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\documentclass[finalversion]{FPSAC2025}
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\usepackage{xargs}
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\usepackage{paralist}
\usepackage{picins}
\usepackage{dsfont}

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\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
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\newcommand{\bb}[1]{\mathbb{#1}} % bb letters
\let\ced\c % cedille
\renewcommand{\cal}[1]{\mathcal{#1}} % cal letters
\newcommand{\f}[1]{\mathfrak{#1}} % frak letters
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% math commands
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\newcommand{\bigset}[2]{\big\{ #1 \;\big|\; #2 \big\}} % big set notation
\newcommand{\Bigset}[2]{\Big\{ #1 \;\Big|\; #2 \Big\}} % Big set notation
\newcommand{\setangle}[2]{\left\langle #1 \;\middle|\; #2 \right\rangle} % set notation
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% operators
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\DeclareMathOperator{\ninv}{ninv} % non-inversions

% others
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\newcommand{\Eg}{\textit{E.g.}~} % exempli gratia
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\newcommand{\viceversa}{\textit{vice versa}} % vice versa
\newcommand{\versus}{\textit{vs.}~} % versus
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\newcommand{\red}[1]{{\color{red} #1}} % red

% lattices
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\newcommand{\join}{\vee} % join
\newcommand{\bigMeet}{\bigwedge} % meet
\newcommand{\bigJoin}{\bigvee} % join
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\newcommandx{\projUp}[1][1={}]{\smash{\pi^\uparrow_{#1}}} % up projection map
\newcommand{\con}{\mathrm{con}} % congruence

% specific
\newcommand{\s}{\b{s}} % s
\newcommandx{\Fan}[1][1=n]{\mathcal{F}(#1)} % fan
\newcommand{\polytope}[1]{\mathds{#1}} % font polytopes
\newcommandx{\Perm}[1][1=d]{\polytope{P}\mathrm{erm}(#1)} % permutahedron
\newcommandx{\Asso}[1][1=d]{\polytope{A}\mathrm{sso}(#1)} % associahedron
\newcommandx{\Zono}[1][1=\s]{\polytope{Z}\mathrm{ono}(#1)} % zonotope
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\newcommandx{\quotientFoam}[1][1=\equiv]{\cal{F}_{#1}} % quotient foam
\newcommandx{\shard}[1][1=\alpha]{\Sigma_{#1}} % shard
\newcommandx{\shardPolytope}[1][1=\alpha]{\polytope{SP}_{#1}} % shard polytope
\newcommandx{\localShardPolytope}[2][1=\alpha, 2=\b{q}]{\polytope{S}_{#1}^{#2}} % local shard polytope
\newcommandx{\quotientope}[1][1=\equiv]{\polytope{Q}_{#1}} % quotientope
\newcommandx{\shardoplex}[1][1=\alpha]{\polytope{S}_{#1}} % shardoplex
\newcommandx{\quotientoplex}[1][1=\equiv]{\polytope{Q}_{#1}} % quotientoplex
\newcommandx{\sTrees}[1][1=\s]{\b{\Gamma}_{#1}} % s-tree grid
\newcommandx{\sTrunks}[1][1=\s]{\b{\Lambda}_{#1}} % s-trunk grid
\newcommand{\altmatch}{\mu} % alternate matching
\newcommandx{\altmatchset}[1][1=\alpha]{\mathcal{M}_{#1}} % set of alternate matchings of an s-arc
\newcommand{\charvect}{\b{\chi}} % characteristic vector
\newcommandx{\tree}[1][1=T]{{\mathsf{#1}}} % tree
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\newcommand{\trunk}{\tree[T]} % trunk
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\newcommand{\closedFiber}[1]{\bar{\polytope{F}}_{#1}} % fiber insertion algorithm

% permutrees
\newcommand{\decoration}{{\b{\delta}}} % decoration
\newcommand{\Bdecoration}{\delta^\textsc{b}} % decoration
\newcommand{\includeSymbol}[1]{\ensuremath{%
	\mathchoice
		{\raisebox{-.7mm}{\includegraphics[height=2.2ex]{#1}}}	
		{\raisebox{-.7mm}{\includegraphics[height=2.2ex]{#1}}}
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		{\raisebox{-.5mm}{\includegraphics[height=1ex]{#1}}}
}}
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\newcommand{\upDownCirc}{\includeSymbol{updown}}
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\graphicspath{{figures/}}

%\usepackage{todonotes}
\newcommand{\vincent}[1]{\todo[color=blue!30]{\rm #1 \\ \hfill --- V.}}

%% define your title in the usual way
\title{The \texorpdfstring{$\s$}{s}-permutahedron and its lattice quotients}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author{\texorpdfstring{Eva Philippe\addressmark{1,2} \and Vincent Pilaud\thanks{\href{mailto:vincent.pilaud@ub.edu}{vincent.pilaud@ub.edu}. Supported by the French--Austrian grant PAGCAP (ANR-21-CE48-0020 \& FWF I 5788), by the Spanish--German project COMPOTE (AEI PCI2024-155081-2 \& DFG 541393733), and by the Spanish grants PID2022-137283NB-C21, 2021 SGR 00697 and CEX2020-001084-M.}\addressmark{2}}{Eva Philippe and Vincent Pilaud}}

%% then use \addressmark to match authors to institutions here
\address{
	\addressmark{1}Sorbonne Université, Paris, France \\ 
	\addressmark{2}Universitat de Barcelona \& Centre de Recerca Matemàtica, Barcelona, Spain
}

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\received{\today}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{
For a tuple~$\s$ of non-negative integers, the $\s$-weak order is a lattice on $\s$-trees, generalizing the weak order on permutations.
We describe its join irreducible elements, its canonical join representations, and its forcing order in terms of combinatorial objects, generalizing the arcs, non-crossing arc diagrams, and subarc order for the weak order.
We then extend the theory of shards and shard polytopes to construct geometric realizations of the $\s$-weak order and all its lattice quotients as polyhedral complexes, generalizing the quotient fans and quotientopes of the weak order.
}

%% put your French abstract here, or comment this out if you don't have one
\resume{
Pour un uplet~$\s$ d'entiers positifs ou nuls, le $\s$-ordre faible est un treillis sur les $\s$-arbres, généralisant l'ordre faible sur les permutations.
Nous décrivons ses éléments sup irréductibles, ses représentations sup canoniques et son ordre de forçage par des objets combinatoires, généralisant les arcs, les diagrammes d'arcs non croisés et l'ordre des sous-arcs pour l'ordre faible.
Nous étendons ensuite la théorie des tessons et des polytopes de tessons pour construire des réalisations géométriques du $\s$-ordre faible et de tous ses treillis quotients sous forme de complexes polyédraux, généralisant les éventails quotients et les quotientopes de l'ordre faible.
}

%% put your keywords here, or comment this out if you don't have them yet
%\keywords{math, maths, mathematics}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{squotientopes.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction}

The structure of permutations and associations of an $n$-element set is a classical topic of algebraic and geometric combinatorics.
In combinatorics, it is encoded by the Cayley graph of permutations under simple transpositions and the rotation graph on binary trees.
In lattice theory, it materializes in the lattice morphism from the weak order on permutations to the Tamari lattice on binary trees.
In polyhedral geometry, it appears in the braid arrangement and the sylvester fan, and their polar permutahedron and associahedron.
See \cite{PilaudSantosZiegler} for a survey on these connections and their influence in mathematics.

This prototype has motivated the study of all lattice congruences of the weak order, pioneered by N.~Reading~\cite{Reading-LatticeCongruences}.
Combinatorially, he provided an elegant combinatorial model for the lattice theory of the weak order in~\cite{Reading-arcDiagrams}.
Namely, the join irreducible permutations are encoded as certain arcs wiggling around the vertical axis, the canonical join representations of the permutations are encoded by non-crossing arc diagrams, and the forcing order on join irreducible permutations is encoded by the subarc order on these arcs.
Geometrically, he showed in~\cite{Reading-HopfAlgebras} that coarsening the braid fan according to the equivalence classes of any congruence of the weak order always yields a complete polyhedral fan.
These quotient fans were shown to be normal fans of so-called quotientopes in~\cite{PilaudSantos-quotientopes}.
Later, the problem was revisited in~\cite{PadrolPilaudRitter} using the theory of shard~polytopes. %, a family of polytopes indexed by the arcs.

