Generalized recursive atom ordering and equivalence to CL-shellability

Bj\"orner and Wachs introduced CL-shellability as a technique for studying the topological structure of order complexes of partially ordered sets (posets). They also introduced the notion of recursive atom ordering, and they proved that a finite bounded poset is CL-shellable if and only if it admits a recursive atom ordering. In this paper, a generalization of the notion of recursive atom ordering is introduced. A finite bounded poset is proven to admit such a generalized recursive atom ordering if and only if it admits a traditional recursive atom ordering. This is also proven equivalent to admitting a CC-shelling (a type of shelling introduced by Kozlov) with a further property called self-consistency. Thus, CL-shellability is proven equivalent to self-consistent CC-shellability. As an application, the uncrossing posets, namely the face posets for stratified spaces of planar electrical networks, are proven to be dual CL-shellable.


Introduction
This paper introduces a new tool for studying the topological structure of order complexes of finite partially ordered sets (posets).This tool, called generalized recursive atom ordering, is a relaxation of the fundamental and widely used recursive atom ordering technique introduced by Björner and Wachs in [BW83].Any recursive atom ordering (a notion that is reviewed in Section 2) of a finite bounded poset gives rise to a lexicographic shelling for the poset, thereby yielding the result that the order complex of the poset is either homotopy equivalent to a wedge of spheres or contractible.
We establish a number of fundamental properties of these generalized recursive atom orderings (GRAOs), including the property that any generalized recursive atom ordering may be transformed into a traditional recursive atom ordering (RAO) by a process we introduce called the atom reordering process.Since GRAOs are easier to construct than RAOs, this may give a useful new pathway to proving a poset is CL-shellable.These generalized recursive atom orderings further allow us to prove that several different forms of lexicographic shellability are all equivalent to each other, by which we mean that a finite bounded poset admits any one of these types of lexicographic shelling if and only if it admits each of the others.One might expect this to imply the stronger statement that any instance of any one of these types of lexicographic shelling is also an instance of any other of these types of lexicographic shelling, but this is not always true.For instance, one may deduce that not every "self consistent CC-shelling" is a CL-shelling from the fact that not every generalized recursive atom ordering is a recursive atom ordering; this latter fact is stated more precisely in Remark 3.3, and an example demonstrating this latter fact appears in Figure 2.These equivalence results 6.2 5.8 6.3 5.11 5.8 [Ko] [ BW83,2.3] [ BW83,3.2] 5.3

2.20
Figure 1.Implications and where they are proven.
clarify the hierarchy of different techniques for proving that a finite bounded poset is lexicographically shellable.Figure 1 gives a schematic of many of the implications proven in this paper, and it points readers (by way of the labels on the implication arrows) to where each result is proven in this paper or elsewhere in the literature.
The importance of the notion of recursive atom ordering stems from the fact that a finite bounded poset admits a recursive atom ordering if and only if it is CL-shellable.ELshellability and CL-shellability are the original and predominant techniques for proving that posets are shellable.EL-shellability was first introduced by Björner in [Bj80].It was generalized to the notion of CL-shellability by Björner and Wachs in [BW83] when they observed that their idea for a way to shell Bruhat order did not meet the requirements of an ELshelling but nonetheless gave rise to a shelling of a very similar flavor to an EL-shelling.Recursive atom orderings were introduced in [BW83] as an alternative approach to proving CL-shellability.For families of posets with inherent recursive structure, such as the partition lattice, recursive atom ordering can often be the easiest and most natural way to prove these posets are shellable.Both CL-shellability and the related notion of recursive atom ordering were extended to the non-graded case in [BW96] and [BW97].In [Ko], EL-shellability and CL-shellability were generalized to the notions of EC-shellability and CC-shellability, which allowed more flexibility in constructing labelings that could be used to establish lexicographic shellability.Some important examples of posets that have been proven to be lexicographically shellable (or dual lexicographically shellable), in some cases by way of a recursive atom ordering, include Bruhat order ( [BW82]), posets with exponential structures ( [Sa]), supersolvable lattices ( [Bj80]), geometric lattices ( [Bj80]), geometric semilattices ( [WW]), various posets from finite group theory ( [Sh01], [Wo]) and combinatorial commutative algebra ( [PRS])), intersection posets of k-equal subspace arrangements ( [BW96]), and face posets of shellable d-complexes ([Bj84]).
