\gamma} x_2\lessdot_{\gamma}\cdots\lessdot_{\gamma}x_t=v$ in 
	$[u_1,v]_{\gamma}$ which is lexicographically first. Moreover, 
	$\min\{i\mid i\in\alpha_{\gamma}(v)\smallsetminus\alpha_{\gamma}(u_1)\}>1$, since 
	$s_{1}\leq_{\gamma}u_{1}\leq_{\gamma}v$. By definition of our labeling, the label $1$ cannot 
	appear as a label in any chain in the interval $[u_{1},v]_{\gamma}$. 
	On the other hand, it follows from Lemma~\ref{lem:labeling_recursion} that 
	$\lambda_{\gamma}(u,u_1)=1$. Thus, the chain $c:u=x_0\lessdot_{\gamma}x_1
	\lessdot_{\gamma}x_2\lessdot_{\gamma}\cdots\lessdot_{\gamma}x_t=v$ is maximal
	and rising in $[u,v]_{\gamma}$. Suppose that there is another element $u'$ that covers $u$ 
	in $[u,v]_{\gamma}$ such that $\lambda_{\gamma}(u,u')=1$. Then, by definition of 
	$\lambda_{\gamma}$, it follows that $s_{1}$ appears in the $\gamma$-sorting word of $u'$. In 
	particular, since $s_{1}$ is initial in $\gamma$, we deduce that $s_{1}\leq_{\gamma} u'$. Therefore 
	$u'$ is above both $s_{1}$ and $u$ in $\mathcal{C}_{\gamma}$. By the uniqueness of joins and the 
	definition of $u_1$ it follows that $u_1=u'$. Thus, $c$ is the lexicographically 
	smallest maximal chain in $[u,v]_{\gamma}$. Finally, Lemma~\ref{lem:i_0} implies that $c$ is the 
	unique maximal rising chain.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:infinite_shellability}]
	This follows by definition from Theorem~\ref{thm:el_labeling}. 
\end{proof}

\begin{remark}
	In the case where $W$ is finite and crystallographic, Ingalls and Thomas have shown in 
	\cite{ingalls09noncrossing}*{Theorem~4.17} that $\mathcal{C}_{\gamma}$ is trim. Trimness is a lattice 
	property that generalizes distributivity to ungraded lattices. Then, by definition of trimness, it follows 
	that $\mathcal{C}_{\gamma}$ is left-modular, meaning that there exists a maximal chain 
	$c:x_{1}\lessdot_{\gamma} x_{2}\lessdot_{\gamma}\cdots\lessdot_{\gamma} x_{n}$ satisfying
	$(y\vee_{\gamma} x_{i})\wedge_{\gamma} z = y\vee_{\gamma}(x_{i}\wedge_{\gamma}z)$,
	for all $y<_{\gamma}z$ and $i\in\{1,2,\ldots,n\}$. According to \cite{liu99left}, this property
	yields another EL-labeling of $\mathcal{C}_{\gamma}$, defined by
	\begin{displaymath}
		\xi(y,z)=\min\{i\mid y\vee_{\gamma} x_{i}\wedge_{\gamma} z=z\},
	\end{displaymath}
	for all $y,z\in L$ with $y\lessdot_{\gamma}z$. It is not hard to show that this labeling is 
	structurally different from our labeling. (The difference between the two labelings can already be observed
	in the Cambrian lattice depicted in Figure~\ref{fig:camb_a3}.)
\end{remark}

\begin{remark}
  \label{rem:tree}
	In the case where $W$ is finite and $\gamma=s_{1}s_{2}\cdots s_{n}$, \cite{reading07clusters}*{Remark~2.1} 
	states that the $\gamma$-sortable elements constitute a spanning tree of the Hasse diagram of 
	$\mathcal{C}_{\gamma}$, which is rooted at the identity. The edges of this spanning tree correspond to 
	covering relations $u\lessdot_{\gamma}v$ in $\mathcal{C}_{\gamma}$ such that the $\gamma$-sorting word of $u$ 
	is a prefix of the $\gamma$-sorting word of $v$. This spanning tree is 
	related to the labeling $\lambda_{\gamma}$ in the following way: let $w\in W$, with $\ell_{S}(w)=k$, 
	and let $(i_{0},i_{1},\ldots,i_{k-1})$ be the sequence of edge-labels of the unique rising chain in 
	$[\varepsilon,w]_{\gamma}$. In view of Theorem~\ref{thm:el_labeling}, and 
	\cite{reading07clusters}*{Remark~2.1}, we notice that the unique path from $\varepsilon$ to $w$ in 
	the spanning tree of $\mathcal{C}_{\gamma}$ corresponds to the unique rising chain in 
	$[\varepsilon,w]_{\gamma}$. Hence, the $\gamma$-sorting word of $w$ is 
	$s_{i_{0}}s_{i_{1}}\cdots s_{i_{k-1}}$, where $s_{i_{j}}$ is the $i_{j}$-th letter of 
	$\gamma^{\infty}$, and the length of the rising chain in $[\varepsilon,w]_{\gamma}$ is
	precisely $\ell_{S}(w)$. Moreover, it follows from the proof of Theorem~\ref{thm:el_labeling} 
	that the length of the unique rising chain in an interval $[u,v]_{\gamma}$ equals 
	$\ell_S(v)-\ell_S(u)$. 
	