%\begin{figure}[b]
%	\capstart
%	\centerline{\includegraphics[scale=.8]{120-weakOrder} \qquad \includegraphics[scale=.8]{120-foam}}
%	\caption{The $(1,2,0)$-permutahedron (left) and the $(1,2,0)$-foam (right).}
%	\label{fig:mainSWeakOrder}
%\end{figure}

In~\cite{CeballosPons-sWeakOrderI,CeballosPons-sWeakOrderII}, C.~Ceballos and V.~Pons introduced the $\s$-weak order on $\s$-trees for an \mbox{$n$-tuple~$\s$} of non-negative integers, generalizing the classical weak order for~$\s \!=\! (1, \dots, 1)$.
An $\s$-tree is a rooted plane tree on~$[n]$, where the node~$i$ has $s_i+1$ children, all either leaves or nodes~$j > i$. % (note that we have changed the conventions of~\cite{CeballosPons-sWeakOrderI} to make inductive arguments more transparent).
These $\s$-trees are ordered by inequalities between their inversion numbers, generalizing the definition of the weak order on permutations by inclusion of their inversion sets.
See \Cref{sec:sWeakOrder}.
%See \Cref{fig:mainSWeakOrder}\,(left).
\cite[Section~1]{CeballosPons-sWeakOrderI} proves that the $\s$-weak order is a lattice, describes its meets and joins, and establishes some lattice properties, in particular congruence uniformity.
\cite[Section~2]{CeballosPons-sWeakOrderI} introduces the \mbox{$\s$-Tamari} lattice, a sublattice of the $\s$-weak order (also a lattice quotient when~$\s$ contains no~$0$), which is isomorphic to the $\nu$-Tamari lattice of~\mbox{\cite{PrevilleRatelleViennot, CeballosPadrolSarmiento-geometryNuTamari}}  for well-chosen $\s$ and~$\nu$.
\cite[Conjecture~3.1.2]{CeballosPons-sWeakOrderII} conjectures that the Hasse diagram of the $\s$-weak order can be realized as an orientation of the skeleton of a polyhedral subdivision of the permutahedron, in the same spirit~\cite{CeballosPadrolSarmiento-geometryNuTamari} realizes the \mbox{$\nu$-Tamari} lattice.
When~$\s$ contains no~$0$, this conjecture was proved in~\cite{DLeonMoralesPhilippeTamayoYip} using flow~polytopes.
%In this paper, we study the $\s$-weak order combining perspectives from lattice theory and polyhedral geometry.

Combining perspectives from combinatorics, lattice theory, polyhedral geometry and tropical geometry, we not only settle this conjecture for any~$\s$ (including with some~$0$), but actually provide geometric realizations of all lattice quotients of the $\s$-weak order. %, generalizing the quotient fans~\cite{Reading-HopfAlgebras} and quotientopes~\mbox{\cite{PilaudSantos-quotientopes,PadrolPilaudRitter}}.

\begin{theorem}
\label{thm:main}
For any tuple $\s$ of non-negative integers (including with some~$0$), and any lattice congruence~$\equiv$ of the $\s$-weak order~$W_\s$, the lattice quotient~$W_\s/{\equiv}$ is realized geometrically as
\begin{compactitem}
\item the dual graph of the quotient foam~$\quotientFoam$, a polyhedral complex extending quotient fans~\cite{Reading-HopfAlgebras},
\item the graph of the quotientoplex~$\quotientoplex$, a polytopal complex extending quotientopes~\mbox{\cite{PilaudSantos-quotientopes,PadrolPilaudRitter}}.
\end{compactitem}
\end{theorem}

%To achieve this result, we first describe the join irreducibles, the canonical join representations, and the forcing order of the $\s$-weak order in terms of $\s$-arcs, $\s$-arc diagrams and subarc order (\Cref{sec:canonicalRepresentations,sec:quotients}).
%An $\s$-arcs is an arc of~\cite{Reading-arcDiagrams} together with an integer bounded by~$\s$, and the notions of crossing and subarc are slightly more involved, but similar to that of~\cite{Reading-arcDiagrams}.
%We then construct the quotient foams using $\s$-shards, some polyhedral cones associated to the $\s$-arcs (\Cref{sec:quotientFoams}, and the quotientoplexes as Minkowski sums of shardoplexes, some polytopal complexes associated to the $\s$-arcs (\Cref{sec:quotientoplexes}).

To achieve this result, we first describe the join irreducibles of the $\s$-weak order in terms of $\s$-arcs.
An $\s$-arc is an arc of~\cite{Reading-arcDiagrams} together with an integer bounded by~$\s$.
We extend the notions of crossings and subarcs of~\cite{Reading-arcDiagrams} to describe the canonical join representations of $\s$-trees as non-crossing $\s$-arc diagrams, and the forcing order of the $\s$-weak order as the subarc order on $\s$-arcs.
Hence, the congruence lattice of the $\s$-weak order is isomorphic to the downset lattice of the subarc order on $\s$-arcs.
\mbox{See \Cref{sec:canonicalRepresentations,sec:quotients}}.

To each $\s$-arc, we then associate an $\s$-shard~$\shard$ (a polyhedral cone generalizing the shards of~\cite{Reading-posetRegions}) and a shardoplex~$\shardoplex$ (a polytopal complex constructed from and generalizing the shard polytopes of~\cite{PadrolPilaudRitter}).
%We then define the $\s$-shards, some polyhedral cones associated to the $\s$-arcs and generalizing the shards of~\cite{Reading-posetRegions}.
For a congruence~$\equiv$ of the $\s$-weak order with $\s$-arcs~$\cal{A}_\equiv$, the Hasse diagram of the quotient~$W_\s/{\equiv}$ is then the oriented dual graph of the quotient foam~$\quotientFoam$ whose union of walls is~$\bigcup_{\alpha \in \cal{A}_\equiv} \shard$, and the oriented graph of the quotientoplex~$\quotientoplex$ obtained as the Minkowski sum~$\sum_{\alpha \in \cal{A}_\equiv} \shardoplex$.
See \Cref{sec:quotientFoams,sec:quotientoplexes}.

Due to space limitation, this extended abstract omits many details and proofs of~\cite{PhilippePilaud}.
%We invite the reader to consult the long version~\cite{PhilippePilaud}.

%\begin{figure}[t]
%	\capstart
%	\centerline{\includegraphics[scale=.8]{120-congruenceLattice} \quad \includegraphics[scale=.8]{120-quotientFoamLattice} \quad \includegraphics[scale=.8]{120-quotientopeLattice}}
%	\vspace{-.2cm}
%	\caption{The congruence lattice of the $(1,2,0)$-weak order, where each congruence~$\equiv$ is replaced by its $\s$-arc downset (left), its quotient foam~$\quotientFoam$ (middle) and its quotientoplex~$\quotientoplex$~(right).}
%	\vspace{-.2cm}
%	\label{fig:mainCongruenceLattice}
%\end{figure}

%\paragraph{Lattice theory.}
%
%We first generalize the combinatorial description of~\cite{Reading-arcDiagrams} for the lattice theory of the weak order in terms of arcs, non-crossing arc diagrams, and subarcs.
%We define an \defn{$\s$-arc} as an arc of~\cite{Reading-arcDiagrams} together with an integer bounded by~$\s$.
%We then extend to $\s$-arcs the combinatorial notions of \defn{crossings} and of \defn{subarcs} to obtain the next statements.
%
%\begin{theorem}
%\label{thm:main1}
%There is a bijection from join irreducible $\s$-trees to $\s$-arcs, and from all $\s$-trees to non-crossing $\s$-arc diagrams, which encodes the canonical join representations in the $\s$-weak~order. % (\Cref{thm:canonicalJoinRepresentation}).
%\end{theorem}
%
%\begin{theorem}
%The forcing order on join irreducible $\s$-trees is isomorphic to the subarc order on $\s$-arcs. % (\Cref{thm:forcingOrder}), 
%%so that the congruence lattice of the $\s$-weak order is isomorphic to the distributive lattice of subsets of $\s$-arcs closed under subarcs. % (\Cref{coro:sWeakOrderCongruenceLattice}).
%\end{theorem}
%
%We exploit\,\Cref{thm:main1}\,(iii) to define relevant congruences of the $\s$-weak order and their quotients, generalizing the Tamari lattice~\cite{Tamari}, the Cambrian lattices~\cite{Reading-CambrianLattices, LangePilaud, ChatelPilaud}, the permutree lattices~\cite{PilaudPons-permutrees}, and the simple congruences of the weak order~\cite{HoangMutze, DemonetIyamaReadingReitenThomas, BarnardNovelliPilaud}.
%%This opens appealing conjectures about the combinatorics of these congruences.
%
%\paragraph{Polyhedral geometry.}
%
%We then construct geometric realizations of the $\s$-weak order and all its lattice quotients.
%Generalizing the shards of~\cite{Reading-posetRegions}, we associate to each $\s$-arc a polyhedral cone called \defn{$\s$-shard}.
%We use these $\s$-shards to define the \defn{$\s$-foam}, generalizing the braid fan.
%
%\begin{theorem}
%\label{thm:main2}
%\end{theorem}
%
%We then use the $\s$-foam~$\sFoam$ to construct polyhedral realizations of all lattice quotients of the $\s$-weak order, generalizing the quotient fans for the quotients of the weak order.
%See \Cref{fig:mainCongruenceLattice}\,(middle).
%This gives our third battery of results.
%
%\begin{theorem}
%\label{thm:main3}
%\textsf{Lattice quotients of the $\s$-weak order are realized by coarsenings of the $\s$-foam:}
%\begin{compactenum}[(i)]
%\item For any congruence~$\equiv$ of the $\s$-weak order, glueing together the maximal cells of the $\s$-foam~$\sFoam$ corresponding to $\s$-trees in the same congruence class of~$\equiv$ defines a complete polyhedral complex~$\quotientFoam$ (\Cref{prop:quotientFoam2}).
%\item The union of the walls of the quotient foam~$\quotientFoam$ coincides with the union of the $\s$-shards of the $\s$-arcs corresponding to~$\equiv$ (\Cref{prop:quotientFoam1}), where the \defn{$\s$-shards} are affine cones associated to the $\s$-arcs, generalizing the shards of~\cite{Reading-posetRegions}.
%\item The oriented dual graph of the quotient foam~$\quotientFoam$ is isomorphic to the Hasse diagram of the quotient of the $\s$-weak order by~$\equiv$ (\Cref{prop:quotientFoam3}).
%\end{compactenum}
%\end{theorem}
%
%Finally, we extend the theory of shard polytopes and quotientopes~\cite{PilaudSantos-quotientopes, PadrolPilaudRitter}.
%We define the \defn{shardoplex}~$\shardoplex$ associated to any \mbox{$\s$-arc~$\alpha$}, a polyhedral complex whose geometry is tuned to take care of the $\s$-shard of~$\alpha$.
%We then use Minkowski sums of shardoplexes to construct the \defn{quotientoplex}~$\quotientoplex$ of any congruence~$\equiv$ of the $\s$-weak order, a polyhedral complex realizing the quotient of the $\s$-weak order by~$\equiv$.
%See \Cref{fig:mainCongruenceLattice}\,(right).
%This yields our fourth battery of results.
%
%\begin{theorem}
%\label{thm:main4}
%\textsf{Lattice quotients of the $\s$-weak order are realized as Minkowski sums of shardoplexes:}
%\begin{compactenum}[(i)]
%\item For each $\s$-arc~$\alpha$, the union of the walls of the dual polyhedral complex of the shardoplex~$\shardoplex$ contains the $\s$-shard~$\shard$ and is contained in the $\s$-shards of the subarcs of~$\alpha$ (\Cref{prop:shard_trophyp}).
%\item For any congruence~$\equiv$ of the $\s$-weak order, the Hasse diagram of the quotient of the $\s$-weak order by~$\equiv$ is isomorphic to the oriented skeleton of the quotientoplex~$\quotientoplex$ (\Cref{thm:quotientoplex2}).
%\item The quotientoplex~$\quotientoplex$ is a polytopal subdivision of a quotientope~$\quotientope[\tilde\equiv]$ for the congruence~$\tilde\equiv$ of the weak order obtained by projecting~$\equiv$ (\Cref{prop:quotientoplex3}).
%\end{compactenum}
%\end{theorem}
%
%%We note that the proofs of \Cref{thm:main4} are mainly based on tropical geometry~\cite{Joswig21} (in fact, tropical geometry is convenient even to define the notion of dual polyhedral complex).
%
%Applying \Cref{thm:main4} to the trivial congruence provides a definitive answer to the geometric conjecture of C.~Ceballos and V.~Pons~\cite[Conjecture~3.1.2]{CeballosPons-sWeakOrderII} (which was already partially answered using flow polytopes in~\cite{DLeonMoralesPhilippeTamayoYip}, when~$\s$ contains no~$0$).
%
%\begin{corollary*}
%For any tuple~$\s$ of non-negative integers, the Hasse diagram of the $\s$-weak order is isomorphic to the oriented skeleton of a polytopal subdivision of a graphical zonotope (\Cref{coro:quotientoplexTrivial}).
%\end{corollary*}