In Section 2, we review background, including some key ideas of Björner and Wachs that we will build upon later in the paper.We establish some fundamental properties of GRAOs and of the atom reordering process in Sections 3 and 4, respectively.In particular, Section 4 presents one of the main results of the paper, Theorem 4.11; this result establishes that a finite bounded poset admits a recursive atom ordering (RAO) if and only if it admits a generalized recursive atom ordering (GRAO).Figure 2 gives an example highlighting the subtlety of Theorem 4.11.It shows a small example of a poset P endowed with a GRAO that is not an RAO (on the left in the figure) and the same poset endowed with an RAO (on the right in the figure).A key ingredient to the proof of Theorem 4.11 is an algorithm that transforms any GRAO into an RAO on the same poset P , a procedure we call the "atom reordering process" (see Algorithm 4.3).This procedure transforms the GRAO shown on the left in Figure 2 into the RAO shown on the right in Figure 2.
Sections 5 and 6 establish a link that is not necessarily an equivalence between finite bounded posets that admit a GRAO and those that are CC-shellable.This link led us to introduce a variation on the notion of CC-shellability in Section 5.1, namely TCL-shellability (see Definition 5.7).
Many of the results proven throughout the paper are tied together in Theorem 6.4, a result which shows that several different notions of lexicographic shellability are equivalent to each other.In this result, we specify eight different versions of lexicographic shellability and show that a finite bounded poset admits any one of these types of lexicographic shelling if and only if it admits all of the others.
In Section 7, we apply our results to deduce that a class of posets previously shown to be CC-shellable in [HK] is in fact CL-shellable.That is, we prove that the dual posets to the uncrossing orders (conjectured to be lexicographically shellable by Lam in [La14a]) are CL-shellable.These uncrossing orders arise naturally as face posets of stratified spaces of planar electrical networks (see e.g.[Ke], [La14a], and references therein).The fact that they are shellable posets combines with Lam's result from [La14a] that they are Eulerian posets to imply that they are CW posets, i.e. face posets of regular CW complexes with finitely many cells.Thus, the shellability of uncrossing orders provides an important step in understanding the topological structure of these spaces of planar electrical networks.
The paper concludes with further results, observations, and open questions in Section 8.

Background
Let P denote a partially ordered set (poset).All posets throughout this paper are assumed to be finite and bounded but are not assumed to be graded.For background on posets, poset topology, and shellability beyond what appears below, we refer readers to [St12], [Wa], [Bj95], [St96], and [Zi].
A cover relation u⋖v in a poset P is an order relation u < v with the further requirement that there does not exist any z ∈ P with u < z < v.In this case, we say that v covers u.
A poset P is bounded if it has both a unique least element (often denoted 0) and a unique greatest element (often denoted 1).A closed interval, denoted [u, v], in a poset P is the subposet consisting of all elements z ∈ P such that u ≤ z ≤ v.
The atoms of a bounded poset P (resp.a closed interval [u, v]) are those elements a in P (resp.[u, v]) that cover 0 (resp.u).Likewise the coatoms of a bounded poset P (resp.a closed interval [u, v]) are the elements in P (resp.[u, v]) covered by 1 (resp.v).
A chain in a poset P is a totally ordered subset u 1 < • • • < u r of P .The length of a chain u 1 < • • • < u r is the number r − 1 of order relations in the chain.The length of a poset P , is the length of the longest chain in P .A chain is maximal in P if no additional elements of P may be inserted in it.A chain We will also make the convention of sometimes referring to the maximal chains of a closed interval [u 1 , u k ] as the saturated chains of [u 1 , u k ].If all maximal chains in a poset P are of the same length, the poset is said to be graded.Definition 2.1.The order complex of a poset P , denoted ∆(P ), is the abstract simplicial complex whose k−faces are the chains of length k of P .In particular, this means the vertices are the chains consisting of single elements of P .Note that a face σ in ∆(P ) is contained in another face τ in ∆(P ) if and only if the chain corresponding to σ is contained in the chain corresponding to τ .
