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\title{The Canonical Complex of the Weak Order}

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\author[D.~Albertin and V.~Pilaud]{%
	Doriann Albertin\thanks{\href{mailto:doriann.albertin@u-pem.fr}{doriann.albertin@u-pem.fr}}\addressmark{1} \and %
	%\stepcounter{footnote}%
	Vincent Pilaud\thanks{\href{mailto:vincent.pilaud@lix.polytechnique.fr}{vincent.pilaud@lix.polytechnique.fr}. Supp.~by ANR CAPPS~17\,CE40\,0018 and CHARMS~19\,CE40\,0017.}\addressmark{2}%
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	\addressmark{1}LIGM, Université Gustave Eiffel, CNRS, ESIEE Paris, Marne-la-Vall\'ee, France \\ %
	\addressmark{2}CNRS \& LIX, \'Ecole Polytechnique, Palaiseau, France%
}

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%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{
We study the canonical complex of a finite semidistributive lattice~$L$, a simplicial complex which encodes each interval~$[x,y]$ of~$L$ by recording simultaneously the canonical join representation of~$x$ and the canonical meet representation of~$y$, and behaves properly with respect to lattice quotients of~$L$.
We then describe combinatorially the canonical complex of the weak order on permutations in terms of semi-crossing arc bidiagrams, formed by the superimposition of two non-crossing arc diagrams of N.~Reading.
Finally, we provide an algorithm to describe the Kreweras maps in any lattice quotient of the weak order in terms of semi-crossing arc bidiagrams.
}

%% put your French abstract here, or comment this out if you don't have one
\resumetitle{R\'esum\'e}
\resume{
Nous \'etudions le complexe canonique d'un treillis semidistributif fini~$L$, un complexe simplicial qui encode chaque intervalle~$[x,y]$ de~$L$ par la repr\'esentation canonique par supremums de~$x$ et la repr\'esentation canonique par infimums de~$y$, et qui se comporte bien par rapport aux treillis quotients de~$L$.
Nous d\'ecrivons ensuite combinatoirement le complexe canonique de l'ordre faible sur les permutations en termes de bidiagrammes d'arcs avec des semi-croisements, form\'es par la superposition de deux diagrammes d'arcs sans croisement de N.~Reading.
Finalement, nous pr\'esentons un algorithme pour d\'ecrire les fonctions de Kreweras dans tout treillis quotient de l'ordre faible en termes de bidiagrammes d'arcs.
}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{semidistributive lattices, canonical join representations, weak order, non-crossing arc diagrams, Kreweras complement}

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\begin{document}

\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\section{Introduction}

\medskip
A finite lattice~$L$ is join semidistributive when any element admits a canonical join representation (see \cref{subsec:recollectionLattices} or \cite{FreeseNation} for definitions).
This enables us to define the canonical join complex of~$L$ \cite{Barnard,Reading-arcDiagrams}, whose vertices are the join irreducible elements of~$L$ and whose simplices are the canonical join representations in~$L$.
When $L$ is both join and meet semidistributive, it thus admits both a canonical join complex and a canonical meet complex which are actually isomorphic flag simplicial complexes \cite{Barnard}.

We define the canonical complex of a finite semidistributive lattice~$L$, a larger flag simplicial complex where the canonical join complex and the canonical meet complex live and interact.
More precisely, its vertex set is the disjoint union of the set of join irreducible elements of~$L$ with the set of meet irreducible elements of~$L$, and its simplices are the disjoint unions~$J \sqcup M$ of a canonical join representation~$J$ in~$L$ with a canonical meet representation~$M$ in~$L$ such that~$\bigJoin J \le \bigMeet M$.
In other words, each interval~$[x,y]$ in~$L$ contributes to a simplex of the canonical complex given by the disjoint union of the canonical join representation of~$x$ with the canonical meet representation of~$y$.
This provides a model for the intervals of~$L$ compatible with lattice quotients.
Namely, the canonical complex of a quotient~$L/{\equiv}$ is the subcomplex of the canonical complex of~$L$ induced by the join and meet irreducibles of~$L$ uncontracted by the congruence~$\equiv$.

We then study the combinatorics of the canonical complex of the weak order.
N.~Rea\-ding showed in~\cite{Reading-arcDiagrams} that join irreducible permutations correspond to certain arcs, and that canonical join representations of permutations correspond to non-crossing arc diagrams.
% In other words, the canonical join complex of the weak order is the clique complex on the non-crossing relation among arcs.
We show that the elements of the canonical complex can be interpreted as semi-crossing arc bidiagrams, defined as pairs~$\delta_\join \sqcup \delta_\meet$ of non-crossing arc diagrams where only certain types of crossings are allowed between an arc of~$\delta_\join$ and an arc of~$\delta_\meet$.
It follows that the canonical complex of any quotient of the weak order is isomorphic to a subcomplex of the semi-crossing complex induced by arcs contained in an upper ideal of the subarc order.
%We then provide explicit direct bijections between the semi-crossing arc bidiagrams and the weak order interval posets of G.~Chatel, V.~Pilaud and V.~Pons~\cite{ChatelPilaudPons}, which are both in bijection with the intervals of the weak order.
Finally, we provide an algorithm to describe the Kreweras maps in any lattice quotient of the weak order in terms of semi-crossing arc bidiagrams, generalizing the classical Kreweras complement on non-crossing partitions.

Many details and all proofs omitted in this extended abstract can be found in~\cite{AlbertinPilaud}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The canonical complex of a semidistributive lattice}
\label{sec:canonicalComplex}

%This section deals with canonical meet and join representations in a finite semidistributive lattice and its quotients.
%We start with a recollection on join semidistributive lattices, their canonical join representations, their canonical join complexes, their Kreweras maps, and their lattice congruences (\cref{subsec:recollectionLattices}).
%We then define the canonical complex of a semidistributive lattice~$L$ which encodes the intervals of~$L$ and contains both the canonical join complex and the canonical meet complex of~$L$ (\cref{subsec:canonicalComplex}).

%%%%%%%%%%%%%%%%

\subsection{Recollection on lattices}
\label{subsec:recollectionLattices}

We start by a quick recollection on semidistributive lattices, Kreweras maps and lattice congruences.
All the material covered here is classical, see for instance~\cite{Barnard,FreeseNation,Reading-arcDiagrams,Reading-PosetRegionsChapter}.
%See \eg~\cite{FreeseNation, Reading-PosetRegionsChapter, Reading-arcDiagrams, Barnard}.
%Following \mbox{\cite[Exm.~10]{Barnard}}, we illustrate this section with the case of distributive lattices.

\paragraph{Join representations and semidistributive lattices.}

Consider a finite lattice~$(L, \le, \join, \meet)$ where $\join$ is the join and $\meet$ is the meet.
We see $\join$ and $\meet$ as internal binary operators on~$L$ and try to factorize the elements of~$L$ in some canonical way as products of irreducibles.
%It is first important to understand the irreducible elements for~$\join$ and~$\meet$.

\begin{definition}
An element~$x \in L$ is called \defn{join} (resp.~\defn{meet}) \defn{irreducible} if it covers (resp.~is covered by) a unique element denoted~$x_\star$ (resp.~$x^\star$).
We denote by $\JI$ (resp.~$\MI$) the subposet of~$L$ induced by the set of join (resp.~meet) irreducible elements of~$L$.
\end{definition}

\begin{definition}
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 in~$L$ are antichains of~$L$, and are ordered by containement of the lower sets of their elements (\ie~$J \le J'$ if and only if for any~$y \in J$ there exists~$y' \in J'$ such that~$y \le y'$ in~$L$).
The \defn{canonical join representation} of~$x$, denoted~$\CJR(x)$, is the minimal irredundant join representation of~$x$ for this order, when it exists.
\end{definition}

Note that when it exists, $\CJR(x)$ is an antichain of~$\JI$.
The following statement characterizes the lattices where canonical join representations exist.

\begin{proposition}[{\cite[Theorem~2.24, Theorem~2.56]{FreeseNation}}]
\label{prop:semidistributive}
A finite lattice~$L$ is \defn{join semidistributive} when the following equivalent conditions hold:
\begin{compactenum}[(i)]
\item $x \join y = x \join z$ implies $x \join (y \meet z) = x \join y$ for any~$x, y, z \in L$,
\item for any cover relation~$x \lessdot y$ in~$L$, the set \(K_\join(x,y) \eqdef \set{z \in L}{z \not\le x \text{ but } z \le y}\) has a unique minimal element~$k_\join(x,y)$ (which is then automatically join irreducible),
\item any element of~$L$ admits a canonical join representation.
\end{compactenum}
Moreover, the canonical join representation of~$y \in L$ is~$\CJR(y) = \set{k_\join(x, y)}{x \lessdot y}$.
\end{proposition}

Note that in a finite join semidistributive lattice~$L$, we can associate to any meet irreducible element~$m$ of~$L$ a join irreducible element~$\kappa_\join(m) \eqdef k_\join(m, m^\star)$ of~$L$.
Moreover, the existence of canonical join representations enable us to consider the following complex, initially defined  for the weak order in \cite{Reading-arcDiagrams} and studied in general in~\cite{Barnard}.
See \cref{fig:CCs}.

\begin{definition}
The \defn{canonical join complex}~$\CJC$ of a finite join semidistributive lattice~$L$ is the simplicial complex on~$\JI$ whose faces are the canonical join representations in~$L$.
\end{definition}

%\begin{figure}
%	\centerline{\includegraphics[scale=.8]{CJCs-CMCs-short}}
%	\caption{Some semidistributive lattices and their canonical join (red) and meet (blue) complexes. The letters label all join or meet irreducible elements, and we denote by~$x_\join$ (resp.~$x_\meet$) the element~$x$ when it is considered as a join (resp.~meet) irreducible. Note that we always consistently color joinands in red and meetands in blue. The right lattice is distributive while the left one is only semidistributive.}
%	\label{fig:CJCs-CMCs}
%\end{figure}

The \defn{meet semidistributivity}, the maps~$K_\meet$, $k_\meet$ and~$\kappa_\meet$, the \defn{canonical meet representation}~$\CMR(x)$ and the \defn{canonical meet complex}~$\CMC$ are all defined dually.
A lattice~$L$ is \defn{semidistributive} if it is both meet and join semidistributive.
%In this case, the maps~$\kappa_\join$ and~$\kappa_\meet$ define inverse bijections between~$\MI$ and~$\JI$, and the complexes $\CJC$ and~$\CMC$ behave particularly nicely.

\begin{proposition}[{\cite[Theorem~2 \& Corollary~5]{Barnard}}]
\label{prop:canonicalJoinComplex}
If~$L$ is a finite semidistributive lattice, then
\begin{compactenum}[(i)]
\item $\CJC$ and~$\CMC$ are flag simplicial complexes (\ie their minimal non-faces are edges, or equivalently they are the clique complexes of their graphs),
\item the maps~$\kappa_\join$ and~$\kappa_\meet$ induce inverse isomorphisms between $\CMC$ and~$\CJC$.
\end{compactenum}
\end{proposition}

%In fact, it was proved in~\cite[Theorem~2]{Barnard} that~$\CJC$ is flag if and only if $L$ is semidistributive.
%We will not use the ``only if'' direction in this paper.

