EL-Shelling on Comodernistic Lattices

TL;DR

This paper establishes that comodernistic lattices are EL-shellable by proving their recursive atom orderings are independent of roots, simplifying the understanding of their combinatorial structure.

Contribution

It proves the equivalence of EL-shellability and root-independent recursive atom ordering, and demonstrates this for comodernistic lattices, including order congruence lattices.

Findings

Comodernistic lattices are EL-shellable.

Recursive atom orderings can be independent of roots.

Simplified EL-shelling for order congruence lattices.

Abstract

We prove the equivalence of EL-shellability and the existence of recursive atom ordering independent of roots. We show that a comodernistic lattice, as defined by Schweig and Woodroofe, admits a recursive atom ordering independent of roots, therefore is EL-shellable. We also present and discuss a simpler EL-shelling on one of the most important classes of comodernistic lattice, the order congruence lattices.

Full text

Department of Mathematics and Statistics, Washington University in St. Louis

1. Introduction

Modernistic and Comodernistic lattices are two large classes of finite lattices with shellable order complexes. Schweig and Woodroofe defined and studied these lattices in [2] and showed that a wide range of lattices are either modernistic or comodernistic, including subgroup lattices of finite solvable groups, supersolvable and left-modular lattices, semi-modular lattices, $k$-equal partition lattices, order congruence lattices, and others. They proved in [2] that comodernistic lattices are CL-shellable, which implies that the order complexes of modernistic and comodernistic lattices are shellable.

In this paper, we show that comodernistic lattices are EL-shellable, as defined in [1]. This can be viewed as a generalization of Woodroofe’s result on subgroup lattices. After Shareshian showed in [3] that subgroup lattices of solvable groups are CL-shellable, Woodroofe proved a stronger result in [5] that these lattices are in fact EL-shellable.

It is shown in [1] that a poset is CL-shellable if and only if it admits a recursive atom ordering. We will prove a similar result for EL-shellable posets in order to prove our main result.

Recall that a recursive atom ordering of $P$ assigns to each pair $(x,r)$ with x$$\in$$P$$\backslash$$\{\hat{1}\} and a maximal chain $r$ in $[\hat{0},x]$ an ordering of the atoms in $[x,\hat{1}]$

Lemma 1.1**.**

If $P$ admits a recursive atom ordering such that, for each pair $(x,r)$, the atom ordering of $[x,\hat{1}]$ does not depend on $r$, then $P$ is EL-shellable.

The following is our main result.

Theorem 1.2**.**

Comodernistic lattices are EL-shellable.

We will further study the class of order congruence lattices after proving Theorem 1.2. We will show that these lattices are EL-shellable using integer labels without invoking Theorem 1.2.

In the next section, we will include necessary definitions concerning modernistic and comodernistic lattices. For readers unfamiliar with EL-shellability and CL-shellability, we recommend [1] for more information.

2. Preliminaries

We call a simplicial complex $\Delta$ $shellable$ if there is an ordering of its facets $F_{1},F_{2},\dots,F_{t}$ such that $(\bigcup_{i=1}^{k}F_{i})\cap F_{k+1}$ is pure and $(dim(F_{k})-1)$-dimensional for $k=1,2,\dots,t-1$. A poset $P$ is $shellable$ if its order complex $\Delta P$ is.

For a poset $P$, an edge-labeling is a map from the edge set of the Hasse diagram of $P$ to some label poset $\Lambda$. An edge-labeling is an $EL-shelling$ if for every interval $[x,y]$ of $P$, there exists a unique weakly increasing maximal chain lexicographically preceding all other maximal chains in $[x,y]$ in a fixed linear extension of the poset $\Lambda$. A chain-edge labeling is a map from the set of all pairs $(c,e)$ to the label set $\Lambda$, where $c$ is a maximal chain of $P$ and $e$ is an edge in $c$, such that $(c,e)$ and $(c^{\prime},e)$ get the same label if $c$ and $c^{\prime}$ coincide from $\hat{0}$ to $e$. A poset $P$ is $CL-shellable$ if it admits a chain-edge labeling such that for every interval $[x,y]$ and maximal chain $r$ in $[\hat{0},x]$, in the rooted interval $[x,y]_{r}$, there exists a unique weakly increasing maximal chain lexicographically preceding all other maximal chains in a fixed linear extension of the label poset $\Lambda$.

Clearly, EL-shellability implies CL-shellability. More importantly, both EL-shellability and CL-shellability imply shellabiliy.

Theorem 2.1**.**

[1, Proposition 2.3]** EL-shellability $\Rightarrow$ CL-shellability $\Rightarrow$ Shellability.

