Coordinatization of join-distributive lattices
G\'abor Cz\'edli
Join-distributive lattices are finite, meet-semidistributive, and semimodular lattices. They are the same as Dilworth's lattices in 1940, and many alternative definitions and equivalent concepts have been discovered or rediscovered since then. Let $L$ be a join-distributive lattice of length $n$, and let $k$ denote the width of the set of join-irreducible elements of $L$. A result of P.\,H.~Edelman and R.\,E.~Jamison, translated from Combinatorics to Lattice Theory, says that $L$ can be described by $k-1$ permutations acting on the set $\nset$. We prove a similar result within Lattice Theory: there exist $k-1$ permutations acting on $\nset$ such that the elements of $L$ are coordinatized by $k$-tuples over $\set{0,\dots,n}$, and the permutations determine which $k$-tuples are allowed. Since the concept of join-distributive lattices is equivalent to that of antimatroids and convex geometries, our result offers a coordinatization for these combinatorial structures.
20121012_czedli_coordinatization-of-join-distributive-lattices.tex
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\begin{document}
\title[Coordinatization of join-distributive lattices]
{Coordinatization of join-distributive lattices}
%
\author[G.\ Cz\'edli]{G\'abor Cz\'edli}
\email{czedli@math.u-szeged.hu}
\urladdr{http://www.math.u-szeged.hu/$\sim$czedli/}
\address{University of Szeged\\Bolyai Institute\\
Szeged, Aradi v\'ertan\'uk tere 1\\HUNGARY 6720}

%% Thanks
\thanks{This research was supported by the NFSR of Hungary (OTKA), grant numbers  K77432 and
K83219, and by  T\'AMOP-4.2.1/B-09/1/KONV-2010-0005}

%% Dedication 

\subjclass[2010]{Primary 06C10; secondary 05E99 and 52C99}
\nothing{
05E99 Algebraic combinatorics (1991-now) None of the above, but in this section 
;
52C99  Discrete Geometry  (1991-now) None of the above, but in this section} 

\keywords{Semimodular lattice, diamond-free lattice, planar lattice, Jordan-H\"older permutation, antimatroid, convex geometry, anti-exchange closure}


\date{October 12, 2012; earlier versions:   August 17, 2012,  September 9, 2012}

\begin{abstract} 
\emph{Join-distributive lattices} are finite, meet-semidistributive, and semimodular lattices. 
They are the same as Dilworth's lattices in 1940, and many alternative definitions and equivalent concepts have
been discovered or rediscovered since then. Let $L$ be a join-distributive lattice of length $n$, and let $k$ denote the width of the set of join-irreducible elements of $L$. A result of P.\,H.~Edelman and R.\,E.~Jamison, translated from Combinatorics to Lattice Theory, says that $L$ can be described by $k-1$ permutations acting on the set $\nset$. 
We prove a similar result within Lattice Theory: there exist $k-1$ permutations acting on $\nset$ such that the elements of $L$ are coordinatized by $k$-tuples over $\set{0,\dots,n}$, and the permutations determine which $k$-tuples are allowed.  
Since the concept  of join-distributive lattices is equivalent to that of antimatroids and  convex geometries, 
our result offers a coordinatization for these combinatorial structures. 
\end{abstract}



\maketitle

\section{Introduction}
In 1940, \init{R.\,P.~}Dilworth~\cite{r:dilworth40} introduced an important class of finite lattices. Recently, these lattices are called   join-distributive. The concept of 
antimatroids, which are particular greedoids of \init{B.~}Korte and \init{L.~}Lov\'asz~\cite{kortelovasz81} and \cite{kortelovasz83}, and that of convex geometries were introduced only much later by \init{P.\,H.~}Edelman and \init{R.\,E.~}Jamison \cite{jamison}, \cite{edelman}, and \cite{edeljam}. 
Join-distributive lattices,  antimatroids, and convex geometries are equivalent concepts in a natural way, see Section~\ref{secantimatr}.
 
Hence, though the majority of the paper belongs to Lattice Theory,  the result we prove can also be interesting in Combinatorics. 
Note that there were a lot of discoveries and rediscoveries of join-distributive lattices and the corresponding combinatorial structures; see \init{B.~}Monjardet~\cite{monjardet} and \init{M.~}Stern~\cite{stern} for surveys.

Although there are very deep coordinatization results in Lattice Theory, see \init{J.~}von Neumann~\cite{neumann}, \init{C.~}Herrmann~\cite{herrmann}, and \init{F.~}Weh\-rung~\cite{wehrung} for example, our investigations were motivated  by simple ideas that go back to Descartes. Namely, let $B$ be a  subset  of a  $k$-dimensional Euclidian space $V$,  and  let $\tuple{ v_1,\dots,v_n} \in V^k$ be an orthonormal basis. Then the system $\tuple{ V;v_1,\dots,v_k}$ is represented by $\tuple{ \mathbb R^k; e_1\ldots, e_k}$, where $e_1=\tuple{1, 0,\dots,0} $, \dots, $e_k=\tuple{ 0,\dots,0,1}$, and $B$ corresponds to a subset of $\mathbb R^k$ given by a set of equations, provided $B$ is a ``nice'' subset of $V$. While one can easily describe the relation between two  orthonormal bases of $V$, the analogous task for join-distributive lattices seems to be too hard. This is why we consider the lattice-theoretic counterpart of $\tuple{ V;v_1,\dots,v_k}$ rather than that of $V$. 

Next, instead of $B$, consider a join-distributive lattice $L$ of length $n$. Assume that the width of $\Jir L$, the poset (= partially ordered set) of join-irreducible elements of $L$, equals $k$. Then we can chose $k$ maximal chains, $C_1,\ldots, C_k$, in $L$ such that $\Jir L\subseteq C_1,\ldots, C_k$. These chains will correspond to the vectors $v_i$ above. The direct product $D=C_1\times \cdots\times C_k$, which happens to be the $k$-th direct power of the chain $\set{0<1<\dots<n}$, will play the role of $V$. We know that there is a join-embedding  $\phi\colon L \to D$. If we  describe $\phi(L)$ within $D$ by a simple set of equations, then we obtain a satisfactory description, a coordinatization,  of $L$. These equations will be defined by means of some permutations;  $k-1$ permutations will suffice.  The case $k=2$ was settled, partly rediscovered, in \init{G.~}Cz\'edli and \init{E.\,T.~}Schmidt~\cite{czgschperm}; the case $k > 2$ requires a more complex approach.

\subsection*{On a satellite paper}
After an earlier version of the present paper, available at \texttt{http://arxiv.org/abs/1208.3517}, Kira Adaricheva pointed out that the main result here is closely related to   \init{P.\,H.~}Edelman and  \init{R.\,E.~}Jamison~\cite[Theorem 5.2]{edeljam}, which is formulated for convex geometries. This connection is analyzed in 
\init{K.~}Adaricheva and \init{G.~}Cz\'edli~\cite{adarichevaczg}, which serves as a satellite paper. It appears from \cite{adarichevaczg} that our coordinatization result and the Edelman-Jamison description can mutually be  derived from each other in less than a page. 
However, we feel that the present, almost self-contained, longer  approach still makes sense by the following reasons. 

First,  it exemplifies how Lattice Theory can be applied to other fields of mathematics.
%
Second, not only the methods  and the motivations of \cite{edeljam} and the present paper are entirely different, the results are not exactly the same; see \cite{adarichevaczg} for comparison. 
Note that our coordinatization is equivalent to a  representation of a join-distributive lattice $L$ as a meet-homomorphic image of the direct power of a chain, while \cite{edeljam} represents $L$ as a join-sublattice of the powerset lattice of the same chain.  While \cite{edeljam} belongs to Combinatorics, 
the coordinatization result is a logical ``step'' in a chain of purely lattice theoretical papers, starting from  \init{G.~}Gr\"atzer and \init{E.~}Knapp~\cite{gratzerknapp} and \init{G.~}Gr\"atzer and \init{J.\,B.~}Nation~\cite{gratzernation}, and including, among others,  \init{G.~}Cz\'edli, \init{L.~}Ozsv\'art, and \init{B.~}Udvari~\cite{czgolub}, \init{G.~}Cz\'edli and \init{E.\,T.~}Schmidt~\cite{czgschsomeres}, \cite{czgschperm}, and also the paper \init{G.~}Cz\'edli and \init{E.\,T.~}Schmidt~\cite{czgschjh}, which gives another application of Lattice Theory.
%
Third, our method motivates a new characterization of join-distributive lattices, see \cite{edeljam}, and implies some known characterizations, see Remark~\ref
{rwMsRdh}. Fourth, it is not yet clear which approach will be better to attack the problem before Example~\ref{exMple}.



\subsection*{Target}
Let $L$ be a join-distributive lattice of length $n$, and let $C_1,\ldots, C_k$ be maximal chains of $L$ such that $\Jir L\subseteq C_1\cup\dots\cup C_k$. The collection of isomorphism classes of  systems $\langle L;C_1,\ldots,C_k\rangle$ is denoted by $\Lat nk$. The symmetric group of degree $n$, which consists of all 
$\nset \to \nset $ permutations, is denoted by $S_n$. Our goal is to establish a bijection between $\Lat nk$ and $S_n^{k-1}$. If $k$ is small compared to $n$, then this bijection gives a very economic way to describe $\langle L;C_1,\ldots,C_k\rangle$ and, consequently, $L$ with few data. 
Our coordinatization, that is the bijection,
can easily be translated to the language of convex geometries and antimatroids. 


\subsection*{Outline} Section~\ref{prelim} contains the lattice theoretic prerequisites, and recalls the known characterizations of join-distributive lattices. Trajectories, which represent the main tool used in the paper,  were introduced for the planar case in \init{G.~}Cz\'edli and \init{E.\,T.~}Schmidt~\cite{czgschjh}. Section~\ref{trajsection} is devoted to trajectories in arbitrary join-distributive lattices. With the help of trajectories, we develop a new approach to Jordan-H\"older permutations in Section~\ref{JHsection}. Our main result, the coordinatization theorem for join-distributive lattices, is formulated in Section~\ref{secmainres}.  This theorem is proved in Section~\ref{sectproofs}. 
Section~\ref{secantimatr} surveys antimatroids and convex geometries briefly. It also translates our coordinatization theorem to the language of Combinatorics.



\section{Preliminaries}\label{prelim}
The objective of this section is to give various descriptions for the lattices  the present paper deals with. 
The \emph{length} of an $(n+1)$-element chain is $n$, while the length of a lattice $L$, denoted by $\length L$, is the supremum of $\{\length C: C$ is a chain of $L\}$. A lattice is \emph{trivial} if it consists of a single element. Let us agree that all lattices in this paper are either finite, or they are explicitely assumed to be of  finite length. 
As usual, $\prec$ stands for the covering relation:  
$x\prec y$ means that the interval $[x,y]$ is 2-element.  If $0\prec a$, then $a$ is an \emph{atom}. A lattice $L$ is \emph{semimodular} if $x\prec y$ implies $x\vee z\prec y\vee z$, for all $x,y,z\in L$. 
An element is \emph{meet-irreducible} if it has exactly one cover. 
The poset of these elements of $L$ is denoted by $\Mir L$. Note that $1\notin \Mir L$ and $0\notin \Jir L$. Since $L$ is of finite length, each element $x\in L$ is of the form $x=\bigwedge Y$ for some $Y\subseteq \Mir L$. Note that $Y=\emptyset$ if{f} $x=1$. The equation $x=\bigwedge Y$ is an \emph{irredundant meet-decomposition} of $x$ if  $Y\subseteq \Mir L$ and   $x\neq \bigwedge Y'$ for every proper subset $Y'$ of $Y$. If each $x\in L$ has only one irredundant meet-decomposition, then we say that $L$ is a lattice \emph{with unique meet-irreducible decompositions}.
A \emph{diamond} of $L$ is a 5-element modular but not distributive sublattice $M_3$ of $L$.  A diamond consists of its top, its bottom, and the rest of its elements form an antichain. If no such sublattice exists, then $L$ is \emph{diamond-free}. 
If $S$ is a sublattice of $L$ such that, for all $x,y\in S$, $x\prec_S y$ implies $x\prec_L y$, then $S$ is a \emph{cover-preserving sublattice} of $L$. If $S$ is a nonempty subset of $L$ such that $x\vee y\in S$ for all $x,y\in S$, then $S$ is a \emph{join-subsemilattice} of $L$. 
%
For $x\in L$, the join of all covers of $x$ is denoted by $\upstar x$. An important property of $L$ is that $[x,\upstar x]$ is distributive for all $x\in L$.  
If, for all $x,y,z\in L$,  $x\wedge y=x\wedge z$ implies $x\wedge y=x\wedge (y\vee z)$, then $L$ is \emph{meet-semidistributive}. If $\Jir L$ is the union of two chains, then $L$ is \emph{slim}.
%
The next statement is known and  gives a good understanding of join-distributive lattices within Lattice Theory. For further characterizations, see  Section~\ref{secantimatr} here, see \init{S.\,P.~}Avann~\cite{avann1}, which is recalled in 
 \init{P.\,H.~}Edelman~\cite[Theorem 1.1]{edelmanproc}, and see also 
\init{M.~}Stern~\cite[Theorem 7.2.27]{stern}.