\begin{example}[Distributive lattices]
\label{exm:kappaDistributiveLattice}
The name semidistributivity comes from the well understood class of distributive lattices.
A lattice~$L$ is \defn{distributive} if ${x \join (y \meet z) = (x \join y) \meet (x \join z)}$ for any~$x, y, z \in L$.
%Note that the dual condition $x \meet (y \join z) = (x \meet y) \join (x \meet z)$ for any~${x, y, z \in L}$ is actually equivalent to the primal one.
The fundamental theorem for distributive lattices affirms that $L$ is distributive if and only if it is isomorphic to the lattice of lower sets of its join irreducible poset~$P$.
In other words, any antichain of join irreducible elements in~$P$ forms a canonical join representation in~$L$.
Consider, for an antichain~$A$ of~$P$, the two lower sets~$j_A \eqdef \set{x \in P}{x \le y \text{ for some } y \in A}$ and~$m^A \eqdef \set{x \in P}{x \not\ge y \text{ for all } y \in A}$.
Said differently, $A$ is the set of maximal elements of~$j_A$ and the set of minimal elements of~$P \ssm m^A$.
For~$y \in P$, we abbreviate~$j_{\{y\}}$ into~$j_y$ and~$m^{\{y\}}$ into~$m^y$.
%Then for any~$y \in P$, the lower set~$j_{\{y\}}$ is join irreducible as it covers only~$j_{\{y\}} \ssm \{y\}$.
Then 
\begin{compactitem}
\item the join (resp.~meet) irreducibles of~$L$ are precisely the lower sets~$j_y$ (resp.~$m^y$) for~$y \in P$, 
\item the map~$\kappa_\join$ (resp.~$\kappa_\meet$) is given by $\kappa_\join(m^y) = j_y$ (resp.~$\kappa_\meet(j_y) = m^y$),
\item the canonical join representation of~$j_A$ is $\CJR(j_A) = \bigset{j_y}{y \in A}$ and the canonical meet representation of~$m^A$ is~$\CMR(m^A) = \bigset{m^y}{y \in A}$.
\item the canonical join and meet complexes~$\CJC$ and~$\CMC$ are both (isomorphic to) the clique complex on the incomparability graph of~$P$.
\end{compactitem}
See \cite[Exm.~10]{Barnard}.
\end{example}

\paragraph{Kreweras maps.}
\label{subsubsec:Kreweras}

In a semidistributive lattice~$L$, each element has both a canonical join representation and a canonical meet representation.
It is natural to consider the maps that exchange the canonical join representations with the canonical meet representations.

\begin{definition}
The \defn{Kreweras maps}~$\eta_\join : \CMC \to \CJC$ and~$\eta_\meet : \CJC \to \CMC$ are defined by~$\eta_\join(M) \eqdef \CJR \big( \bigMeet M \big)$ and~$\eta_\meet(J) \eqdef \CMR \big( \bigJoin J \big)$.
\end{definition}

% Note that some authors call Kreweras maps the compositions~$\eta_\join \circ \kappa_\meet : \CJC \to \CJC$ and $\eta_\meet \circ \kappa_\join : \CMC \to \CMC$, see for instance~\cite{Barnard}.
The Kreweras maps are related to the Kreweras complement on non-crossing partitions for the Tamari lattice (see \cref{exm:KrewerasComplement}) and to rowmotion for distributive lattices.

\begin{example}[Distributive lattices]
\label{exm:KrewerasDistributiveLattice}
With the notations of~\cref{exm:kappaDistributiveLattice}, for an antichain~$A$ in~$P$, we denote by $\row_\join(A)$ the set of maximal elements of~$m^A$ and by $\row_\meet(A)$ the set of minimal elements of~$P \ssm j_A$.
In other words, we have~$m^A = j_{\row_\join(A)}$ and~$j_A = m_{\row_\meet(A)}$.
Hence, by \cref{exm:kappaDistributiveLattice}, the Kreweras maps~$\eta_\join$ and~$\eta_\meet$ are given by~$\eta_\join(\set{m^y}{y \in A}) = \set{j_y}{y \in \row_\join(A)}$ and~$\eta_\meet(\set{j_y}{y \in A}) = \set{m^y}{y \in \row_\meet(A)}$.
See \cite[Rem.~32]{Barnard}.
\end{example}

\paragraph{Lattice congruences.}

We now discuss quotients of the lattice $L$, considered as an algebraic structure with two internal binary operators~$\join$ and~$\meet$.
%We thus need equivalence relations on~$L$ that respects~$\join$ and~$\meet$.

\begin{definition}
A \defn{congruence}~$\equiv$ on~$L$ 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.
\end{definition}

\begin{definition}
The \defn{lattice quotient}~$L/{\equiv}$ is the lattice structure on the congruence classes, where for any two congruence classes~$X$ and~$Y$, 
\begin{compactitem}
\item the order is given by~$X \le Y$ if and only if $x \le y$ for some representatives~$x \in X$ and~$y \in Y$,
\item the join~$X \join Y$ (resp.~meet~$X \meet Y$) is the congruence class of~$x \join y$ (resp.~$x \meet y$) for any representatives~$x \in X$ and~$y \in Y$.
\end{compactitem}
\end{definition}

The lattice quotient~$L/{\equiv}$ is isomorphic to the subposet of~$L$ induced by the minimal (or maximal) elements in their congruence classes.
It is a join (resp.~meet) subsemilattice of~$L$ but may fail to be a sublattice of~$L$.
We now consider all congruences of~$L$.

\begin{definition}
The \defn{congruence lattice}~$\con(L)$ is the set of all congruences of~$L$ ordered by refinement.
\end{definition}

The congruence lattice~$\con(L)$ is a distributive lattice where the meet is the intersection of relations and the join is the transitive closure of union of relations.
For any join irreducible element~$j \in \JI$, we denote by~$\con(j)$ the unique minimal congruence of~$L$ that \defn{contracts}~$j$, that is with~$j_\star \equiv j$.
It turns out that~$\con(j)$ is join irreducible in~$\con(L)$ and that all join irreducible congruences in~$\con(L)$ are of this form.
Hence, any congruence of~$L$ is completely determined by the set of join irreducible elements of~$L$ that it contracts.
We denote by~$\UJI$ the set of join irreducible elements of~$L$ uncontracted by~$\equiv$.
Not all subsets of join irreducible elements of~$L$ are of the form~$\UJI$ for some congruence~$\equiv$ of~$L$.
The possible subsets are governed by the following relation.

\begin{definition}
For~$j,j' \in \JI$, we say that~$j$ \defn{forces}~$j'$, and write~$j \succcurlyeq j'$, if $\con(j) \ge \con(j')$, that is if any congruence contracting~$j$ also contracts~$j'$.
\end{definition}

The forcing relation is a preorder~$\preccurlyeq$ (\ie a transitive and reflexive, but not necessarily antisymmetric, relation) on $\JI$, whose upper sets correspond to the congruences of~$L$.
%Our next statement characterizes the sets~$\UJI$ as the upper sets of~$\preccurlyeq$ (\ie subsets~$J \subseteq \JI$ such that~$j \succcurlyeq j'$ and~$j \in J$ implies~$j' \in J$).

\begin{proposition}[{\cite[Proposition~9-5.16]{Reading-PosetRegionsChapter}}]
The following conditions are equivalent for~$J \subseteq \JI$:
\begin{compactitem}
\item $J$ is an upper set of the forcing preorder (\ie $j \succcurlyeq j'$ and~$j \in J$ implies~$j' \in J$).
\item $J = \UJI$ for some congruence~$\equiv$ of~$L$.
\end{compactitem}
\end{proposition}

The set~$\UJI$ characterizes~$\equiv$ and enables to understand the elements of~$L$ which are minimal in their congruence classes and their canonical representations as follows.

\begin{proposition}[{\cite[Proposition~9-5.29]{Reading-PosetRegionsChapter} and \cite[Proposition~16]{AlbertinPilaud}}]
\label{prop:canonicalJoinRepresentationsQuotient}
Let~$\equiv$ be a congruence of a finite join semidistributive lattice~$L$.
Then
\begin{compactenum}[(i)]
\item an element~$x \in L$ is minimal in its congruence class if and only if $\CJR(x) \subseteq \UJI$,
\item the quotient~$L/{\equiv}$ is join semidistributive and the canonical joinands of a congruence~class~$X$ in~$L/{\equiv}$ are the congruence classes of the canonical joinands of the minimal element~in~$X$,
\item for any~$x \in L$, the lower ideal of~$L$ generated by~$\CJR(x)$ contains~$\CJR \big( \projDown[\equiv](x) \big)$,
\item the canonical join complex~$\CJC[L/{\equiv}]$ of the quotient~$L/{\equiv}$ is isomorphic to the subcomplex~${\CJC[\equiv]}$ of the canonical join complex~$\CJC$ of~$L$ induced by~$\UJI$.
\end{compactenum}
\end{proposition}

%\cref{prop:canonicalJoinRepresentationsQuotient} translates as follows to the canonical join complex~$\CJC$.
%
%\begin{proposition}
%\label{prop:canonicalJoinComplexQuotient}
%Let~$\equiv$ be a congruence on a finite join semidistributive lattice~$L$.
%Then the canonical join complex~$\CJC[L/{\equiv}]$ of the quotient~$L/{\equiv}$ is isomorphic to the subcomplex~${\CJC[\equiv]}$ of the canonical join complex~$\CJC$ of~$L$ induced by~$\UJI$.
%\end{proposition}

%We will need the following statement relating the canonical join representation of an element~$x$ of~$L$ with the canonical join representation of the minimal element~$\projDown[\equiv](x)$ in its equivalence class.
%It ensures that we can look for~$\CJR \big( \projDown[\equiv](x) \big)$ among the antichains of join irreducible elements of~$L$ uncontracted by~$\equiv$ and below a join irreducible element of~$\CJR(x)$.
%Unfortunately, it remains difficult in general to describe~$\CJR \big( \projDown[\equiv](x) \big)$ because not all such antichains define a canonical representation of~$L$.
%%We provide a proof here as we have not found this statement explicitly in the literature.
%
%\begin{proposition}
%\label{prop:canonicalJoinRepresentationProjection}
%Let~$\equiv$ be a congruence of a finite join semidistributive lattice~$L$.
%For any~$x \in L$, the lower ideal of~$L$ generated by~$\CJR(x)$ contains~$\CJR \big( \projDown[\equiv](x) \big)$.
%\end{proposition}

%\begin{proof}
%The proof works by induction on the size of the interval~$[\projDown[\equiv](x), x]$. 
%The statement is immediate if~$\projDown[\equiv](x) = x$.
%Otherwise, there exists~$j \in \CJR(x) \ssm \UJI$.
%Let~$y \eqdef \bigJoin \big( \CJR(x) \symdif \{j, j_\star\} \big)$, where~$\symdif$ denotes the symmetric difference.
%Since~$j \equiv j_\star$, we have~$x \equiv y$ and thus~$\projDown[\equiv](x) = \projDown[\equiv](y)$.
%Since~$y$ has a join representation strictly contained in the lower ideal of~$L$ generated by~$\CJR(x)$, we have~$y < x$ so that~$[\projDown[\equiv](y), y]$ is strictly contained in~$[\projDown[\equiv](x), x]$.
%By induction hypothesis, we thus obtain that~$\CJR \big( \projDown[\equiv](y) \big)$ is contained in the lower ideal of~$L$ generated by~$\CJR(y)$.
%Moreover, observe that~$J \eqdef \big( \CJR(x) \ssm \{j\} \big) \cup \CJR(j_\star)$ is a join representation for~$y$, so that~$\CJR(y)$ is contained in the lower ideal of~$L$ generated by~$J$, which is itself contained in the lower ideal of~$L$ generated by~$\CJR(x)$ (since any element of~$\CJR(j_\star)$ is lower than~$j_\star$ and thus than~$j$).
%We conclude that~$\CJR \big( \projDown[\equiv](x) \big)$ is indeed in the lower ideal of~$L$ generated by~$\CJR(x)$.
%\end{proof}