%\section{Quotients of the classical weak order}
%
%\begin{definition}[{\cite[Definitions~39~\&~40]{PadrolPilaudRitter}}]
%\label{def:shardPolytopes}
%Fix a classical arc~$\alpha \eqdef (i, j, A, B)$.
%An \defn{$\alpha$-alternating matching} $\altmatch$ is a sequence $i \le i_1 < j_1 < i_2 < j_2 < \dots < i_q < j_q \le j$ such that~$i_p \in \{i\} \cup A$ and~$j_p \in \{j\} \cup B$ for all~$p \in [q]$.
%Its \defn{characteristic vector} is~$\charvect_{\altmatch} \eqdef \sum_{p \in [q]} \b{e}_{i_p} - \b{e}_{j_p}$. 
%We denote by $\altmatchset$ the set of all $\alpha$-alternating matchings. 
%The \defn{shard polytope} of~$\alpha$ is the convex hull~$\shardPolytope$ of the characteristic vectors of all $\alpha$-alternating matchings~$\altmatch$ in $\altmatchset$.
%%
%%\begin{figure}
%%	\capstart
%%	\centerline{\includegraphics[scale=.9]{shardPolytopes}}
%%	\caption{The shard polytopes of the four arcs of the form~$(1, 4, A, B)$. The vertices of the shard polytope of~$\alpha \eqdef (i, j, A, B)$ are labeled by the corresponding $\alpha$-alternating matchings, where we use solid dots~$\bullet$ for elements in~$\{i\} \cup A$ and hollow dots~$\circ$ for elements in~$B \cup \{j\}$. The corresponding vertex coordinates are directly read replacing~$\bullet$ by~$1$ and~$\circ$ by~$-1$. For instance, the vertex labeled~\raisebox{-.1cm}{\rotatebox{90}{\scalebox{.6}{${\bullet \cdot \cdot \,\circ}$}}} has coordinates~$(1,0,0,-1)$. Adapted from~\cite[Figure~10]{PadrolPilaudRitter}.}
%%	\label{fig:shardPolytiopes}
%%\end{figure}
%\end{definition}
%
%\begin{proposition}[{\cite[Proposition~48]{PadrolPilaudRitter}}]
%\label{prop:shardPolytopes}
%For any arc~$\alpha$, the union of the walls of the normal fan of the shard polytope~$\shardPolytope$
%\begin{compactitem}
%\item contains the shard~$\shard[\alpha]$ corresponding to~$\alpha$,
%\item is contained in the union of the shards~$\shard[\beta]$ over all subarcs~$\beta$ of~$\alpha$.
%\end{compactitem}
%\end{proposition}
%
%\begin{theorem}[{\cite[Corollary~50]{PadrolPilaudRitter}}]
%\label{thm:quotientopes}
%For any congruence~$\equiv$ of the weak order, the quotient fan~$\quotientFan$ is the normal fan of the \defn{quotientope}~$\quotientope$, obtained as the Minkowski sum of (any positive scaling of) the shard polytopes~$\shardPolytope$, over all arcs~$\alpha$ in~$\cal{A}_\equiv$.
%\end{theorem}

\section{The \texorpdfstring{$\s$}{s}-weak order}
\label{sec:sWeakOrder}

We first recall some definitions and results from~\cite{CeballosPons-sWeakOrderI} (with slightly different conventions).
We fix an $n$-tuple~$\s \eqdef (s_1, \dots, s_n)$ of non-negative integers (note that we allow~${s_i = 0}$).
%The following definitions are illustrated in \Cref{fig:sTrees,fig:weakOrders}\,(left) and \Cref{exm:sTrees}.


\parpic(2.5cm,3.5cm)(-5pt, 135pt)[r][b]{\includegraphics[scale=1.1]{sTree}}{
\begin{definition}[{\cite[Section~1.2]{CeballosPons-sWeakOrderI}}]
An \defn{$\s$-tree} is a rooted plane tree with internal nodes bijectively labeled by~$[n]$ such that the node~$j$ has~$s_j+1$ children which are all either leaves or nodes larger than~$j$.
\end{definition}

\begin{definition}[{\cite[Definition~1.3]{CeballosPons-sWeakOrderI}}]
\label{def:positions}
For an $\s$-tree~$\tree$ and~${1 \le i < j \le n}$,~the \defn{position}~$\pos(\tree, i, j) \in [0, s_i]$ is the minimum of~$s_i$ and the number of outgoing edges of~$i$ to the right of the increasing path from the root of~$\tree$~to~$j$.
\end{definition}

For instance, the tree~$\tree_\circ$ on the right is a $(1,2,2,0,2,2,1,1)$-tree, where $\pos(\tree_\circ, 1, 4) \!=\! 0$, $\pos(\tree_\circ, 1, 5) \!=\! 1$, $\pos(\tree_\circ, 2, 5) \!=\! 2$ and~${\pos(\tree_\circ, 5, 7) \!=\! 2}$.
The following definition is illustrated in \Cref{fig:weakOrders}\,(left) for~$\s = (2,1,0)$.
}

\begin{definition}[{\cite[Definition~1.9]{CeballosPons-sWeakOrderI}}]
\label{def:sWeakOrder}
The \defn{$\s$-weak order}~$W_\s$ is the partially ordered set of $\s$-trees given by~$\tree \le \tree'$ if and only if~$\pos(\tree, i, j) \le \pos(\tree', i, j)$ for all~$1 \!\le\! i \!<\! j \!\le\! n$.
%See \Cref{fig:weakOrders}\,(left).
\end{definition}

\begin{theorem}[{\cite[Theorems~1.21 \& 1.40]{CeballosPons-sWeakOrderI}}]
\label{thm:sWeakOrderLattice}
The $\s$-weak order is a congruence uniform lattice.
\end{theorem}

%\begin{example}
%\label{exm:sTrees}
%\Cref{fig:sTrees} represents some~$\s$-trees for $\s = (1,2,2,0,2,2,1,1)$.
%The $\s$-tree~$\tree$ on the left has~$\pos(\tree, 1, 4) = 0$, $\pos(\tree, 1, 5) = 1$, $\pos(\tree, 2, 5) = 2$ and~$\pos(\tree, 5, 7) = 2$. The $4$ $\s$-trees on the right cover the $\s$-tree~$\tree$ in the $\s$-weak order.
%%
%\begin{figure}[b]
%	\capstart
%	\centerline{\includegraphics[scale=.85]{sTree} \;\, \includegraphics[scale=.85]{descents} \;\, \includegraphics[scale=.85]{nonCrossingsArcDiagram} \;\, \includegraphics[scale=.85]{rotation1} \;\, \includegraphics[scale=.85]{rotation2} \;\, \includegraphics[scale=.85]{rotation3} \;\, \includegraphics[scale=.85]{rotation4}}
%	\caption{A $(1,2,2,0,2,2,1,1)$-tree~$\tree$, its descents in red, its non-crossing arc diagram, and the covers of~$\tree$ obtained by rotating its descents~$(1,5)$, $(2,4)$, $2,6$, $(5,7)$, and $(5,8)$.}
%	\label{fig:sTrees}
%\end{figure}
%\end{example}