\begin{proposition}\label{Dlwltcs}
For a finite lattice $L$,  the following properties are equivalent.
\begin{enumeratei}
\item\label{Dlwltcsa} $L$ is join-distributive, that is, semimodular and meet-semidistributive
%. That is, $L$ is join-distributive. 
\item\label{Dlwltcsb} $L$ has unique meet-irreducible decompositions. 
\item\label{Dlwltcsc} For each $x\in L$, the interval $[x,\upstar x]$ is distributive.
\item\label{Dlwltcsd} For each $x\in L$, the interval $[x,\upstar x]$ is boolean.
\item\label{Dlwltcse} The length of each  maximal chain of $L$ equals $|\Mir L|$.
\item\label{neWjsldSb} $L$ is semimodular and diamond-free.
\item\label{neWjsldSc} $L$ is semimodular and has no cover-preserving diamond sublattice.
\item\label{neWjsldSd} $L$ is a cover-preserving join-subsemilattice of a finite distributive lattice.
\end{enumeratei}
\end{proposition}


Now, we explain how Proposition~\ref{Dlwltcs} can be extracted from the literature. 
The equivalence of \eqref{Dlwltcsb} and \eqref{Dlwltcsc} above was proved by \init{R.\,P.~}Dilworth~\cite{r:dilworth40}. 
\init{D.~}Armstrong~\cite[Theorem 2.7]{armstrong}  states  \eqref{Dlwltcsa} $\iff$ \eqref{Dlwltcsb} $\iff$ \eqref{Dlwltcsc} by extracting it from  
\init{K.~}Adaricheva, \init{V.\,A.~}Gorbunov, and \init{V.\,I.~}Tumanov~\cite[Theorems 1.7 and 1.9]{r:adarichevaetal}, where the dual statement is given. 
We know \eqref{Dlwltcsc} $\iff$ \eqref{Dlwltcse} $\iff$  \eqref{neWjsldSc}  from \init{M.~}Stern~\cite[Theorem 7.2.27]{stern}, who attributes it to \init{S.\,P.~}Avann~\cite{avann1} and \cite{avann2}.
The implications \eqref{Dlwltcsa} $\then$ \eqref{neWjsldSb} and 
\eqref{neWjsldSb} $\then$ \eqref{neWjsldSc}
 are trivial.  \init{H.~}Abels~\cite[Theorem 3.9]{abels} contains \eqref {neWjsldSc} $\iff$ \eqref {neWjsldSd}. 
Next, as the fourth sentence in \init{P.\,H.~}Edelman~\cite[Section 3]{edelman} points out, 
\eqref{Dlwltcsc} $\iff$ \eqref{Dlwltcsd} is practically trivial; the argument runs as follows. Assume that $[x,\upstar x]$ is distributive. Let $a_1,\ldots,a_t$ be the covers of $x$. They are independent in $[x,\upstar x]$ by distributivity and \init{G.~}Gr\"atzer~\cite[Theorem 360]{GGLT}. Hence they generate a boolean sublattice $B$ of length $t$ and size $2^t$, and $[x,\upstar x]$ is also of length $t$. Since  $|\Jir{([x,\upstar x])}|=\length{([x,\upstar x])}=t$ by \cite[Corollary 112]{GGLT}, we obtain from \cite[Theorem 107]{GGLT} that $|[x,\upstar x]|\leq 2^t$. Thus $[x,\upstar x]=B$ is boolean. 


\begin{remark}\label{rwMsRdh}
The proof of our coordinatization result  offers an alternative way to the implication \eqref {neWjsldSc} $\then$ \eqref {Dlwltcsa}, even for lattices of finite length. 
\end{remark}

\begin{remark}\label{reMrkdh} If $L$ is a lattice of finite length, then each of  conditions 
\eqref{Dlwltcsa}, \eqref{Dlwltcsc}, \dots, \eqref{neWjsldSd} implies that $L$ is finite. (Condition \eqref{Dlwltcsb} has not been investigated from this aspect.)  This follows from  either from  Propositions~\ref{Dlwltcs} and \ref{almostmain}, or from Proposition~\ref{Dlwltcs} and  \init{H.~}Abels~\cite[Theorem 3.9]{abels}; see also Corollary~\ref{idsehGH}. 
\end{remark}





\section{Trajectories}\label{trajsection}
The general assumption in Sections~\ref{trajsection} is that $L$ semimodular lattice of finite length and without cover-preserving diamonds. A \emph{prime interval} is a 2-element interval. 
A \emph{covering square} of $L$ is a cover-preserving 4-element boolean sublattice $S=\set{a\wedge b, a,b,a\vee b}$. 
The prime intervals $[a\wedge b, a]$ and $[b, a\vee b]$ are
\emph{opposite sides} of $S$, and so are the prime intervals 
$[a\wedge b, b]$  and $[a, a\vee b]$. The set of prime intervals of $L$ is denoted by $\print L$. If two prime intervals are opposite sides of the same covering square, then they are \emph{consecutive}. As in \init{G.~}Cz\'edli and \init{E.\,T.~}Schmidt~\cite{czgschjh}, the transitive reflexive closure of this consecutiveness relation on $\print L$ is an equivalence relation, and the blocks of this equivalence relation are the \emph{trajectories} of $L$. The collection of all trajectories of $L$ is denoted by $\traj L$. 

\begin{lemma}\label{trajrestlem}
Let $L$ be a semimodular lattice of finite length, having no cover-preserving diamond, and let  $S$ be a cover-preserving join-subsemilattice of $L$. Then the following two statements hold.

$\textup{(A)}$ For each $R\in \traj S$, there is a unique  $T\in\traj L$ such that $R\subseteq T$. 

$\textup{(B)}$  Let $\kappa\colon\traj L\cup\set{\emptyset}\to \traj S\cup\set{\emptyset}$, defined by $\kappa(T)=T\cap\print S$. Then $\kappa$ is a surjective map. If $\set{0_L,1_L}\subseteq S$, then $\kappa$ is a bijection.
\end{lemma}


\begin{proof}[Proof of Lemma~\ref{trajrestlem}$\textup{(A)}$] Denoting the meet in $S$ by $\wedge_S$, let $\set{a\wedge_S b,a,b,a\vee b}$ be a covering square of $S$. Then $a$ and $b$ are incomparable, and both cover $a\wedge_S b$ in $L$ since $S$ is a cover-preserving subset of $L$. This yields $a\wedge_S b=  a\wedge_L b$, and we conclude that the covering squares of $S$ are also covering squares of $L$.  This implies part (A) of the lemma.
\end{proof}

If $L_1$ and $L_2$ are lattices, $\phi\colon L_1\to L_2$ is join-homomorphism, and $\phi(x)\preceq\phi(y)$ holds for all $x,y\in L_1$ with $x\preceq y$, then $\phi$ is a \emph{cover-preserving} join-homomorphism. The kernel of a cover-preserving join-homomorphism is a \emph{cover-preserving join-congruence}. For a join-congruence $\Theta\subseteq L^2$ and a covering square $S=\set{a\wedge b, a,b,a\vee b}$, $S$ is a \emph{$\Theta$-forbidden covering square} if the $\Theta$-blocks $\blokk a\Theta$,  $\blokk b\Theta$, and  $\blokk {(a\wedge b)}\Theta$ are pairwise distinct but $\blokk {(a\vee b)}\Theta$ equals $\blokk a\Theta$ or  $\blokk b\Theta$. The following easy lemma was proved in \init{G.~}Cz\'edli and \init{E.\,T.~}Schmidt~\cite{czgschjh} and \cite[Lemma 6]{czgschsomeres}.

\begin{lemma}\label{thforbIDnn} Let $\Theta$ be a join-congruence of a semimodular lattice of finite length. Then $\Theta$ is cover-preserving if{f} $L$ has no $\Theta$-forbidden covering square.
\end{lemma}



The initial idea of the present paper is formulated in the next lemma.



\begin{lemma}\label{trajlemma}
Let  $L$ be a semimodular lattice of finite length such that $L$ contains no cover-preserving diamond. Then, for each maximal chain $C$ of $L$ and for each trajectory $T$ of $L$, $T$ contains exactly one prime interval of $C$. 
\end{lemma}

\begin{proof} Take a prime interval $\inp\in T$, and pick a maximal chain $D$ of $L$ such that $\inp\in\print D$. Let $S=\jgen{C\cup D}$, the join-subsemilattice generated by $C\cup D$. We know from \init{G.~}Cz\'edli and \init{E.\,T.~}Schmidt~\cite[Lemma 2.4]{czgschjh} that $S$ is a cover-preserving 0,1-join-subsemilattice of $L$, and it is a slim semimodular lattice. (But $S$ is not a sublattice of $L$ in general.) It follows from \cite[Lemmas 2.4 and 2.8]{czgschjh} that there are a unique $\inq\in\print C$ and a unique trajectory $R$ of $S$  such that $\inp$ and $\inq$ belong $R$.  By Lemma~\ref{trajrestlem}(A), $L$ has a trajectory $R'$ such that $R\subseteq R'$. Since $p\in R'\cap T$, we obtain that $T=R'$ contains $\inq$. This proves the existence part. 

Note that, instead of \cite{czgschjh}, one could use  \init{H.~}Abels~\cite[Corollary 3.3]{abels}
to prove the  existence part. Actually, some ideas of \cite{abels} were rediscovered in  \cite{czgschjh}. However, since the concept of trajectories comes from \cite{czgschjh}, it is more convenient to reference \cite{czgschjh}.

We prove the uniqueness by contradiction. Suppose that $a\prec b\leq c\prec d$ such that $[a,b]$ and $[c,d]$ belong to the same trajectory $T$ of $L$. Then there exists a sequence 
$[a,b]=[x_0,y_0]$, $[x_1,y_1]$, \dots, $[x_t,y_t]=[c,d]$
of prime intervals such that $H_u=\set{x_{u-1}, y_{u-1}, x_u,y_u}$ is a covering square for $u\in \tset $. 
Pick a maximal chain $0=b_0\prec b_1\prec\dots\prec b_s=b$ in the interval $[0,b]$, and consider the join-homomorphisms $\psi_m\colon L\to L$, defined by $\psi_m(z)=b_m\vee z$, for $m\in\set{0,\ldots,s}$. Let $\Theta_m$ stand for $\Ker {\psi_m}$, for $m\in\set{0,\ldots,s}$. By semimodularity, the $\psi_m$ are cover-preserving join-homomorphisms. 
%
Since  $\psi_s(x_0)=b=\psi_s(y_0)$ but    $\psi_s(x_t) =c  \neq d= \psi_s(y_t)$, there is a smallest $i\in\set{1,\ldots,t}$ such that $\psi_s(x_i)\neq \psi_s(y_i)$. 
That is, $\pair{x_{i-1}}{y_{i-1}}\in\Theta_s$ but $\pair{x_{i}}{y_{i}}\notin\Theta_s$. Clearly, $x_{i}$ is the bottom of  the covering square $H_i$, while $y_{i-1}$ is its top. 

The restriction of a relation $\rho$ to a subset $X$ will be denoted by $\restrict\rho X$. The equality relation on $X$ is denoted by $\omega_X$ or $\omega$.  
Since $\restrict{\Theta_0}{H_i} =\omega_{H_i}$ but $\pair{x_{i-1}}{y_{i-1}} \in  \restrict{\Theta_s}{H_i}$, there is a smallest $j$ such that $\pair{x_{i-1}}{y_{i-1}}\in  \restrict{\Theta_j}{H_i}$. 
However, $\pair{x_{i}}{y_{i}}\notin  \restrict{\Theta_j}{H_i}$
since otherwise $b_j\vee x_{i}= \psi_j(x_{i}) = \psi_j(y_{i}) = b_j\vee y_{i}$ together with $b_j\leq b_s$ would imply  $\psi_s(x_{i}) = \psi_s(y_{i})$, which would contradict 
 $\pair{x_{i}}{y_{i}}\notin\Theta_s$. 