% The lattice~$L$ is called \defn{congruence uniform} if the map~$j \mapsto \con(j_\star,j)$ (resp.~$m \mapsto \con(m, m^\star)$) is a bijection between the join (resp.~meet) irreducible elements of~$L$ and that of~$\con(L)$.
% In other words, when~$L$ is congruence uniform, the congruence lattice~$\con(L)$ is isomorphic to the set of lower sets of the \defn{forcing order} on join irreducibles of~$L$, defined by~$j \prec j'$ if $\con(j'_\star, j')$ refines $\con(j_\star, j)$.
Dual statements hold using meets instead of joins, and we denote by~$\UMI$ the meet irreducible elements of~$L$ uncontracted by~$\equiv$, and by~$\CMC[\equiv]$ the subcomplex of~$\CMC$ induced by~$\UMI$ for a congruence~$\equiv$ on~$L$.
Due to \cref{prop:canonicalJoinRepresentationsQuotient}\,(iv), we will always work with the subcomplexes~$\CJC[\equiv]$ and~$\CMC[\equiv]$ rather than with the complexes~$\CJC[L/{\equiv}]$ and~$\CMC[L/{\equiv}]$.
%
When the lattice~$L$ is semidistributive, the two sets~$\UJI$ and~$\UMI$ and the two subcomplexes~$\CJC[\equiv]$ and~$\CMC[\equiv]$ are connected by the maps~$\kappa_\join$ and~$\kappa_\meet$.
%We provide a proof here as we have not found this statement explicitly in the literature.

\begin{proposition}
\label{prop:uncontractedElements}
Let~$\equiv$ be a congruence on a finite semidistributive lattice~$L$.
Then we have ${\UJI = \kappa_\join \big( \UMI \big)}$ and~$\UMI = \kappa_\meet \big( \UJI \big)$.
Hence, the maps~$\kappa_\join$ and~$\kappa_\meet$ induce inverse isomorphisms between the subcomplexes~$\CMC[\equiv]$ and~$\CJC[\equiv]$.
\end{proposition}

%\begin{proof}
%Let~$m \in \MI$ and~$j = \kappa_\join(m)$.
%By definition, we have~$m \join j = m^\star$ and~$m \join j_\star = m$.
%Hence, $j \equiv j_\star$ implies that~$m = m \join j_\star \equiv m \join j = m^\star$.
%In other words, $\kappa_\join \big( \UMI \big) \subseteq \UJI$.
%By symmetry, we have~$\kappa_\meet \big( \UJI \big) \subseteq \UMI$.
%Since~$\kappa_\join$ and~$\kappa_\meet$ are reversed bijections, this yields equalities.
%The last sentence of the statement thus follows from \cref{prop:canonicalJoinComplex}\,(ii).
%\end{proof}

\begin{example}[Distributive lattices]
\label{exm:congruencesDistributiveLattice}
In a distributive lattice~$L$, there is no forcing at all.
Hence, any subset of join irreducible elements of~$L$ defines a congruence of~$L$.
In other words, with the notations of~\cref{exm:kappaDistributiveLattice}, any subset~$Y$ of~$P$ defines a congruence~$\equiv_Y$ with~${\UJI[\equiv_Y] \! = \! \set{j_y}{y \in Y}}$ and~${\UMI[\equiv_Y] = \set{m^y}{y \in Y}}$.
The lattice quotient~$L/{\equiv}$ is again distributive and isomorphic to the lattice of lower ideals of the restriction of the poset~$P$ to~$Y$.
\end{example}

%%%%%%%%%%%%%%%%

\subsection{The canonical complex}
\label{subsec:canonicalComplex}

%We still consider in this section a semidistributive lattice~$L$.
We now define another complex that connects the canonical join complex~$\CJC$ to the canonical meet complex~$\CMC$ using intervals of~$L$.
See \cref{fig:CCs}.

\begin{definition}
\label{def:canonicalComplex}
The \defn{canonical complex}~$\CC$ of a finite semidistributive lattice~$L$ is the simplicial complex whose
ground set is the \textbf{disjoint} union~$\JI \sqcup \MI$, and whose faces are the \textbf{disjoint} unions~$J \sqcup M$ where~$J \in \CJC$ while $M \in \CMC$ and~$\bigJoin J \le \bigMeet M$.
%\begin{compactitem}
%%\item ground set is the \textbf{disjoint} union~$\JI \sqcup \MI$ of the sets of join irreducible and of meet irreducible elements of~$L$, and
%%\item faces are the \textbf{disjoint} unions~$J \sqcup M$ where~$J \in \CJC$ is a canonical join representation, $M \in \CMC$ is a canonical meet representation, and~$\bigJoin J \le \bigMeet M$.
%\item ground set is the \textbf{disjoint} union~$\JI \sqcup \MI$, and
%\item faces are the \textbf{disjoint} unions~$J \sqcup M$ where~$J \in \CJC$, $M \in \CMC$, and~$\bigJoin J \le \bigMeet M$.
%\end{compactitem}
\end{definition}

\begin{figure}[b]
	\centerline{\includegraphics[scale=.85]{CCs-short}}
	\caption{The canonical complexes of two semidistributive lattices. The corresponding join (resp.~meet) canonical complexes are highlighted in red (resp.~blue). Since the canonical complexes are flag by \cref{prop:canonicalComplex}, it is sufficient to represent their graphs. The letters label all join or meet irreducible elements, and we denote by~$x_\join$ (resp.~$x_\meet$) the element~$x$ when it is considered as a join (resp.~meet) irreducible.}
	% The vertices of the canonical complex are positioned so that the map $\kappa$ of \cref{rem:symmetryKappa} acts by central symmetry.}
	\label{fig:CCs}
\end{figure}

To avoid any confusion, let us insist that an element appears twice in~$\CC$ if it is both join and meet irreducible.
%We will thus always write the faces of~$\CC$ explicitly as disjoint unions.
%In our pictures, we denote by~$x_\join$ (resp.~$x_\meet$) and color red (resp.~blue) the element~$x \in L$ considered as a join (resp.~meet) irreducible.
We now gather some relevant properties of~$\CC$.

\begin{proposition}
\label{prop:canonicalComplex}
For a finite semidistributive lattice~$L$, 
\begin{compactenum}[(i)]
\item the canonical join (resp.~meet) complex $\CJC$ (resp.~$\CMC$) is the subcomplex of the canonical complex~$\CC$ induced by~$\JI$ (resp.~$\MI$),
\item the faces of the canonical complex~$\CC$ are in bijection with the intervals of~$L$,
\item the canonical complex~$\CC$ is a flag simplicial complex,
\item for any~$j \in \JI$, the pair~$\{j, \kappa_\meet(j)\}$ is not in~$\CC$, so that the canonical complex $\CC$ can be embedded in the boundary of the $|\JI|$-dimensional cross-polytope,
\item for any congruence~$\equiv$ on~$L$, the canonical complex~$\CC[L/{\equiv}]$ of the quotient~$L/{\equiv}$ is isomorphic to the subcomplex~$\CC[\equiv]$ of the canonical complex~$\CC$ of~$L$ induced by the disjoint union~$\UJI \sqcup \UMI$ of the join and meet irreducible elements uncontracted~by~$\equiv$.
\end{compactenum}
\end{proposition}

%\begin{example}
%The canonical complex of the boolean lattice on~$[n]$ is isomorphic to the boundary of the $n$-dimensional cross-polytope.
%\end{example}

\begin{example}[Distributive lattices]
\label{exm:canonicalComplexDistributiveLattice}
With the notations of~\cref{exm:kappaDistributiveLattice}, we have~$j_y \subseteq m^z$ if and only if~$y \not\ge z$.
Hence, the canonical complex~$\CC$ is the clique complex of the graph whose vertex set is made of two copies~$P_\join$ and~$P_\meet$ of~$P$ and whose edge set is the union of two copies~$I_\join$ and~$I_\meet$ of the incomparability graph of~$P$ with the edges~$\{y_\join, z_\meet\}$ for~$y \not\ge z$ in~$P$.
\end{example}