Another notion commonly used in the context of lexicographic shellability is recursive atom ordering.

Definition 2.2**.**

[4, Definition 4.2.1] A poset $P$ is said to admit a recursive atom ordering if the length of $P$ is $1$ or if the length of $P$ is greater than $1$ and there is an ordering $a_{1},a_{2},\dots,a_{t}$ of the atoms of P which satisfies:

1. (1)

For all $j=1,2,\dots,t$, $[a_{j},\hat{1}]$ admits a recursive atom ordering in which the atoms of $[a_{j},\hat{1}]$ that come first in the ordering are those that cover some $a_{i}$, where $i<j$. 2. (2)

For all $i<j$, if $a_{i},a_{j}<y$ then there is a $k<j$ and an atom $z$ of $a_{j}$ such that $z<y$ and $a_{k}<z$.

Bjöner and Wachs have proved that CL-shellability is equivalent to the existence of a recursive atom ordering.

Theorem 2.3**.**

[4, Theorem 4.2.2]** A bounded poset $P$ admits a recursive atom ordering if and only if $P$ is CL-shellable.

Next we introduce modernistic and comodernistic lattices. Let $L$ denote a lattice. Recall that an element $m$ in $L$ is $left-modular$ if for any $x<y$ in $L$, we have $(x\vee m)\wedge y=x\vee(m\wedge y)$. A lattice $L$ is $modernistic$ if for every interval of $L$, there exists a left-modular atom in that interval. A lattice is $comodernistic$ if it is the dual of a modernistic lattice. That is, there exists a left-modular coatom in every interval. Schweig and Woodroofe proved that comodernistic lattices are CL-shellable [2].

Theorem 2.4**.**

[2, Theorem 1.2]** If $L$ is a comodernistic lattice, then $L$ has a CL-labeling

As stated above in Theorem 1.2, we will show that comodernistic lattices are EL-shellable. To show that comodernistic lattices are EL-shellable, we need the notion of a sub-M-chain, which can be viewed as an analogy of an M-chain in a left-modular lattice.

Definition 2.5**.**

[2] A maximal chain $\hat{0}=m_{0}\lessdot m_{1}\lessdot\dots\lessdot m_{n}=\hat{1}$ in $L$ is a sub-M-chain if for every $i$, the element $m_{i}$ is left-modular in the interval $[\hat{0},m_{i+1}]$.

We also list here two lemmas from [2] that will help prove our main theorem. We refer readers to [2] for the proofs of the lemmas.

Lemma 2.6**.**

[2, Lemma 3.1]** Let $L$ be a lattice with a sub-M-chain m of length $n$. Then no chain of $L$ has length greater than $n$.

Lemma 2.7**.**

[2, Lemma 2.12]** Let $m$ be a coatom of the lattice $L$. Then $m$ is left-modular in $L$ if and only if for every $y$ such that $y\nleq m$ we have $m\wedge y\lessdot y$.

3. Proof of Lemma 1.1

Let $P$ be a CL-shellable poset with an induced recursive atom ordering such that for each x$$\in$$P$$\backslash$$\{\hat{1}\}, all orderings of the atoms of $[x,\hat{1}]$ are the same, as in Lemma 1.1. We will construct an EL-labeling for $P$. For every edge $e=[x,y]$ in the Hasse diagram of $P$, we define a $(\kappa_{e}+1)$-tuple, where $\kappa_{e}$ is the number of maximal chains in $[\hat{0},x]$. The last coordinate records the edge itself. That is, for $e=[x,y]$, let the last coordinate of the label of $e$ simply be the 2-tuple (x, y). Now we consider the first $\kappa_{e}$ coordinates. These are indexed by the roots in $[x,y]$. Call these the $\kappa_{e}$ large coordinates of the label. In each large coordinate, we place a $2$-tuple, whose first entry is the union of the corresponding root and the edge itself, and the second entry is the label induced by the root in the CL-shelling of $P$. Let us call these $2$-tuples the small coordinates. We order the $\kappa_{e}$ large coordinates according to the original CL-labeling of $P$. That is, if $C$ is lexicographically the $k^{th}$ maximal chain in $[\hat{0},x]$ according to the CL-labeling, the $k^{th}$ large coordinate of $e$ consists of $C\cup e$ and the label induced by $C$.

Now we define a partial order on the labeling set. Suppose $e=[x,y]$ and $e^{\prime}=[x^{\prime},y^{\prime}]$ are two edges in the Hasse diagram labeled as above. Then we say $e\leq e^{\prime}$ if $y<x^{\prime}$, or if $y=x^{\prime}$ and for some large coordinate $e_{m}$ of $e$, there exists a large coordinate $e^{\prime}_{n}$ of $e^{\prime}$ such that $e_{m}\leq e^{\prime}_{n}$, by which we mean that the root (first small coordinate) of $e_{m}$ is contained in the root of $e^{\prime}_{n}$, and the label (second small coordinate) in $e_{m}$ is less than or equal to the label in $e^{\prime}_{n}$.