%\begin{example}
%\Cref{fig:weakOrders} represents the $(2,1,0)$-weak order.
%
%\begin{figure}[t]
%	\capstart
%	\centerline{\includegraphics[scale=.85]{210-weakOrder} \hspace{1.5cm} \includegraphics[scale=.85]{210-arcDiagrams}}
%	\caption{The $\s$-weak order on $\s$-trees and on $\s$-arc diagrams, for $\s = (2,1,0)$.}
%	\label{fig:weakOrders}
%\end{figure}
%\end{example}

%\begin{definition}[{\cite[Definition~1.24]{CeballosPons-sWeakOrderI}}]
%An \defn{ascent} (resp.~\defn{descent}) of an $\s$-tree~$\tree$ is a pair~$(i,j)$ of nodes of~$\tree$ such that 
%\begin{compactitem}
%\item $i$ is the greatest ancestor of~$j$ such that the increasing path from~$i$ to~$j$ in~$\tree$ takes the leftmost (resp.~rightmost) outgoing edge at each node, except~at~node~$i$,
%\item either~$s_j = 0$ or the leftmost (resp.~rightmost) edge of~$j$ is a leaf.
%\end{compactitem}
%\end{definition}

\begin{definition}[{\cite[Definition~1.24]{CeballosPons-sWeakOrderI}}]
A \defn{descent} of an $\s$-tree~$\tree$ is a pair~$(i,j)$ of nodes where
\begin{compactitem}
\item $i$ is an ancestor of~$j$ and the increasing path from~$i$ to~$j$ takes the rightmost outgoing edge at each node, except~at~node~$i$,
\item either~$s_j = 0$ or the rightmost edge of~$j$ is a leaf.
\end{compactitem}
\end{definition}

For instance,\,our\,$(1,2,2,0,2,2,1,1)$-tree\,$\tree_\circ$\,has descents\,$(1,5)$,\,$(2,4)$,\,$(2,6)$,\,$(5,7)$,\,$(5,8)$.

%\begin{definition}[{\cite[Definition~1.30]{CeballosPons-sWeakOrderI}}]
%\label{def:rotation}
%Consider an ascent (resp.~descent)~$(i,j)$ in an $\s$-tree~$\tree$.
%Let~$r$ be the parent of~$j$ (note that~$r$ might be~$i$).
%Let~$j \to s$ be the rightmost (resp.~leftmost) outgoing edge of~$j$ (note that~$s$ might be a leaf).
%Let~$p \to q$ be last edge whose source is smaller than~$j$ along the rightmost (resp.~leftmost) increasing path leaving~$i$ through the edge immediately to the left (resp.~right) of the path from~$i$ to~$j$ (note that~$q$ might be a leaf).
%The \defn{left} (resp.~\defn{right}) \defn{rotation} of~$(i,j)$ transforms~$\tree$ by replacing the edges~$r \to j$, $j \to s$ and $p \to q$ by new edges~$r \to s$, $j \to q$ and~$p \to j$ respectively.
%\end{definition}

\parpic(3.8cm,3.8cm)(-5pt, 133pt)[r][b]{\includegraphics[scale=1.15]{rotation}}{
\begin{definition}[{\cite[Definition~1.30]{CeballosPons-sWeakOrderI}}]
\label{def:rotation}
Pick a descent~$(i,j)$ in an \mbox{$\s$-tree~$\tree$}.
Let~$p$ be the parent of~$j$ ($p$ might be~$i$).
Let~$j \to q$ be the leftmost outgoing edge of~$j$ ($q$ might be a leaf).
Let~$r \to s$ be last edge whose source is smaller than~$j$ along the leftmost increasing path leaving~$i$ through the edge immediately to the right of the path from~$i$ to~$j$ (note that~$s$ might be a leaf).
The \defn{right rotation} of~$(i,j)$ transforms~$\tree$ by replacing the edges~$p \to j$, $j \to q$ and $r \to s$ by new edges~$p \to q$, $j \to s$ and~$r \to j$ respectively. %, as illustrated on the right.
\end{definition}

%\begin{proposition}[{\cite[Theorem~1.32]{CeballosPons-sWeakOrderI}}]
%\label{prop:sWeakOrderCoverRelations}
%The $\s$-trees which cover (resp.~are covered by) an $\s$-tree~$\tree$ in the $\s$-weak order~$W_\s$ are precisely those obtained by rotating an ascent (resp.~descent) of~$\tree$.
%\end{proposition}

\begin{proposition}[{\cite[Theorem~1.32]{CeballosPons-sWeakOrderI}}]
\label{prop:sWeakOrderCoverRelations}
The $\s$-trees covered by an $\s$-tree~$\tree$ in the $\s$-weak order~$W_\s$ are precisely those obtained by the right rotation of a descent of~$\tree$.
\end{proposition}

\begin{remark}
When $\s = (1, \dots, 1)$, the $\s$-trees are just the increasing binary trees, in bijection with permutations of~$[n]$.
Namely, the permutation~$\pi_{\tree}$ corresponding to an increasing tree~$\tree$ is obtained by reading its nodes in infix order.
Moreover, $\pos(\tree, i, j) = 1$ if and only if~$(j,i)$ is an inversion of~$\pi_{\tree}$, so that the $\s$-weak order is the classical weak order.
More generally, if~$\s$ contains no~$0$, then the $\s$-trees are in bijection with \defn{Stirling $\s$-permutations}, \ie permutations of the multiset~$\{1^{s_1}, \dots, n^{s_n}\}$ avoiding the pattern~$121$~\cite{CeballosPons-sWeakOrderI}.
\end{remark}
}

%\begin{example}
%The left $\s$-tree of \Cref{fig:sTrees} has descents $(1,5)$, $(2,4)$, $(2,6)$, $(5,7)$, and $(5,8)$.
%The $\s$-trees obtained by rotation of these descents are represented on the right of \Cref{fig:sTrees}.
%\end{example}

%\begin{figure}[t]
%	\capstart
%	\centerline{\raisebox{.5cm}{\includegraphics[scale=.6]{120-weakOrder}} \hspace{1.5cm} \includegraphics[scale=.6]{210-weakOrder}}
%	\caption{The $\s$-weak order~$W_\s$ for $\s = (1,2,0)$ (left) and~$\s = (2,1,0)$ (right).}
%	\label{fig:weakOrders}
%\end{figure}

\section{Canonical representations in the \texorpdfstring{$\s$}{s}-weak order}
\label{sec:canonicalRepresentations}

\begin{figure}[t]
	\capstart
	\centerline{\includegraphics[scale=.85]{210-weakOrder} \hspace{1.5cm} \includegraphics[scale=.85]{210-arcDiagrams}}
	\caption{The $\s$-weak order on $\s$-trees and on $\s$-arc diagrams, for $\s = (2,1,0)$.}
	\label{fig:weakOrders}
\end{figure}

We now develop lattice properties of the $\s$-weak order.
Generalizing the arcs and non-crossing arc diagrams of~\cite{Reading-arcDiagrams}, we describe the join irreducible $\s$-trees and the canonical join representations in the $\s$-weak order in terms of the following combinatorial objects.

\begin{definition}
An \defn{$\s$-arc} is a quintuple~$(i, j, A, B, r)$ where~$1 \le i < j \le n$, the sets~$A$ and~$B$ form a partition of~$\set{k \in {]i,j[}}{s_k \ne 0}$, and~$r \in [s_i]$.
\end{definition}

%Note that the first entry~$i$ of an $\s$-arc must satisfy~$s_i \ne 0$ (otherwise, we have no choice for~$r$).
%Graphically, we place~$n$ points on the vertical axis, where the $i$th point is round if~$s_i \ne 0$ and square if~$s_i = 0$.
%Graphically, the $\s$-arc~$(i, j, A, B, r)$ is represented by a curve wiggling around the vertical axis, starting at~$i$ and ending at~$j$, passing on the right of the points in~$A$ and on the left of the points in~$B$, and with a label~$r$ close~to~it.
%See \Cref{fig:weakOrders}\,(right).