%%%%%%%%%%%%%%%%

%\subsection{Kreweras maps}
%\label{subsec:Kreweras}
%
%We have seen in \cref{prop:canonicalJoinComplex}\,(ii) that the canonical join complex~$\CJC$ and the canonical meet complex~$\CMC$ of a semidistributive lattice~$L$ are connected via the maps~$\kappa_\join$ and~$\kappa_\meet$.
%We now define an alternative connection between~$\CJC$ and~$\CMC$.
%
%\subsubsection{Kreweras maps}
%\label{subsec:KreverasMaps}
%
%In a semidistributive lattice~$L$, each element has both a canonical join representation and a canonical meet representation.
%It is natural to consider the maps that exchange the canonical join representation with the canonical meet representation of the same element.
%
%\begin{definition}
%The \defn{Kreweras maps}~$\eta_\join : \CMC \to \CJC$ and~$\eta_\meet : \CJC \to \CMC$ are defined by
%\[
%\eta_\join(M) \eqdef \CJR \big( \bigMeet M \big)
%\qquad\text{and}\qquad
%\eta_\meet(J) \eqdef \CMR \big( \bigJoin J \big).
%\]
%\end{definition}
%
%% $\kappa_\join^\star : \CMC \to \CMC$ the map defined by~$\kappa_\join^\star(M) \eqdef \CMR \big( \bigJoin_{m \in M} \kappa_\join(m) \big)$.
%Note that some authors call Kreweras maps the compositions~$\eta_\join \circ \kappa_\meet : \CJC \to \CJC$ and $\eta_\meet \circ \kappa_\join : \CMC \to \CMC$, see for instance~\cite{Barnard}.
%As will be discussed in \cref{exm:KrewerasComplement}, the Kreweras maps for the Tamari lattice are closely related to the classical Kreweras complement on non-crossing partitions.
%\vincent{todo}
%For the moment, we recall that the Kreweras maps for the distributive lattices are related to rowmotion.
%
%\begin{example}[Distributive lattices]
%\label{exm:KrewerasDistributiveLattice}
%With the notations of~\cref{exm:kappaDistributiveLattice}, for an antichain~$A$ in~$P$, we denote by $\row_\join(A)$ the set of maximal elements of~$m^A$ and by $\row_\meet(A)$ the set of minimal elements of~$P \ssm j_A$.
%In other words, we have~$m^A = j_{\row_\join(A)}$ and~$j_A = m_{\row_\meet(A)}$.
%Hence, by \cref{exm:canonicalComplexDistributiveLattice}, the Kreweras maps~$\eta_\join$ and~$\eta_\meet$ are given by
%\[
%\eta_\join(\set{m^y}{y \in A}) = \set{j_y}{y \in \row_\join(A)}
%\qquad\text{and}\qquad
%\eta_\meet(\set{j_y}{y \in A}) = \set{m^y}{y \in \row_\meet(A)}.
%\]
%See \cite[Rem.~32]{Barnard}.
%\end{example}
%
%For a congruence~$\equiv$ on~$L$, we have seen that the quotient~$L/{\equiv}$ is still semidistributive, and thus admits its own Kreweras maps.
%The goal of this section is to express the Kreweras maps in the quotient~$L/{\equiv}$ in terms of that in~$L$.
%While this is difficult in general, we will consider a special family of lattices in which a simple description exists.
%
%\subsubsection{Canonical coherent lattices}
%\label{subsubsec:canonicalCoherent}
%
%At the moment, consider a congruence~$\equiv$ on a join semidistributive lattice~$L$.
%Recall that we denote by~$\CJR(x)$ the canonical join representation of~$x \in L$, by~$\projDown[\equiv](x)$ the minimum element in the congruence class of~$x \in L$, and by~$\UJI$ the set of join irreducible elements of~$L$ uncontracted by~$\equiv$.
%We start by an observation on the canonical representations of~$\projDown[\equiv](x)$.
%
%\begin{proposition}
%\label{prop:canonicalJoinRepresentationProjection}
%For any~$x \in L$, the lower ideal of~$L$ generated by~$\CJR(x)$ contains~$\CJR \big( \projDown[\equiv](x) \big)$.
%\end{proposition}
%
%\begin{proof}
%The proof works by induction on the size of the interval~$[\projDown[\equiv](x), x]$. 
%The statement is immediate if~$\projDown[\equiv](x) = x$.
%Otherwise, there exists~$j \in \CJR(x) \ssm \UJI$.
%Let~$y \eqdef \bigJoin \big( \CJR(x) \symdif \{j, j_\star\} \big)$, where~$\symdif$ denotes the symmetric difference.
%Since~$j \equiv j_\star$, we have~$x \equiv y$ and thus~$\projDown[\equiv](x) = \projDown[\equiv](y)$.
%Since~$y$ has a join representation strictly contained in the lower ideal of~$L$ generated by~$\CJR(x)$, we have~$y < x$ so that~$[\projDown[\equiv](y), y]$ is strictly contained in~$[\projDown[\equiv](x), x]$.
%By induction hypothesis, we thus obtain that~$\CJR \big( \projDown[\equiv](y) \big)$ is contained in the lower ideal of~$L$ generated by~$\CJR(y)$.
%Moreover, observe that~$J \eqdef \big( \CJR(x) \ssm \{j\} \big) \cup \CJR(j_\star)$ is a join representation for~$y$, so that~$\CJR(y)$ is contained in the lower ideal of~$L$ generated by~$J$, which is itself contained in the lower ideal of~$L$ generated by~$\CJR(x)$ (since any element of~$\CJR(j_\star)$ is lower than~$j_\star$ and thus than~$j$).
%We conclude that~$\CJR \big( \projDown[\equiv](x) \big)$ is indeed in the lower ideal of~$L$ generated by~$\CJR(x)$.
%\end{proof}
%
%\cref{prop:canonicalJoinRepresentationProjection} ensures that we can look for~$\CJR \big( \projDown[\equiv](x) \big)$ among the antichains of join irreducible elements of~$L$ uncontracted by~$\equiv$ and below a join irreducible element of~$\CJR(x)$.
%Unfortunately, it remains difficult in general to describe~$\CJR \big( \projDown[\equiv](x) \big)$ because not all such antichains define a canonical representation of~$L$.
%The following natural condition on the lattice~$L$ fixes this problem and thus enables to describe precisely~$\CJR \big( \projDown[\equiv](x) \big)$ in terms of~$\CJR(x)$.
%
%\begin{definition}
%\label{def:joinCoherent}
%A join semidistristributive lattice is \defn{join canonical coherent} if $J \in \CJC$ implies~${J' \in \CJC}$ for any antichains~$J, J'$ of~$\JI$ such that~$J'$ is contained in the lower ideal of~$L$ generated by~$J$.
%\end{definition}
%
%% In other words, $L$ is join coherent when $\CJC$ forms a lower ideal in the distributive lattice defined by the lower ideals of the restriction of~$L$ to~$\JI$.
%
%\begin{remark}\label{rem:coherent}
%It follows from \cref{prop:canonicalJoinComplex}\,(i) that a semidistributive lattice~$L$ is join canonical coherent if and only if $\{j_1, j_2\} \in \CJC$ implies $\{j'_1, j'_2\} \in \CJC$ for any~$j_1, j_2, j'_1, j'_2 \in \JI$ (not necessarily distinct) such that~$j_1 \ge j'_1$ and~$j_2 \ge j'_2$.
%\end{remark}
%
%Note that the condition of \cref{def:joinCoherent} is not connected to congruence normality or congruence uniformity.
%In particular, it is not stable by the interval doubling construction of~\cite{Day}.
%See for instance \cref{fig:noncoherent}\,(left).
%
%\begin{figure}
%	\centerline{\includegraphics{doublableNonCoherent}}
%	\caption{A lattice (left), its restriction to its join irreducible elements (middle), and its poset of antichains of join irreducible elements (right), where the canonical join representations are highlighted in red. As the red canonical join representations do not for a lower ideal of the poset of antichains of join irreducible elements, the lattice is not canonical coherent. In contrast, removing~$e$ in all pictures shows that the hexagon lattice is join canonical coherent.}
%	\label{fig:noncoherent}
%\end{figure}
%
%\begin{proposition}
%\label{prop:canonicalJoinRepresentationProjectionJoinCoherent}
%If~$L$ is a join canonical coherent, then~$\CJR \big( \projDown[\equiv](x) \big)$ is the set of maximal elements of~$\set{j' \in \UJI}{j' \le j \text{ for some } j \in \CJR(x)}$.
%\end{proposition}
%
%\begin{proof}
%Let~$J \eqdef \CJR(x)$, let~$J'$ be the set of maximal elements of~$\set{j' \in \UJI}{j' \le j \text{ for some } j \in J}$ and let~$J'' \eqdef \CJR \big( \projDown[\equiv](x) \big)$.
%By \cref{prop:canonicalJoinRepresentationProjection}, $J'' \subseteq \set{j' \in \UJI}{j' \le j \text{ for some } j \in J}$.
%By maximality of the elements of~$J'$, we thus obtain that $J''$ is contained in the lower ideal of~$L$ generated by~$J'$, which is contained in that generated by~$J$.
%% Hence the lower ideal of~$L$ generated by~$J$ contains that generated by~$J'$ which contains that generated by~$J''$.
%Therefore, $\bigJoin J'' \le \bigJoin J' \le \bigJoin J$.
%Since~$\bigJoin J'' \equiv \bigJoin J'$, we obtain that~$\bigJoin J' \equiv \bigJoin J$.
%Moreover, we have~$J' \subseteq \UJI$ by definition and~$J' \in \CJC$ since $J \in \CJC$ and~$L$ is join canonical coherent.
%Hence we obtain that~$J' = \CJR \big( \projDown[\equiv](x) \big)$.
%\end{proof}
%
%Dual statements to \cref{prop:canonicalJoinRepresentationProjection,prop:canonicalJoinRepresentationProjectionJoinCoherent} hold for~$\projUp[\equiv](x)$ in a meet canonical coherent meet semidistributive lattice.
%In the sequel, we will focus on lattices which are both join and meet canonical coherent.
%
%\begin{definition}
%\label{def:coherent}
%We say that a semidistributive lattice~$L$ is \defn{canonical coherent} when it is both join and meet canonical coherent.
%\end{definition}
%
%\begin{example}[Distributive lattices]
%All distributive lattices are canonical coherent, since any antichain of join irreducible elements forms a canonical join representation.
%\end{example}
%
%\subsubsection{Kreweras maps in quotients of coherent lattices}
%\label{subsubsec:KrewerasCoherent}
%
%We now assume that~$L$ is a canonical coherent semidistributive lattice.
%In that situation, \cref{prop:canonicalJoinRepresentationProjectionJoinCoherent} enables us to compute the Kreweras maps in~$L/{\equiv}$ from that in~$L$.
%Namely, identify the canonical join complex~$\CJC[L/{\equiv}]$ with the subcomplex~$\CJC[\equiv]$, and the canonical meet complex~$\CMC[L/{\equiv}]$ with the subcomplex~$\CMC[\equiv]$ as in \cref{prop:canonicalJoinComplexQuotient}.
%Denote the Kreweras maps in~$L$ by~$\eta_\join$ and~$\eta_\meet$, and the Kreweras maps in~$L/{\equiv}$ by~$\eta_\join^\equiv$ and~$\eta_\meet^\equiv$. % (see \cref{subsec:KreverasMaps}).
%%Then it follows from \cref{prop:canonicalJoinRepresentationProjectionJoinCoherent} that, if~$M \in \CMC[\equiv]$, then $\eta_\join^\equiv(M)$ is the set of maximal elements of~$\set{j' \in \UJI}{j' \le j \text{ for some } j \in \eta_\join(M)}$ and, if~$J \in \CJC[\equiv]$, then $\eta_\meet^\equiv(J)$ is the set of minimal elements of~$\set{m' \in \UMI}{m' \ge m \text{ for some } m \in \eta_\meet(J)}$.
%
%\begin{proposition}
%\label{prop:KrewerasMapsCoherent}
%Consider a canonical coherent distributive lattice~$L$.
%If~$M \in \CMC[\equiv]$, then $\eta_\join^\equiv(M)$ is the set of maximal elements of~$\set{j' \in \UJI}{j' \le j \text{ for some } j \in \eta_\join(M)}$ and, if~${J \in \CJC[\equiv]}$, then $\eta_\meet^\equiv(J)$ is the set of minimal elements of~$\set{m' \in \UMI}{m' \ge m \text{ for some } m \in \eta_\meet(J)}$.
%\end{proposition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Semi-crossing arc bidiagrams}
\label{sec:bidiagrams}

%In this section, we apply the lattice theoretic results presented in \cref{sec:canonicalComplex} to the classical weak order on permutations.
%We first recall that the canonical join and meet representations of the weak order can be encoded as non-crossing arc diagrams as defined by N.~Reading in~\cite{Reading-arcDiagrams} (\cref{subsec:NCADs}).
%We then briefly study the restriction of the weak order on join or meet irreducibles in terms of arcs (\cref{subsec:weakOrderArcs}).
%We then describe the intervals and the canonical complex of the weak order in terms of semi-crossing arc diagrams (\cref{subsec:SCABs}).
%%We then provide direct bijections between the semi-crossing arc bidiagrams and the weak order interval posets of~\cite{ChatelPilaudPons} (\cref{subsec:WOIPs}).
%Finally, we provide an algorithm to compute the Kreweras maps in any lattice quotient of the weak order, generalizing the classical Kreweras complement on non-crossing partitions (\cref{subsec:KrewerasBidiagrams}).

