Poset EdgeLabellings and Left Modularity
Abstract.
It is known that a graded lattice of rank is supersolvable if and only if it has an ELlabelling where the labels along any maximal chain are exactly the numbers without repetition. These labellings are called ELlabellings, and having such a labelling is also equivalent to possessing a maximal chain of left modular elements. In the case of an ungraded lattice, there is a natural extension of ELlabellings, called interpolating labellings. We show that admitting an interpolating labelling is again equivalent to possessing a maximal chain of left modular elements. Furthermore, we work in the setting of an arbitrary bounded poset as all the above results generalize to this case. We conclude by applying our results to show that the lattice of nonstraddling partitions, which is not graded in general, has a maximal chain of left modular elements.
1. Introduction
An edgelabelling of a poset is a map from the edges of the Hasse diagram of to . Our primary goal is to express certain classical properties of in terms of edgelabellings admitted by . The idea of studying edgelabellings of posets goes back to [12]. An important milestone was [3], where A. Björner defined ELlabellings, and showed that if a poset admits an ELlabelling, then it is shellable and hence CohenMacaulay. We will be interested in a subclass of ELlabellings, known as ELlabellings. In [13], R. Stanley introduced supersolvable lattices and showed that they admit ELlabellings. Examples of supersolvable lattices include distributive lattices, the lattice of partitions of , the lattice of noncrossing partitions of and the lattice of subgroups of a supersolvable group (hence the terminology). It was shown in [9] that a finite graded lattice of rank is supersolvable if and only if it admits an ELlabelling. In many ways, this characterization of lattice supersolvability in terms of edgelabellings serves as the starting point for our investigations.
For basic definitions concerning partially ordered sets, see [14]. We will say that a poset is bounded if it contains a unique minimal element and a unique maximal element, denoted and respectively. All the posets we will consider will be finite and bounded. A chain of a poset is said to be maximal if it is maximal under inclusion. We say that is graded if all the maximal chains of have the same length, and we call this length the rank of . We will write if covers in and if either covers or equals . The edgelabelling of is said to be an ELlabelling if for any in ,

there is a unique unrefinable chain such that , and

the sequence of labels of this chain (referred to as the increasing chain from to ), when read from bottom to top, lexicographically precedes the labels of any other unrefinable chain from to .
This concept originates in [3]; for the case where is not graded, see [4, 5]. If is graded of rank with an ELlabelling , then is said to be an ELlabelling if the labels along any maximal chain of are all distinct and are elements of . In other words, for every maximal chain of , the map sending to is a permutation of . Note that the second condition in the definition of an ELlabelling is redundant in this case.
Example 1.1.
Any finite distributive lattice has an ELlabelling. Let be a finite distributive lattice of rank . By the Fundamental Theorem of Finite Distributive Lattices [2, p. 59, Thm. 3], that is equivalent to saying that , the lattice of order ideals of some element poset . Let be a linear extension of , i.e., any bijection labelling the vertices of that is orderpreserving (if in then ). This labelling of the vertices of defines a labelling of the edges of as follows. If covers in , then the order ideal corresponding to is obtained from the order ideal corresponding to by adding a single element, labeled by , say. Then we set . This gives us an ELlabelling for . Figure 1 shows a labelled poset and its lattice of order ideals with the appropriate edgelabelling.
A finite lattice is said to be supersolvable if it contains a maximal chain, called an Mchain of , which together with any other chain in generates a distributive sublattice. We can label each such distributive sublattice by the method described in Example 1.1 in such a way that the Mchain is the unique increasing maximal chain. As shown in [13], this will assign a unique label to each edge of and the resulting global labelling of is an ELlabelling.
There is also a characterization of lattice supersolvability in terms of left modularity. Given an element of a finite lattice , and a pair of elements , it is always true that
(1) 
The element is said to be left modular if, for all , equality holds in (1). Following A. Blass and B. Sagan [6], we will say that a lattice itself is left modular if it contains a left modular maximal chain, that is, a maximal chain each of whose elements is left modular. (One might guess that we should define a lattice to be left modular if all of its elements are left modular, but this is equivalent to the definition of a modular lattice.) As shown in [13], any Mchain of a supersolvable lattice is always a left modular maximal chain, and so supersolvable lattices are left modular. Furthermore, it is shown by L. S.C. Liu [7] that if is a finite graded lattice with a left modular maximal chain , then has an ELlabelling with increasing maximal chain . In turn, as shown in [9], this implies that is supersolvable, and so we conclude the following.
Theorem 1.
Let be a finite graded lattice of rank . Then the following are equivalent:

has an ELlabelling,

is left modular,

is supersolvable.
It is shown in [13] that if is uppersemimodular, then is left modular if and only if is supersolvable. Theorem 1 is a considerable strengthening of this. Here we used ELlabellings to connect left modularity and supersolvability. It is natural to ask for a more direct proof that (2) implies (3); such a proof has recently been provided by the second author in [15].
Our goal is to generalize Theorem 1 to the case when is not graded and, moreover, to the case when is not necessarily a lattice. We now wish to define natural generalizations of ELlabellings and of maximal left modular chains.
Definition 1.2.
An ELlabelling of a poset is said to be interpolating if, for any , either

or

the increasing chain from to , say , has the properties that its labels are strictly increasing and that and .
Example 1.3.
The reader is invited to check that the labelling of the nongraded poset shown in Figure 2 is an interpolating ELlabelling.
In fact, the poset shown is the socalled “Tamari lattice” . For all positive integers , there exists a Tamari lattice with elements, where , the th Catalan number. More information on the Tamari lattice can be found in [5, §9], [6, §7] and the references given there, and in [7, §3.2], where this interpolating ELlabelling appears. The Tamari lattice is shown to have an ELlabelling in [5] and is shown to be left modular in [6].
If is graded of rank and has an interpolating labelling in which the labels on the increasing maximal chain reading from bottom to top are , then we can check (cf. Lemma 3.2) that is an ELlabelling.
Our next step is to define left modularity in the nonlattice case. Let and be elements of . We know that and have at least one common upper bound, namely . If the set of common upper bounds of and has a least element, then we denote it by . Similarly, if and have a greatest common lower bound, then we denote it by .
Now let and be elements of with . Consider the set of common lower bounds for and that are also greater than or equal to . Clearly, is in this set. If this set has a greatest element, then we denote it by and we say that is welldefined (in ). We see that is welldefined in the poset shown in Figure 3, even though is not. Similarly, let and be elements of with . If the set has a least element, then we denote it by and we say that is welldefined (in ). We will usually be interested in expressions of the form and . The reader that is solely interested in the lattice case can choose to ignore the subscripts and superscripts on the meet and join symbols.
Definition 1.4.
An element of a poset is said to be viable if, for all in , and are welldefined. A maximal chain of is said to be viable if each of its elements is viable.
Example 1.5.
The poset shown in Figure 3 is certainly not a lattice but the reader can check that the increasing maximal chain is viable.
Definition 1.6.
A viable element of a poset is said to be left modular if, for all in ,
A maximal chain of is said to be left modular if each of its elements is viable and left modular, and is said to be left modular if it possesses a left modular maximal chain.
This brings us to the first of our main theorems.
Theorem 2.
Let be a bounded poset with a left modular maximal chain . Then has an interpolating ELlabelling with as its increasing maximal chain.
The proof of this theorem will be the content of the next section. In Section 3, we will prove the following converse result.
Theorem 3.
Let be a bounded poset with an interpolating ELlabelling. The unique increasing chain from to is a left modular maximal chain.
These two theorems, when compared with Theorem 1, might lead one to ask about possible supersolvability results for bounded posets that aren’t graded lattices. This problem is discussed in Section 4. In the case of graded posets, we obtain a satisfactory result, namely Theorem 4. As a consequence, we have given an answer to the question of when a graded poset has an ELlabelling. This has ramifications on the existence of a “good 0Hecke algebra action” on the maximal chains of the poset, as discussed in [9]. However, it remains an open problem to appropriately extend the definition of supersolvability to ungraded posets.
An explicit application of Theorem 3 is the subject of Section 5. As a variation on noncrossing partitions and nonnesting partitions, we define nonstraddling partitions. Ordering the set of nonstraddling partitions of by refinement gives a poset, denoted , that is generally a nongraded lattice. We define an edgelabelling for that is analogous to the usual ELlabelling for the lattice of partitions of . In order to show that is left modular, we then prove that is an interpolating ELlabelling.
2. Proof of Theorem 2
Throughout this section, we suppose that is a bounded poset with a left modular maximal chain . We want to show that has an interpolating ELlabelling. Our approach will be as follows: we will begin by specifying an edgelabelling for such that is an increasing chain with respect to . We will then prove a series of lemmas which build on the viability and left modularity properties. These culminate with Proposition 2.6 which, roughly speaking, gives a more local definition for . We will then be ready to show that is an ELlabelling and is, furthermore, an interpolating ELlabelling.
We choose a label set of natural numbers. (For most purposes, we can let .) We define an edgelabelling on by setting for if
It is easy to see that is welldefined. We will refer to it as the labelling induced by and the label set . When is a lattice, this labelling appears, for example, in [7, 16]. As in [7], we can give an equivalent definition of as follows.
Lemma 2.1.
Suppose in . Then if and only if
Proof.
That is immediate from the definition of . By left modularity, if and only if and In other words, and . It follows that . ∎
Lemma 2.2.
Suppose that in and let . Then is welldefined and equals . Similarly, is welldefined and equals .
Proof.
It is routine to check that, in , is the least common upper bound for and , and that, in , is the greatest common lower bound lower bound for and . ∎
Lemma 2.3.
Suppose that in and . Let in . Then and are welldefined elements of and are equal.
Proof.
Lemma 2.4.
Suppose and are viable and that is left modular in .