Let us first check that this is a well-defined partial order. Obviously we have reflexivity. Antisymmetry is satisfied because if $e<e^{\prime}$, $x$ cannot be above or equal $y^{\prime}$. Transitivity holds because if $e<e^{\prime}$ and $e^{\prime}<e^{\prime\prime}$, $y<x^{\prime\prime}$. So this is indeed a partial order.

We now check that this edge-labeling gives an EL-shelling of $P$. For any interval $[x,y]$, let $C$ be the weakly-increasing chain in the original CL-labeling (with respect to any roots). We claim that the new edge-labeling on $P$ makes $C$ the unique weakly increasing and lexicographic first maximal chain in the interval.

The fact that $C$ is weakly-increasing follows from the consistency of the CL-labeling. Let $C$ be weakly increasing in $[x,y]_{r}$ for all roots $r$. Suppose there exists some $C^{\prime}=\{c_{0}\lessdot c_{1}\lessdot\dots\lessdot c_{k}\}$ in $[x,y]$ that is also weakly increasing in the new edge-labeling. Then for each $0<i<k$, there exists a root $r_{i}$ such that $c_{i}$ is the first atom in $[c_{i-1},c_{i+1}]_{r_{i}}$. Notice that the assumption on the atom orderings implies that whether labels of two consecutive edges in $[c_{i-1},c_{i+1}]$ are weakly increasing is independent of roots. Hence $C^{\prime}$ must be weakly increasing in some rooted interval, which would further imply that there are two weakly increasing maximal chains in one rooted interval. This contradicts to $P$ being CL-shellable. A similar argument shows that $C$ must be the lexicographic first maximal chain of the interval. ∎

4. Proof of Theorem 1.2

Let $L$ be a comodernistic lattice. We fix a sub-M-chain $\textbf{m}=\{\hat{0}=m_{0}\lessdot{m_{1}}\lessdot{\dots}\lessdot{m_{n}}=\hat{1}\}$ of $L$ and prove Theorem 1.2 by induction on height. For each atom $a$ of $L$, label $[\hat{0},a]$ by the index of the minimal element of m that is larger than or equal to $a$ in the partial order. Suppose we have labeled all elements of height less than $k$. For $y$ of height $k$, we pick an element $z$ covered by $y$ and define a sub-M-chain $\textbf{m}_{y}$ in $[y,\hat{1}]$ as follows. Let $w$ be the minimal element above $y$ in $\textbf{m}_{z}$, where $\textbf{m}_{z}$ has been chosen by induction, and pick some sub-M-chain of $[y,w]$. Then $\textbf{m}_{y}$ is the sub-M-chain of $[y,w]$ followed by the rest of the sub-M-chain of $\textbf{m}_{z}$. We label the atom edges of $[y,\hat{1}]$ using $\textbf{m}_{y}$ the same way we label atom edges of $L$ with m.

Now we show that the edge labeling defined above induces a recursive atom ordering independent of roots as in Lemma 1.1. We claim that any atom ordering that is consistent with the edge labeling is a recursive atom ordering. That is, if for a fixed linear extension, the label of $[x,y]$ precedes the label of $[x,y^{\prime}]$, where $y$ and $y^{\prime}$ both cover $x$, then $y$ precedes $y^{\prime}$ in the atom ordering of $x$. And if the label of $[x,y]$ equals the label of $[x,y^{\prime}]$, we can either have $y$ precedes $y^{\prime}$ or $y^{\prime}$ precedes $y$. We prove this claim by induction on coheight. It is obvious for all intervals $[x,\hat{1}]$ where $x$ is a coatom of $L$. Suppose the claim stands for all $[x,\hat{1}]$ of length less than $k$. For any $[x,\hat{1}]$ of length $k$, consider any atom $a_{i}$ of the interval.

$[a_{i},\hat{1}]$ admits a recursive atom ordering by induction. For any two atoms $b_{j}$ and $b_{k}$ of $[a_{i},\hat{1}]$, suppose $j<k$ and $b_{k}$ covers some $a_{l}$ with $l<i$. We need to show that $b_{j}$ covers some $a_{m}$ for $m<i$. Since $b_{k}$ covers $a_{i}$ and $a_{l}$ with $l<i$, $b_{k}\in[x,w]$ where $w$ is the minimal element above $a_{i}$ in the sub-M-chain $\textbf{m}_{x}$ of $[x,\hat{1}]$. Therefore $b_{j}<w$ since $j<k$. Let $w^{\prime}$ be the maximal element in $\textbf{m}_{x}$ below $w$ and not above $b_{j}$. Then $b_{j}\wedge w^{\prime}$ is an atom in $[x,\hat{1}]$ prior to $a_{i}$ and covered by $b_{j}$.