	In view of Theorem~\ref{thm:el_labeling}, we can carry out the same construction even in the case of 
	infinite Coxeter groups. 
\end{remark}

\section{Applications}
  \label{sec:applications}
In \cite{reading05lattice}, Reading investigated, among other things, the topological properties of open
intervals in so-called \emph{fan posets}. A fan poset is a certain partial order defined on the maximal 
cones of a complete fan of regions of a real hyperplane arrangement. For a finite Coxeter group $W$ and a
Cambrian congruence $\theta$, the \emph{Cambrian fan $\mathcal{F}_{\theta}$} is the complete fan induced by 
certain cones in the Coxeter arrangement $\mathcal{A}_{W}$ of $W$. More precisely, each such cone is a 
union of regions of $\mathcal{A}_{W}$ which correspond to elements of $W$ lying in the same congruence 
class of $\theta$. It is the assertion of \cite{reading05lattice}*{Theorem~1.1} that a Cambrian lattice of 
$W$ is the fan poset associated to the corresponding Cambrian fan. The following theorem is a concatenation
of \cite{reading05lattice}*{Theorem~1.1} and \cite{reading05lattice}*{Propositions~5.6~and~5.7}. In fact, 
Propositions~5.6 and 5.7 in \cite{reading05lattice} imply this result for a much larger class of fan posets. 

\begin{theorem}
  \label{thm:finite_topology}
	Let $W$ be a finite Coxeter group and let $\gamma\in W$ be a Coxeter element. Every open interval in 
	the Cambrian lattice $\mathcal{C}_{\gamma}$ is either contractible or spherical.
\end{theorem}

It is well-known that the reduced Euler characteristic of the order complex
of an open interval $(x,y)$ in a poset equals $\mu(x,y)$, see for instance
\cite{stanley97enumerative}*{Proposition~3.8.6}. Hence, it follows immediately from
Theorem~\ref{thm:finite_topology} that for $\gamma$-sortable elements $x$ and $y$ in a finite Coxeter 
group $W$ satisfying $x\leq_{\gamma} y$, we have $\lvert\mu(x,y)\rvert\leq 1$, as was already remarked in 
\cite{reading06cambrian}*{pp.\;4-5}. In light of Proposition~\ref{prop:mobius} and 
Theorem~\ref{thm:el_labeling}, we can extend this statement to compute the M\"obius function of closed 
intervals in the Cambrian semilattice $\mathcal{C}_{\gamma}$, by counting the falling maximal chains with respect to 
the labeling defined in \eqref{eq:labeling}, as our next theorem shows.

\begin{theorem}
  \label{thm:cambrian_mobius}
	Let $W$ be a (possibly infinite) Coxeter group and $\gamma\in W$ a Coxeter element. For 
	$u,v\in C_{\gamma}$ with $u\leq_{\gamma} v$, we have $|\mu(u,v)|\leq 1$. 
\end{theorem}
\begin{proof}
	In view of Proposition~\ref{prop:mobius} it is enough to show that the interval $[u,v]_{\gamma}$ 
	has at most one maximal falling chain. We use similar arguments as in 
	the proof of Theorem~\ref{thm:el_labeling} and proceed by induction on length and rank. Again, 
	we may assume that $\ell_{S}(v)=k\geq 3$ and that $W$ is a Coxeter group of rank $n\geq 2$, since 
	the result is trivial otherwise. Suppose that the induction hypothesis is true for all parabolic 
	subgroups of $W$ with rank $<n$ and suppose that for every closed interval $[u',v']_{\gamma}$ of 
	$\mathcal{C}_{\gamma}$ with $\ell_S(v')<k$, there exists at most one falling maximal chain. We will show that
	there is at most one maximal falling chain in the interval $[u,v]_{\gamma}$ as well. 
	Let $\gamma=s_{1}s_{2}\cdots s_{n}$. We distinguish two cases: (1) $s_{1}\not\leq_{\gamma} v$ and (2) 
	$s_{1}\leq_{\gamma} v$. 

	(1) The result follows directly by induction on the rank of $W$ by following the steps 
	of case (1) in the proof of Theorem~\ref{thm:el_labeling}.

	(2a) Suppose in addition that $s_{1}\leq_{\gamma} u$. The result follows directly by induction on the 
	length of $v$ by following the steps of case (2a) in the proof of Theorem~\ref{thm:el_labeling}.
	