%%%%%%%%%%%%%%%%

\subsection{Non-crossing arc diagrams}
\label{subsec:NCADs}

\enlargethispage{.2cm}
We consider the set~$\f{S}_n$ of permutations of $[n] \eqdef \{1, \dots, n\}$.
An \defn{inversion} of a permutation $\sigma \in \f{S}_n$ is a pair $(u,v)$ with~$1 \le u < v \le n$ and~$\sigma^{-1}(u) > \sigma^{-1}(v)$. % (in other words, $u$ is smaller than~$v$ but $u$ appears after~$v$ in~$\sigma$).
The \defn{weak order} is the lattice on~$\f{S}_n$ defined by inclusion of inversion sets. %, illustrated in \cref{fig:weakOrder}.
Note that a cover relation corresponds to the swap of two values~$\sigma_i$ and~$\sigma_{i+1}$ at consecutive positions.
The swap is increasing in the weak order if $i$ is an \defn{ascent} \ie $\sigma_i < \sigma_{i+1}$, and decreasing if $i$ is a \defn{descent} \ie $\sigma_i > \sigma_{i+1}$.
It is classical that the weak order is semidistributive.
We now describe its join (resp.~meet) irreducible elements and its canonical join (resp.~meet) representations in terms of the arcs and non-crossing arc diagrams introduced by N.~Reading~in~\cite{Reading-arcDiagrams}.

\begin{definition}[\cite{Reading-arcDiagrams}]
\label{def:NCAD}
An \defn{arc} is a quadruple~$(a,b,A,B)$ where $1 \le a < b \le n$ and $A \sqcup B$ forms a partition of~$]a,b[ \eqdef \{a+1, \dots, b-1\}$.
Two arcs~$\alpha \eqdef (a, b, A, B)$ and~$\alpha' \eqdef (a', b', A', B')$ \defn{cross} if there are~$u \ne v$ with ${u \in (A' \cup \{a', b'\}) \cap (B \cup \{a, b\})}$ and ${v \in (A \cup \{a, b\}) \cap (B' \cup \{a', b'\})}$.
A \defn{non-crossing arc diagram} (\defn{NCAD}) is a collection of pairwise non-crossing arcs.
The \defn{non-crossing complex} is the clique complex of the non-crossing relation on all arcs.
\end{definition}

\begin{remark}
Visually, an arc~$(a, b, A, B)$ is represented by an $x$-monotone curve wiggling around the horizontal axis, starting at~$a$ and ending at~$b$, and passing above points of $A$ and below points of $B$. 
Two arcs cross if they cross in their interior or start at the same point or end at the same point (but they do not cross if one ends where the other starts).
See \cref{fig:diagExamples}.% for illustrations of arcs and of non-crossing arc diagrams.
\end{remark}

Observe that the join (resp.~meet) irreducible elements of the weak order are precisely the permutations with exactly one descent (resp.~ascent).
Hence, we associate to an arc~$\alpha \eqdef (a, b, A, B)$ with~$A \eqdef \{a_1 < \dots < a_k\}$ and~$B \eqdef \{b_1 < \dots < b_\ell\}$
\begin{compactitem}
\item a join irreducible $\b{\sigma}_\join(\alpha) \eqdef [1, \dots, (a-1), a_1, \dots, a_k, b, a, b_1, \dots, b_\ell, (b+1), \dots, n]$, 
\item a meet irreducible $\b{\sigma}_\meet(\alpha) \eqdef [n, \dots, (b+1), a_k, \dots, a_1, a, b, b_\ell, \dots, b_1, (a-1), \dots, 1]$.
\end{compactitem}
%where we use the one-line notation of permutations~$\sigma = [\sigma_1, \dots, \sigma_n]$.

Conversely, consider a permutation~$\sigma \in \f{S}_n$ represented by its permutation table formed by dots at coordinates~$(\sigma_i,i)$ for~$i \in [n]$.
Draw segments between consecutive dots~$(\sigma_i, i)$ and~$(\sigma_{i+1}, i+1)$, colored red for a descent~$\sigma_i > \sigma_{i+1}$ and blue for an ascent~$\sigma_i < \sigma_{i+1}$.
Finally, flatten the picture vertically to the horizontal line, allowing segments to bend but not to pass points.
The resulting picture is the superimposition of a set~$\b{\delta}_\join(\sigma)$ of red arcs and a set~$\b{\delta}_\meet(\sigma)$ of blue arcs.
See \cref{fig:diagExamples}.
More formally, $\b{\delta}_\join(\sigma) \eqdef \set{\b{\alpha}_\join(\sigma, i)}{\sigma_i < \sigma_{i+1}}$ and $\b{\delta}_\meet(\sigma) \eqdef \set{\b{\alpha}_\meet(\sigma, i)}{\sigma_i > \sigma_{i+1}}$ 
where
\begin{align*}
\b{\alpha}_\join(\sigma, i) & \eqdef (\sigma_i, \sigma_{i+1}, \set{\sigma_j}{j < i \text{ and } \sigma_i < \sigma_j < \sigma_{i+1}}), \set{\sigma_j}{j > i+1 \text{ and } \sigma_i < \sigma_j < \sigma_{i+1}}), \\ % \quad\text{when } \sigma_i < \sigma_{i+1} \\
\b{\alpha}_\meet(\sigma, i) & \eqdef (\sigma_{i+1}, \sigma_i, \set{\sigma_j}{j < i \text{ and } \sigma_i > \sigma_j > \sigma_{i+1}}, \set{\sigma_j}{j > i+1 \text{ and } \sigma_i > \sigma_j > \sigma_{i+1} })). %\quad\text{when } \sigma_i > \sigma_{i+1}.
\end{align*}

\begin{proposition}[\cite{Reading-arcDiagrams}]
\label{prop:NCAD}
The map~$\b{\delta}_\join$ (resp.~$\b{\delta}_\meet$) is a bijection between the permutations and the non-crossing arc diagrams.
Moreover, the canonical join (resp.~meet) representation of~$\sigma \in \f{S}_n$ is~$\CJR(\sigma) = \set{\b{\sigma}_\join(\alpha_\join)}{\alpha_\join \in \b{\delta}_\join(\sigma)}$ (resp.~$\CMR(\sigma) = \set{\b{\sigma}_\meet(\alpha_\meet)}{\alpha_\meet \in \b{\delta}_\meet(\sigma)}$).
Hence, the canonical join (resp.~meet) complex of the weak order is isomorphic to the non-crossing complex.
\end{proposition}

\pagebreak

\begin{figure}
	\centerline{\includegraphics[scale=.85]{arcBidiagrams}}
	\caption{NCADs and SCABs of the permutations $2537146$, $2531746$, $2513746$, and $2513476$. The first line represents the table~$(\sigma(i), i)$ of a permutation~$\sigma$ with ascents in blue and descents in red, the second line is the join diagram $\b{\delta}_\join(\sigma)$, the third line is the meet diagram $\b{\delta}_\meet(\sigma)$, and the fourth line is the superimposition~$\b{\delta}_\join(\sigma) \sqcup \b{\delta}_\meet(\sigma)$.}
	\label{fig:diagExamples}
\end{figure}

By construction, the non-crossing arc diagrams are adapted to the maps~$\kappa_\join$ and~$\kappa_\meet$ and to quotients of the weak order.
%First, our next statement says that the maps~$\kappa_\join$ and~$\kappa_\join$ only change the colors of the arcs.
%We provide a proof as we have not found it explicitly in~\cite{Reading-arcDiagrams}.

\begin{proposition}
\label{prop:kappaChangesColor}
$\kappa_\join(\b{\sigma}_\meet(\alpha)) = \b{\sigma}_\join(\alpha)$ and~$\kappa_\meet(\b{\sigma}_\join(\alpha)) = \b{\sigma}_\meet(\alpha)$ for any arc~$\alpha$.
\end{proposition}

%\begin{proof}
%Let~$\alpha \eqdef (a, b, A, B)$, let~$\sigma \eqdef \b{\sigma}_\meet(\alpha) = [n, \dots, (b+1), a_k, \dots, a_1, a, b, b_\ell, \dots, b_1, (a-1), \dots, 1]$ and let~$\sigma^\star = [n, \dots, (b+1), a_k, \dots, a_1, b, a, b_\ell, \dots, b_1, (a-1), \dots, 1]$ denote the only element covering~$\sigma$.
%Any permutation~$\tau$ with~$\tau \not\le \sigma$ but~$\tau \le \sigma^\star$ must have all elements of~$A$ before~$b$ before~$a$ before all elements of~$B$.
%Therefore, the minimal such permutation is clearly $\b{\sigma}_\join(\alpha) = [1, \dots, (a-1), a_1, \dots, a_k, b, a, b_1, \dots, b_\ell, (b+1), \dots, n]$.
%\end{proof}

\begin{proposition}[\cite{Reading-arcDiagrams}]
\label{prop:subarcs}
For any two arcs~$\alpha \eqdef (a, b, A, B)$ and~$\alpha' \eqdef (a', b', A', B')$, the join irreducible $\b{\sigma}_\join(\alpha)$ forces the join irreducible~$\b{\sigma}_\join(\alpha')$ if and only if~$\alpha$ is a \defn{subarc} of~$\alpha'$, meaning that ${a' \le a < b \le b'}$ and~$A \subseteq A'$ while~$B \subseteq B'$.
Hence, to each upper ideal~$I$ of the subarc order corresponds a lattice congruence~$\equiv_I$ of the weak order, and the canonical join (resp.~meet) complex of the quotient of the weak order by~$\equiv_I$ is isomorphic to the non-crossing complex on~$I$.
%The lattice congruences of the weak order are thus in bijection with the lower ideals of the subarc order.
%The canonical join (resp.~meet) complex of a quotient of the weak order is isomorphic to the non-crossing complex on the corresponding lower ideal of the subarc order.
\end{proposition}

\begin{remark}
%\begin{figure}[b]
%	\centerline{\includegraphics[scale=.5]{forcing35}}
%	\caption{The subarc order (or forcing order) on arcs of sizes $3$ to $5$.}
%	\label{fig:forcing35}
%\end{figure}
%%
Visually, $\alpha$ is a subarc of~$\alpha'$ if the endpoints of~$\alpha$ are weakly in between the endpoints of~$\alpha'$, and $\alpha$ follows $\alpha'$ between its endpoints.
%The subarc order on arcs of size~$3$ to~$5$ is represented in \cref{fig:forcing35}.
\end{remark}

\begin{example}
The prototypical congruence of the weak order is the \defn{sylvester congruence}~$\equiv_\textrm{sylv}$~\cite{LodayRonco,HivertNovelliThibon-algebraBinarySearchTrees}, corresponding to the upper ideal of the subarc order given by all up arcs~$(a, b, {]a,b[}, \varnothing)$.
%which can be defined equivalently as
%\begin{compactitem}
%\item the fiber of the binary search tree insertion (inserting a permutation from right to left),
%\item the congruence where each class is the set of linear extensions of a binary tree (labeled in inorder and oriented toward its root),
%\item the transitive closure of the rewriting rule~$UacVbW \equiv UcaVbW$ for~$1 \le a < b < c \le n$,
%\item the congruence corresponding to the upper ideal of the subarc order given by all up arcs~$(a, b, {]a,b[}, \varnothing)$ (or equivalently, generated by the long up arc~$(1, n, {]1,n[}, \varnothing)$.
%\end{compactitem}
%Hence, a permutation is minimal (resp.~maximal) in its class if and only if it avoids the pattern~$312$~(resp.~$132$).
%\begin{compactitem}
%\item a permutation is minimal (resp.~maximal) in its class when it avoids the pattern~$312$~(resp.~$132$),
%\item the canonical representations of the sylvester classes correspond to non-crossing sets of up arcs, also known as non-crossing partitions.
%\end{compactitem}
The quotient of the weak order by the sylvester congruence is (isomorphic to) the classical \defn{Tamari lattice}~\cite{Tamari,HuangTamari}, whose elements are the binary trees on~$n$ nodes and whose cover relations are rotations in binary trees.
The canonical representations in the Tamari lattice correspond to non-crossing sets of up arcs, also known as \defn{non-crossing partitions}.
%The sylvester congruence and the Tamari lattice are illustrated in \cref{fig:sylvesterCongruence} for~$n = 4$.
%The sylvester congruence was extended in~\cite{Reading-CambrianLattices} to Cambrian congruences and in~\cite{PilaudPons-permutrees} to permutree congruences.
\end{example}

%%%%%%%%%%%%%%%%

\subsection{Weak order on arcs}
\label{subsec:weakOrderArcs}

We now compare join or meet irreducible elements in the weak order in terms of arcs.
%For this, we observe that the inversions of~$\b{\sigma}_\join(\alpha)$ and~$\b{\sigma}_\meet(\alpha)$ are easily read on the arc~$\alpha$.

\begin{lemma}
\label{lem:arcInversions}
For an arc $\alpha \eqdef (a, b, A, B)$ and~$u < v$, the pair~$(u, v)$ is an inversion of~$\b{\sigma}_\join(\alpha)$ (resp.~of~$\b{\sigma}_\meet(\alpha)$) if and only if~$u \in B \cup \{a\}$ and $v \in A \cup \{b\}$ (resp.~$u \notin A \cup \{a\}$ or~${v \notin B \cup \{b\}}$).
\end{lemma}