When we say that a poset P has a topological property (such as shellability or homotopy equivalence to a wedge of spheres), we mean that ∆(P ) has this topological property.
The dual of a poset P , denoted P * , has the same elements as P with u ≤ v in P * if and only if v ≤ u in P .
Remark 2.2.Virtually everything in this paper has a dual version for the simple reason that a poset and its dual have the same order complex.This fact enables any poset theoretic technique to be applied to the dual poset to derive the same consequence regarding the order complex.There are indeed posets where this is a helpful thing to do (e.g. the uncrossing orders as discussed in Section 7).We leave it to the interested reader to fill in the dual versions of our results.
Definition 2.3.Given any simplicial complex K, its face poset P (K) consists of the faces of K with order relation σ ≤ τ if and only if the set of vertices in σ is a subset of the set of vertices in τ .The closure of a face τ , denoted τ , is the set of faces σ such that σ ≤ τ in P (K).
The augmented face poset P (K) is P (K) with a maximal element 1 adjoined if P (K) does not already have a unique maximal element, and P (K) = P (K) otherwise.
Definition 2.4.A simplicial complex is shellable if there is a total order F 1 , . . ., F k on its maximal faces (known as facets) such that F j ∩ (∪ i<j F i ) is a pure, codimension one subcomplex of F j for each j ≥ 2. Such a facet ordering is known as a shelling.Definition 2.5.A poset P is said to be shellable if its order complex ∆(P ) is shellable.
Next we review various types of edge and chain-edge labelings of a finite poset P .These labelings will induce lexicographic shellings for ∆(P ), namely shellings obtained by taking the facets of ∆(P ) in order according to the lexicographic (i.e.dictionary) order of the label sequences of the corresponding maximal chains of P , breaking ties in any manner.Let E(P ) be the set of edges in the Hasse diagram of a finite poset P , that is, the pairs x, y ∈ P such that x ⋖ y.An edge labeling of P is a map λ : E(P ) → Q for Q a poset.Quite often, Q is the integers with their usual order.
We say that x ⋖ y ⋖ z is an ascent with respect to the edge labeling λ if λ(x, y) < Q λ(y, z).Any x ⋖ y ⋖ z that is not an ascent with respect to λ is said to be a descent.A maximal chain in a finite poset P (or more generally in a closed interval [u, v] in P ) is an ascending chain if it consists entirely of ascents.
Definition 2.6 ([Bj80], [BW97]).An edge labeling of a finite, bounded poset P is called an EL-labeling (for edge lexicographical labeling) if for every interval [x, y] ∈ P the following conditions are both met: (i) There is a unique ascending maximal chain c in [x, y].
(ii) The label sequence associated to c lexicographically precedes the label sequences associated to every other maximal chain in [x, y].A finite, bounded poset admitting such a labeling is EL-shellable.
Remark 2.7.In this paper, when we say that we take the lexicographic order on maximal chains of P , we mean that we take the maximal chains in order according to the lexicographic order on their label sequences.
Björner first introduced EL-labelings and EL-shellability in the graded case in [Bj80].This was generalized by Björner and Wachs to the not necessarily graded case when they introduced the notion on nonpure shellability in [BW96] and [BW97].The following fundamental result from [Bj80] explains the usage of the term EL-shellablility.
Theorem 2.8 ([Bj80], Theorem 2.3).If P is a finite, bounded, graded poset with an ELlabeling, then the lexicographic order of the maximal chains of P is a shelling order for the corresponding facets of ∆(P ).Now we turn to a generalization of EL-shellability due to Björner and Wachs (see [BW83]) in which edge labelings are replaced by more general chain-edge labelings (defined next).In this context, we replace E(P ) by the set E * (P ) defined as follows: Definition 2.9.Let P be a finite bounded poset and let Q be any poset.A chain-edge labeling (or CE-labeling) of P is a map λ : E * (P ) → Q that satisfies the following condition: If two maximal chains coincide along their first d edges, then they have the same labels as each other on these edges.In other words, if c is a maximal chain 0 = To see a naturally arising example of a chain-edge labeling that is not an edge labeling, we refer readers to the chain-edge labeling due to Björner and Wachs for the dual poset to Bruhat order.This labeling is reviewed just prior to Proposition 8.4.Björner and Wachs proved in [BW83] that it is a CL-labeling, a notion defined shortly.