	(2b) Suppose now that $s_{1}\not\leq_{\gamma} u$. Thus, it follows from Lemma~\ref{lem:i_0} that the 
	label $1$ occurs in every maximal chain from $u$ to $v$, and that a maximal 
	chain $u=x_0\lessdot_{\gamma}x_1\lessdot_{\gamma}\cdots\lessdot_{\gamma}x_{t-1}\lessdot x_t=v$ of 
	$[u,v]_{\gamma}$ can be falling only if $\lambda_{\gamma}(x_{t-1},v)=1$. Hence, if there is no 
	element $v_1\in (u,v)_{\gamma}$, with $v_1\lessdot v$ satisfying $\lambda_{\gamma}(v_1,v)=1$, then 
	the interval $[u,v]_{\gamma}$ has no maximal falling chain, which means that $\mu(u,v)=0$. Otherwise, 
	consider the interval $[u,v_1]_{\gamma}$. By the choice of $v_1$, it follows that every maximal 
	falling chain in $[u,v_{1}]_{\gamma}$ can be extended to a maximal falling chain in the interval 
	$[u,v]_{\gamma}$. Conversely, every maximal falling chain in $[u,v]_{\gamma}$ can be restricted to a 
	maximal falling chain in $[u,v_{1}]_{\gamma}$. Therefore, since $\ell_S(v_1)<\ell_S(v)$, we deduce
	from the induction hypothesis that the interval $[u,v_1]_{\gamma}$ has at most one maximal falling 
	chain. Thus $|\mu(u,v)|\leq 1$. 
\end{proof}

In addition Propositions~5.6 and 5.7 in \cite{reading05lattice} characterize the open 
intervals in a (finite) Cambrian lattice which are contractible, and those which are spherical in the 
following way: an interval $[u,v]_{\gamma}$ in $\mathcal{C}_{\gamma}$ is called \emph{nuclear} if the 
join of the upper covers of $u$ (in $[u,v]_{\gamma}$) is precisely $v$. Reading showed that the nuclear intervals are 
precisely the spherical intervals. With the help of our labeling, we can generalize this characterization
to infinite Coxeter groups. 

\begin{theorem}
  \label{thm:mobius_nuclear}
	Let $u,v\in C_{\gamma}$ with $u\leq_{\gamma}v$ and let $k$ denote the number of atoms of the interval 
	$[u,v]_{\gamma}$. Then, $\mu(u,v)=(-1)^{k}$ if and only if $[u,v]_{\gamma}$ is nuclear.
\end{theorem}

For the proof of Theorem~\ref{thm:mobius_nuclear}, we need the following lemma.

\begin{lemma}
  \label{lem:nuclear_sub}
	Let $\gamma=s_{1}s_{2}\cdots s_{n}$, and let $u,v\in C_{\gamma}$ with $u\leq_{\gamma}v$. 
	Suppose further that $s_{1}\not\leq_{\gamma}u$, while $s_{1}\leq_{\gamma}v$. 
	Then the following are equivalent:
	\begin{enumerate}
	 \item The interval $[u,v]_{\gamma}$ is nuclear.
	 \item There exists an element $v'\in [u,v]_{\gamma}$ satisfying 
	 $s_{1}\not\leq_{\gamma}v'\lessdot_{\gamma}v$, and the interval $[u,v']_{\gamma}$ is nuclear.
	\end{enumerate}
\end{lemma}
\begin{proof}
	Let $A=\{w\in C_{\gamma}\mid u\lessdot_{\gamma}w\leq_{\gamma}v\}$ be the set of
	atoms of the interval $[u,v]_{\gamma}$. Since $s_{1}\leq_{\gamma}v$ and $u\leq_{\gamma}v$, we conclude from 
	Theorem~\ref{thm:joins_meets} that the join $s_{1}\vee_{\gamma}u$ exists, and we set $z=s_{1}\vee_{\gamma}u$. 
	It follows from Lemma~\ref{lem:joincov} that $u\lessdot_{\gamma}z$, and hence $z\in A$.
	We set $A_z=A\setminus\{z\}$ and remark that if $w\in A_z$, then $s_{1}\not\leq_{\gamma} w$. 
	(Indeed, suppose that there exists some $z'\in A_z$ with $s_{1}\leq_{\gamma}z'$. Since $u\lessdot_{\gamma}z'$, 
	this implies $s_{1}\vee_{\gamma}u\leq_{\gamma}z'$, and hence $z\leq_{\gamma} z'$. Since $z$ and $z'$ 
	both cover $u$, this implies $z=z'$, which contradicts $z\notin A_z$.) In particular we have 
	$A_z\subseteq W_{\langle s_{1}\rangle}$. (Indeed, suppose that
	there is some $z'\in A_{z}$ with $\ell_{S}(z')=k$ and $z'\notin w_{\langle s_{1}\rangle}$. This means that the 
	$\gamma$-sorting word of $z'$ contains the letter $s_{1}$ at least once. Since $z'\in C_{\gamma}$ 
	this means that $s_{1}$ must occur in the first position of the $\gamma$-sorting word of $z'$. Then
	$\ell_{S}(z')=k=(k-1)+1=\ell_{S}(s_{1}z')+\ell_{S}(s_{1})$ which implies $s_{1}\leq_{S}z'$, and with 
	\eqref{eq:order} follows $s_{1}\leq_{\gamma}z'$ which is a contradiction.)
	