Next, to simplify our notation, let
let  
\begin{equation*}
\begin{aligned}
\alpha&=\psi_{j-1}(x_{i}),\cr 
\alpha'&=\psi_{j}(x_{i}),
\end{aligned}
\quad 
\begin{aligned}
 \delta  &= \psi_{j-1}(y_{i-1}),\cr 
 \delta' &= \psi_{j}(y_{i-1}),
\end{aligned}
\quad 
\begin{aligned}
\set{\beta,\gamma} &=\set{\psi_{j-1}(y_{i}), 
 \psi_{j-1}(x_{i-1})} , \cr
 \set{\beta',\gamma'} &=\set{\psi_{j}(y_{i}), 
 \psi_{j}(x_{i-1})}
\end{aligned}
\end{equation*} 
such that $\beta'=\beta\vee b_j$ and $\gamma'=\gamma\vee b_j$. 
By the minimality of $j$, $|\set{\alpha,\beta,\gamma,\delta}|=4$. Hence, $\set{\alpha,\beta,\gamma,\delta}=\psi_{j-1}(H_i)$ is a covering square with bottom $\alpha$ and top $\delta$ in the filter $\filter {b_{j-1}}=[b_{j-1},1]$ since $\psi_{j-1}$ is a cover-preserving join-homomorphism. Consider the cover-preserving join-homomorphism $\phi\colon \filter {b_{j-1}} \to \filter {b_{j-1}}$, defined by $\phi(z)=b_j\vee z$. Denote the height function on $\filter {b_{j-1}}$ by $h$. 
Since $b_j$ is an atom of the filter $\filter b_{j-1}$, semimodularity implies that $h(z)\leq h(\phi(z)) \leq h(z)+1$ holds for all $z\in \filter {b_{j-1}}$. By definitions, $\phi(\alpha)=\alpha'$, $\phi(\beta)=\beta'$, $\phi(\gamma)=\gamma'$, and  $\phi(\delta)=\delta'$, and we also  have $\gamma'=\delta'$ and $\alpha'\neq\beta'$. Let $\Phi=\Ker\phi$. It is a cover-preserving join-congruence, and Lemma~\ref{thforbIDnn} yields that $\set{\beta,\gamma}\subseteq \blokk{\delta}\Phi$ but $\alpha\notin \blokk{\delta}\Phi$.
Thus $\alpha'\neq\beta'=\gamma'=\delta'$. Actually, $\alpha'\prec\beta'=\gamma'=\delta'$  since $\phi$ is cover-preserving. 

Since $\beta\neq\gamma$, we have $\pair{\beta'}{\gamma'}\neq \pair{\beta}{\gamma}$. Hence we can assume $\beta'\neq\beta$, and we obtain $\beta\prec \beta'$. Using 
$h(\gamma)+1=h(\beta)+1=h(\beta')=h(\gamma')$, we obtain $\gamma\prec \gamma'$. Now $h(\delta')=h(\beta')= h(\beta)+1=h(\delta)$, together with $\delta\leq\delta'$, yields $\delta'=\delta$. We have $\alpha'\preceq \beta'=\delta'=\delta$ since $\phi$ is cover-preserving. Hence $\alpha'\neq\alpha$, and we obtain $\alpha\prec \alpha'$.  

The previous relations imply $b_j\leq\delta$, $b_j\not\leq \alpha$,  $b_j\not\leq \beta$, and  $b_j\not\leq \gamma$. Let $p=\alpha\vee b_j$. Since $b_j$ is an atom in the filter $\filter {b_{j-1}}$, we have $\alpha\prec p$. We obtain $p\not\leq \beta$ from $b_j\not\leq \beta$, and we obtain $p\not\leq \gamma$ similarly. Hence $\beta$, $p$, and $\gamma$ are three different covers of $\alpha$ in the interval $[\alpha,\delta]$ of length 2. Thus $\set{\alpha, \beta, p,\gamma,\delta}$ is a cover-preserving diamond of $L$, which is a contradiction.
\end{proof}

Now we are in the position to complete the proof of Lemma~\ref{trajrestlem}.




\begin{proof}[Proof of Lemma~\ref{trajrestlem}$\textup{(B)}$] 
Let $C$ be a map from $\print S$ to the set of maximal chains of $S$ such that, for every $\inp\in\print S$,  $\inp\in\print {C(\inp)}$. 
%Let $D$ be a map from the set of maximal chains of $S$ to the set of maximal chains of $L$ such that $X\subseteq D(X)$ for all maximal chains $X$ of $S$. 

To show that $\kappa$ is a map from $\traj L$ to $\traj S$, let  $T\in\traj L$ such that $T\cap\print S\neq\emptyset$. Pick a prime interval $\inp\in T\cap\print S$, and let $R\in\traj S$ be the unique trajectory containing $\inp$. At present, we know that 
\begin{equation}\label{oasiwWE}
\text{$T\in\traj L$, $R\in\traj S$, and $p\in T\cap R\cap\print S$,
}
\end{equation}
and this will be the only assumption on $T$ and $R$ we use in the rest of the present paragraph. 
By Lemma~\ref{trajrestlem}$\textup{(A)}$, there is an $R'\in \traj L$ such that $R\subseteq R'$. Since $\inp\in T\cap R'$, we have $R\subseteq R'=T$, which implies $R\subseteq\kappa(T)$. To show the converse inclusion, let $\inq\in\kappa(T)=T\cap\print S$. Applying the existence part of  Lemma~\ref{trajlemma} to $S$, we obtain an $\inr\in R\cap \print{C(\inq)}$. 
Since $R\subseteq T$, both $\inr$ and $\inq$ belong to $T$. Thus the uniqueness part of  Lemma~\ref{trajlemma} gives $\inq=\inr\in R$. Hence, $R=\kappa(T)$, and $\kappa$ is a map from $\traj L$ to $\traj S$.

Since \eqref{oasiwWE} implies $R=\kappa(T)$,   $\,\kappa$ is surjective. 

Finally, let $\set{0_L,1_L}\subseteq S$. Then $S$ contains a maximal chain $X$ of $L$. Assume that $T_1,T_2\in\traj L$ such that $\kappa(T_1)=\kappa(T_2)$. By Lemma~\ref{trajlemma}, there is a unique $\inq\in T_1\cap \print X$. 
We have $q\in T_1\cap\print S= \kappa(T_1)=\kappa(T_2)\subseteq T_2$. Hence $T_1\cap T_2\neq\emptyset$, and we conclude $T_1=T_2$. Consequently, $\kappa$ is injective, and we obtain that it is  bijective.
\end{proof}



\section{Jordan-H\"older permutations}\label{JHsection}
Any two maximal chains of a semimodular lattice of length $n$ determine a so-called  Jordan-H\"older permutation on the set $\set{1,\dots,n}$. This was first stated by \init{R.\,P.~}Stanley~\cite{stanley}; see also \init{H.~}Abels~\cite{abels} for further developments. Independently, the same permutations emerged in \init{G.~}Gr\"atzer and \init{J.\,B.~}Nation~\cite{gratzernation}. The Jordan-H\"older permutations were rediscovered in \init{G.~}Cz\'edli and \init{E.\,T.~}Schmidt~\cite{czgschjh}, and were successfully applied to add a uniqueness part to the classical Jordan-H\"older Theorem for groups. As an excuse for this rediscovery, note that some results we need here were proved in \cite{czgschjh} and the subsequent \init{G.~}Cz\'edli and \init{E.\,T.~}Schmidt~\cite{czgschperm}. 
In \cite{czgschperm}, there are three equivalent definitions for Jordan-H\"older permutations.  Here we combine the treatment given in \cite{czgschjh} and \cite{czgschperm} with Lemma~\ref{trajlemma}. As opposed to \init{H.~}Abels~\cite{abels}, we always assume that $L$ has no cover-preserving diamond.  

\begin{definition}\label{deFperm}
Let $L$ be a semimodular lattice  of length $n$, and assume that $L$ has no cover-preserving diamond. 
Let $C=\set{0=c_0\prec c_1\prec \dots \prec c_n=1}$ and $D=\set{0=d_0\prec d_1\prec \dots \prec d_n=1}$ be maximal chains of $L$. Then the \emph{Jordan-H\"older permutation}  $\perm CD\colon\nset \to \nset $ is  defined by $\perm CD(i)=j$ if{f}
$[c_{i-1},c_i]$ and $[d_{j-1},d_j]$ belong to the same trajectory of $L$.
\end{definition}

It is clear from Lemma~\ref{trajlemma} that $\perm CD$ above is a permutation. The next lemma shows that our definition of $\perm CD$ is the same as that of \init{H.~}Abels~\cite[(3.1)]{abels}, provided $L$ has  no cover-preserving diamond. 

\begin{lemma}\label{ujHpJpw} Let $L$, $C$,  and $D$ be as in Definition~\ref{deFperm}. Then, for $i\in\nset $,  
\[\perm CD(i)=\min\set{j: c_{i-1}\vee d_j=c_i\vee d_j}\text.
\]
The permutation  $\perm CD$ equals the identity permutation $\id$ if{f} $\,C=D$. We have $c_{i-1}\vee d_j=c_i\vee d_j$ if{f} $\,\perm CD(i)\leq j$.  
\end{lemma}

\begin{proof}If follows from Lemma~\ref{trajrestlem} that $\perm CD(i)$ can be computed in $\jgen{C\cup D}$. Hence the first part of the statement can be extracted from  \init{G.~}Cz\'edli and \init{E.\,T.~}Schmidt~\cite[Definition 2.5]{czgschperm} or, more easily, from  \init{G.~}Cz\'edli, \init{L.~}Ozsv\'art, and \init{B.~}Udvari~\cite[Definition 2.4]{czgolub}. Now that we know that our $\perm CD$ is the same as defined in \init{H.~}Abels~\cite{abels}, the middle part follows from \cite[3.5.(a)]{abels}. It also follows from \cite[Theorem 3.3]{czgschperm} or \cite[Lemma 7.2]{czgolub}. The last part is a trivial consequence of the first part.
\end{proof}



While the following lemma needs a proof in \init{H.~}Abels~\cite[Theorem 3.9(f)]{abels}, Definition~\ref{deFperm} in our setting makes it obvious. We compose permutations and maps from right to left, that is, $(f\circ g)(x)= f(g(x))$. 

\begin{lemma}\label{permcat}
Let $L$ be a semimodular lattice of finite length and without cover-preserving diamonds, and let $C$, $D$, and $E$ be maximal chains of $L$. Then the following hold.
\begin{enumeratei}
\item\label{permcata} $\perm CC=\id$, the identity map. 
\item\label{permcatb} $\perm CD=\perm DC ^{-1}$
\item\label{permcatc}$\perm DE\circ\perm CD=\perm CE$.
\end{enumeratei}
\end{lemma}
Equivalently, the lemma above asserts that the maximal chains of $L$ with singleton hom-sets hom$\textup(C,D)=\set{\perm CD}$ form a category, namely, a groupoid. We do not recall further details since although this category (equipped with the weak Bruhat order) determines $L$ by \init{D.\,S.~}Herscovici~\cite{herscovici},
it is rather large and complicated for our purposes. The best way for coordinatization is offered by Lattice Theory.