%\begin{remark}
%The number of~$\s$-arcs is $\sum_{1 \le i < j \le n} s_i \, 2^{\# \set{k \in {]i,j[}}{s_k \ne 0}}$.
%\end{remark}

\begin{definition}
\label{def:non-crossing}
Consider two $\s$-arcs~$\alpha \eqdef (i, j, A, B, r)$ and~$\alpha' \eqdef (i', j', A', B', r')$.
Assume without loss of generality that~$j \le j'$ (otherwise, exchange~$\alpha$ and~$\alpha'$).
Then~$\alpha$ and~$\alpha'$ are \defn{non-crossing} if~$j < j'$, and one of the following holds:
\begin{compactenum}
\item\label{non-crossing_item1} $j \le i'$,
\item\label{non-crossing_item2} $i < i' < j$ and~$i' \in A$ and~$j \notin A'$ and $A' \cap {]i,j[} \subseteq A \cap {]i',j'[}$,
\item\label{non-crossing_item3} $i < i' < j$ and~$i' \in B$ and~$j \notin B'$ and $A' \cap {]i,j[} \supseteq A \cap {]i',j'[}$,
\item\label{non-crossing_item4} $i = i'$ and~$r < r'$ and~$j \notin A'$ and $A' \cap {]i,j[} \subseteq A \cap {]i',j'[}$,
\item\label{non-crossing_item5} $i = i'$ and~$r = r'$ and~$s_j = 0$ and $A' \cap {]i,j[} = A \cap {]i',j'[}$,
\item\label{non-crossing_item6} $i = i'$ and~$r > r'$ and~$j \notin B'$ and $A' \cap {]i,j[} \supseteq A \cap {]i',j'[}$,
\item\label{non-crossing_item7} $i' < i$ and~$i \in A'$ and~$j \notin B'$ and $A' \cap {]i,j[} \supseteq A \cap {]i',j'[}$,
\item\label{non-crossing_item8} $i' < i$ and~$i \in B'$ and~$j \notin A'$ and $A' \cap {]i,j[} \subseteq A \cap {]i',j'[}$.
\end{compactenum}
A \defn{non-crossing $\s$-arc diagram} is a set~$\delta$ of pairwise non-crossing $\s$-arcs.
%The \defn{non-crossing $\s$-arc complex} is the (flag) simplicial complex of non-crossing $\s$-arc diagrams.
\end{definition}

Graphically, consider~$n$ points on the vertical axis, labeled by $[n]$ from bottom to top.
The $\s$-arc~$(i, j, A, B, r)$ is represented by a curve wiggling around these points, starting at~$i$ and ending at~$j$, passing on the right of the points in~$A$ and on the left of the points in~$B$  (we do not care if it passes on the left or right of the points in~$\set{k \in {]i,j[}}{s_k = 0}$), and with a label~$r$ close~to~it.
Two arcs are non-crossing if the corresponding curves are not crossing in their interior, plus some boundary conditions at their endpoints (the starting points may coincide according to rules~4, 5, 6 of \Cref{def:non-crossing}, while the ending points must be distinct).
See \Cref{fig:weakOrders}\,(right), and the figure on the bottom of this page.

\begin{remark}
When $\s = (1, \dots, 1)$, the $\s$-arcs are the original arcs of~\cite{Reading-arcDiagrams}.
An arc is a quadruple~$(i, j, A, B)$ where~$1 \le i < j \le n$ and~${]i,j[} = A \sqcup B$, or equivalently a curve joining~$i$ to~$j$ and passing right of~$A$ and left of~$B$.
Two arcs are non-crossing if they do not cross in their interior and have distinct starting points and distinct ending points.
\end{remark}

Let~$(L, \le, \join)$ be a join semilattice.
A \defn{join representation} of~${x \in L}$ is a subset~$J \subseteq L$ such that~$x = \bigJoin J$.
Such a representation is \defn{irredundant} if~$x \ne \bigJoin J'$ for any strict subset~$J' \subsetneq J$.
The irredundant join representations of~$x \in L$ are ordered by
% containment of the downsets of their elements, \ie
$J \le J'$ if and only if for any~$y \in J$ there is~${y' \in J'}$~with~$y \le y'$.
The \defn{canonical join representation} of~$x$ is the unique minimal irredundant join representation of~$x$ for this order when it exists.
%Its elements are the \defn{canonical joinands} of~$x$.
Note that the canonical joinands form an antichain of \defn{join irreducible} elements of~$L$ (those which cover a single element).
Canonical join representations always exist if and only if~$L$ is \defn{join semidistributive} (that is,~$x \join y = x \join z$ implies $x \join (y \meet z) = x \join y$ for any~$x, y, z \in L$).~It follows from \Cref{thm:sWeakOrderLattice} and~\cite{Day} that the $\s$-weak order is join semidistributive.
%
We now provide bijections from join irreducible $\s$-trees to $\s$-arcs, and from all $\s$-trees to non-crossing $\s$-arc diagrams, which enable to \mbox{read the canonical join representations in the $\s$-weak order.}


%\begin{figure}[t]
%	\capstart
%	\centerline{\raisebox{.5cm}{\includegraphics[scale=.6]{120-arcDiagrams}} \hspace{1.5cm} \includegraphics[scale=.6]{210-arcDiagrams}}
%	\caption{The $\s$-weak order~$W_\s$ for $\s = (1,2,0)$ (left) and~$\s = (2,1,0)$ (right), where each $\s$-tree~$\tree$ is replaced by its non-crossing $\s$-arc diagram~$\delta_\join(\tree)$.}
%	\label{fig:sWeakOrderArcDiagrams}
%\end{figure}

%\vspace{-.2cm}
%\parpic(1.3cm,3cm)(-5pt, 100pt)[r][b]{\includegraphics[scale=.8]{joinIrreducibleTrees}}{
%\begin{definition}
%\label{def:sArc2sTree}
%For~$J \subseteq [n]$, the \defn{right $\s$-comb} with nodes~$J$ is the increasing tree on~$J$, where each node~$j \in J$ has~$s_j+1$ outgoing edges, which are all leaves except the rightmost.
%For an $\s$-arc~$\alpha \eqdef (i, j, A, B, r)$, we define~$\tree_{\!\join}(\alpha)$ as the \mbox{$\s$-tree} obtained by attaching the right $\s$-comb with nodes~$A \cup \{j\}$ to the \mbox{$(r+1)$-st} rightmost leaf of~$i$ in the right \mbox{$\s$-comb} with nodes~$[n] \ssm (A \cup \{j\})$.
%See example of~$\tree_{\!\join}(\alpha)$ on the right for~${\s = (2, 3, 1, 0, 1, 0, 0)}$ and~$\alpha = (2, 7, \{3\}, \{5\}, 2)$.
%\end{definition}
%}

%\begin{definition}
%\label{def:sArc2sTree}
%For a join irreducible $\s$-tree~$\tree$ with descent~$(i,j)$, let~$\alpha_\join(\tree) \eqdef (i, j, A, B, r)$, where~$r \eqdef \pos(\tree, i, j) \in [s_i]$, $A$ is the set of nodes along the path from~$i$ to~$j$ (but distinct from~$i$ and~$j$), and let~$B$ be the complement of~$A$ in~$\set{k \in {]i,j[}}{s_k \ne 0}$.
%\end{definition}

\begin{definition}
\label{def:sTree2sArcDiagram}
For a descent~$(i,j)$ of an $\s$-tree~$\tree$, let~$r \eqdef\! \pos(\tree, i, j)$ and~$A$ (resp.~$B$)~the~set of nodes~$i \!<\! k \!<\! j$ with~${s_k \ne 0}$ and weakly on the left (resp.~strictly on the right) of the~path from~$i$ to~$j$.
Define~\defn{$\alpha_\join(\tree, i, j)$}$\eqdef (i, j, A, B, r)$ and~\defn{$\delta_\join(\tree)$}$\eqdef \set{\alpha_\join(\tree, i, j)}{(i,j) \text{ descent of } \tree}$.
%Define similarly~\defn{$\alpha_\meet(\tree, i, j)$} for an ascent~$(i,j)$ of~$\tree$, and~\defn{$\delta_\meet(\tree)$} $\eqdef \set{\alpha_\meet(\tree, i, j)}{(i,j) \text{ ascent of } \tree}$.
%%
\end{definition}

%\begin{proposition}
%\label{prop:canonicalJoinRepresentation}
%The canonical join (resp.~meet) representation of an~$\s$-tree~$\tree$ is~$\tree = \bigJoin_{\alpha \in \delta_\join(\tree)} \tree_{\!\join}(\alpha)$ (resp.~$\tree = \bigMeet_{\alpha \in \delta_\meet(\tree)} \tree_{\!\meet}(\alpha)$).
%%In other words, the map~$\alpha_\join$ (resp.~$\alpha_\meet$) induces an isomorphism from the canonical join (resp.~meet) complex of the $\s$-weak order~$W_\s$ to the non-crossing $\s$-arc complex.
%\end{proposition}

%\begin{proposition}
%\label{prop:sArc2sTree}
%The map~$\tree_{\!\join}$ is a bijection from the $\s$-arcs to the join irreducible $\s$-trees. % in the $\s$-weak order.
%\end{proposition}

%\begin{proposition}
%\label{prop:sTree2sArc}
%The map~$\alpha_\join$ is a bijection from join irreducible $\s$-trees to $\s$-arcs.
%\end{proposition}

\begin{theorem}
\label{thm:canonicalJoinRepresentation}
The map~$\alpha_\join$ is a bijection from join irreducible $\s$-trees to $\s$-arcs.
The map~$\delta_\join$ is a bijection from all $\s$-trees to non-crossing $\s$-arc diagrams.
The canonical join representation of an~$\s$-tree~$\tree$ is~$\tree = \bigJoin_{\alpha \in \delta_\join(\tree)} \alpha_{\!\join}^{-1}(\alpha)$. %, where~$\tree_\join$ is the inverse of~$\alpha_\join$.
%See \Cref{fig:weakOrders}\,(right).
%In other words, the map~$\alpha_\join$ induces an isomorphism from the canonical join complex of the $\s$-weak order~$W_\s$ to the non-crossing $\s$-arc complex.
\end{theorem}
 
\parpic(5cm,3cm)(-5pt, 110pt)[r][b]{\includegraphics[scale=1.1]{descents}\;\;\includegraphics[scale=1.1]{nonCrossingsArcDiagram}}{
As illustrated on the right, the non-crossing $\s$-arc diagram~$\delta_\join(\tree)$ of an $\s$-tree~$\tree$ can be obtained graphically by drawing the path joining~$i$ to~$j$ in~$\tree$ for each descent~$(i,j)$ of~$\tree$, perturbing all these paths so that they pass slightly to the right of their intermediate nodes, and flattening the picture horizontally, allowing the arcs to bend but not to cross nor to pass a node.
}

%\begin{example}
%The left of \Cref{fig:sTrees} illustrates this graphical construction of~$\delta_\join(\tree)$~from~$\tree$.
%\end{example}

\medskip
For instance, \Cref{fig:weakOrders}\,(right) represents the $\s$-weak order where the $\s$-trees are replaced by their non-crossing $\s$-arc diagrams, for~$\s = (2,1,0)$.