	(1)$\Rightarrow$(2) Suppose that $[u,v]_{\gamma}$ is nuclear and let $v'=\bigvee_{\gamma} A_z$. 
	Again, Theorem~\ref{thm:joins_meets} ensures that $v'$ exists and that it satisfies 
	$u\leq_{\gamma}v'\leq_{\gamma}v$. Since $A_z\subseteq W_{\langle s_{1}\rangle}$, it follows from 
	Proposition~\ref{prop:commuting_operations} that $v'=\bigvee_{\gamma} A_{z}\in W_{\langle s_{1}\rangle}$
	which means that $s_{1}\not\leq_{\gamma} v'$, and $A_z$ is thus the set of atoms 
	of the interval $[u,v']_{\gamma}$. Hence, $[u,v']_{\gamma}$ is nuclear.
	It remains to show that $v'\lessdot_{\gamma}v$. It follows from $u\leq_{\gamma}v'$ and the 
	associativity of $\vee_{\gamma}$ that
	\begin{displaymath}
		v={\bigvee}_{\!\gamma} A=z\vee_{\gamma}\left({\bigvee}_{\!\gamma} A_{z}\right)=z\vee_{\gamma}v'
		  =(s_{1}\vee_{\gamma}u)\vee_{\gamma}v'=s_{1}\vee_{\gamma}(u\vee_{\gamma}v')=s_{1}\vee_{\gamma}v'.
	\end{displaymath}
	From above, we know that $s_{1}\not\leq_{\gamma}v'$ and we can apply Lemma~\ref{lem:joincov} which 
	implies immediately that $v'\lessdot_{\gamma}s_{1}\vee_{\gamma}v'=v$. 

	(2)$\Rightarrow$(1) Suppose now that there exists an element $v'\in [u,v]_{\gamma}$ satisfying 
	$s_{1}\not\leq_{\gamma}v'\lessdot_{\gamma}v$, and suppose that the interval $[u,v']_{\gamma}$ is nuclear. 
	Let $A'$ denote the set of atoms of $[u,v']_{\gamma}$. Since $s_{1}\not\leq_{\gamma}v'$ and 
	$s_{1}\leq_{\gamma} z$, it follows that $z\notin A'$, thus $A'\subseteq A_z$. Furthermore, from 
	$s_{1}\leq_{\gamma}v$, from $v'\lessdot_{\gamma}v$, and from Lemma~\ref{lem:joincov}, we deduce that 
	$s_{1}\vee_{\gamma}v'=v$. Now we have 
	\begin{displaymath}
		z\vee_{\gamma}v'=(s_{1}\vee_{\gamma}u)\vee_{\gamma}v'
		  =s_{1}\vee_{\gamma}(u\vee_{\gamma}v')=s_{1}\vee_{\gamma}v'=v,
	\end{displaymath}
	since $u\leq_{\gamma}v'$. Thus, we can write $v=\bigvee_{\gamma}\bigl(A'\cup\{z\}\bigr)$. Finally, we 
	show that $v=\bigvee_{\gamma} A$. Let $z'\in A\setminus A'$. Since $z'\leq_{\gamma}v$, it follows that 
	\begin{displaymath}
		{\bigvee}_{\!\gamma}\bigl(A'\cup\{z,z'\}\bigr)
		  ={\bigvee}_{\!\gamma}\bigl(A'\cup\{z\}\bigr)\vee_{\gamma}z'
		  =v\vee_{\gamma}z'=v,
	\end{displaymath}
	and hence $v=\bigvee_{\gamma} A$. This implies that $[u,v]_{\gamma}$ is nuclear. 
\end{proof}

We remark that under the hypothesis of Lemma~\ref{lem:nuclear_sub}, the element $v'=\bigvee A_z$ constructed in 
the part (1)$\Rightarrow$(2) of the proof is the unique element in $[u,v]_{\gamma}$ satisfying condition (2). 
The uniqueness of $v'$ is a consequence of the uniqueness of the join $\bigvee_{\gamma} A_z$. 


\begin{proof}[Proof of Theorem~\ref{thm:mobius_nuclear}]
	In view of Proposition~\ref{prop:mobius}, we need to show that $[u,v]_{\gamma}$ has a falling chain 
	if and only if $[u,v]_{\gamma}$ is nuclear. We use similar arguments as in the proof of 
	Theorem~\ref{thm:el_labeling} and proceed by induction on length and rank. Again we may assume that
	$\ell_{S}(v)=k\geq 3$ and that $W$ is a Coxeter group of rank $n\geq 2$, since the result is trivial
	otherwise. Suppose that the induction hypothesis is true for all parabolic subgroups of $W$ with 
	rank $<n$ and suppose that for every closed interval $[u',v']_{\gamma}$ of $\mathcal{C}_{\gamma}$ with 
	$\ell_{S}(v')<k$ there exists a falling maximal chain if and only if $[u',v']_{\gamma}$ is nuclear.
	Let $\gamma=s_{1}s_{2}\cdots s_{n}$. We distinguish two cases: (1) $s_{1}\not\leq_{\gamma}v$ 
	and (2) $s_{1}\leq_{\gamma}v$.
	