To give a short illustration of the strength of Lemma~\ref{permcat}, we prove the following corollary even if it is known; see  Remark~\ref{reMrkdh}.


\begin{corollary}\label{idsehGH} If $L$ is a semimodular lattice of finite length and $L$ has no cover-preserving diamonds, then $L$ is finite.
\end{corollary}

\begin{proof} Pick a maximal chain $C$ in $L$. 
By Lemma~\ref{permcat},  if $\perm CD=\perm CE$, then $\id=\perm CD \circ \perm CD^{-1}=\perm CE \circ\perm DC=\perm DE$, and Lemma~\ref{ujHpJpw} implies 
 $D=E$. Hence, $L$ has only finitely many maximal chains. Thus $L$ is finite.
\end{proof}




\section{The main result}\label{secmainres}
We always assume that $n$ belongs to $\mathbb N=\set{1,2,3,\dots}$ and, unless otherwise specified, $k\in\set{2,3,\ldots}$. For a structure $\alg A$, the class of structures isomorphic to $\alg A$ is denoted by $\isomcl{\alg A}$. 
As usual, $S_n$ stands for the group of permutations of the set $\nset $. Consider the class 
\begin{equation*}
\begin{aligned}
\Lat nk =\{&\isomcl \langle L;C_1,\ldots,C_k\rangle : L\text{  is a join-distributive lattice of length }n\text{,  and }\cr
&C_1,\dots,C_k  \text{ are maximal chains of }L\text{ such that  }
\Jir L\subseteq C_1\cup\dots C_k  \}\text.
\end{aligned}
\end{equation*}
 We define a map 
\begin{equation}
\maplp\colon \Lat nk\to S_n^{k-1}
\,\text{ by }\, \isomcl\langle L;C_1,\ldots,C_k\rangle \mapsto \langle \perm{C_1}{C_2}, \perm{C_1}{C_3},\dots, \perm{C_1}{C_k}\rangle\text.
\end{equation}
%
If $\vpi=\langle  \pi_{2},\dots,\pi_{k}\rangle\in S_n^{k-1}$, then by the corresponding \emph{extended vector} we mean the
$k^2$-tuple 
$
\langle \pi_{ij}: i,j\in\kset  \rangle ,
$
where $\pi_{ij}=\pi_{1j}\circ\pi_{1i}^{-1}$. In general, 
\begin{equation}\label{kiInmd}
\text{$\pi_{ij}$ is always understood as $\pi_{1j}\circ\pi_{1i}^{-1}$,}
\end{equation}
even if  this is not emphasized all the time. 

\begin{definition}\label{defeligl}
By an \emph{eligible $\vpi$-tuple} we mean a $k$-tuple $\vec x=\langle x_1,\ldots, x_k\rangle \in\set{0,1,\ldots,n}^k$ such that $\pi_{ij}(x_i+1 )\geq x_j+1$ holds for all $i,j\in \kset $ such that $x_i<n$. The set of eligible $\vpi$-tuples is denoted by $\elpi$. 
It is a poset with respect to the componentwise order: $\vec x\leq \vec y$ means that, for all $i\in \kset $,  $x_i\leq y_i$.
\end{definition}

For $i\in \kset $, an eligible $\vpi$-tuple $\vec x$ is \emph{initial in its $i$-th component} if for all $\vec y\in\elpi$, 
$x_i=y_i$ implies $\vec x\leq \vec y$. Let $\lancpi i$ be the set of all eligible $\vpi$-tuples that are initial in their $i$-th component. Now we are in the position to define a map
\begin{equation}
\mappl \colon S_n^{k-1}\to  \cdf nk
\,\text{ by }\, \vpi\mapsto \isomcl\langle \elpi; \lancpi 1,\dots,\lancpi k  \rangle\text.
\end{equation}

It is not obvious that $\mappl(\vpi)\in \cdf nk$, but we will prove it soon. 
Now, we can formulate our main result as follows.

\begin{theorem}\label{thmmain}
The maps $\maplp$ and $\mappl$ are reciprocal bijections between 
$\Lat nk$ and $S_n^{k-1}$.
\end{theorem}


This theorem gives the desired coordinatization since each 
$\isomcl\langle L;C_1,\ldots,C_k\rangle$ from $\Lat nk$ is described by its $\maplp$-image, that is, by $k-1$ permutations. The elements of $\langle L;C_1,\ldots,C_k\rangle$ correspond to $k$-tuples over $\set{0,\ldots,n}$, and the $k-1$ permutations  specify which $k$-tuples occur.

\begin{problem}
It would be desirable to characterize those pairs
 $\pair{\vpi}{\vsigma}$ 
 of $(k-1)$-tuples of permutations of $S_n$ for which the lattice part of  $\mappl(\vpi)$ (without the $k$ chains) coincides with that of $\mappl(\vsigma)$.  
However, in spite of the theory developed in  \init{D.\,S.~}Herscovici~\cite{herscovici},  we do not expect an elegant characterization. While the requested characterization for $k=2$ is known from  \init{G.~}Cz\'edli and \init{E.\,T.~}Schmidt~\cite{czgschperm} and it was used in \init{G.~}Cz\'edli, \init{L.~}Ozsv\'art, and \init{B.~}Udvari~\cite{czgolub},
even the case $k=3$ seems to be quite  complicated; this is witnessed by the following example.
\end{problem}
%
\begin{example}\label{exMple} Let $L=\set{0,a,b,c,a\vee b, a\vee c, b\vee c,1}$ be the 8-element boolean lattice. Consider the maximal chains $C_1=\set{0,a,a\vee b, 1}$,  $C_2=\set{0,b,a\vee b, 1}$, $C_3=\set{0,c,b\vee c, 1}$, and also the maximal chains $C_1'=\set{0,a,a\vee b, 1}$, $C_2'=\set{0,b,b\vee c, 1}$, and $C_3'=\set{0,c,a\vee c, 1}$. Then
\[\begin{aligned}
\perm{C_1}{C_2}&=\begin{pmatrix}1&2&3\cr 2&1&3\end{pmatrix},\qquad
\perm{C_1}{C_3}=\begin{pmatrix}1&2&3\cr 3&2&1\end{pmatrix},\cr
%
\perm{C_1'}{C_2'}&=\begin{pmatrix}1&2&3\cr 3&1&2\end{pmatrix},\qquad
\perm{C_1'}{C_3'}=\begin{pmatrix}1&2&3\cr 2&3&1\end{pmatrix}\text.
\end{aligned}
\] 
Let $\vpi=\pair{\perm{C_1}{C_2}}{\perm{C_1}{C_3}}$ and $\vsigma=\pair{\perm{C_1'}{C_2'}}{\perm{C_1'}{C_3'}}$. These two vectors  look very different  since $\vpi$ consists of transpositions while $\vsigma$ does not; furthermore, the second component of $\vsigma$ is the inverse of the first component, while this is not so for $\vpi$. However, by construction and Theorem~\ref{thmmain}, $\vpi$ and $\vsigma$ determine the same lattice $L$.
\end{example}


\section{Proving the main result}\label{sectproofs}
Some of the auxiliary statements we will prove are valid under a seemingly weaker assumption than requiring $\isomcl L\in \Lat nk$. Therefore, in accordance with Remark~\ref{reMrkdh}, we prove a (seemingly) stronger statement with  little extra effort. To do so, consider the class 
%
\begin{equation*}
\begin{aligned}
\cdf nk =\{\isomcl \langle L;C_1,\ldots,C_k\rangle : L\text{ is semimodular of length } n\text{, }L \text{ contains}&\cr
\text{no cover-preserving diamond,   and }  C_1,\ldots, C_k \text { are maximal chains of $L$.}\}& 
\end{aligned}
\end{equation*}
This notation comes from  ``Cover-preserving Diamond-Free''. 
Although we know from Proposition~\ref{Dlwltcs} and Remark~\ref{reMrkdh} that  
$\cdf nk$ equals $\Lat nk$, we will give a new  proof for  this equality. We will use only the obvious  $\Lat nk \subseteq \cdf nk$. Keeping the notation of the original map, we extend its range as follows:  
\begin{align*}
\maplp\colon \cdf nk\to S_n^{k-1}
\text{, where }\, \isomcl\langle L;C_1,\ldots,C_k\rangle \mapsto \langle \perm{C_1}{C_2}, \perm{C_1}{C_3},\dots, \perm{C_1}{C_k}\rangle\text.
\end{align*}
Our aim in this section is to prove the following statement, which implies Theorem~\ref{thmmain} and harmonizes with Remark~\ref{reMrkdh}.

\begin{proposition}\label{almostmain}\ 

$\textup{(A)}$ $\cdf nk=\Lat nk$.

$\textup{(B)}$ 
The maps $\maplp$ and $\mappl$ are reciprocal bijections. 
\end{proposition}

The proof of Proposition~\ref{almostmain} will need the following lemma.



\begin{lemma}\label{siodsGz} If $\vpi\in S_n^{k-1}$, then $\mappl(\vpi)\in \Lat nk$.
\end{lemma}

%
%The proof of our coordinatization result  offers an alternative way to the implication \eqref {neWjsldSc} $\then$ \eqref {Dlwltcsa} for lattices of finite length. 




\begin{proof} Convention \eqref{kiInmd} should be kept in mind. 
Clearly, $\langle n,\ldots,n\rangle\in\elpi$. Hence $\elpi$ has a top element. Let $\vec x=\langle x_1,\ldots,x_k\rangle$ and $\vec y=\langle y_1,\ldots,y_k\rangle$ belong to $\elpi$, and let $z_i=x_i\wedge y_i=\min\set{x_i,y_i}$, for $i\in\kset $.
Assume that $j\in\kset $ such that $z_j<n$. Since $\vec x,\vec y\in\elpi$, we have $\pi_{jt}(x_j+1)\geq x_t+1$ and $\pi_{jt}(y_j+1)\geq y_t+1$, for every $t\in\kset $. Let, say, $z_j=x_j$. We obtain 
$\pi_{jt}(z_j+1) = \pi_{jt}(x_j+1)\geq x_t+1 \geq z_t+1$. Hence $\vec z=\langle z_1,\ldots,z_k\rangle\in\elpi$, and we conclude that $\elpi$ is a lattice.

If we had $\vec x=\tuple{ n,x_2,\ldots,x_k}\in\elpi$ and $x_i\neq n$ for some $i\in\set{2,\ldots,k}$, then $\pi_{i1}(x_i+1)\leq n < x_1+1$ would contradict $\vec x\in\elpi$. Hence $\langle n,\ldots,n\rangle$ is the only vector in $\elpi$ with first component $n$, and we conclude that this vector belongs to $\lancpi 1$. Next, assume that $\vec y=\langle y_1,\ldots,y_k\rangle$ and 
$\vec z=\langle z_1,\ldots,z_k\rangle$ belong to $\lancpi 1$ such that $y_1\leq z_1$. Since $\elpi$ is meet-closed, 
$\vec y\wedge \vec z=\langle y_1,y_2\wedge z_2,\ldots,y_k\wedge z_k\rangle\in\elpi$. Since $\vec y$ is initial in its first component, 
$\vec y\leq \vec y\wedge \vec z$, which gives $\vec y\leq \vec z$. Thus $\lancpi 1$ is a chain. 


A vector $\vec x$ in $\nset ^k$ is a \emph{$\vpi$-orbit}, if $\pi_{ij}(x_i)=x_j$ for all $i,j\in\kset$.
Equivalently, if it is of the form 
\begin{equation}\label{slVCbk} \vec x=\tuple{ \pi_{i1}(b),\dots,\pi_{ik}(b)} \rangle
\end{equation}
for some $i\in\kset $ and $b\in \nset $.   A $\vpi$-orbit need not belong to $\elpi$.
A vector $\vec y=\langle y_1,\ldots,y_k \rangle$ in $\set{0,\ldots,n-1}^k$ is  \emph{ suborbital with respect to $\vpi $} if 
\begin{equation}\label{suboRbital}
\text{
$\langle y_1+1,\ldots, y_k+1\rangle$ is a $\vpi$-orbit.}
\end{equation} 
Clearly, suborbital vectors belong to $\elpi$. For each $b\in \set{0,\dots,k-1}$,  \eqref{slVCbk}, applied for $i=1$, shows that  there exists a vector $\vec x\in \elpi$ whose first component is $b$. Let $\vec y$ be the meet of all these vectors. Clearly, $y_1=b$ and $y\in \lancpi 1$. This shows that $\lancpi 1$ is a chain of length $n$, and so are the $\lancpi i$ for $i\in\kset $. 