\section{Quotients of the \texorpdfstring{$\s$}{s}-weak order}
\label{sec:quotients}

A \defn{congruence}~$\equiv$ on a finite lattice~$(L, \le, \meet, \join)$ is an equivalence relation on~$L$ such that $x \equiv x'$ and $y \equiv y'$ implies $x \join y \equiv x' \join y'$ and $x \meet y \equiv x' \meet y'$.
%Equivalently, the equivalence classes are intervals, and the maps~$\projDown[\equiv]$ and~$\projUp[\equiv]$ sending an element to the minimum and maximum elements in its congruence class are order preserving.
The \defn{lattice quotient}~$L/{\equiv}$ is the lattice where the join~$X \join Y$ (resp.~meet~$X \meet Y$) of two congruence classes is the congruence class of the join~$x \join y$ (resp.~meet~$x \meet y$) of any representatives~$x \in X$ and~${y \in Y}$.
%The \defn{congruence lattice} of~$L$ is the refinement lattice on the congruences of~$L$.
%It is a lattice where the meet is the intersection of relations and the join is the transitive closure of union of relations.
A congruence~$\equiv$ is determined by its contracted join irreducibles, \ie its relations~$j \equiv j_\star$ where~$j$ covers only~$j_\star$.
%A join irreducible element~$j \in L$ (covering only~$j_\star \in L$) is \defn{contracted} by~$\equiv$ if~$j \equiv j_\star$, and the set of contracted join irreducible elements of~$L$ determines~$\equiv$.
If any congruence contracting~$j$ also contracts~$j'$, then~$j$ \defn{forces}~$j'$.
This defines a preorder on join irreducibles of~$L$ whose lattice of downsets is isomorphic to the \defn{congruence lattice} of~$L$.
This preposet is a poset when~$L$~is congruence uniform.
Understanding the congruences of~$L$ then boils down to understanding the forcing poset on join irreducibles of~$L$.
Generalizing the subarc order of~\cite{Reading-arcDiagrams}, we now describe the forcing order of the $\s$-weak order in terms of the subarc order on $\s$-arcs.
%See \Cref{fig:sSubarcOrder}.

\begin{definition}
\label{def:subarc}
Consider two $\s$-arcs~$\alpha \eqdef (i, j, A, B, r)$ and~$\alpha' \eqdef (i', j', A', B', r')$. 
We say that $\alpha$ is a \defn{subarc} of $\alpha'$ if all the following conditions hold:

\parpic(5.3cm,3cm)(-5pt, 73pt)[r][b]{\includegraphics[scale=.8]{210-subarcOrder}}{
\begin{compactitem}
\item $i'\leq i < j \leq j'$,
\item $A\subseteq A'$ and $B\subseteq B'$,
\item if $s_j=0$ then $j=j'$,
\item if $i'=i$ then $r=r'$,
\item if $i'<i$ then $i\in A'$ and $r=1$, or $i\in B'$ and $r=s_i$.
\end{compactitem}}
%\begin{figure}[ht]
%	\capstart
%	\centerline{\includegraphics[scale=.8]{120-subarcOrder} \hspace{1.5cm} \includegraphics[scale=.8]{210-subarcOrder}}
%	\caption{The subarc order on $\s$-arcs for $\s = (1,2,0)$ (left) and~$\s = (2,1,0)$ (right).}
%	\label{fig:sSubarcOrder}
%\end{figure}
\end{definition}


\begin{remark}
When~$\s = (1, \dots, 1)$, an arc~$(i,j,A,B)$ is a subarc of an arc~$(i',j',A',B')$ if and only if~$i' \le i < j \le j'$ and~$A\subseteq A'$ and $B\subseteq B'$, as defined in~\cite{Reading-arcDiagrams}.
\end{remark}

\begin{theorem}
\label{thm:forcingOrder}
The forcing order on join irreducible $\s$-trees is the subarc order on $\s$-arcs:
%if~$\alpha$ and~$\alpha'$ are two~$\s$-arcs, then $\tree_{\!\join}(\alpha)$ forces $\tree_{\!\join}(\alpha')$ if and only if $\alpha$ is a subarc of $\alpha'$.
If $\tree$ and~$\tree'$ are two join irreducible $\s$-trees, then~$\tree$ forces $\tree'$ if and only if~$\alpha_\join(\tree)$ is a subarc of~$\alpha_\join(\tree')$.
\end{theorem}

\begin{corollary}
\label{coro:sWeakOrderCongruenceLattice}
The congruence lattice of the $\s$-weak order~$W_\s$ is isomorphic to the lattice of downsets of the subarc order on $\s$-arcs.
We denote by~$\cal{A}_\equiv$ the downset of a congruence~$\equiv$.
\end{corollary}

\begin{remark}
\label{rem:specialCongruences}
We can exploit \Cref{coro:sWeakOrderCongruenceLattice} to define relevant congruences and quotients:
\begin{compactitem}
\item The \defn{$\s$-Tamari lattice} is the quotient of the $\s$-weak order by the \defn{$\s$-sylvester congruence}~$\equiv_\mathrm{sylv}$ where~$\cal{A}_{\equiv_\mathrm{sylv}}$ is the set of \defn{right $\s$-arcs}~$(i, j, \set{k \in {]i,j[}}{s_k \ne 0}, \varnothing, r)$ for some~$1 \le i < j \le n$ and~$r \in [s_i]$.
%An $\s$-tree~$\tree$ is minimal in its $\s$-sylvester congruence class if and only if~$\pos(\tree, a, b) \le \pos(\tree, a, c)$ for all~$1 \le a < b < c \le n$ with~$s_b \ne 0$.
When~$\s$~contains~no~$0$, this quotient was considered in~\cite[Section~2.4]{CeballosPons-sWeakOrderI} in connection to the $\nu$-Tamari lattice~\cite{PrevilleRatelleViennot, CeballosPadrolSarmiento-geometryNuTamari}.

\item For any $\s$-arc~$\alpha$, the \defn{$\alpha$-Cambrian lattice} is the quotient of the $\s$-weak order by the \defn{$\alpha$-Cambrian congruence}~$\equiv_\alpha$, where~$\cal{A}_{\equiv_\alpha}$ is the downset of subarcs of~$\alpha$. See~\cite{Reading-CambrianLattices}.
%As the classical (type~$A$) Cambrian lattices~\cite{Reading-CambrianLattices}, up to isomorphism), the cover graph of the $\alpha$-Cambrian lattice and the face lattices of the $\alpha$-Cambrian foam~$\quotientFoam[\equiv_\alpha]$ and quotientoplex~$\quotientoplex[\equiv_\alpha]$ (see \Cref{sec:quotientFoams,sec:quotientoplexes}) only depend on the endpoints of~$\alpha$.

\item For a map~$\decoration : \set{i \in [n]}{s_i \ne 0} \to \Decorations$, the  \defn{$\decoration$-permutree lattice} is the quotient of the $\s$-weak order by the \defn{$\decoration$-per\-mutree congruence}~$\equiv_\decoration$, where~$\cal{A}_{\equiv_\decoration}$ is the set of arcs which do not pass on the right of a point~$j$ with~$\decoration(j) \in \{\upCirc, \upDownCirc\}$ nor on the left of the points~$j$ with~$\decoration(j) \in \{\downCirc, \upDownCirc\}$. See~\cite{PilaudPons-permutrees}.

\item A congruence~$\equiv$ is \defn{regular} if any minimal (in subarc order) arc~$(i,j,A,B)$ of the complement of the set~$\cal{A}_\equiv$ is either a left arc (${A = \varnothing}$) or a right arc ($B = \varnothing$).~See~\cite{HoangMutze}.
\end{compactitem}
\end{remark}

\section{Quotient foams}
\label{sec:quotientFoams}

We now realize the $\s$-weak order and its lattice quotients as dual graphs of polyhedral complexes.
Generalizing the shards of~\cite{Reading-posetRegions}, we first associate codimension~$1$ polyhedral cones to the $\s$-arcs, and use them to generalize the braid fan.
See \Cref{fig:foams}\,(left).

\begin{figure}[b]
	\capstart
	\centerline{\includegraphics[scale=.82]{210-foam} \hspace{-.8cm} \includegraphics[scale=.82]{210-sylvesterQuotientFoam}}
	\caption{The $\s$-foam and its sylvester quotient foam, for $\s = (2,1,0)$. See \Cref{fig:MinkowskiSums}.}
	\label{fig:foams}
\end{figure}

\begin{definition}
\label{def:sShard}
The \defn{$\s$-shard} of an $\s$-arc~$\alpha \!\eqdef\! (i, j, A, B, r)$ is the polyhedron~$\shard$ of~$\R^n$ given~by
\begin{compactitem}
\item the equality~$x_i - x_j = r - 1 + \sum_{k \in B} \max(0, s_k-1)$,
\item the inequalities~$x_i - x_a \ge r - 1 + \sum_{k \in B \cap {]i,a[}} \max(0, s_k-1)$ for all~$a \in A$, and
\item the inequalities~$x_i - x_b \le r - 1 + \sum_{k \in B \cap {]i,b[}} \max(0, s_k-1)$ for all~$b \in B$.
\end{compactitem}
\end{definition}