	(1) The result follows directly by induction on the rank of $W$ by following the steps of case 
	(1) in the proof of Theorem~\ref{thm:cambrian_mobius}. 

	(2a) Suppose in addition that $s_{1}\leq_{\gamma}u$. The result follows directly by induction on the 
	length of $v$ by following the steps of case (2a) in the proof of Theorem~\ref{thm:cambrian_mobius}.
	
	(2b) Suppose now that $s_{1}\not\leq_{\gamma}u$. If $[u,v]_{\gamma}$ is nuclear, then
	Lemma~\ref{lem:nuclear_sub} implies that there exists a unique element $v'\in \mathcal{C}_{\gamma}$ with
	$u\leq_{\gamma}v'\lessdot_{\gamma}v$ such that $[u,v']_{\gamma}$ is nuclear, and 
	$s_{1}\not\leq_{\gamma}v'$. Thus, we can apply induction on the rank of $W$ and obtain a maximal falling 
	chain $c':u=x_{0}\lessdot_{\gamma}x_{1}\lessdot_{\gamma}\cdots\lessdot_{\gamma}x_{t-1}=v'$. 
	Lemma~\ref{lem:i_0} implies that $1\notin\lambda_{\gamma}(c')$, and 
	Lemma~\ref{lem:labeling_recursion} implies that 
	$\lambda_{\gamma}(v',v)=1$. Thus, the chain $c:u=x_{0}\lessdot_{\gamma}x_{1}\lessdot_{\gamma}\cdots
	\lessdot_{\gamma}x_{t-1}\lessdot_{\gamma}x_{t}=v$ is a falling maximal chain in $[u,v]_{\gamma}$, 
	and Theorem~\ref{thm:cambrian_mobius} implies its uniqueness.

	Conversely, suppose that there exists a maximal falling chain 
	$c:u=x_{0}\lessdot_{\gamma}x_{1}\lessdot_{\gamma}\cdots\lessdot_{\gamma}x_{t}=v$ in $[u,v]_{\gamma}$. In view 
	of Lemma~\ref{lem:i_0}, we notice that $\lambda_{\gamma}(x_{t-1},v)=1$, which implies 
	$s_{1}\not\leq_{\gamma}x_{t-1}$. Clearly $\ell_{S}(x_{t-1})<k$ and the chain 
	$c':u=x_{0}\lessdot_{\gamma}x_{1}\lessdot_{\gamma}\cdots\lessdot_{\gamma}x_{t-1}$ is falling, 
	thus by induction we can conclude that the interval $[u,x_{t-1}]_{\gamma}$ is nuclear. 
	Since $s_{1}\not\leq_{\gamma}x_{t-1}\lessdot_{\gamma} v$, it follows from Lemma~\ref{lem:nuclear_sub} 
	that $[u,v]_{\gamma}$ is nuclear. This completes the proof of the theorem. 
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:cambrian_topology}]
	Theorem~\ref{thm:infinite_shellability} implies that every closed interval $[u,v]_{\gamma}$ of 
	$\mathcal{C}_{\gamma}$ is EL-shellable. Theorem~5.9 in \cite{bjorner96shellable} states that the dimension 
	of the $i$-th homology group of the order complex of $(u,v)_{\gamma}$ corresponds to the number of 
	falling chains in $[u,v]_{\gamma}$ having length $i+2$. Theorem~\ref{thm:cambrian_mobius} implies 
	that there is at most one falling chain in $[u,v]_{\gamma}$. Hence, either all homology groups of 
	the order complex of $(u,v)_{\gamma}$ have dimension $0$ (then, $(u,v)_{\gamma}$ is contractible) or 
	there exists exactly one homology group of dimension $1$ (then, $(u,v)_{\gamma}$ is spherical). 
	Finally, the characterization of the spherical intervals  is an immediate consequence of 
	Theorem~\ref{thm:mobius_nuclear}. 
\end{proof}

\begin{remark}
	In the case of finite Coxeter groups, the statements of Theorems~\ref{thm:infinite_shellability} and
	\ref{thm:cambrian_topology} can be generalized straightforwardly to the increasing flip order of subword complexes
	for so-called \emph{realizing words}, as was pointed out to us by Stump (private conversation). In
	\cite{pilaud12brick}*{Section~5.3}, Pilaud and Stump define an acyclic, 
	directed, edge-labeled graph on the facets of the subword complex, the so-called 
	\emph{increasing flip graph}. The transitive closure of this graph is then a partial order, the 
	\emph{increasing flip order}. In the case of realizing words, the Hasse diagram of the increasing 
	flip order coincides with the increasing flip graph which then yields two natural edge-labelings of this 
	poset. One can show that these labelings are indeed EL-labelings and that every interval has at most one 
	falling chain with respect to either of these labelings. This has recently been done in \cite{pilaud12el}.
	