If then for any in we have .

If then for any in we have .
Proof.
We prove (a); (b) is similar. Assume, seeking a contradiction, that for some . Now and . It follows that .
Now . Therefore, . So
which is a contradiction. ∎
Lemma 2.5.
The elements of of the form form a left modular maximal chain in .
Proof.
We are now ready for the last, and most important, of our preliminary results. Let be an interval in . We call the maximal chain of from Lemma 2.5 the induced left modular maximal chain of . One way to get a second edgelabelling for would be to take the labelling induced in by this induced maximal chain. We now prove that, for a suitable choice of label set, this labelling coincides with .
Proposition 2.6.
Let be a bounded poset, a left modular maximal chain and the corresponding edgelabelling with label set . Let , and define by saying
Let . Let be the labelling of induced by its induced left modular maximal chain and the label set . Then agrees with restricted to the edges of .
Proof.
Proof of Theorem 2..
We now know that the induced left modular chain in has (strictly) increasing labels, say . Our first step is to show that it is the only maximal chain with (weakly) increasing labels. Suppose that is the induced chain and that is another chain with increasing labels.
If then and the result is clear. Suppose . By Proposition 2.6, we may assume that the labelling on is induced by the induced left modular chain . In particular, we have that where . Let be the least number such that . Then it is clear that . Note that this is the smallest label that can occur on any edge in . Since the labels on the chain are assumed to be increasing, we must have . It follows that and since , we must have . Thus, by induction, the two chains coincide. We conclude that the induced left modular maximal chain is the only chain in with increasing labels.
It also has the lexicographically least set of labels. To see this, suppose that is another chain in . We assume that since, otherwise, we can just restrict our attention to . We have , where since . Hence . This gives that is an ELlabelling. (That is an ELlabelling was already shown in the lattice case in [7, 16].)
Finally, we show that it is an interpolating ELlabelling. If is not the induced left modular maximal chain in , then let be the induced left modular maximal chain. We have that where
since . Therefore, . Also, where
since . Therefore, , as required. ∎
3. Proof of Theorem 3
We suppose that is a bounded poset with an interpolating ELlabelling . Let be the increasing chain from to and let . We will begin by establishing some basic facts about interpolating labellings. These results will enable us to show certain meets and joins exist by looking at the labels that appear along particular increasing chains. We will thus show that the are viable. We will finish by showing that the are left modular, again by looking at the labels on increasing chains.
Let . Suppose that, for some , we have . Then the “basic replacement” at takes the given chain and replaces the subchain by the increasing chain from to . The basic tool for dealing with interpolating labellings is the following wellknown fact about ELlabellings.
Lemma 3.1.
Let . Successively perform basic replacements on this chain, and stop when no more basic replacements can be made. This algorithm terminates, and yields the increasing chain from to .
Proof.
At each step, the sequence of labels on the new chain lexicographically precedes the sequence on the old chain, so the process must terminate, and it is clear that it terminates in an increasing chain. ∎
We now prove some simple consequences of this lemma.
Lemma 3.2.
Let be the chain . Then the labels on all occur on the increasing chain from to and are all different. Furthermore, all the labels on the increasing chain from to are bounded between the lowest and highest labels on .
Proof.
That the labels on the given chain all occur on the increasing chain follows immediately from Lemma 3.1 and the fact that after a basic replacement, the labels on the old chain all occur on the new chain. Similar reasoning implies that the labels on the increasing chain are bounded between the lowest and highest labels on .
That the labels are all different again follows from Lemma 3.1. Suppose otherwise. By repeated basic replacements, one obtains a chain which has two successive equal labels, which is not permitted by the definition of an interpolating labelling. ∎
Lemma 3.3.
Let such that there is some chain from to all of whose labels are in . Then . Conversely, if , then all the labels on any chain from to are in .
Proof.