%\begin{proof}
%Immediate from the definition~$\b{\sigma}_\join(\alpha) \eqdef [1, \dots, (a-1), a_1, \dots, a_k, b, a, b_1, \dots, b_\ell, (b+1), \dots, n]$ and~$\b{\sigma}_\meet(\alpha) \eqdef [n, \dots, (b+1), a_k, \dots, a_1, a, b, b_\ell, \dots, b_1, (a-1), \dots, 1]$.
%\end{proof}

\begin{corollary}
\label{coro:weakOrderArcs}
For any two arcs~$\alpha \eqdef (a, b, A, B)$ and~$\alpha' \eqdef (a', b', A', B')$, we have
\begin{compactenum}[(i)]
\item $\b{\sigma}_\join(\alpha) \le \b{\sigma}_\join(\alpha')$ if and only if $a \in B' \cup \{a'\}$, $b \in A' \cup \{b'\}$, $A \subseteq A'$ and~$B \subseteq B'$,
\item $\b{\sigma}_\meet(\alpha) \le \b{\sigma}_\meet(\alpha')$ if and only if $a' \in B \cup \{a\}$, $b' \in A \cup \{b\}$, $A' \subseteq A$ and~$B' \subseteq B$,
\item $\b{\sigma}_\join(\alpha) \le \b{\sigma}_\meet(\alpha')$ if and only if there is no~$u < v$ such that~${u \in (A' \cup \{a'\}) \cap (B \cup \{a\})}$ and ${v \in (A \cup \{b\}) \cap (B' \cup \{b'\})}$.
\end{compactenum}
\end{corollary}

%\begin{proof}
%Immediate from \cref{lem:arcInversions}.
%\end{proof}

\begin{remark}
\label{rem:weakOrderArcs}
\cref{fig:induction35} shows the weak order on arcs defined by~$\alpha \le \alpha'$ if $\b{\sigma}_\join(\alpha) \le \b{\sigma}_\join(\alpha')$.
Visually, $\alpha \le \alpha'$ if $\alpha$ is a subarc of~$\alpha'$ which starts weakly below~$\alpha'$ and ends weakly above~$\alpha'$.
Note that~$\alpha \eqdef (a, b, A, B)$ covers at most two arcs, namely~${(\min B, b, A \cap {]\min b, b[}, B \ssm \min B)}$ and ${(a, \max A, A \ssm \max A, B \cap {]a, \max A[})}$ when they are defined.
Similar remarks hold for the order defined by~$\b{\sigma}_\meet(\alpha)$ instead of~$\b{\sigma}_\join(\alpha)$.
%
\begin{figure}[b]
	\centerline{\includegraphics[scale=.25]{induction}}
%	\centerline{\includegraphics[scale=.25]{induction_noclip}}
	\caption{The weak orders of size $3$ (top left), $4$ (top right), and $5$ (bottom) restricted to their join irreducibles represented by the corresponding arcs.}
	\label{fig:induction35}
\end{figure}
\end{remark}

\begin{remark}
As illustrated in \cref{fig:induction35}, the weak order on join irreducible of~$\f{S}_n$ has interesting enumerative properties. Let us just mention here that it has
\begin{compactitem}
\item $2^n - n - 1$ elements (permutations with a single descent, or arcs) \OEIS{A000295},
\item $2^{n+1} - n^2 - n - 2$ cover relations (in bijection with arcs of size $n+1$ crossing the horizontal axis, or with subsets of $[n+1]$ crossing their complement) \OEIS{A324172},
\item $n(n+1)2^{n-2}$ intervals (including the singletons) \OEIS{A001788}.
\end{compactitem}
\end{remark}

%\begin{lemma}
%\label{lem:compareArcs}
%%An arc~$\alpha_\join$ crosses up an arc~$\alpha_\meet$ if and only if~${\b{\sigma}_\join(\alpha_\join) \not\le \b{\sigma}_\meet(\alpha_\meet)}$ in the weak order.
%For any two arcs~$\alpha_\join \eqdef (a_\join, b_\join, A_\join, B_\join)$ and $\alpha_\meet \eqdef (a_\meet, b_\meet, A_\meet, B_\meet)$, we have \linebreak ${\b{\sigma}_\join(\alpha_\join) \le \b{\sigma}_\meet(\alpha_\meet)}$ if and only if there is no~$u < v$ such that~${u \in (A_\meet \cup \{a_\meet\}) \cap (B_\join \cup \{a_\join\})}$ and ${v \in (A_\join \cup \{b_\join\}) \cap (B_\meet \cup \{b_\meet\})}$.
%%\begin{compactitem}
%%\item $\sigma_\join(\alpha_\join) \le \sigma_\meet(\alpha_\meet)$ in the weak order,
%%\item there is no~$1 \le u < v \le n$ such that~$u \in \overline A_\meet \cap \overline B_\join$ and~$v \in \overline A_\join \cap \overline B_\meet$.
%%\end{compactitem}
%\end{lemma}
%
%\begin{proof}
%By \cref{def:SCAB,lem:arcInversions}, we have $\alpha_\join$ crosses up~$\alpha_\meet$ $\iff$ there is~$u < v$ with ${u \in (A_\meet \cup \{a_\meet\}) \cap (B_\join \cup \{a_\join\})}$ and ${v \in (A_\join \cup \{b_\join\}) \cap (B_\meet \cup \{b_\meet\})}$ $\iff$ there is a pair~$(u, v)$ which is an inversion of~$\b{\sigma}_\join(\alpha_\join)$ but not of~$\b{\sigma}_\meet(\alpha_\meet)$ $\iff$ ${\b{\sigma}_\join(\alpha_\join) \not\le \b{\sigma}_\meet(\alpha_\meet)}$.
%\vincent{clean the proof}
%\end{proof}

%%%%%%%%%%%%%%%%

\subsection{Semi-crossing arc bidiagrams}
\label{subsec:SCABs}

We now describe the canonical complex of the weak order as defined in \cref{subsec:canonicalComplex} in terms of the following combinatorial objects, illustrated in \cref{fig:bidExamples}.

\begin{figure}[b]
	\centerline{\includegraphics[scale=.5]{bidExamples}}
	\caption{SCABs of the weak order intervals $[2531746, 2531746]$, $[2531746, 2537146]$, $[2513476, 2537146]$ and $[5264137, 6574231]$.}
	\label{fig:bidExamples}
\end{figure}

\begin{definition}
\label{def:SCAB}
% An arc~$\alpha_\join \eqdef (a_\join, b_\join, A_\join, B_\join)$ \defn{crosses up} an arc~$\alpha_\meet \eqdef (a_\meet, b_\meet, A_\meet, B_\meet)$ if there is~${u < v}$ with ${u \in (A_\meet \cup \{a_\meet\}) \cap (B_\join \cup \{a_\join\})}$ and ${v \in (A_\join \cup \{b_\join\}) \cap (B_\meet \cup \{b_\meet\})}$.
%A \defn{semi-crossing arc bidiagram} (or \defn{SCAB} for short) is a disjoint union~$\delta_\join \sqcup \delta_\meet$ of two non-crossing arc diagrams such that there is no arc~${\alpha_\join \in \delta_\join}$ crosses up an arc ${\alpha_\meet \in \delta_\meet}$.
A \defn{semi-crossing arc bidiagram} (\defn{SCAB}) is a pair~$(\delta_\join, \delta_\meet)$ of non-crossing arc diagrams such that for any arcs~$\alpha_\join \eqdef (a_\join, b_\join, A_\join, B_\join) \in \delta_\join$ and $\alpha_\meet \eqdef (a_\meet, b_\meet, A_\meet, B_\meet) \in \delta_\meet$, there is no~$u < v$ with ${u \in (A_\meet \cup \{a_\meet\}) \cap (B_\join \cup \{a_\join\})}$ and ${v \in (A_\join \cup \{b_\join\}) \cap (B_\meet \cup \{b_\meet\})}$.
The \defn{semi-crossing complex} is the simplicial complex whose ground set contains two copies~$\alpha_\join$ and~$\alpha_\meet$ of each arc~$\alpha$ and whose simplices are all semi-crossing arc bidiagrams.
\end{definition}

\begin{remark}
% Visually, $\alpha_\join$ crosses up $\alpha_\meet$ when either~$\alpha_\join$ and $\alpha_\meet$ cross in their interiors with~$\alpha_\join$ going up and $\alpha_\meet$ going down goes down at the crossing, or $\alpha_\join$ and~$\alpha_\meet$ end at the same point with~$\alpha_\join$ arriving below~$\alpha_\meet$ at this point.
Visually, a semi-crossing arc bidiagram~$\delta_\join \sqcup \delta_\meet$ is a collection of arcs such that
\begin{compactitem}
\item no two arcs of~$\delta_\join$ (resp.~of~$\delta_\meet$) cross in their interior, or start or end at the same points,
\item no two arcs~$\alpha_\join \in \delta_\join$ and~$\alpha_\meet \in \delta_\meet$ cross in their interiors with~$\alpha_\join$ going up and $\alpha_\meet$ going down at the crossing, or start at the same point with~$\alpha_\join$ leaving above~$\alpha_\meet$, or end at the same point with~$\alpha_\join$ arriving below~$\alpha_\meet$ at this point.
\end{compactitem}
%We call the bidiagrams~$\delta_\join \sqcup \delta_\meet$ semi-crossing as $\delta_\join$ and~$\delta_\meet$ are themselves non-crossing, but the arcs of~$\delta_\join$ may cross down the arcs of~$\delta_\meet$.
\end{remark}

%\begin{remark}
%Before going further, we report in \cref{table:SCABs} on the number of semi-crossing arc bidiagrams~${\delta_\join \sqcup \delta_\meet}$ according to the cardinalities~$|\delta_\join|$ and~$|\delta_\meet|$ for~$n = 2$ to~$6$.
%
%\begin{table}
%	\begin{gather*}
%	\begin{array}[t]{l|cc}
%		  & 0 & 1 \\
%		\hline
%		0 & 1 & 1 \\ 
%		1 & 1 & 0
%	\end{array}
%	\qquad
%	\begin{array}[t]{l|ccc}
%		  0 & 1 & 2 \\
%		\hline
%		0 & 1 & 4 & 1 \\
%		1 & 4 & 6 & 0 \\ 
%		2 & 1 & 0 & 0
%	\end{array}
%	\qquad
%	\begin{array}[t]{l|cccc}
%		  & 0 & 1 & 2 & 3 \\
%		\hline
%		0 & 1 & 11 & 11 & 1 \\ 
%		1 & 11 & 54 & 24 & 0 \\
%		2 & 11 & 24 & 2 & 0 \\ 
%		3 & 1 & 0 & 0 & 0
%	\end{array}
%	\qquad
%	\begin{array}[t]{l|ccccc}
%		  & 0 & 1 & 2 & 3 & 4 \\
%		\hline
%		0 & 1 & 26 & 66 & 26 & 1 \\
%		1 & 26 & 300 & 420 & 80 & 0 \\ 
%		2 & 66 & 420 & 320 & 20 & 0 \\ 
%		3 & 26 & 80 & 20 & 0 & 0 \\ 
%		4 & 1 & 0 & 0 & 0 & 0
%	\end{array}
%%	\quad
%%	\begin{array}[t]{l|cccccc}
%%		  & 0 & 1 & 2 & 3 & 4 & 5 \\
%%		\hline
%%		0 & 1 & 57 & 302 & 302 & 57 & 1 \\
%%		1 & 57 & 1340 & 4145 & 2505 & 240 & 0 \\
%%		2 & 302 & 4145 & 8270 & 3035 & 120 & 0 \\
%%		3 & 302 & 2505 & 3035 & 562 & 5 & 0 \\ 
%%		4 & 57 & 240 & 120 & 5 & 0 & 0 \\ 
%%		5 & 1 & 0 & 0 & 0 & 0 & 0
%%	\end{array}
%	\end{gather*}
%	\caption{The number of semi-crossing arc bidiagrams~${\delta_\join \sqcup \delta_\meet}$ according to the cardinalities~$|\delta_\join|$ and~$|\delta_\meet|$ for~$n = 2$ to~$6$.}
%	\label{table:SCABs}
%\end{table}
%
%In these tables, observe that
%\begin{compactitem}
%\item the first row (resp.~column) corresponds to the intervals~$[\sigma, w_\circ]$ (resp.~$[e, \sigma]$) for the permutations~$\sigma$ of~$[n]$ with $k$ ascents (resp.~descents) and are thus counted by the Eulerian numbers~\OEIS{A008292},
%\item the last row (resp.~column) corresponds to the single interval~$[w_\circ, w_\circ]$ (resp.~$[e, e]$).
%\end{compactitem}
%\end{remark}