	It is the statement of \cite{pilaud12brick}*{Corollary~6.31} that the Cambrian lattices of finite
	Coxeter groups correspond to the increasing flip order of special subword complexes. In addition, 
	the construction of \cite{pilaud12brick} as briefly described in the previous paragraph provides a 
	nice geometric description of the statements of Theorems~\ref{thm:infinite_shellability} and
	\ref{thm:cambrian_topology}. We remark that both EL-labelings of Pilaud and Stump are structurally different 
	from our labeling. This is pointed out in \cite{pilaud12el}*{Example~5.12} for the so-called positive labeling, 
	and it can be seen for the so-called negative labeling in the Cambrian lattice of the symmetric group 
	$\mathfrak{S}_{4}$ with respect to the Coxeter element $\gamma=(2\;3)(1\;2)(3\;4)$.
\end{remark}

We conclude this section with a short example of an infinite Coxeter group.

\begin{example}
	Consider the affine Coxeter group $\tilde{A}_{2}$, which is generated by the set
	$\{s_{1},s_{2},s_{3}\}$ satisfying $(s_{1}s_{2})^{3}=(s_{1}s_{3})^{3}=(s_{2}s_{3})^{3}=\varepsilon$,
	as well as $s_{1}^{2}=s_{2}^{2}=s_{3}^{2}=\varepsilon$. Consider the Coxeter element 
	$\gamma=s_{1}s_{2}s_{3}$. Figure~\ref{fig:camb_affine_a2} shows the sub-semilattice of the 
	Cambrian semilattice $\mathcal{C}_{\gamma}$ consisting of all $\gamma$-sortable elements of $\tilde{A}_{2}$ of 
	length $\leq 7$. We encourage the reader to verify Theorem~\ref{thm:el_labeling} and 
	Theorem~\ref{thm:cambrian_mobius}. 
	
	\begin{figure}
		\centering
		\begin{tikzpicture}\scriptsize
			\def\x{1};
			\def\y{1.5};
			%%
			\draw(4*\x,1*\y) node(v1){$\varepsilon$};
			\draw(1*\x,2*\y) node(v2){$s_{3}$};
			\draw(5*\x,2*\y) node(v3){$s_{1}$};
			\draw(8*\x,2*\y) node(v4){$s_{2}$};
			\draw(2*\x,3*\y) node(v5){$s_{1}s_{3}$};
			\draw(6.25*\x,3*\y) node(v6){$s_{2}s_{3}$};
			\draw(7.25*\x,3*\y) node(v7){$s_{1}s_{2}$};
			\draw(.5*\x,4*\y) node(v8){$s_{1}s_{3}\vert s_{1}$};
			\draw(4*\x,4*\y) node(v9){$s_{2}s_{3}\vert s_{2}$};
			\draw(6*\x,4*\y) node(v10){$s_{1}s_{2}s_{3}$};
			\draw(9.5*\x,4*\y) node(v11){$s_{1}s_{2}\vert s_{1}$};
			\draw(2.5*\x,5*\y) node(v12){$s_{1}s_{2}s_{3}\vert s_{2}$};
			\draw(7.5*\x,5*\y) node(v13){$s_{1}s_{2}s_{3}\vert s_{1}$};
			\draw(6*\x,6*\y) node(v14){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}$};
			\draw(9.5*\x,6*\y) node(v15){$s_{1}s_{2}s_{3}\vert s_{1}s_{3}$};
			\draw(4*\x,7*\y) node(v16){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}\vert s_{1}$};
			\draw(7*\x,7*\y) node(v17){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}s_{3}$};
			\draw(5*\x,8*\y) node(v18){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}s_{3}\vert s_{1}$};
			\draw(8*\x,8*\y) node(v19){$s_{1}s_{2}s_{3}\vert s_{1}s_{2}s_{3}\vert s_{2}$};
			%%
			\draw(v1) -- (v2) node[fill=white] at (2.5*\x,1.5*\y){\tiny $3$};
			\draw(v1) -- (v3) node[fill=white] at (4.5*\x,1.5*\y){\tiny $1$};
			\draw(v1) -- (v4) node[fill=white] at (6*\x,1.5*\y){\tiny $2$};
			\draw(v2) -- (v8) node[fill=white] at (.75*\x,3*\y){\tiny $1$};
			\draw(v2) -- (v9) node[fill=white] at (2.9*\x,3.25*\y){\tiny $2$};
			\draw(v3) -- (v5) node[fill=white] at (3.5*\x,2.5*\y){\tiny $3$};
			\draw(v3) -- (v7) node[fill=white] at (6.125*\x,2.5*\y){\tiny $2$};
			\draw(v4) -- (v6) node[fill=white] at (7.125*\x,2.5*\y){\tiny $3$};
			\draw(v4) -- (v11) node[fill=white] at (8.75*\x,3*\y){\tiny $1$};
			\draw(v5) -- (v8) node[fill=white] at (1.25*\x,3.5*\y){\tiny $4$};
			\draw(v5) -- (v12) node[fill=white] at (2.25*\x,4*\y){\tiny $2$};
			\draw(v6) -- (v9) node[fill=white] at (5.125*\x,3.5*\y){\tiny $5$};
			\draw(v7) -- (v10) node[fill=white] at (6.625*\x,3.5*\y){\tiny $3$};
			\draw(v7) -- (v11) node[fill=white] at (8.375*\x,3.5*\y){\tiny $4$};
			\draw(v10) -- (v12) node[fill=white] at (4.25*\x,4.5*\y){\tiny $5$};
			\draw(v10) -- (v13) node[fill=white] at (6.75*\x,4.5*\y){\tiny $4$};
			\draw(v11) -- (v15) node[fill=white] at (9.5*\x,5*\y){\tiny $3$};
			\draw(v12) -- (v16) node[fill=white] at (3.25*\x,6*\y){\tiny $4$};
			\draw(v13) -- (v14) node[fill=white] at (6.75*\x,5.5*\y){\tiny $5$};
			\draw(v13) -- (v15) node[fill=white] at (8.5*\x,5.5*\y){\tiny $6$};
			\draw(v14) -- (v16) node[fill=white] at (5*\x,6.5*\y){\tiny $7$};
			\draw(v14) -- (v17) node[fill=white] at (6.5*\x,6.5*\y){\tiny $6$};
			\draw(v15) -- (v19) node[fill=white] at (8.75*\x,7*\y){\tiny $5$};
			\draw(v17) -- (v18) node[fill=white] at (6*\x,7.5*\y){\tiny $7$};
			\draw(v17) -- (v19) node[fill=white] at (7.5*\x,7.5*\y){\tiny $8$};
		\end{tikzpicture}
		\caption{The first seven ranks of an $\tilde{A}_{2}$-Cambrian semilattice, with the labeling
		  as defined in \eqref{eq:labeling}.}
		\label{fig:camb_affine_a2}
	\end{figure}
\end{example}