Now for any two atoms $a_{i}$ and $a_{j}$ with $i<j$ of $[x,\hat{1}]$, we need to find an atom of $[a_{j},\hat{1}]$ that covers some $a_{k}<a_{i}\vee a_{j}$ with $k<j$. Consider the first atom $b$ in $[a_{j},\hat{1}]$ that sits below $a_{i}\vee a_{j}$. Let $w$ be the minimal element of $\textbf{m}_{x}$ that sits above $a_{i}\vee a_{j}$. Then $w$ is the minimal element of $\textbf{m}_{x}$ that sits above $a_{j}$ since $i<j$. Let $w^{\prime}$ be the element in $\textbf{m}_{x}$ covered by $w$. Then $b\wedge w^{\prime}$ is an atom in $[x,\hat{1}]$ that is prior to $a_{j}$ ∎

5. Order Congruence Lattices

The order congruence lattice $\mathcal{O}(P)$ of a poset $P$ is the set of all equivalence classes of level set partitions from $P$ to $\mathbb{Z}$. That is, the set of all weakly order preserving maps from $P$ to $\mathbb{Z}$, where two such maps are considered equivalent if they induce the same partition on $P$.

For example, the order congruence lattice on a totally ordered set is a boolean lattice. The order congruence lattice on a set of pairwise incomparable elements is isomorphic to a partition lattice. In general, order congruence lattice of any poset can be considered as in between the boolean lattice and the partition lattice.

Schweig and Woodroofe proved in [2] that order congruence lattices are comodernistic, therefore CL-shellable. We here present a different proof where any linear extension of $P$ gives a sub-M-chain and an EL-shelling on $\mathcal{O}(P)$.

Fix a linear extension of $P=\{z_{1},z_{2},\dots,z_{n}\}$. For an element in $\mathcal{O}(P)$ with $k$ blocks, we can index each block as follows. First we take the smallest element in the linear extension of each block as representative. We assign indices $1$ through $k$ to the blocks preserving the order of their representatives. And we use these indices to construct a sub-M-chain and EL-shelling as follows.

Let $C$ be the maximal chain $\hat{0}=c_{1}\lessdot c_{1}\lessdot\dots\lessdot c_{n}=\hat{1}$ where $c_{k}$ is obtained by having the first $k$ elements ($z_{1}$ through $z_{k}$) in one block, and then one block each for the remaining elements.

Proposition 5.1**.**

$C$* is a sub-M-chain which induces an EL-shelling on $\mathcal{O}(P)$.*

Proof.

For any $k$ and $x\in[\hat{0},c_{k}]$, we need to show either $x<c_{k-1}$ or $x\wedge c_{k-1}\lessdot x$. If $z_{k}$ in $x$ is a block by itself, $x<c_{k-1}$. Otherwise, $x\wedge c_{k-1}$ is obtained by isolating $z_{k}$ to a single block from $x$. Hence $x\wedge c_{k-1}$ is covered by $x$ and $C$ is a sub-M-chain.

Now we show that it induces an EL-shelling on $\mathcal{O}(P)$. Consider the following edge-labeling. For any edge $[x,y]$, if $y$ is obtained from $x$ by merging the $i^{th}$ block and the $j^{th}$ block where $i<j$, we assign $j$ to $[x,y]$. This is a well-defined edge-labeling. Now we show that it is an EL-shelling.

Consider any interval $[x,y]$ in $\mathcal{O}(P)$. Notice that if $y$ consists of $k$ blocks, then any edge in the interval can be viewed as a merge within one of the $k$ blocks of $y$. The lexicographically first maximal chain is obtained by consecutively merging the smallest indexed two sub-blocks in the same block of $y$. This is a weakly increasing chain. Next we check that this is the unique weakly increasing chain of the interval. We show that any other chain must either violate the merging order within a block of $y$ or among blocks of $y$. Suppose we merge two sub-blocks within a block of $y$ that are not the two smallest possible blocks to merge. Merging with a smaller sub-block later will result in a smaller edge label, in which case the maximal chain cannot be weakly increasing. Suppose we missed a merge within a smaller block of $y$. The edge obtained by completing that missed merge will again create a smaller label hence the maximal chain cannot be weakly increasing. ∎