%We now connect the semi-crossing arc bidiagrams with the canonical complex of the weak order using \cref{coro:weakOrderArcs}.

\begin{proposition}
\label{prop:SCAB}
The map~$[\sigma, \tau] \mapsto \b{\delta}_\join(\sigma) \sqcup \b{\delta}_\meet(\tau)$ is a bijection between the intervals of the weak order on~$\f{S}_n$ and the semi-crossing arc bidiagrams.
Hence, the canonical complex of the weak order is isomorphic to the semi-crossing complex.
\end{proposition}

%\begin{proof}
%By \cref{prop:NCAD}, $\b{\delta}_\join(\sigma)$ and~$\b{\delta}_\meet(\tau)$ are both non-crossing arc diagrams, and correspond to the canonical join representation of~$\sigma$ and the canonical meet representation of~$\tau$ respectively.
%By \cref{coro:weakOrderArcs}\,(iii), $\sigma \le \tau$ implies that~$\b{\delta}_\join(\sigma) \sqcup \b{\delta}_\meet(\tau)$ is a semi-crossing arc bidiagram, and corresponds to the canonical representation of~$[\sigma, \tau]$.
%\end{proof}

\begin{figure}
	\centerline{\includegraphics[scale=1]{CCS3}}
	\caption{The weak order on~$\f{S}_3$ with permutations labeled by semi-crossing arc bidiagrams, and the canonical complex of $\mathfrak{S}_3$ with join and meet irreducible permutations labeled by arcs.}
	\label{fig:CCS3}
\end{figure}

Some semi-crossing arc bidiagrams of intervals are illustrated in \cref{fig:bidExamples}, and the canonical complex of the weak order on~$\f{S}_3$ is illustrated in \cref{fig:CCS3}.
The central symmetry corresponds to the maps~$\kappa_\join$ and~$\kappa_\meet$, which just corresponds to the exchange of color of the arcs by~\cref{prop:kappaChangesColor}.
Let us insist again here that this combinatorial model for the intervals of the weak order is adapted to the study of its quotients.
%The following statement follows from \cref{prop:kappaChangesColor,prop:subarcs}.

\begin{proposition}
For any lower ideal~$I$ of the subarc order, the canonical complex of the quotient of the weak order by~$\equiv_I$ is isomorphic to the subcomplex of the semi-crossing complex induced by~$\set{\alpha_\join}{\alpha \in I} \sqcup \set{\alpha_\meet}{\alpha \in I}$.
\end{proposition}

\begin{remark}
Observe that the semi-crossing arc bidiagram corresponding to a singleton~$[\sigma, \sigma]$ is just a path alternating between increasing blue steps and decreasing red steps corresponding to the ascents and descents of~$\sigma$.
See \cref{fig:diagExamples,}.
In general, we conjecture that the semi-crossing arc bidiagram corresponding to an inclusion minimal interval in a lattice quotient of the weak order does not contain any crossing in the interior of an arc.
\end{remark}

%%%%%%%%%%%%%%%%

%\subsection{Weak order interval posets}
%\label{subsec:WOIPs}
%
%Following a classical result of A.~Bj\"orner and M.~Wachs~\cite[Theorem~6.8]{BjornerWachs}, G.~Chatel, V.~Pilaud and V.~Pons already studied in~\cite{ChatelPilaudPons} a family of posets corresponding to the intervals of the weak order.
%
%\pagebreak
%
%\begin{proposition}[{\cite[Theorem~6.8]{BjornerWachs} \& \cite[Proposition~26]{ChatelPilaudPons}}]
%\label{prop:WOIP}
%The following conditions are equivalent for a poset~$\less$ on~$[n]$:
%\begin{compactitem}
%\item the linear extensions of $\less$ form an interval~$[\sigma, \tau]$ of the weak order,
%\item there are~$\sigma \le \tau$ in the weak order such that the increasing relations of~$\less$ are the non-inversions of~$\tau$ and the decreasing relations are the inversions of~$\sigma$,
%\item $a \less c$ implies~$a \less b$ or $b \less c$, and $a \more c$ implies $a \more b$ or $b \more c$ for all~$1 \le a < b < c \le n$.
%\end{compactitem}
%Such a poset is called a \defn{weak order interval poset} (or \defn{WOIP} for short).
%\end{proposition}
%
%Although they are specific to the weak order and do not behave well with respect to its quotients, the WOIPs are combinatorial objects in bijection with intervals of the weak order, and thus with SCABs.
%It is thus relevant to provide direct explicit bijections between~SCABs and~WOIPs, which are illustrated in \cref{fig:WOIP_SCAB}.
%We need the following definitions, illustrated in \cref{fig:WOIP_SCAB}:
%
%\begin{figure}
%	\centerline{\includegraphics[scale=.5]{WOIP_SCAB}}
%	\caption{The bijection between SCABs (top) and WOIPs (bottom). We represent a WOIP $\less$ with arcs joining $a<b$ from above (resp.~below) in blue (resp.~red) when $a\less b$ (resp.~$a\more b$).}
%	\label{fig:WOIP_SCAB}
%\end{figure}
%
%\medskip\noindent
%\textsf{From SCABs to WOIPs.}
%For any arc~$\alpha \eqdef (a, b, A, B)$, we denote by~$\b{\less}_\alpha$ the set of increasing relations~$u \; {\b{\less}_\alpha} \, v$ (by~$\b{\more}_\alpha$ the set of decreasing relations~$u \; {\b{\more}_\alpha} \, v$) where~${u < v}$ with~$u \in A \cup \{a\}$ and~$v \in B \cup \{b\}$.
%For a NCAD~$\delta$, we denote by~$\b{\less}_\delta$ (resp.~$\b{\more}_\delta$) the transitive closure of the union of the decreasing relations~$\set{\b{\more}_{\alpha_\join}}{\alpha_\join \in \delta_\join}$ (resp.~of the increasing relations~$\set{\b{\less}_{\alpha_\meet}}{\alpha_\meet \in \delta_\meet}$).
%
%\medskip\noindent
%\textsf{From WOIPs to SCABs.}
%Fix a WOIP~${\more} \sqcup {\less}$ where~$\more$ denote the decreasing relations and $\less$ denote the increasing relations.
%We say that an increasing (resp.~decreasing) cover relation~$a \lesscover b$ (resp.~$a \morecover b$) is \defn{maximal} if there is no cover relation~$a' \lesscover b$ (resp.~$a' \morecover b$) with~${a' < a}$ or~$a \lesscover b'$ (resp.~$a \morecover b'$) with~${b < b'}$.
%To a maximal decreasing (resp.~increasing) cover relation~$a \morecover b$ (resp.~$a \lesscover b$), we associate the arc~$\b{\alpha}(a \morecover b) \eqdef (a, b, {]a,b[} \cap \downIdeal[a], {]a,b[} \cap \upIdeal[b])$ (resp.~${\b{\alpha}(a \lesscover b) \eqdef (a, b, {]a,b[} \cap \downIdeal[b], {]a,b[} \cap \upIdeal[a])}$), where~$\downIdeal[x]$ and~$\upIdeal[x]$ denote the lower and upper ideals of~$\less$ generated by~$x$.
%Finally, we denote by~$\b{\delta}(\more)$ (resp.~$\b{\delta}(\less)$) the set of arcs~$\b{\alpha}(a \morecover b)$ (resp.~$\b{\alpha}(a \lesscover b)$) for all maximal decreasing (resp.~increasing) cover relations of~$\less$.
%
%\begin{proposition}
%The maps~$\delta_\join \sqcup \delta_\meet \mapsto {\b{\more}_{\delta_\join}} \sqcup {\b{\less}_{\delta_\meet}}$ and~${\more} \sqcup {\less} \mapsto \b{\delta}(\more) \sqcup \b{\delta}(\less)$ are inverse bijections between SCABs and WOIPs.
%\end{proposition}
%
%\begin{proof}
%Consider an interval~$[\sigma, \tau]$ of the weak order corresponding to a SCAB~$\delta_\join \sqcup \delta_\meet$.
%For any arc~$\alpha_\join$, the decreasing relations of~$\b{\more}_{\alpha_\join}$ are precisely the inversions of~$\sigma_\join(\alpha_\join)$ by \cref{lem:arcInversions}.
%Hence, the decreasing relations of~$\b{\more}_{\delta_\join}$ are precisely the inversions of~$\sigma = \bigJoin \set{\sigma_\join(\alpha_\join)}{\alpha_\join \in \delta_\join}$ (since the inversion set of a join is the transitive closure of the union of the inversion sets of the joinands).
%Similarly, the increasing relations of~$\b{\less}_{\delta_\meet}$ are precisely the non-inversions of~$\tau = \bigMeet \set{\sigma_\meet(\alpha_\meet)}{\alpha_\meet \in \delta_\meet}$.
%Hence~${\b{\more}_{\delta_\join}} \sqcup {\b{\less}_{\delta_\meet}}$ is indeed the WOIP of the interval~$[\sigma, \tau]$ by \cref{prop:WOIP}\,(ii).
%Finally, to see that the two maps are inverse to each other, we just need to observe that the relations~$a \, {\b{\less}_{\delta_\meet}} \, b$ created out of the extremities of the arcs~$(a, b, A, B)$ of~$\delta_\meet$ are precisely the maximal cover relations of~$\b{\less}_{\delta_\meet}$ (and similarly for~$\b{\more}_{\delta_\join}$). 
%\end{proof}
%
%%\subsubsection{From SCABs to WOIPs}
%%
%%For any arc~$\alpha \eqdef (a, b, A, B)$, we denote by~$\b{\less}_\alpha$ the set of increasing relations~$u \; {\b{\less}_\alpha} \, v$ (by~$\b{\more}_\alpha$ the set of decreasing relations~$u \; {\b{\more}_\alpha} \, v$) where~${u < v}$ with~$u \in A \cup \{a\}$ and~$v \in B \cup \{b\}$.
%%For a NCAD~$\delta$, we denote by~$\b{\less}_\delta$ (resp.~$\b{\more}_\delta$) the transitive closure of the union of the decreasing relations~$\set{\b{\more}_{\alpha_\join}}{\alpha_\join \in \delta_\join}$ (resp.~of the increasing relations~$\set{\b{\less}_{\alpha_\meet}}{\alpha_\meet \in \delta_\meet}$).
%%
%%\begin{proposition}
%%\label{prop:SCAB2WOIP}
%%The map~$\delta_\join \sqcup \delta_\meet \mapsto {\b{\more}_{\delta_\join}} \sqcup {\b{\less}_{\delta_\meet}}$ is a bijection from SCABs to WOIPs.
%%\end{proposition}
%%
%%\begin{proof}
%%Consider an interval~$[\sigma, \tau]$ of the weak order corresponding to a SCAB~$\delta_\join \sqcup \delta_\meet$.
%%For any arc~$\alpha_\join$, the decreasing relations of~$\b{\more}_{\alpha_\join}$ are precisely the inversions of~$\sigma_\join(\alpha_\join)$ by \cref{lem:arcInversions}.
%%Hence, the decreasing relations of~$\b{\more}_{\delta_\join}$ are precisely the inversions of~$\sigma = \bigJoin \set{\sigma_\join(\alpha_\join)}{\alpha_\join \in \delta_\join}$ (since the inversion set of a join is the transitive closure of the union of the inversion sets of the joinands).
%%Similarly, the increasing relations of~$\b{\less}_{\delta_\meet}$ are precisely the non-inversions of~$\tau = \bigMeet \set{\sigma_\meet(\alpha_\meet)}{\alpha_\meet \in \delta_\meet}$.
%%Hence~${\b{\more}_{\delta_\join}} \sqcup {\b{\less}_{\delta_\meet}}$ is indeed the WOIP of the interval~$[\sigma, \tau]$ by \cref{prop:WOIP}\,(ii).
%%\end{proof}
%%
%%\begin{figure}
%%	\centerline{\includegraphics[scale=.55]{WOIP_SCAB}}
%%	\caption{The bijection between SCABs (top) and WOIPs (bottom). We represent a WOIP $\less$ by $n$ dots, and an arc joining $a<b$ from above (resp.~below) in blue (resp.~red) whenever $a\less b$ (resp.~$a\more b$).}
%%	\label{fig:WOIP_SCAB}
%%\end{figure}
%%
%%\subsubsection{From WOIPs to SCABs}
%%
%%Fix a WOIP~${\more} \sqcup {\less}$ where~$\more$ denote the decreasing relations and $\less$ denote the increasing relations.
%%We say that an increasing (resp.~decreasing) cover relation~$a \lesscover b$ (resp.~$a \morecover b$) is \defn{maximal} if there is no cover relation~$a' \lesscover b$ (resp.~$a' \morecover b$) with~${a' < a}$ or~$a \lesscover b'$ (resp.~$a \morecover b'$) with~$b < b'$.
%%To a maximal decreasing (resp.~increasing) cover relation~$a \morecover b$ (resp.~$a \lesscover b$), we associate the arc~$\b{\alpha}(a \morecover b) \eqdef (a, b, {]a,b[} \cap \downIdeal[a], {]a,b[} \cap \upIdeal[b])$ (resp.~${\b{\alpha}(a \lesscover b) \eqdef (a, b, {]a,b[} \cap \downIdeal[b], {]a,b[} \cap \upIdeal[a])}$), where~$\downIdeal[x]$ and~$\upIdeal[x]$ denote the lower and upper ideals of~$\less$ generated by~$x$.
%%Finally, we denote by~$\b{\delta}(\more)$ (resp.~$\b{\delta}(\less)$) the set of arcs~$\b{\alpha}(a \morecover b)$ (resp.~$\b{\alpha}(a \lesscover b)$) for all maximal decreasing (resp.~increasing) cover relations of~$\less$.
%%
%%\begin{proposition}
%%\label{prop:WOIP2SCAB}
%%The map~${\more} \sqcup {\less} \mapsto \b{\delta}(\more) \sqcup \b{\delta}(\less)$ is a bijection from WOIPs to SCABs.
%%\end{proposition}
%%
%%\begin{proof}
%%This map is the inverse from that of \cref{prop:SCAB2WOIP}.
%%Indeed, observe that the relations~$a \, {\b{\less}_{\delta_\meet}} \, b$ created out of extremities of the arcs~$(a, b, A, B)$ of~$\delta_\meet$ are precisely the maximal cover relations of~$\b{\less}_{\delta_\meet}$ (and similarly for~$\b{\more}_{\delta_\join}$). 
%%\end{proof}
%%
%%Note that we have actually shown the following statement.
%%
%%\begin{proposition}
%%The maps~$\delta_\join \sqcup \delta_\meet \mapsto {\b{\more}_{\delta_\join}} \sqcup {\b{\less}_{\delta_\meet}}$ of \cref{prop:SCAB2WOIP} and~${\more} \sqcup {\less} \mapsto \b{\delta}(\more) \sqcup \b{\delta}(\less)$ of \cref{prop:WOIP2SCAB} are inverse bijections.
%%\end{proposition}