%We use these $\s$-shards to define the following generalization of the braid fan.

\begin{definition}
The \defn{$\s$-foam}~$\sFoam$ is the polyhedral complex of~$\R^n$ whose maximal cells are the closures of the connected components of the complement of the union of all $\s$-shards.
\end{definition}

\begin{theorem}
\label{thm:sFoam}
The Hasse diagram of the $\s$-weak order is isomorphic to the dual graph of the $\s$-foam~$\sFoam$ oriented in the direction~$\b{\omega} \eqdef (1, 2, \dots n) - (n, \dots, 2, 1) = \sum_{i \in [n]} (2i-n-1) \, \b{e}_i$.
\end{theorem}

\begin{remark}
When~$\s = (1, \dots, 1)$, the shard of an arc~$(i,j,A,B)$ is the polyhedral cone defined by~$x_b \le x_i = x_j \le x_a$ for all~$a \in A$ and~$b \in B$.
The union of the shards is the braid arrangement, and the $\s$-foam is the classical braid fan.
\end{remark}

%\begin{figure}[ht]
%	\capstart
%	\centerline{\includegraphics[scale=.6]{120-foam} \hspace{1cm} \includegraphics[scale=.6]{210-foam}}
%	\caption{The $\s$-foam~$\sFoam$ for $\s = (1,2,0)$ (left) and~$\s = (2,1,0)$ (right).}
%	\label{fig:foams}
%\end{figure}

\begin{remark}
The cells of the $\s$-foam~$\sFoam$ can also be obtained as the closures of the fibers of an insertion algorithm, generalizing the insertion algorithm of permutations in increasing trees.
This perspective is the most convenient to prove \Cref{thm:sFoam}, but its presentation is more technical, and unnecessary here.
We refer to~\cite[Section~1]{PhilippePilaud} for details.
\end{remark}

\begin{proposition}
The vertices of~$\sFoam$ form a $(s_0 \times \dots \times s_{n-1})$-grid~$\smash{\sTrunks \eqdef \!\!\!\!\displaystyle\prod\limits_{i \in [n-1]} \!\![s_i] (\b{e}_1 - \b{e}_{i+1})}$.
\end{proposition}

Generalizing~\cite{Reading-HopfAlgebras}, we can then either glue some regions of the $\s$-foam or delete some $\s$-shards according to a congruence of the $\s$-weak order, as illustrated in \Cref{fig:foams}\,(right).

\begin{definition}
\label{def:quotientFoam}
For any congruence~$\equiv$ of the $\s$-weak order, the \defn{$\equiv$-quotient foam}~$\quotientFoam$ is the complete polyhedral complex defined by the following equivalent descriptions:
\begin{compactenum}[(i)]
\item the maximal cells of~$\quotientFoam$ are obtained by glueing together the maximal cells of the $\s$-foam~$\sFoam$ corresponding to $\s$-trees in the same congruence class of~$\equiv$,
\item the union of the walls of~$\quotientFoam$ is the union of the $\s$-shards~$\shard$ of the $\s$-arcs~$\alpha$~in~$\cal{A}_\equiv$.
\end{compactenum}
\end{definition}

\begin{theorem}
For any congruence~$\equiv$ of the $\s$-weak order~$W_\s$, the two descriptions of \Cref{def:quotientFoam}~coincide and define a polyhedral complex~$\quotientFoam$, whose dual graph, oriented in the direction~$\b{\omega}$, is isomorphic to the  Hasse diagram of the quotient~$W_\s / {\equiv}$ .
\end{theorem}

\begin{remark}
When~$\s = (1, \dots, 1)$, the $\s$-quotient foams are the quotient fans of~\cite{Reading-HopfAlgebras}.
\end{remark}

%\begin{example}
%See \Cref{fig:foams,fig:sQuotientFoamLattice} for $(2,1,0)$-quotient foams.
%\end{example}


%\begin{figure}[h!]
%	\capstart
%	\centerline{\includegraphics[scale=.6]{120-sylvesterQuotientFoam} \hspace{.5cm} \includegraphics[scale=.6]{210-sylvesterQuotientFoam}}
%	\caption{The quotient foam~$\equiv_\mathrm{sylv}$ for the $\s$-sylvester congruence of the $\s$-weak order of \Cref{subsubsec:sylvester}, for $\s = (1,2,0)$ (left) and~$\s = (2,1,0)$ (right).}
%	\label{fig:sylvesterQuotientFoams}
%\end{figure}

%\begin{figure}[h!]
%	\capstart
%	\centerline{\raisebox{1.2cm}{\includegraphics[scale=.6]{120-quotientFoamLattice}} \hspace{.5cm} \includegraphics[scale=.6]{210-quotientFoamLattice}}
%	\caption{The congruence lattice of~$W_\s$ for $\s = (1,2,0)$ (left) and~$\s = (2,1,0)$ (right), where each congruence~$\equiv$ is replaced by its quotient foam~$\quotientFoam$. See also \Cref{fig:sQuotientopeLattice}.}
%	\label{fig:sQuotientFoamLattice}
%\end{figure}

\begin{figure}[ht!]
	\capstart
	\centerline{\includegraphics[scale=1.1]{210-quotientFoamLattice}}
	\caption{The congruence lattice of the~$(2,1,0)$-weak order, where each congruence~$\equiv$ is replaced by its quotient foam~$\quotientFoam$. See also \Cref{fig:sQuotientopeLattice}.}
	\label{fig:sQuotientFoamLattice}
\end{figure}

%\begin{figure}[h!]
%	\capstart
%	\centerline{\includegraphics[scale=.7]{210-quotientFoamLattice} \hspace{.1cm} \includegraphics[scale=.7]{210-quotientopeLattice}}
%	\caption{The congruence lattice of the~$(2,1,0)$-weak order, where each congruence~$\equiv$ is replaced by its quotient foam~$\quotientFoam$ (left) and by its quotientoplex~$\quotientope$ (right).}
%	\label{fig:sQuotientFoamLattice}
%\end{figure}

\section{Quotientoplexes}
\label{sec:quotientoplexes}

We now realize the $\s$-weak order and its lattice quotients as graphs of polytopal complexes.
Our construction relies on the shard polytopes of~\cite{PadrolPilaudRitter}, which are elementary pieces whose Minkowski sums enable to construct quotientopes for all lattice congruences of the~weak~order.
We first recall their definition.

\begin{definition}[{\cite[Definitions 39 \& 40]{PadrolPilaudRitter}}]
\label{def:shardPolytopes}
Fix an original arc~$\alpha \!\eqdef\! (i, j, A, B)$.
An \defn{$\alpha$-alternating matching} $\altmatch$ is a sequence $i \le i_1 < j_1 < i_2 < j_2 < \dots < i_q < j_q \le j$ such that~$i_p \in \{i\} \cup A$ and~$j_p \in \{j\} \cup B$ for all~$p \in [q]$.
Its \defn{characteristic vector} is~$\charvect_{\altmatch} \eqdef \sum_{p \in [q]} \b{e}_{i_p} - \b{e}_{j_p}$. 
%We denote by $\altmatchset$ the set of all $\alpha$-alternating matchings. 
The \defn{shard polytope}~$\shardPolytope$ is the convex hull of the characteristic vectors of all $\alpha$-alternating matchings. %~$\altmatch$ in $\altmatchset$.
%
%\begin{figure}
%	\capstart
%	\centerline{\includegraphics[scale=.9]{shardPolytopes}}
%	\caption{The shard polytopes of the four arcs of the form~$(1, 4, A, B)$. The vertices of the shard polytope of~$\alpha \eqdef (i, j, A, B)$ are labeled by the corresponding $\alpha$-alternating matchings, where we use solid dots~$\bullet$ for elements in~$\{i\} \cup A$ and hollow dots~$\circ$ for elements in~$B \cup \{j\}$. The corresponding vertex coordinates are directly read replacing~$\bullet$ by~$1$ and~$\circ$ by~$-1$. For instance, the vertex labeled~\raisebox{-.1cm}{\rotatebox{90}{\scalebox{.6}{${\bullet \cdot \cdot \,\circ}$}}} has coordinates~$(1,0,0,-1)$. Adapted from~\cite[Fig.~10]{PadrolPilaudRitter}.}
%	\label{fig:shardPolytiopes}
%\end{figure}

%For any congruence~$\equiv$ of the weak order, the \defn{quotientope}
\end{definition}

%\begin{theorem}[{\cite[Coro.~50]{PadrolPilaudRitter}}]
%\label{thm:quotientopes}
%For any congruence~$\equiv$ of the weak order, the quotient fan~$\quotientFan$ is the normal fan of the \defn{quotientope}~$\quotientope$, obtained as the Minkowski sum of (any positive scaling of) the shard polytopes~$\shardPolytope$, over all arcs~$\alpha$ in~$\cal{A}_\equiv$.
%\end{theorem}

Given an $\s$-arc~$\alpha$, we now associate a face~$\localShardPolytope$ of a shard polytope to each vertex~$\b{q} \in \sTrunks$ of the $\s$-foam, and consider the polytopal complex formed by the collection~$(\localShardPolytope)_{\b{q} \in \sTrunks}$.