Next, let $B(\vpi)$ denote the set of suborbital vectors with respect to $\vpi$.  
We have $B(\vpi)\subseteq \elpi$. 
By \eqref{slVCbk}, each $b\in\nset $ is the $i$-th component of exactly one $\vpi$-orbit. Therefore, 
\begin{equation}\label{dkcBGb}
|B(\vpi)|=n\text.
\end{equation}
We assert that 
\begin{equation}\label{msBxG}
\begin{aligned}
&\text{for each $\vec x\in\elpi$, there is a unique $U\subseteq B(\vpi)$ such}\cr
&\text{that 
$\vec x=\bigwedge U$ is an irredundant meet decomposition.}
\end{aligned}
\end{equation}
Let $\vec x=\tupk x\in \elpi\setminus\set{\tuple{n,\ldots,n}}$, and let $I=\set{i: x_i<n}$. 
 For $i\in I$, let 
\[\vec y^{\,(i)}=\tuple{\pi_{i1}(x_i+1)-1,\dots,\pi_{ik}(x_i+1)-1}\text.
\]
It belongs to $B(\vpi)$, whence  $\vec y^{\,(i)}\in\elpi$. Since $\vec x$ is also in $\elpi$, for $j\in\kset $ we have
$x_j=x_j+1-1\leq \pi_{ij}(x_i+1)-1= y_j^{(i)}$. Hence $\vec x\leq \vec y^{(i)}$, and we conclude 
$\vec x\leq \bigwedge\set{ \vec y^{\,(i)}: i\in I }$. The converse inequality also holds since $x_i=n$ for $i\notin I$ and $y_i^{(i)}=\pi_{ii}(x_i+1)-1 = x_i$ for $i\in I$. That is, $\vec x = \bigwedge\set{ \vec y^{\,(i)}: i\in I }$. Since the meetands here are not necessarily distinct, we let 
\[\text{
$J=\{j\in I:\vec y^{\,(i)}\neq  \vec y^{\,(j)}$ for all $i<j$ such
that $i\in I\}$.
}\] 
Clearly, $\vec x = \bigwedge\set{ \vec y^{\,(j)}: j\in J }$.
We assert that if  $V\subseteq B(\vpi)$ and  $\vec x=\bigwedge V$, then 
$\set{\vec y^{\,(j)}: j\in J}\subseteq V$.  To show this, let  $j\in J$. Since the $j$-th components of the vectors of $V$ form a chain and the meet of these components equals $x_j$, there exists a $\vec u\in V$ such that $u_j=x_j$. The $j$-th component of $\vec y^{\,(j)}$ is also $x_j$ by definition. Since any two orbital vectors with a common $j$-th component are equal, we obtain 
$\vec y^{\,(j)}=\vec u\in V$. Thus we obtain $\set{\vec y^{\,(j)}: j\in J}\subseteq V$. Since the $\vec y^{\,(j)}$, for $j\in J$,  are pairwise distinct, we can take $U=\set{\vec y^{\,(j)}: j\in J}$, and we conclude \eqref{msBxG}.

Now, let $\vec a\in\Mir{(\elpi)}$. Since $\vec a= \ibvec 1\wedge\dots \wedge \ibvec s$ for appropriate $ \ibvec 1,\ldots, \ibvec s$ in $B(\vpi)$ by \eqref{msBxG}, we obtain $s=1$ and $\vec a= \ibvec 1\in B(\vpi)$. That is, $\Mir{(\elpi)}\subseteq B(\vpi)$. For the sake of contradiction, suppose $B(\vpi)\setminus\Mir{(\elpi)}\neq \emptyset$, and let $\vec b\in B(\vpi)\setminus\Mir{(\elpi)}$. 
Observe that $U'=\set{ \vec b}$ gives an irredundant meet-decomposition according to \eqref{msBxG}. 
There are a minimal $t\in\mathbb N$ and $\iwvec 1,\ldots,\iwvec t\in\Mir{(\elpi)}$ such that $\vec b=\iwvec 1\wedge\dots\wedge \iwvec t$. This meet is irredundant by the minimality of $t$, and $t\geq 2$ since $\vec b\neq\tuple{n,\ldots,n}$ and $\vec b\notin \Mir{(\elpi)}$. However, $\iwvec 1,\ldots,\iwvec t\in B(\vpi)$ since $\Mir{(\elpi)}\subseteq B(\vpi)$. Thus $U''=\set{\iwvec 1,\ldots,\iwvec t}$ also gives an irredundant meet-decomposition according to \eqref{msBxG}. This is a contradiction since $U'\neq U''$. Hence,
\begin{equation}\label{mlpbP}
\Mir{(\elpi)} = B(\vpi),
\end{equation}
and $\elpi$ has unique meet-irreducible decompositions. Thus we conclude from Proposition~\ref{Dlwltcs} that 
 $\elpi$ is join-distributive. 


Next, we have to show that $\lancpi i$ is a maximal chain for $i\in\kset $. Since $|\lancpi i|=n+1$, it suffices to show that $\elpi$ is of length at most $n$. To prove this by contradiction, suppose  $H=\set{h_0\prec h_1\prec \dots\prec h_{n+1}}$ is a chain of $\elpi$. Let $W_i=\Mir{(\elpi)}\cap \filter{h_i}$. Clearly, $W_0 \supsetneq W_1 \supsetneq \dots \supsetneq W_{n+1}$, which contradicts the fact that $|\Mir{\elpi}|=|B(\vpi)|=n$ by \eqref{dkcBGb} and \eqref{mlpbP}. 


Finally, let $\vec x=\tupk x\in \elpi$. For $i\in\kset$, let $\vec y^{\,(i)}\in\elpi$ be the smallest vector whose $i$-th component is $x_i$. Clearly, $\vec y^{\,(i)}\in \lancpi i$, $\vec y^{\,(i)}\leq \vec x$, and $\vec y^{\,(1)}\vee\dots\vee \vec y^{\,(k)}=\vec x$. Hence $\Jir{(\elpi)}\subseteq \lancpi 1\cup\dots\cup \lancpi k$.
\end{proof}



\begin{lemma}[Roof Lemma]\label{rooflemma} If  $k\in \mathbb N$, $\isomcl{\alg L}$ belongs to $\cdf nk$,  $x_1,\ldots,x_k\in L$,  $x_1\vee \dots\vee x_k=1$, and $\inp_i\in\print{\filter{x_i}}$ for $i\in\kset$, then $\inp_1,\ldots,\inp_k$ cannot belong to the same trajectory of $L$. 
\end{lemma}

\begin{proof}
We can assume that $1\notin \set{x_1,\ldots,x_k}$ since otherwise $\print{\filter{x_i}}=\emptyset$ for some $i\in\kset$, and the statement trivially holds. Therefore, we can also assume that $k>1$.
 
First, we deal with $k=2$. For the sake of contradiction, suppose $\inp_1=[u_1,v_1]$ and $\inp_2=[u_2,v_2]$ belong to the same trajectory $T$. 
For $i\in\set{1,2}$, pick a maximal chain $X_i$ containing $x_i$ such that  $\inp_i\in\print {X_i}$. Let $M=\jgen{X_1\cup X_2}$, the join-subsemilattice generated by $X_1\cup X_2$. 
We know from \init{G.~}Cz\'edli and \init{E.\,T.~}Schmidt~\cite[Lemma 2.4]{czgschjh} that $M$ is a cover-preserving 0,1-join-subsemilattice of $L$, and it is a slim semimodular lattice. Its length, $n$, is the same as that of $L$.
We obtain from Lemma~\ref{trajrestlem} that $\inp_1$ and $\inp_2$ belong to the same trajectory $T$ of $M$.
By \cite[Lemma 2.2]{czgschjh}, $M$ is a planar lattice, and it has a planar diagram whose left boundary chain is $X_1$.  The trajectories of planar semimodular lattices join-generated by two chains are well-understood. By \cite[Lemma 2.9]{czgschjh}, there is an interval $\inp_3=[u_3,v_3]$ such that both $\inp_1$ and $\inp_2$ are  up-perspective to $\inp_3$.
This means that $u_i=v_i\wedge u_3$ and $v_3=v_i\vee u_3$, for $i\in\set{1,2}$. Hence $1=x_1\vee x_2\leq u_1\vee u_2\leq u_3<v_3\leq 1$, which is a contradiction that proves the statement for  $k=2$.

Next, to proceed by induction, assume that $k>2$ and the lemma holds for smaller values. To obtain a contradiction, suppose that $\inp_1,\ldots,\inp_k$ belong to the same trajectory $T$ of $L$. 
Let $y_1=x_2\vee\cdots\vee x_{k}$. We can assume $y_1\neq 1$ since otherwise $x_1$ can be omitted and the induction hypothesis applies. Pick a maximal chain $U_1$ of $L$ such that $y_1\in U_1$. 
By Lemma~\ref{trajlemma}, there is a unique $\inq_1\in T\cap\print{U_1}$. We assert that $\inq_1\in\print{\filter {y_1}}$. Suppose not, and let $R =T\cap \print{\ideal{y_1}}$. It is nonempty by Lemma~\ref{trajlemma}. Hence, it is a trajectory of  $\ideal{y_1}$ by Lemma~\ref{trajrestlem}. Thus the induction hypothesis, together with Lemma~\ref{trajlemma}, yields an $i\in\set{2,\ldots,k}$ and a prime interval $\inr_i$  such that $\inr_i\in  \print{\ideal{x_i}}\cap R  \subseteq \print{\ideal{x_i}}\cap T$. Since we can clearly take a maximal chain $V$ of $L$ through $x_i$  such that $\inp_i,\inr_i\in\print{V}$, we obtain a contradiction by  Lemma~\ref{trajlemma}. Thus we conclude $\inq_1\in\print{\filter{y_1}}$.

Since $1\in\kset$ in the argument above does not play any special role, we obtain that $T$ contains a prime interval $\inq_2$ in the filter $\filter{y_2}$ generated by $y_2=x_1\vee x_3\vee\dots\vee x_k$. This contradicts the induction hypothesis since  $y_1\vee y_2=1$.
\end{proof}  



Let $\alg L=\tuple{L;C_1,\ldots,C_k}\in \cdf nk$. We denote the elements of $C_i$ as follows:
\[ 
C_i=\set{0=c_0^{\,(i)} \prec c_1^{\,(i)} \prec \dots \prec c_n^{\,(i)} =1 }\text.
\]
A vector
$\vec x\in \nnset^k$ is called \emph{$\alg L$-maximal} if for all $i\in \kset$, we have 
\[ c_{x_1}^{\,(1)}\vee \cdots\vee c_{x_k}^{\,(k)} <  
c_{x_1}^{\,(1)}\vee \cdots\vee c_{x_{i-1}}^{\,(i-1)}\vee c_{x_i+1}^{\,(i)}\vee  c_{x_{i+1}}^{\,(i+1)}\vee \cdots\vee c_{x_k}^{\,(k)}\text.
\]


\begin{lemma}\label{maxEliglma} Let $\isomcl\alg L=\isomcl\tuple{L;C_1,\ldots,C_k}\in \cdf nk$. Let $\vpi$ denote its $\xi$-image, and let $\vec x\in \nnset^k$. Then $\vec x$ is $\alg L$-maximal if{f} it is an eligible $\vpi$-tuple.
\end{lemma}


\begin{proof} 
To prove the ``only if'' part, let $\vec x=\tupk x$ be an $\alg L$-maximal vector, and let $i\neq j\in\kset$. By $\alg L$-maximality, 
\begin{align*}
  c_{x_i}^{\,(i)}  \vee c_{x_j}^{\,(j)}&\vee\bigvee\bigset{ c_{x_t}^{\,(t)}: t\in\kset\setminus\set{i,j} } \cr
< {}  
&c_{x_i+1}^{\,(i)}\vee c_{x_j}^{\,(j)}\vee\bigvee\bigset{ c_{x_t}^{\,(t)}: t\in\kset\setminus\set{i,j} } \text.
\end{align*}
Hence $c_{x_i}^{\,(i)}  \vee c_{x_j}^{\,(j)}  <  c_{x_i+1}^{\,(i)}\vee c_{x_j}^{\,(j)}$. Thus,  by Lemma~\ref{ujHpJpw}, 
$\perm{C_i}{C_j}(x_i+1) > x_j$. Therefore, $\vec x$ is  an eligible $\vpi$-tuple.