We begin by proving the first statement. By Lemma 3.2, the labels on the increasing chain from to are in . Find the increasing chain from to . Let be the element in that chain such that all the labels below it on the chain are in , and those above it are in . Again, by Lemma 3.2, the increasing chain from to has all its labels in , and the increasing chain from to has all its labels in . Thus is on the increasing chain from to , and so . But by construction . So .
To prove the converse, observe that by Lemma 3.2, no label can occur more than once on any chain. But since every label in occurs on the increasing chain from to , no label from among that set can occur on any edge below . ∎
The obvious dual of Lemma 3.3 is proved similarly:
Corollary 3.4.
Let such that there is some chain from to all of whose labels are in . Then . Conversely, if , then all the labels on any chain from to are in .
We are now ready to prove the necessary viability properties.
Lemma 3.5.
and are welldefined for any and for .
Proof.
We will prove that is welldefined. The proof that is welldefined is similar. Let be the maximum element on the increasing chain from to such that all labels on the increasing chain between and are in . Clearly and, by Lemma 3.3, .
Suppose . It follows that all labels from to are in . Consider the increasing chain from to . There exists an element on this chain such that all the labels on the increasing chain from to are in and all the labels on the increasing chain from to are in . Therefore, is on the increasing chain from to and, in fact, . Also, we have that . We conclude that is the greatest common lower bound for and . ∎
Lemma 3.6.
, after we delete repeated elements, is the increasing chain in . Hence, is welldefined for . Similarly, is welldefined.
Proof.
From the previous proof, we know that is the maximum element on the increasing chain from to such that all labels on the increasing chain between and are in . The first assertion follows easily from this.
Now apply Lemma 3.5 to the bounded poset . It has an obvious interpolating labelling induced from the interpolating labelling of . Recall that our definition of the existence of only requires it to be welldefined in . The result follows. ∎
We conclude that the increasing maximal chain of is viable. It remains to show that it is left modular.
Proof of Theorem 3..
Suppose that is not left modular for some . Then there exists some pair such that . Set , and . Observe that while . So the picture is as shown in Figure 4.
By Lemma 3.3, the labels on the increasing chain from to are less than or equal to . Consider the increasing chain from to . Let be the first element along the chain. If , then by Lemma 3.3, , contradicting the fact that . Thus the labels on the increasing chain from to are all greater than . Dually, the labels on the increasing chain from to are less than or equal to . But now, by Lemma 3.2, the labels on the increasing chain from to must be contained in the labels on the increasing chain from to , and also from to . But there are no such labels, implying a contradiction. We conclude that the are all left modular. ∎
We have shown that if is a bounded poset with an interpolating labelling , then the unique increasing maximal chain is a left modular maximal chain. By Theorem 2, then induces an interpolating ELlabelling of . We now show that this labelling agrees with for a suitable choice of label set, which is a special case of the following proposition.
Proposition 3.7.
Let and be two interpolating ELlabellings of a bounded poset . If and agree on the increasing chain from to , then and coincide.
Proof.
Let be the maximal chain with the lexicographically first labelling among those chains for which and disagree. Since is not the increasing chain from to , we can find an such that . Let be the result of the basic replacement at with respect to the labelling . Then the label sequence of lexicographically precedes that of , so and agree on . But using the fact that and are interpolating, it follows that they also agree on . Thus they agree everywhere. ∎
4. Generalizing Supersolvability
Suppose is a bounded poset. For now, we consider the case of being graded of rank . We would like to define what it means for to be supersolvable, thus generalizing Stanley’s definition of lattice supersolvability. A definition of poset supersolvability with a different purpose appears in [16] but we would like a more general definition. In particular, we would like to be supersolvable if and only if has an ELlabelling. For example, the poset shown in Figure 3, while it doesn’t satisfy V. Welker’s definition, should satisfy our definition. We need to define, in the poset case, the equivalent of a sublattice generated by two chains.
Suppose has a viable maximal chain . Thus and are welldefined for and in . Given any chain of , we define to be the smallest subposet of satisfying the following two conditions:

and are contained in ,

If in and and are in , then so are and for any in .
Definition 4.1.
We say that a bounded poset is supersolvable with Mchain if is a viable maximal chain and is a distributive lattice for any chain of .
Since distributive lattices are graded, it is clear that a poset must be graded in order to be supersolvable. We now come to the main result of this section.
Theorem 4.
Let be a bounded graded poset of rank . Then the following are equivalent:

has an ELlabelling,

is left modular,

is supersolvable.
Proof.
Observe that for a graded poset, Lemma 3.2 implies that an interpolating labelling is an ELlabelling, and the converse is obvious. Thus, Theorems 2 and 3 restricted to the graded case give us that .
Our next step is to show that and together imply . Suppose is a bounded graded poset of rank with an ELlabelling. Let denote the increasing maximal chain of . We also know that is viable and left modular and induces the same ELlabelling. Given any maximal chain of , we define to be the closure of in under basic replacements. In other words, is the smallest subposet of which contains and and which has the property that, if and are in with , then the increasing chain between and is also in . It is shown in [9, Proof of Thm. 1] that is a distributive lattice. There is a lattice but the proof of distributivity doesn’t use this fact. Now consider . We will show that there exists a maximal chain of such that . Let be the maximal chain of which contains and which has increasing labels between successive elements of . The only idea we need is that, for in , the increasing chain from to is given by , where we delete repeated elements. This follows from Lemma 2.5 since the induced left modular chain in has increasing labels. It now follows that , and hence is a distributive lattice.
Finally, we will show that . We suppose that is a bounded supersolvable poset with Mchain . Suppose in and let be the chain . For any in , is welldefined in (because is assumed to be viable) and equals the usual join of and in the lattice . The same idea applies to , and . Since is distributive, we have that
in and so is left modular in . ∎
Remark 4.2.
We know from Theorem 1 that a graded lattice of rank is supersolvable if and only if it has an ELlabelling. Therefore, it follows from Theorem 4 that the definition of a supersolvable poset when restricted to graded lattices yields the usual definition of a supersolvable lattice. (Note that this is not a priori obvious from our definition of a supersolvable poset.)
Remark 4.3.
The argument above for the equality of and holds even if is not graded. However, in the ungraded case, it is certainly not true that is distributive. The search for a full generalization of Theorem 1 thus leads us to ask what can be said about in the ungraded case. Is it a lattice? Can we say anything even in the case that is a lattice?
5. Nonstraddling partitions
Let denote the lattice of partitions of the set into blocks, where we order partitions by refinement: if and are partitions of we say that if every block of is contained in some block of . Equivalently, covers in if is obtained from by merging two blocks of . Therefore, is graded of rank . is shown to be supersolvable in [13] and hence has an ELlabelling, which we denote be . In fact, it will simplify our discussion if we use the label set for , rather than the label set . We choose the Mchain, and hence the increasing maximal chain for , to be the maximal chain consisting of the bottom element and those partitions of whose only nonsingleton block is , where . In the literature, is often defined in the following form, which can be shown to be equivalent. If is obtained from by merging the blocks and , then we set
For any , we will say that is a block minimum in if for some block of . In particular, we see that is the unique block minimum in that is not a block minimum in .
Recall that a noncrossing partition of is a partition with the property that if some block contains and and some block contains and with , then . Again, we can order the set of noncrossing partitions of by refinement and we denote the resulting poset by . This poset, which can be shown to be a lattice, has many nice properties and has been studied extensively. More information can be found in R. Simion’s survey article [11] and the references given there. Since is a subposet of , we can consider restricted to the edges of . It was observed by Björner and P. Edelman in [3] that this gives an ELlabelling for and we can easily see that this ELlabelling is, in fact, an ELlabelling (once we subtract 1 from every label).