%%%%%%%%%%%%%%%%

\subsection{Kreweras maps in quotients of the weak order}
\label{subsec:KrewerasBidiagrams}

We finally describe the Kreweras maps defined in~\cref{subsubsec:Kreweras} in all quotients of the weak order in terms of semi-crossing arc bidiagrams.
For this, we first connect the canonical join representation of a permutation to the canonical join representation of the minimal element in its class for a given congruence, improving on \cref{prop:canonicalJoinRepresentationsQuotient}\,(iii). %, as illustrated in \cref{fig:Kreweras}.
We call weak order on arcs the order~$\alpha \le \alpha'$ if $\b{\sigma}_\join(\alpha) \le \b{\sigma}_\join(\alpha')$ (see \cref{subsec:weakOrderArcs,fig:induction35}).

\begin{proposition}
\label{prop:Kreweras}
Consider an upper ideal~$I$ of the subarc order and a permutation~$\sigma$.
Let~$X$ be the intersection of~$I$ with the lower ideal generated by the non-crossing arc diagram~$\b{\delta}_\join(\sigma)$ in the weak order on arcs.
Let~$Y$ be the set of arcs~$(a, b, A, B)$ of~$X$ such that there is~$a < p < b$ such that both arcs~$(a, p, A \cap {]a,p[}, B \cap {]a,p[})$ and~$(p, b, A \cap {]p,b[}, B \cap {]p,b[})$ belong to~$X$.
Then the non-crossing arc diagram~$\b{\delta}_\join \big( \projDown[\equiv_I](\sigma) \big)$ is the set of maximal elements of~$X \ssm Y$ in the weak order on arcs.
\end{proposition}

%\begin{proof}
%By \cref{prop:canonicalJoinRepresentationsQuotient,prop:canonicalJoinRepresentationProjection}, $\b{\delta}_\join \big( \projDown[\equiv_I](\sigma) \big)$ is a non-crossing arc diagram contained in~$X$.
%Among all options, we need to choose the non-crossing arc diagram with maximal join.
%This implies that the arcs of~$Y$ cannot appear in $\b{\delta}_\join \big( \projDown[\equiv_I](\sigma) \big)$ since
%\[
%\b{\sigma}_\join(a, b, A, B) \le \b{\sigma}_\join(a, p, A \cap {]a,p[}, B \cap {]a,p[}) \join \b{\sigma}_\join(p, b, A \cap {]p,b[}, B \cap {]p,b[})
%\]
%for any arc~$(a, b, A, B)$ and any~$a < p < b$ by \cref{lem:arcInversions}.
%We finally claim that any two incomparable elements in~$X \ssm Y$ cannot cross.
%Hence, the maximal elements of~$X \ssm Y$ form a non-crossing arc diagram, which must therefore be~$\b{\delta}_\join \big( \projDown[\equiv_I](\sigma) \big)$.
%To prove the claim, consider two arcs~$\alpha \eqdef (a, b, A, B)$ and~$\alpha' \eqdef (a', b', A', B')$ in~$X$ which are incomparable and crossing.
%Assume for instance that there are~$u < v$ such that~${u \in (A \cup \{a\}) \cap (B' \cup \{a'\})}$ and ${v \in (A' \cup \{b'\}) \cap (B \cup \{b\})}$.
%We can moreover assume that~$\alpha$ and~$\alpha'$ agree on~$]u,v[$, meaning that~$A \cap {]u,v[} = A' \cap {]u,v[}$ and~$B \cap {]u,v[} = B' \cap {]u,v[}$.
%Since $\alpha$ and~$\alpha'$ are incomparable, this implies that~$u \ne a$ or~$v \ne b$.
%Hence, at least two of the three arcs
%\[
%(a, u, A \cap {]a, u[}, B \cap {]a, u[}),
%\qquad
%(u, v, A' \cap {]u, v[}, B' \cap {]u, v[})
%\quad\text{and}\quad
%(u, b, A \cap {]u, b[}, B \cap {]u, b[})
%\]
%are non-trivial (not reduced to a single point).
%Moreover, they also belong to $X$, which implies that~$\alpha$ is in~$Y$.
%\end{proof}

This enables us to compute the Kreweras maps in quotients of the weak order directly on non-crossing arc diagrams.
For this, we extend the notations of~\cref{subsubsec:Kreweras} to quotients and transport them to non-crossing arc diagrams.
For an upper ideal~$I$ of the subarc order, each equivalence class of~$\equiv_I$ is an interval~$[x,y]$ of the weak order and thus corresponds to two non-crossing arc diagrams~$\delta_\join \eqdef \b{\delta}_\join(x)$ and~$\delta_\meet \eqdef \b{\delta}_\meet(y)$.
We denote by~$\eta^I_\join$ and~$\eta^I_\meet$ the two opposite maps defined by~$\eta^I_\join(\delta_\meet) = \delta_\join$ and~$\eta^I_\meet(\delta_\join) = \delta_\meet$.
We just write~$\eta_\join$ and~$\eta_\meet$ when~$I$ is the set of all arcs.
Note that~$\eta_\join = \b{\delta}_\join \circ \b{\delta}_\meet^{-1}$ and~$\eta_\meet = \b{\delta}_\meet \circ \b{\delta}_\join^{-1}$ are easily computed from the descriptions of the maps~$\b{\delta}_\join$ and $\b{\delta}_\meet$ (see \cref{subsec:NCADs}) and of their inverses (see~\cite{Reading-arcDiagrams}).
\cref{prop:Kreweras} enables to compute~$\eta^I_\join$ and~$\eta^I_\meet$  in general.

\begin{corollary}
\label{coro:Kreweras}
Consider an upper ideal~$I$ of the subarc order and a non-crossing arc diagram~$\delta_\meet$ with all arcs in~$I$.
Then the non-crossing arc diagram~$\eta^I_\join(\delta_\meet)$ is obtained from~$\eta_\join(\delta_\meet)$ by applying the algorithm of \cref{prop:Kreweras}.
\end{corollary}

\begin{example}
\label{exm:KrewerasComplement}
When~$I$ is the upper ideal of up arcs corresponding to the sylvester congruence, the description of \cref{coro:Kreweras} can be translated to the classical description of the Kreweras complement of a non-crossing partition, obtained by shifting the points and connecting the points in the same connected component.
See \cref{fig:KrewerasComplement}.

\begin{figure}[h]
	\centerline{\includegraphics[scale=.58]{kreweras_classique}}
	\caption{Classical Kreweras complement on non-crossing partitions.}
	\label{fig:KrewerasComplement}
\end{figure}
\end{example}

%\begin{figure}
%	\centerline{\includegraphics[scale=.9]{bidTable}}
%	\caption{Illustrations of computations of~$\b{\delta}_\join \big( \projDown[\equiv_I](\sigma) \big)$ and $\eta^I_\join(\delta_\meet)$ in the trivial congruence, the descent congruence, the sylvester congruence, and a generic congruence.}
%	\label{fig:Kreweras}
%\end{figure}

\acknowledgements{We are grateful to J.-C. Novelli for many discussions and suggestions on this paper, and to S.~Giraudo for a careful proofreading of a preliminary version.}

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