To prove the converse implication by contradiction, suppose that $\vec x=\tupk x$ is an eligible $\vpi$-tuple but it is not $\alg L$-maximal. Let, say, the $k$-th component of $\vec x$ violate $\alg L$-maximality, and let 
\begin{equation}\label{sWodBjT}
\text{
$u=c_{x_1}^{\,(1)}\vee\dots\vee c_{x_{k-1}}^{\,(k-1)}$.}
\end{equation} 
We have $u\vee c_{x_k}^{\,(k)}=u\vee c_{x_k+1}^{\,(k)}$.  For $i\in \set{1,\ldots,k-1}$, extend $(C_i\cap \ideal {c_{x_i}^{\,(i)}})\cup\set u$ to a maximal chain $U_i$ of $L$. We denote the trajectory of $L$ 
that contains $[c_{x_k}^{\,(k)}, c_{x_k+1}^{\,(k)}]$ by $T$. 
Since $u\vee c_{x_k}^{\,(k)}=u\vee c_{x_k+1}^{\,(k)}$, Lemma~\ref{ujHpJpw} yields  
$\perm {C_k}{U_i}(x_k+1)\leq h(u)$, where $h$ is the height function. Hence, by Definition~\ref{deFperm}, there is a $\inp_i\in\print{U_i}\cap T$ such that $\inp_i$ is below $u$, that is, 
\begin{equation}\label{nsiseoe}
\inp_i\in \print{U_i\cap \ideal u}\cap T \text.
\end{equation}
On the other hand, the eligibility of $\vec x$ gives $\perm{C_k}{C_i}(x_k+1)\geq x_i+1$. Hence, again  by Definition~\ref{deFperm},  $C_i$ contains a prime interval of $T$ above $c_{x_i}^{\,(i)}$. 
Therefore, by Lemma~\ref{trajlemma}, $C_i$ does not contain any prime interval of $T$ below $c_{x_i}^{\,(i)}$. But $C_i\cap \ideal c_{x_i}^{\,(i)}= U_i\cap \ideal c_{x_i}^{\,(i)}$, and we conclude that $  {\print{U_i\cap \ideal c_{x_i}^{\,(i)}}}\cap T=\emptyset$. 
Combining this with \eqref{nsiseoe}, we obtain that $T\cap\print{\ideal u}$, which is a trajectory of $\ideal u$ by Lemma~\ref{trajrestlem}, contains a prime interval  in $\filter c_{x_i}^{\,(i)}$, for 
$i\in\set{1,\ldots,k-1}$. This, together with \eqref{sWodBjT}, contradicts  Lemma~\ref{rooflemma}.
\end{proof}


The following lemma generalizes \init{G.~}Cz\'edli and \init{E.\,T.~}Schmidt~\cite[Lemma 2.3]{czgschperm}. We use the notation preceding Lemma~\ref{maxEliglma}. The set of suborbital vectors is still denoted by $B(\vpi)$, as above \eqref{dkcBGb}. 
For $u\in L$, the \emph{foot} of $u$ is the following vector in $\nnset^k$: 
\[\feet u=\tuple{\foot u1,\ldots,\foot uk}=\Bigl\langle 
\max\set{j: c_{j}^{\,(1)} \leq u },
\dots,
\max\set{j: c_{j}^{\,(k)} \leq u }  
\Bigr\rangle\text.
\]


\begin{lemma}\label{ujsuborblMa}
If $\,\isomcl\alg L\in\cdf nk$, then 
$\set{\feet u: u\in\Mir L} \subseteq  B(\vpi)$, where $\vpi=\maplp(\isomcl{\alg L})$. If $\,\isomcl\alg L\in\Lat nk$, then even  $\set{\feet u: u\in\Mir L} =  B(\vpi)$ holds.
\end{lemma}


\begin{proof}
For $k=2$, the lemma is only a reformulation of \cite[Lemma 2.3]{czgschperm}. Namely, in this case, one takes the trajectory  $[u,\upstar u]$; it contains a unique prime interval $[a_0,a_1]$, and $\foot u1=h(a_0)$; and analogously for $\foot u2$.

Hence, we assume $k>2$.  Let $u\in \Mir L$. We are going to prove  
\begin{equation}\label{scdsie}
\pi_{12}(\foot u1 +1)=\foot u2+1\text.
\end{equation} 
Let $K=\jgen{C_1\cup C_2}$; it is a slim, semimodular, cover-preserving join-subsemi\-lattice of $L$ by \init{G.~}Cz\'edli and \init{E.\,T.~}Schmidt~\cite[Lemma 2.4]{czgschjh}.
Let $u_0$ be the largest element of $K\cap \ideal u$. 

First, if $u_0\in \Mir K$, in particular if $u=u_0$,  then \eqref{scdsie} follows from \cite[Lemma 2.3]{czgschperm}, Lemma~\ref{trajrestlem}, and the argument detailed in the first paragraph of the present proof. 

Second, for the sake of contradiction, suppose that 
\begin{equation}\label{suctNg}
\text{$u_0$ is distinct from $u$ and  $u_0$ is meet-reducible in  $K$.}
\end{equation}
Pick two distinct covers, $a_0$ and $b_0$ of $u_0$ in $K$. 
Let $u_0\prec u_1\prec\dots\prec u_t=u$ be a maximal chain in the interval $[u_0,u]$. The unique cover of $u$ in $L$ is denoted by $\ucsil$. By semimodularity, $u\vee a_0$ and $u\vee b_0$ cover $u$. Hence, $u\vee a_0 = u\vee b_0=\upstar u$, which implies $\set{a_0,b_0}\subseteq \ideal{\upstar u}$.
Since $u_1$ is not in $K$, the elements $a_0,b_0, u_1$ are three distinct atoms in the filter $\filter{u_0}$. If they are not independent, then we can select two of them that are independent in the sense of \init{G.~}Gr\"atzer~\cite[Theorem 380]{GGLT},
and we easily obtain that  $a_0,b_0, u_1$ generate a cover-preserving diamond, which is a contradiction. 
Hence they are independent, and they  generate a cover-preserving boolean sublattice, a \emph{cube} for being brief. Define $a_1=a_0\vee u_1$ and $b_1=b_0\vee u_1$; they belong to $\ideal \ucsil$ since so are $a_0,b_0,u_1$.  Then the cube we have just obtained is 
$\set{u_0,a_0,b_0, a_0\vee b_0, u_1,a_1,b_1, a_1\vee b_1}$, and it is in $\ideal{\upstar u}$. Since $u_1\prec a_1$ and $u_1\prec b_1$, $u_1\neq u\in\Mir L$.

Now, we repeat the procedure within $[{u_1}, \upstar u]$ instead of $[{u_1}, \upstar u]$. If we had, say, $u_2=a_1$, then 
$a_0<a_1=u_2\leq u$ and $a_0\in K$ would contradict the definition of $u_0$. Hence $a_1,b_1,u_2$ are distinct covers of $u_1$. As before, they generate a cube, which is 
$\{u_1, a_1,b_1, u_2,a_1\vee b_1,
a_2=a_1\vee u_2, b_2=b_1\vee u_2, a_2\vee b_2\}$. Since $u_2\prec a_2$ and $u_2\prec b_2$, $u_2\neq u\in\Mir L$. 

And so on. After $t$ steps, we obtain 
$u=u_t\notin\Mir L$, a contradiction. This proves \eqref{scdsie}. We obtain $\pi_{ij}(\foot ui +1)=\foot uj+1$ similarly, and we conclude $\feet u\in B(\vpi)$. This proves the first part of the lemma.


For $u\in\Mir L$, we have $u= c_{\foot u1}^{\,(1)} \vee\dots\vee  c_{\foot uk}^{\,(k)}$ since  $\Jir L\subseteq C_1\cup\dots\cup C_k$. Hence, $\feet u$ determines $u$, 
the map $\Mir L\to \set{\feet u: u\in\Mir L}$, defined by $u\mapsto \feet u$, is a bijection, and $|\set{\feet u: u\in\Mir L}| =|\Mir L|$. Thus
the second part of the lemma follows from the first part, Proposition~\ref{Dlwltcs}\eqref{Dlwltcse}, and \eqref{dkcBGb}. 
\end{proof}

\begin{remark} Since the proof above excludes  \eqref{suctNg}, we conclude that if $u\in\Mir L$, then  
$u=c_{\foot ui}^{\,(i)} \vee  c_{\foot uj}^{\,(j)}$, for all $i\neq j\in\kset$.
\end{remark}

\begin{proof}[Proof of Proposition~\ref{almostmain}] 
Clearly,  $\Lat nk\subseteq \cdf nk$. Hence we obtain  from Lemma~\ref{siodsGz} that $\mappl$ is a map from $S_n^{k-1}$ to $\cdf nk$. 

Let $\isomcl{\alg L}=\isomcl \tuple{L;C_1,\ldots,C_k}\in \cdf nk$. 
Denote $\maplp(\isomcl\alg L)$ by $\vpi$. This makes sense since, clearly, every $\alg L'$ isomorphic to $\alg L$ gives the same $\vpi$.  It is obvious  by Section~\ref{JHsection} that $\vpi\in S_n^{k-1}$.  That is, $\maplp$ is a map from $\cdf nk$ to $S_n^{k-1}$.

Now, for  $\isomcl \alg L\in \cdf nk$ and $\pi=\maplp(\isomcl\alg L)$ above, we  use the notation introduced before Lemma~\ref{maxEliglma}. Since $\elpi$ coincides with the set of $\alg L$-maximal vectors by Lemma~\ref{maxEliglma}, the map $\mu\colon L\to L(\vpi)$, defined by $u\mapsto\feet u$, 
is an order-isomorphism. Thus $\mu$ is a lattice isomorphism. To deal with $\mu(C_i)$, let $x\in C_i$. For any $y\in L$, if $\foot yi=h(x)=\foot xi$, then $x\leq y$ and, hence, $\mu(x)=\feet x\leq \feet y=\mu(y)$. This shows that $\mu(x)$ is  initial in its $i$-th component. Thus $\mu(C_i)=C_i(\vpi)$, for $i\in \kset$. Hence, $\mappl(\vpi)=\isomcl{\alg L}$.  This shows that 
\begin{equation}\label{lsBkcD}
\text{$\mappl \circ\maplp$ is the identity map on $\cdf nk$.}
\end{equation}

Now, we are in the position to prove $\cdf nk=\Lat nk$. The inclusion $\Lat nk\subseteq \cdf nk$ is trivial. Conversely, let $\isomcl{\alg L}$ belong to $\cdf nk$, and denote $\maplp(\isomcl{\alg L})$ by $\vpi$.  By \eqref{lsBkcD}, $\isomcl{\alg L} = \mappl(\vpi)$. Hence Lemma~\ref{siodsGz} yields $\isomcl{\alg L}\in \Lat nk$. This proves $\cdf nk=\Lat nk$.


Next, let $\vpi\in S_n^{k-1}$, and consider $ \mappl(\vpi)= \alg \elpi=\tuple{\elpi;\lancpi 1,\dots,\lancpi k}$. Let $\vec x\in \elpi$.  Since $\mappl(\vpi)\in \Lat nk$ by Lemma~\ref{siodsGz}, the length of $\lancpi 1$ is $n$.  This, together with the fact that no two distinct vectors in 
$\lancpi 1$ have the same first component,
implies that each $t\in\nnset$ is the first component of exactly one vector in $\lancpi 1$. Thus there is a unique $\vec y\in \lancpi 1$ such that $y_1=x_1$. Since $\vec y$ is initial in its first component, it is the largest vector in $\lancpi 1 \cap \ideal{\vec x}$. Clearly, the height of $\vec y\,$ is  $y_1=x_1$. 
Hence, using the notation given before Lemma~\ref{ujsuborblMa}, $\foot {\vec x}1=x_1$, and similarly for other indices. Therefore, $\feet{\vec x}=\vec x$ holds for all $\vec x\in \elpi$.
%
Applying this observation to $B(\vpi)=\{$suborbital vectors with respect to $\vpi\}$, we conclude $\set{ \feet u: u\in B(\vpi)}=B(\vpi)$. Hence, by \eqref{mlpbP},  
\begin{equation}\label{dieGh}
\text{$\set{ \feet u: u\in \Mir{(\elpi})}=B(\vpi)$.
}\end{equation}
%
On the other hand, Lemma~\ref{ujsuborblMa}, 
applied  to $\alg L(\vpi)=\eta(\vpi)$, yields the equality $\set{ \feet u: u\in \Mir{(\elpi})} = B\bigl(\maplp(\mappl(\vpi))\bigr)$. Combining this equality with \eqref{dieGh}, we obtain $B\bigl(   \maplp(\mappl(\vpi))\bigr)=B(\vpi)$. 
%
This means that  $\vpi$ and $\maplp(\mappl(\vpi))$ have exactly the same 
suborbital vectors. Hence, they have the same orbits. Since they are determined by their orbits, we conclude  $\maplp(\mappl(\vpi))=\vpi$. Thus $\maplp\circ\mappl$ is the identity map on $S_n^{k-1}$.
\end{proof}



%\begin{proof}[Proof of Proposition~\ref{neWjsldS}\textup{(B)}] 
%The implication \eqref {neWjsldSc} $\then$ \eqref{neWjsldSa} is a trivial consequence of Theorem~\ref{thmmain}(A). The rest are included in part (A) of Proposition~\ref{neWjsldS}.
%\end{proof}