\section*{Acknowledgements}
The authors would like to thank Nathan Reading for pointing out a weakness in a previous version of the proof
of Theorem~\ref{thm:el_labeling} and for many helpful discussions on the topic. Further thanks go to Hugh Thomas
and Christian Stump for helpful comments on the overlap of our work. We are very grateful to an anonymous referee 
for the careful reading and for many valuable comments on presentation and content of the article. 

%\bibliography{literature}

% \bib, bibdiv, biblist are defined by the amsrefs package.
\begin{bibdiv}
\begin{biblist}

\bib{bjorner80shellable}{article}{
      author={Bj{\"o}rner, Anders},
       title={{Shellable and Cohen-Macaulay Partially Ordered Sets}},
        date={1980},
     journal={Transactions of the American Mathematical Society},
      volume={260},
       pages={159\ndash 183},
}

\bib{bjorner96topological}{collection}{
      author={Bj{\"o}rner, Anders},
       title={{Topological Methods}},
      series={Handbook of Combinatorics},
   publisher={MIT Press},
     address={Cambridge, MA},
        date={1996},
      volume={2},
}

\bib{bjorner05combinatorics}{book}{
      author={Bj{\"o}rner, Anders},
      author={Brenti, Francesco},
       title={{Combinatorics of Coxeter Groups}},
   publisher={Springer},
     address={New York},
        date={2005},
}

\bib{bjorner83lexicographically}{article}{
      author={Bj{\"o}rner, Anders},
      author={Wachs, Michelle~L.},
       title={{On Lexicographically Shellable Posets}},
        date={1983},
     journal={Transactions of the American Mathematical Society},
      volume={277},
       pages={323\ndash 341},
}

\bib{bjorner96shellable}{article}{
      author={Bj{\"o}rner, Anders},
      author={Wachs, Michelle~L.},
       title={{Shellable and Nonpure Complexes and Posets I}},
        date={1996},
     journal={Transactions of the American Mathematical Society},
      volume={348},
       pages={1299\ndash 1327},
}

\bib{bjorner97shellable}{article}{
      author={Bj{\"o}rner, Anders},
      author={Wachs, Michelle~L.},
       title={{Shellable and Nonpure Complexes and Posets II}},
        date={1997},
     journal={Transactions of the American Mathematical Society},
      volume={349},
       pages={3945\ndash 3975},
}

\bib{bott94on}{article}{
      author={Bott, Raoul},
      author={Taubes, Clifford~H.},
       title={{On the Self-Linking of Knots}},
        date={1994},
     journal={Journal of Mathematical Physics},
      volume={35},
       pages={5247\ndash 5287},
}

\bib{chapoton02polytopal}{article}{
      author={Chapoton, Fr{\'e}d{\'e}ric},
      author={Fomin, Sergey},
      author={Zelevinsky, Andrei},
       title={{Polytopal Realizations of Generalized Associahedra}},
        date={2002},
     journal={Canadian Mathematical Bulletin},
      volume={45},
       pages={537\ndash 566},
}

\bib{davey02introduction}{book}{
      author={Davey, Brian~A.},
      author={Priestley, Hilary~A.},
       title={{Introduction to Lattices and Order}},
   publisher={Cambridge University Press},
     address={Cambridge},
        date={2002},
}