\begin{definition}
\label{def:localShardPolytope}
Consider an $\s$-arc~$\alpha \eqdef (i, j, A, B, r)$, the arc~$\tilde\alpha \eqdef (i, j, A, B)$, % denote the corresponding classical arc, 
and a vertex ${\b{q} \in \sTrunks}$ of the $\s$-foam~$\sFoam$.
The \defn{local shard polytope}~$\localShardPolytope$ is the face of the shard polytope~$\shardPolytope[\tilde\alpha]$ maximizing the scalar product with
\(
\sum_{\ell \in {]i,j]}} \big( q_i - q_\ell + r - 1 + \sum_{k \in B \cap {]i,\ell[}} \max(0, s_k-1) \big) \b{e}_\ell.
\)
\end{definition}

\begin{proposition}
\label{prop:shardoplex}
For any $\s$-arc~$\alpha$, the collection of all local shard polytopes~$\localShardPolytope$ for~$\b{q} \in \sTrunks$, together with all their faces, form a polyhedral complex~$\shardoplex$ that we call the \defn{shardoplex} of~$\alpha$.
\end{proposition}

We now construct quotientoplexes as Minkowski sums of shardoplexes, see \Cref{fig:MinkowskiSums}.
%We now use the shardoplexes as basic bricks to construct the quotientoplexes as illustrated in \Cref{fig:MinkowskiSums}

\begin{definition}
\label{def:quotientoplex}
For a congruence~$\equiv$ of the $\s$-weak order~$W_\s$, the \defn{quotientoplex}~$\quotientoplex$ is the polytopal complex obtained as the coordinatewise Minkowski sum~$(\sum_{\alpha \in \cal{A}_\equiv} \shardoplex^{\b{q}})_{\b{q} \in \sTrunks}$ of the shardoplexes~$\shardoplex$ over all $\s$-arcs~$\alpha$ in~$\cal{A}_\equiv$.
%
\begin{figure}[b]
	\capstart
%	\centerline{\includegraphics[scale=1.5]{210-quotientoplex1}}
%	\vspace{-.7cm}
%	\centerline{\includegraphics[scale=1.5]{210-quotientoplex2}}
%	\vspace{-.7cm}
%	\vspace{-.1cm}
	\centerline{\!\!\! \scalebox{.9}{\includegraphics[scale=1.6]{210-quotientoplex3}}}
	\vspace{-.8cm}
	\centerline{\!\!\!\scalebox{.9}{\includegraphics[scale=1.6]{210-quotientoplex4}}}
%	\vspace{-.1cm}
%	\caption{Some quotientoplexes of the $(2,1,0)$-weak order, obtained as Minkowski sums of shardoplexes.}
	\caption{The $\s$-permutahedron (top) and $\s$-associahedron (bottom) obtained as Minkowski sums of shardoplexes, for $\s = (2,1,0)$. See \Cref{fig:foams}.}
	\label{fig:MinkowskiSums}
\end{figure}
\end{definition}

\begin{proposition}
\label{prop:quotientoplex1}
There is an inclusion reversing bijection~$\psi$ from the faces of the quotient foam~$\quotientFoam$ to the faces of the quotientoplex~$\quotientoplex$ such that~$\polytope{F}$ and~$\psi(\polytope{F})$ are orthogonal.
\end{proposition}

\begin{theorem}
\label{thm:quotientoplex2}
For any congruence~$\equiv$ of the $\s$-weak order~$W_\s$, the Hasse diagram of the quotient~$W_\s / {\equiv}$ is isomorphic to the skeleton of the quotientoplex~$\quotientoplex$.
\end{theorem}

%\begin{example}
%See \Cref{fig:MinkowskiSums,fig:sQuotientopeLattice} for $(2,1,0)$-quotientoplexes.
%\end{example}

\begin{remark}
When~$\s = (1, \dots, 1)$, the shardoplexes are the shard polytopes of~\cite{PadrolPilaudRitter}, and the quotientoplexes are the quotientopes of~\cite{PadrolPilaudRitter}.
\end{remark}

\begin{figure}[ht]
	\capstart
	\centerline{\includegraphics[scale=1.1]{210-quotientopeLattice}}
	\caption{The congruence lattice of the~$(2,1,0)$-weak order, where each congruence~$\equiv$ is replaced by its quotientoplex~$\quotientope$. See also \Cref{fig:sQuotientFoamLattice}.}
	\label{fig:sQuotientopeLattice}
\end{figure}

Our next result states that quotientoplexes are polytopal subdivisions of quotientopes.
To be precise, we need to manipulate weighted Minkowski sums of shard polytopes and shardoplexes.
Note that the combinatorics of the resulting quotientopes and quotientoplexes does not depend on these weights, as long as they are positive.

\begin{proposition}
\label{prop:quotientoplex3}
For an $\s$-arc~$\alpha \eqdef (i, j, A, B, r)$, denote by~$\tilde\alpha \eqdef (i, j, A, B)$ the corresponding original arc.
For a congruence~$\equiv$ of the $\s$-weak order~$W_\s$, denote by~$\tilde\equiv$ the corresponding congruence of the weak order~$W_n$, with downset of arcs~$\cal{A}_{\tilde\equiv} \eqdef \set{\tilde\alpha}{\alpha \in \cal{A}_\equiv}$.
Consider~$\b{\lambda} \eqdef (\lambda_\alpha)_{\alpha \in \cal{A}_\equiv}$ with~${\lambda_\alpha > 0}$, and let~$\tilde{\b{\lambda}} \eqdef (\tilde\lambda_{\tilde\alpha})_{\tilde\alpha \in \cal{A}_{\tilde\equiv}}$ with~$\tilde\lambda_{\tilde\alpha} \eqdef \sum_\alpha \lambda_\alpha$ where the sum ranges over all $\s$-arcs~$\alpha$ which project to~$\tilde\alpha$.
Then the quotientoplex~$(\sum_{\alpha \in \cal{A}_\equiv} \lambda_\alpha \, \shardoplex^{\b{q}})_{\b{q} \in \sTrunks}$ is a polytopal subdivision of (a translate of) the quotientope~$\sum_{\tilde\alpha \in \cal{A}_{\tilde\equiv}} \tilde\lambda_{\tilde\alpha} \, \shardPolytope[\tilde\alpha]$.
\end{proposition}

For instance, \Cref{fig:2101-quotientopes} show the $(2,1,0,1)$-permutahedron and the $(2,1,0,1)$-associa\-hedron, which are polytopal subdivisions of a permutahedron and an associahedron.

\begin{figure}[ht]
	\capstart
	\centerline{\includegraphics[scale=.25 ]{2101-permutahedron}\quad\includegraphics[scale=.21 ]{2101-associahedron}}
%	\centerline{\includegraphics[scale=.22]{2101-permutahedron_square}\hspace{.5cm}\includegraphics[scale=.22]{2101-associahedron_square}}
	\caption{The $(2,1,0,1)$-permutahedron and the $(2,1,0,1)$-associahedron.}
	\label{fig:2101-quotientopes}
\end{figure}

Applying \Cref{thm:quotientoplex2,prop:quotientoplex3} to the trivial congruence (where each congruence class contains a single $\s$-tree), we obtain the following statement, answering a question of C.~Ceballos and V.~Pons~\cite{CeballosPons-sWeakOrderI,CeballosPons-sWeakOrderII}.
We note that this question was partially solved in~\cite{DLeonMoralesPhilippeTamayoYip} in the case when~$\s$ contains no~$0$ entry, with a very different method based on a combination of flow polytopes, tropical geometry, and Cayley embedding.

\begin{corollary}
\label{coro:quotientoplexTrivial}
For any $\s$ (including with some~$0$ entries), the Hasse diagram of the $\s$-weak order~$W_\s$ is isomorphic to the oriented skeleton of a polytopal subdivision of a polytope combinatorially equivalent to the zonotope~$\Zono \eqdef \sum_{1 \le i < j \le n} s_i \conv\{\b{e}_i, \b{e}_j\}$.
\end{corollary}

%\begin{figure}[p]
%	\capstart
%	\centerline{\raisebox{1.2cm}{\includegraphics[scale=.8]{120-quotientopeLattice}} \hspace{.5cm} \includegraphics[scale=.8]{210-quotientopeLattice}}
%	\caption{The congruence lattice of~$W_\s$ for $\s = (1,2,0)$ (left) and~$\s = (2,1,0)$ (right), where each congruence~$\equiv$ is replaced by its quotientoplex~$\quotientope$. See also \Cref{fig:sQuotientFoamLattice}.}
%	\label{fig:sQuotientopeLattice}
%\end{figure}

We conclude with some intriguing conjectures on the special congruences discussed in \Cref{rem:specialCongruences}.

\begin{conjecture}
Up to isomorphism, the face lattice of the $\alpha$-Cambrian quotientoplex~$\quotientoplex[\equiv_\alpha]$ % (see \Cref{sec:quotientFoams,sec:quotientoplexes}) 
only depends on the endpoints of~$\alpha$.
\end{conjecture}

\begin{conjecture}
For any fixed~$\s$, changing any~$\upCirc$ to~$\downCirc$ in~$\decoration$ does not affect the cardinality of the $\decoration$-permutree lattice~$W_\s/{\equiv_\decoration}$, although it may change the isomorphism class of the face lattice of the $\decoration$-permutree quotientoplex~$\quotientoplex[\equiv_\decoration]$.
\end{conjecture}

\begin{conjecture}
All faces of the quotientoplex~$\quotientoplex$ are simple polytopes if and only $\equiv$ is regular.
\end{conjecture}

\begin{remark}
When~$\s = (1, \dots, 1)$, these conjectures hold by~\cite{Reading-CambrianLattices}, \cite{PilaudPons-permutrees}, and~\cite{HoangMutze, DemonetIyamaReadingReitenThomas}.
\end{remark}



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