\section{Coordinatizing antimatroids and convex geometries}\label{secantimatr}
The concept of antimatroids is due to \init{R.\,E.~}Jamison-Waldner~\cite{jamison}, who was the first to use the term ``antimatroid''. 
At the same time, an equivalent  complementary concept was introduced by  \init{P.\,H.~}Edelman~\cite{edelman} under the name ``anti-exchange closures''. 
%
There are several ways to define antimatroids, see \init{D.~}Armstrong~\cite[Lemma 2.1]{armstrong}; here we accept the following one. The set of all subsets of a set $E$ is denoted by $P(E)$.
\begin{definition}
A pair $\pair{E}{\alg F}$ is an \emph{antimatroid} if it satisfies the following properties:
\begin{enumeratei}
\item\label{antimatdefa} $E$ is a finite set, and $\emptyset\neq \alg F \subseteq P(E)$;
\item\label{antimatdefb} $\alg F$ is a \emph{feasible set}, that is, for each nonempty $A\in \alg F$, there exists an $x\in A$ such that $A\setminus\set x\in\alg F$;
\item\label{antimatdefc} $\alg F$ is closed under taking unions;
\item\label{antimatdefd} $E=\bigcup\set{A: A\in \alg F}$.  
\end{enumeratei}
\end{definition}

If $\pair{E}{\alg F}$ satisfies \eqref{antimatdefa}, \eqref{antimatdefb}, and \eqref{antimatdefc}, but possibly not \eqref{antimatdefd},  then  the elements of $E\setminus\bigcup\set{A: A\in \alg F}$  are called \emph{dummy points}. Many authors allow dummy points, that is, do not stipulate \eqref{antimatdefd} in the definition of antimatroids. However, this is not an essential difference since a structure $\pair{E}{\alg F}$ satisfying  \eqref{antimatdefa}, \eqref{antimatdefb}, and \eqref{antimatdefc} is clearly characterized by the antimatroid $\pair{\bigcup\set{A: A\in \alg F}}{\alg F}$  (in our sense) and the number $|E\setminus\bigcup\set{A: A\in \alg F}|$ of dummy points. Note that without \eqref{antimatdefd}, the forthcoming Proposition~\ref{pdZT} would fail.


Antimatroids were generalized to more general systems called greedoid by 
\init{B.~}Korte and \init{L.~}Lov\'asz~\cite{kortelovasz81} and \cite{kortelovasz83}. 
A \emph{closure operator} on a set $E$ is an extensive, monotone, and idempotent map $\Phi\colon P(E)\to P(E)$. That is,  $X\subseteq \Phi(X)=\Phi(\Phi(X))\subseteq \Phi(Y)$,   for all $X\subseteq Y\in P(E)$. A \emph{closure system} on $E$ is a nonempty subset of $P(E)$ 
that is closed under taking arbitrary intersections. In particular, a closure system on $E$ always contains the empty intersection, $E$.  There is a well-known bijective correspondence between closure operators and closure systems; see
\init{S.~}Burris and \init{H.\,P.~}Sankappanavar~\cite[I.\S5]{burrissankap}.
The closure system corresponding to a closure operator $\Phi$ consists of the \emph{closed sets}, that is, of the sets $X\in P(E)$ satisfying
$\Phi(X)=X$. The closure operator corresponding to a closure system $\alg C$  is the map $P(E)\to P(E)$, defined by 
$X\to\bigcap\set{Y\in \alg C: X\subseteq Y}$. Now, we define a concept closely related to antimatroids, 
see \init{P.H.~}Edelman~\cite{edelman} and    \init{K.~}Adaricheva, \init{V.\,A.~}Gorbunov, and \init{V.\,I.~}Tumanov~\cite{r:adarichevaetal}. Let us emphasize that \eqref{convgeoc} below is stipulated in \cite{r:adarichevaetal} and also in \init{D.~}Armstrong~\cite{armstrong}.
%
\begin{definition}\label{convgeo}
A pair $\pair{E}{\Phi}$ is a \emph{convex geometry}, also called \emph{anti-exchange system}, if it satisfies the following properties:
\begin{enumeratei}
\item\label{convgeoa} $E$ is a finite set,  and $\Phi\colon P(E)\to P(E)$ is a closure operator. 
\item\label{convgeob} If $\Phi(A)=A\in P(E)$, $x,y\in E$, $x\notin A$, $y\notin A$, $x\neq y$, and $x\in \Phi(A\cup\set y)$, then $y\notin \Phi(A\cup\set x)$. (This is the so-called \emph{anti-exchange property}.)
\item\label{convgeoc}  $\Phi(\emptyset)=\emptyset$.
\end{enumeratei}
For a closures system $\alg G$ on $E$ with corresponding closure operator $\Phi$, $\pair E{\alg G}$ is a \emph{convex geometry}
if so is  $\pair{E}{\Phi}$ in the above sense. In what follows, the notations $\pair{E}{\Phi}$  and $\pair E{\alg G}$ can be used interchangeably for the same mathematical object. The members of $\alg G$ are called \emph{closed sets}. 
\end{definition}

It follows easily from \eqref{convgeob}  that if $\pair E{\alg G}$ is a convex geometry, then 
\begin{equation}\label{adodkF}
\text{for each $B\in \alg G\setminus\set E$, there is an $x\in E\setminus B$ such that $B\cup\set x\in \alg G$.}
\end{equation}
The  following statement is taken from the book \init{B.~}Korte, \init{L.~}Lov\'asz, and \init{L.~}Schrader~\cite[Theorem III.1.3]{korteatalbook}, see also \eqref{adodkF} together with \init{D.~}Armstrong~\cite[Lemma 2.5]{armstrong}.

\begin{proposition}\label{pdZT} Let $E$ be a finite set, and let $\emptyset\neq \alg F\subseteq P(E)$. Then  $\alg A=\pair E{\alg F}$ is an antimatroid if{f} $\dual{\alg A}=\pair E{ \set{E\setminus X: X\in \alg F}}$ is a convex geometry. 
\end{proposition}


Part of the following proposition
was proved by  \init{P.\,H.~}Edelman~\cite[Theorem 3.3]{edelman}, see also \init{D.~}Armstrong~\cite[Theorem 2.8]{armstrong}. The rest can be extracted from \init{K.~}Adaricheva, \init{V.\,A.~}Gorbunov, and \init{V.\,I.~}Tumanov~\cite[proof of Theorem 1.9]{r:adarichevaetal}. Since this extraction is not so obvious, we will give some details for the reader's convenience. Lattices whose duals are join-distributive were called ``join-semidistributive and lower semimodular'' in \cite{r:adarichevaetal}; here we return to the original terminology of \init{P.\,H.~}Edelman~\cite{edelman}, and call them \emph{meet-distributive}.



\begin{lemma}\label{duallatamatEQ} If $L$ is a meet-distributive lattice and  
 $\alg M=\pair E{\alg G}$ is a  convex geometry, then the following three statements hold.
\begin{enumeratei}
\item\label{latamatEQa}  $\pair{\alg G}{\subseteq}$ is a meet-distributive lattice; it is denoted by  $\halomd(\alg M)$.
\item\label{latamatEQb} $\pair {\Jir L}{\set{\Jir L\cap \ideal x: x\in L }}$ is a  convex geometry; it is denoted by $\geom(L)$.
\item\label{latamatEQc} $\halomd\bigl (\geom(L)\bigr)  \cong L$ and $\geom\bigl(\halomd(\alg M) \bigr)\cong \alg M$.
\end{enumeratei}
\end{lemma}

\begin{proof} \eqref{latamatEQa} was proved by \init{P.\,H.~}Edelman~\cite[Theorem 3.3]{edelman}, see the first paragraph in his proof. 
%
The proof of \init{K.~}Adaricheva, \init{V.\,A.~}Gorbunov, and \init{V.\,I.~}Tumanov~\cite[Theorem 1.9]{r:adarichevaetal} contains the statement that $\geom(L)$ is a convex geometry. 
The sixteenth line in the proof of \cite[Theorem 1.9]{r:adarichevaetal} explicitely says $\halomd\bigl (\geom(L)\bigr)  \cong L$.


Now, we are left with the proof of $\geom\bigl(\halomd(\alg M) \bigr)\cong \alg M$.
So let $\alg M$ be a  convex geometry. Its closed sets, the members of $\alg G$, are exactly the elements of  $\halomd(\alg M)$. 
We assert that
\begin{equation}\label{siGk}
\text{for every $A\in \alg G$, $\,A$ is 1-generated if{f} $A\in\Jir{(\halomd(\alg M))}$.
}
\end{equation}
To show this, let $A=\set{a_1,\dots,a_t}\in \alg G$. Assume that $A\in\Jir{(\halomd(\alg M))}$. 
The closure operator corresponding to
 $\alg G$ is denoted by  $\Phi$.
Since $A=\Phi(\set{a_1})\vee \dots\vee \Phi(\set{a_t})$ holds in $\halomd(\alg M)$, the join-irreducibility of $A$ yields $A=\Phi(\set{a_i})$ for some $i\in\set{1,\ldots, t}$. This means that $A$ is 1-generated. We prove the converse implication by way of contradiction. Suppose that $A$ is 1-generated but $A\notin\Jir{(\halomd(\alg M))}$. We have 
$A=\Phi(\set{X}) \vee \Phi(\set{Y})$ for some $X,Y\subset A$ such that $A\neq\Phi(\set{X})$ and $A\neq \Phi(\set{Y})$. 
Clearly, we can pick  elements $b_1,\ldots,b_s\in X\cup Y$ such that 
$A=\Phi(\set{b_1})\vee\dots\vee \Phi(\set{b_s})$, this join is  irredundant (that is, no joinand can be omitted), and $s\geq 2$.
Since $A$ is 1-generated, we can also pick a $c\in A$ such that $A=\Phi(\set c)$.
Since the join we consider is irredundant,  $c\notin \Phi(\set{b_1,\dots, b_{s-1}})$,  $b_s\notin \Phi(\set{b_1,\dots, b_{s-1}})$, and $c\neq b_s$. So we have
\[ c\in \Phi\bigl(\Phi(\set{b_1,\dots, b_{s-1}} \cup \set{b_s}\bigr),\quad \set{c,b_s}\cap \Phi(\set{b_1,\dots, b_{s-1}})=\emptyset,\quad\text{and } c\neq b_s\text.
\]
Thus the anti-exchange property yields that $b_s\notin \Phi\bigl(\Phi(\set{b_1,\dots, b_{s-1}} \cup \set{c}\bigr)$. This  contradicts $b_s\in A=\Phi(\set c) \subseteq \Phi\bigl(\Phi(\set{b_1,\dots, b_{s-1}}  \cup \set{c}  \bigr)$, proving  \eqref{siGk}. 

Next,  if we had $x,y\in E$ such that $\Phi(x)=\Phi(y)$ but $x\neq y$, then 
$y\in\Phi(\set y)=\Phi(\set x)= \Phi(\emptyset \cup \set x)$ together with the analogous  $x\in \Phi(\emptyset \cup \set y)$ 
would contradict the anti-exchange property by   Definition~\ref{convgeo}\eqref{convgeoc}. Thus $\Phi(x)=\Phi(y)$  implies  $x=y$. 
%
Hence, by \eqref{siGk},  for each $A\in\Jir{(\halomd(\alg M))}$, there is a unique $e_A\in E$ such that $A= \Phi(\set{e_A})$. Since $\Phi(\set{e})\in  \Jir{(\halomd(\alg M))}$ also holds for all $e\in E$ by \eqref{siGk}, we have a bijection $\psi\colon  \Jir{(\halomd(\alg M))}\to E$,  defined by $A\mapsto e_A$. Its inverse is denoted by $\eta$; it is defined by $\eta(e)=\Phi(\set e)$.
We assert that $\psi$ is an isomorphism from $\geom\bigl(\halomd(\alg M) \bigr)$ to $\alg M$. To prove this, let $X$ be a closed set of $\geom\bigl(\halomd(\alg M) \bigr)$. This means that $X$ is of the form  $X=\set{A\in \Jir{(\halomd(\alg M))}: A\subseteq B}$ for some $B\in \halomd(\alg M))=\alg G$. Hence, 
\begin{align*}
\psi(X)=\psi\bigl(\set{ \eta(e): \eta(e)\subseteq B }\bigr)= \psi\bigl(\set{ \eta(e): e\in  B } \bigr) = \set{ \psi(\eta(e)): e\in  B } =B\text.
\end{align*}
That is, $\psi$ maps the closed sets $X$ of 
$\geom\bigl(\halomd(\alg M) \bigr)$ to the closed sets of $\alg M$. Since each $B\in \halomd(\alg M))=\alg G$ determines a closed set 
$X=\set{A\in \Jir{(\halomd(\alg M))}: A\subseteq B}$ of $\geom\bigl(\halomd(\alg M) \bigr)$, the calculation above also shows that all $B\in \alg G$ are $\psi$-images of closed sets of $\geom\bigl(\halomd(\alg M) \bigr)$. Thus $\psi$ is an isomorphism.
\end{proof}


Combining Proposition~\ref{pdZT} with the dual of Lemma~\ref{duallatamatEQ}, we easily obtain the following  statement.
It asserts that  join-distributive lattices and antimatroids are essentially the same mathematical objects.


\begin{corollary}\label{latamatEQ} If $L$ is a join-distributive lattice and  
 $\alg A=\pair E{\alg F}$ is an  antimatroid, then the following three statements hold.
\begin{enumeratei}
\item\label{latamatEQa}  $\pair{\alg F}{\subseteq}$ is a join-distributive lattice; it is denoted by  $\halojd(\alg A)$.
\item\label{latamatEQb} $\pair {\Mir L}{\set{\Mir L\setminus \filter x: x\in L }}$ is an antimatroid; it is denoted by $\amat(L)$, 
\item\label{latamatEQc} $\halojd\bigl (\amat(L)\bigr)  \cong L$ and $\amat\bigl(\halojd(\alg A) \bigr)\cong \alg A$.
\end{enumeratei}
\end{corollary}


Now, Corollary~\ref{latamatEQ} allows us to translate Theorem~\ref{thmmain} to a coordinatization of  antimatroids, while the coordinatization of  convex geometries is reduced to that of antimatroids by 
 Proposition~\ref{pdZT}. A brief translation is exemplified by the following corollary; the full translation and the case of convex geometries are omitted.  