\bib{fomin03systems}{article}{
      author={Fomin, Sergey},
      author={Zelevinsky, Andrei},
       title={{$Y$-Systems and Generalized Associahedra}},
        date={2003},
     journal={Annals of Mathematics},
      volume={158},
       pages={977\ndash 1018},
}

\bib{humphreys90reflection}{book}{
      author={Humphreys, James~E.},
       title={{Reflection Groups and Coxeter Groups}},
   publisher={Cambridge University Press},
     address={Cambridge},
        date={1990},
}

\bib{ingalls09noncrossing}{article}{
      author={Ingalls, Colin},
      author={Thomas, Hugh},
       title={{Noncrossing Partitions and Representations of Quivers}},
        date={2009},
     journal={Composition Mathematica},
      volume={145},
       pages={1533\ndash 1562},
}

\bib{jedlicka05combinatorial}{article}{
      author={Jedli{\v{c}}ka, P{\v{r}}emsyl},
       title={{A Combinatorial Construction of the Weak Order of a Coxeter
  Group}},
        date={2005},
     journal={Communications in Algebra},
      volume={33},
       pages={1447\ndash 1460},
}

\bib{liu99left}{thesis}{
      author={Liu, Larry Shu-Chung},
       title={{Left-Modular Elements and Edge-Labellings}},
        type={Ph.D. Thesis},
        date={1999},
}

\bib{loday04realization}{article}{
      author={Loday, Jean-Louis},
       title={{Realization of the Stasheff Polytope}},
        date={2004},
     journal={Archiv der Mathematik},
      volume={83},
       pages={267\ndash 278},
}

\bib{pilaud12brick}{article}{
      author={Pilaud, Vincent},
      author={Stump, Christian},
       title={{Brick Polytopes of Spherical Subword Complexes: A new Approach
  to Generalized Associahedra}},
        date={2012},
      eprint={arXiv:1111.3349},
}

\bib{pilaud12el}{article}{
      author={Pilaud, Vincent},
      author={Stump, Christian},
       title={{EL-Labelings and Canonical Spanning Trees for Subword
  Complexes}},
        date={2012},
      eprint={arXiv:1210.1435},
}

\bib{reading05lattice}{article}{
      author={Reading, Nathan},
       title={{Lattice Congruences, Fans and Hopf Algebras}},
        date={2005},
     journal={Journal of Combinatorial Theory (Series A)},
      volume={110},
       pages={237\ndash 273},
}

\bib{reading06cambrian}{article}{
      author={Reading, Nathan},
       title={{Cambrian Lattices}},
        date={2006},
     journal={Advances in Mathematics},
      volume={205},
       pages={313\ndash 353},
}

\bib{reading07clusters}{article}{
      author={Reading, Nathan},
       title={{Clusters, Coxeter-Sortable Elements and Noncrossing
  Partitions}},
        date={2007},
     journal={Transactions of the American Mathematical Society},
      volume={359},
       pages={5931\ndash 5958},
}

\bib{reading07sortable}{article}{
      author={Reading, Nathan},
       title={{Sortable Elements and Cambrian Lattices}},
        date={2007},
     journal={Algebra Universalis},
      volume={56},
       pages={411\ndash 437},
}

\bib{reading09cambrian}{article}{
      author={Reading, Nathan},
      author={Speyer, David~E.},
       title={{Cambrian Fans}},
        date={2009},
     journal={Journal of the European Mathematical Society},
      volume={11},
       pages={407\ndash 447},
}

\bib{reading11sortable}{article}{
      author={Reading, Nathan},
      author={Speyer, David~E.},
       title={{Sortable Elements in Infinite Coxeter Groups}},
        date={2011},
     journal={Transactions of the American Mathematical Society},
      volume={363},
       pages={699\ndash 761},
}

\bib{simion03type}{article}{
      author={Simion, Rodica},
       title={{A Type-$B$ Associahedron}},
        date={2003},
     journal={Advances in Applied Mathematics},
      volume={30},
       pages={2\ndash 25},
}

\bib{stanley97enumerative}{book}{
      author={Stanley, Richard~P.},
       title={{Enumerative Combinatorics, Vol. 1}},
   publisher={Cambridge University Press},
     address={Cambridge},
        date={1997},
}

\bib{stasheff97from}{article}{
      author={Stasheff, James~D.},
       title={{From Operads to ``Physically'' Inspired Theories}},
        date={1997},
     journal={Contemporary Mathematics},
      volume={202},
       pages={53\ndash 81},
}

\bib{tamari62algebra}{article}{
      author={Tamari, Dov},
       title={{The Algebra of Bracketings and their Enumeration}},
        date={1962},
     journal={Nieuw Archief voor Wiskunde},
      volume={10},
       pages={131\ndash 146},
}

\bib{wachs07poset}{collection}{
      author={Wachs, Michelle~L.},
       title={{Poset Topology: Tools and Applications}},
      series={IAS/Park City Mathematics Series},
   publisher={American Mathematical Society},
     address={Providence, RI},
        date={2007},
      volume={13},
}

\end{biblist}
\end{bibdiv}


\end{document}