We are now ready to state our main definition for this section, which should be compared with the definition above of noncrossing partitions.
Definition 5.1.
A partition of is said to be nonstraddling if whenever some block contains and and some block contains and with , then .
This definition is also very similar to that of nonnesting partitions, as defined by A. Postnikov and discussed in [10, Remark 2] and [1]. The only difference in the definition of nonnesting partitions is that we do not require if there is also an element of between and . So, for example, is a nonnesting partition in but is not a nonstraddling partition. We say that is a straddling partition, that is a straddle, and that the blocks and form a straddle.
Let be the subposet of consisting of those partitions that are nonstraddling. To distinguish the interval in from the interval in , we will use the notation and , respectively. We note that the meet in of two nonstraddling partitions is again nonstraddling, implying that is a meetsemilattice. Since is a top element for , we conclude that is a lattice. On the other hand, is not graded. For example, consider those elements of that cover , as represented in Figure 5(a).
, and are all straddling partitions, so is covered in by . Figure 5(b) shows .
Therefore, unlike and , cannot have an ELlabelling. However, we can ask if it has an interpolating ELlabelling. We see that the following three ways of defining an edgelabelling for are equivalent. Observe that if in , then is obtained from by merging the blocks of into a single block in . We set
second smallest element of  (2)  
smallest block minimum in that is not a block  
minimum in  
smallest edge label of under the edgelabelling . 
See Figure 5(b) for examples. Note that the label set for is and that if , then equals . We see that the chain
is an increasing maximal chain in under .
Theorem 5.
The edgelabelling is an interpolating ELlabelling for .
Applying Theorem 3, we get the following result:
Corollary 5.2.
is left modular.
In preparation for proving Theorem 5, we wish to get a firmer grasp on . Suppose . While the meet of and in is just the meet of and in , the situation for joins is more complicated. The next lemma, crucial to the proof that is an ELlabelling, helps us to understand important types of joins. From now on, unless otherwise specified, with will denote the join of and in . Furthermore, if are block minima in , then will denote the block of with minimum element , and will denote the minimum element for which the elements of are all in a single block. Note that is welldefined, since it is the meet of all those elements of that have the required elements in a single block.
Lemma 5.3.
Suppose are block minima in and that
Then
for any .
In words, this says that if merging the blocks and in requires us to merge all of , then merging any two of these blocks also requires us to merge all of them.
Proof.
The proof is by induction on , with the result being trivially true when . While elementary, the details are a little intricate. To gain a better understanding, the reader may wish to treat the proof as an exercise. If , then by the induction assumption and the hypothesis that , we have
as required. Therefore, it suffices to let .
Since , we know that forms a straddle with . There are two ways in which this might happen.
Suppose we have with and . Suppose in . Then, since , we have that is a straddle in , which contradicts .
Secondly, suppose we have with and . Suppose and . Now . If then has a straddle, so we can assume that and that , with the argument being similar if . If , then is a straddle when we merge blocks and in . Therefore,
(3) 
by the induction assumption. If , then is a straddle when we merge blocks and in , also implying (3). ∎
Lemma 5.4.
Suppose in and that has edge labels under the edgelabelling .

There is exactly one edge of the form with in .

On any unrefinable chain in , the label has to appear.
Proof.
(i) We first prove the existence of . Let be the minimum of the block of containing and set . Suppose . We know is obtained from by merging the blocks