Let $\vpi\in S_n^{k-1}$. As before, the set of suborbital vectors and that of $\vpi$-eligible tuples are denoted by $B(\vpi)$ and $\elpi$, respectively; see Definition~\ref{defeligl}, and see also \eqref{suboRbital},  \eqref{dkcBGb}, and \eqref{mlpbP}.  For $\vec x\in\elpi$, let $U(\vec x)$ denote the set $\set{\vec y\in B(\vpi): \vec x\not\leq \vec y}$. The \emph{convex dimension} of an antimatroid $\alg A$ is
the width of $\Jir{(\halojd(\alg A))}$.

\begin{corollary}  For each $\vpi\in S_n^{k-1}$, $\alg A(\vpi)=\pair{B(\vpi)}{\set{U(\vec x) : x\in\elpi }}$ is an  antimatroid with convex dimension at most $k$. Conversely, for each antimatroid $\alg B$ of convex dimension $k$ on an $n$-element set, there exists a  $\vpi\in S_n^{k-1}$ such that $\alg B$ is isomorphic to $\alg A(\vpi)$.
\end{corollary}


\begin{ackno} The author is indebted to Anna Romanowska for calling his attention to 
\cite{abels} and \cite{herscovici}, and to Kira Adaricheva for her comments that led to  \cite{adarichevaczg}.
\end{ackno}




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\bibitem{adarichevaczg}
    K.~Adaricheva, G. Cz\'edli:
    Notes on the description of join-distributive lattices by permutations, \url{http://arxiv.org/}

\bibitem{r:adarichevaetal}
  K.~Adaricheva, V.A.~Gorbunov, V.I.~Tumanov:
  Join-semidistributive lattices and
convex geometries. 
  Advances in Math. {173},   1--49 (2003)

\bibitem{armstrong}
  D. Armstrong:
  The Sorting Order on a Coxeter Group.
  Journal of Combinatorial Theory Series A 
  \tbf{116}, % Issue 8, November, 
  1285--1305 (2009)


\bibitem{avann1} 
  S.P. Avann: 
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  Math. Ann. \tbf{142}, 345--354 (1961)


\bibitem{avann2} 
  S.P. Avann: 
  Increases in the join-excess function in a lattice.
  Math. Ann. \tbf{154}, 420--426 (1964)


\bibitem{burrissankap}
  S.~Burris, H.P.~Sankappanavar:
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  Springer-Verlag, New York�Berlin ( 1981).   The Millennium Edition:
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\bibitem{czgolub}
   G.~Cz\'edli, L.~Ozsv\'art, B.~Udvari:
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\bibitem{czgschsomeres}
    G.~Cz\'edli, E.T.~Schmidt: 
    Some results on semimodular lattices.
    In: Contributions to General Algebra, vol. {19},  45--56.
 (Proc. Conf. Olomouc,  2010)
Johannes Hein verlag, Klagenfurt (2010) 
%ISBN 978-3-7084-0407-3. 


\bibitem{czgschjh}
   G.~Cz\'edli, E.T.~Schmidt: 
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\bibitem{czgschperm}
   G.~Cz\'edli, E.T.~Schmidt:
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\bibitem{r:dilworth40}
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\bibitem{edelman}
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\bibitem{edelmanproc}
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     Abstract convexity and meet-distributive lattices.
     In: Proc. Conf. Combinatorics and ordered sets (Arcata, Calif., 1985), Contemp. Math. \tbf{57}, 127--150, Amer. Math. Soc., Providence, RI (1986)


\bibitem{edeljam}
  P. H. Edelman,   R. E. Jamison:
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\bibitem{GGLT}
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\bibitem{gratzernation}
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  Algebra Universalis \tbf{66}, 69--79 (2011)


\bibitem{herrmann}
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\bibitem{herscovici}
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  Journal of Algebraic Combinatorics \tbf{7}, 39--51 (1998)

\bibitem{korteatalbook}  
  B. Korte, L. Lov\'asz,  R. Schrader:
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\bibitem{neumann}
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\bibitem{jamison}
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\bibitem{kortelovasz81}
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\bibitem{kortelovasz83}
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   Structural properties of greedoids.
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\bibitem{monjardet}
  B. Monjardet:
  A use for frequently rediscovering a concept.
  Order \tbf{1}, 415--417 (1985)


\bibitem{stanley}
  R.P. Stanley:
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  Algebra Universalis \tbf{2}, 197--217 (1972)


\bibitem{stern}
  M. Stern: 
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its Applications \tbf{73}, Cambridge University Press, 1999

\bibitem{wehrung}
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  Algebra Universalis, \tbf{64}, 49--67 (2010) 
 

\end{thebibliography}
\end{document}



1208.3517.extracted_bib
\begin{thebibliography}{99}

\bibitem{abels}
H. Abels:
The geometry of the chamber system of a semimodular lattice.
Order \tbf{8}, 143--158 (1991)
 
\bibitem{adarichevaczg}
    K.~Adaricheva, G. Cz\'edli:
    Notes on the description of join-distributive lattices by permutations, \url{http://arxiv.org/}

\bibitem{r:adarichevaetal}
  K.~Adaricheva, V.A.~Gorbunov, V.I.~Tumanov:
  Join-semidistributive lattices and
convex geometries. 
  Advances in Math. {173},   1--49 (2003)

\bibitem{armstrong}
  D. Armstrong:
  The Sorting Order on a Coxeter Group.
  Journal of Combinatorial Theory Series A 
  \tbf{116}, % Issue 8, November, 
  1285--1305 (2009)


\bibitem{avann1} 
  S.P. Avann: 
  Application of the join-irreducible excess function to semimodular lattices.
  Math. Ann. \tbf{142}, 345--354 (1961)


\bibitem{avann2} 
  S.P. Avann: 
  Increases in the join-excess function in a lattice.
  Math. Ann. \tbf{154}, 420--426 (1964)


\bibitem{burrissankap}
  S.~Burris, H.P.~Sankappanavar:
  A Course in Universal Algebra, 
  Graduate Texts in Mathematics, vol. 78. 
  Springer-Verlag, New York�Berlin ( 1981).   The Millennium Edition:
  \url{http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html}


\bibitem{czgolub}
   G.~Cz\'edli, L.~Ozsv\'art, B.~Udvari:
   How many ways can two composition series intersect?.
   Discrete Mathematics \tbf{312}, 3523--3536 (2012) 


\bibitem{czgschsomeres}
    G.~Cz\'edli, E.T.~Schmidt: 
    Some results on semimodular lattices.
    In: Contributions to General Algebra, vol. {19},  45--56.
 (Proc. Conf. Olomouc,  2010)
Johannes Hein verlag, Klagenfurt (2010) 
%ISBN 978-3-7084-0407-3. 


\bibitem{czgschjh}
   G.~Cz\'edli, E.T.~Schmidt: 
   The Jordan-H\"older theorem with uniqueness for groups and semimodular lattices.
   Algebra Universalis 66 (2011) 69--79.


\bibitem{czgschperm}
   G.~Cz\'edli, E.T.~Schmidt:
   Intersections of composition series in groups and slim semimodular lattices by permutations.
   \url{http://arxiv.org/abs/1208.4749}\footnote{The author's papers referenced here are also  available from his URL}  


\bibitem{r:dilworth40}
    R.P.~Dilworth:  
    Lattices with unique irreducible decompositions.
    Ann. of Math. (2) \tbf{41},  771--777 (1940)


\bibitem{edelman}
  P. H. Edelman:
  Meet-distributive lattices and the anti-exchange closure. 
  Algebra Universalis \tbf{10}, 290--299 (1980)

\bibitem{edelmanproc}
     Edelman, P.H.:
     Abstract convexity and meet-distributive lattices.
     In: Proc. Conf. Combinatorics and ordered sets (Arcata, Calif., 1985), Contemp. Math. \tbf{57}, 127--150, Amer. Math. Soc., Providence, RI (1986)


\bibitem{edeljam}
  P. H. Edelman,   R. E. Jamison:
  The theory of convex geometries.
  Geom. Dedicata \tbf{19}, 247--271 (1985)


\bibitem{GGLT}
   G.~Gr\"atzer:
   Lattice Theory: Foundation.
   Birkh\"auser  Verlag, Basel, 2011


\bibitem{gratzerknapp}
   G.~Gr\"atzer,  Knapp, E.: 
    Notes on planar semimodular lattices I. Construction. 
  Acta Sci. Math. (Szeged) \tbf{73}, 445--462 (2007)


\bibitem{gratzernation}
  G.~Gr\"atzer, J. B.~Nation:
  A new look at the Jordan-H\"older theorem for semimodular lattices. 
  Algebra Universalis \tbf{66}, 69--79 (2011)


\bibitem{herrmann}
  C. Herrmann:
  On the arithmetic of projective coordinate systems.
  Tran. Amer. Math. Soc. \tbf{284}, 759--785 (1984)

\bibitem{herscovici}
  D. S. Herscovici:
  Semimodular lattices and semibuildings. 
  Journal of Algebraic Combinatorics \tbf{7}, 39--51 (1998)

\bibitem{korteatalbook}  
  B. Korte, L. Lov\'asz,  R. Schrader:
  Greedoids, Algorithms and Combinatorics 4, Springer (1991)

\bibitem{neumann}
  J. von Neumann: 
  Continuous Geometry.
  Princeton Univ. Press, Princeton, N.J. (1960)


\bibitem{jamison}
  R. E.  Jamison-Waldner,
  Copoints in antimatroids.
  Combinatorics, graph theory and computing, Proc. 11th southeast. Conf., Boca Raton/Florida 1980, Vol. II, Congr. Numerantium \tbf{29}, 535--544 (1980) 


\bibitem{kortelovasz81}
  B. Korte,  L. Lov\'asz: 
  Mathematical structures underlying greedy algorithms.
  Fundamentals of computation theory, Proc. int. FCT-Conf., Szeged/Hung. 1981, Lect. Notes Comput. Sci. 117, 205--209 (1981)


\bibitem{kortelovasz83}
   B. Korte,  L. Lov\'asz: 
   Structural properties of greedoids.
   Combinatorica \tbf{3}, 359--374 (1983) 


\bibitem{monjardet}
  B. Monjardet:
  A use for frequently rediscovering a concept.
  Order \tbf{1}, 415--417 (1985)


\bibitem{stanley}
  R.P. Stanley:
  Supersolvable lattices.
  Algebra Universalis \tbf{2}, 197--217 (1972)


\bibitem{stern}
  M. Stern: 
  Semimodular Lattices. Theory and Applications, Encyclopedia of Mathematics and
its Applications \tbf{73}, Cambridge University Press, 1999

\bibitem{wehrung}
  F. Wehrung:
  Coordinatization of lattices by regular rings without unit and Banaschewski functions, 
  Algebra Universalis, \tbf{64}, 49--67 (2010) 
 

\end{thebibliography}