Definition 2.10.If [x, y] is an interval in P and r is a saturated chain from 0 to x, then the pair ([x, y], r) is called a rooted interval with r as the root of this rooted interval.This is denoted by [x, y] r .
Given a chain-edge labeling λ of a finite bounded poset P , λ associates to each maximal chain of P a label sequence as follows.If m = ( 0 = x 0 ⋖x 1 ⋖. . .⋖x n = 1), then the associated label sequence is By definition of a chain-edge labeling, any two maximal chains both containing the same root r from 0 to x and the same saturated chain c in the interval [x, y] will both have the same label sequence assigned to c.We will denote this label sequence by λ r (c) and the individual labels comprising it as λ r (x i , x i+1 ) for i = 0, . . ., n − 1.Sometimes we will write λ(c) for the label sequence and λ(x i , x i+1 ) for the label on the edge Given a label sequence (λ 1 , . . ., λ r ), we say that a pair of consecutive labels λ i , λ i+1 comprises an ascent if and only if λ i < λ i+1 .The pair λ i , λ i+1 comprises a descent otherwise.

Definition 2.12 ([BW83]
).A CE-labeling λ of a finite bounded poset P is called a CLlabeling (for chain-lexicographical labeling) if for every rooted interval [x, y] r in P , (i) there is a unique ascending chain c in [x, y] r and (ii) the label sequence λ r (c) lexicographically precedes the label sequence for every other maximal chain in [x, y] r .If a finite bounded poset P admits a CL-labeling, then P is said to be CL-shellable.
Björner and Wachs proved in [BW82] (resp.[BW96]) that whenever a finite bounded poset P that is graded (resp. is not necessarily graded) admits a CL-labeling, then any linear extension of the lexicographic order on its maximal chains given by the CL-labeling is a shelling order on the corresponding facets of ∆(P ).
One of the primary techniques for proving that a finite bounded poset is CL-shellable is to construct a recursive atom ordering (see Definition 2.14).The notion of recursive atom ordering (RAO) was introduced by Björner and Wachs in [BW83].They extended it to posets that are not necessarily graded in [BW97].Before defining recursive atom ordering, we lay the groundwork with a notion we call chain-atom ordering that will encompass all recursive atom orderings and all generalized recursive atom orderings (defined later) as special cases.
Definition 2.13.A chain-atom ordering Ω of a finite bounded poset P is a choice of ordering on the atoms of each rooted interval [u, 1] r of P .For the rooted interval [u, 1] r , we will denote this ordering of atoms as Ω([u, 1] r ).On the other hand, we denote the restriction of the full chain-atom ordering Ω to a rooted interval [u, 1] r as Ω| [u, 1]r , and more generally we denote the restriction of Ω to [u, v] r as Ω| [u,v]r .We sometimes call Ω| [u,v]r the chain-atom ordering on [u, v] r induced by Ω.
Whenever a finite bounded poset has a recursive atom ordering, defined next, this by definition guarantees the existence of an especially well-behaved type of chain-atom ordering.We will often refer to these especially nice chain-atom orderings themselves as recursive atom orderings.A key place where we will do this is when we introduce a relaxation of the notion of recursive atom ordering in Section 3.
Definition 2.14.A finite bounded poset P admits a recursive atom ordering if P has length 1 or if the atoms of P can be ordered a 1 , a 2 , . . .a t such that: (i) (a) For all j = 1, . . ., t, [a j , 1] admits a recursive atom ordering.(b) For j = 1, the atoms that come first in this recursive atom ordering for [a j , 1] are those that are greater than some atom a k of P for k < j. (ii) For all i < j and y ∈ P satisfying y > a i and y > a j , there exists k < j and z ∈ P such that a j ⋖ z and a k < z ≤ y.
The following theorem of Björner and Wachs from [BW83] (which they extended to the not necessarily graded case in [BW97]) established a very useful relationship between the existence of a recursive atom ordering and of a CL-labeling for any finite bounded poset P .
We sketch one direction of the proof of Theorem 2.15 shortly, since many of the ideas in this argument will be used in other proofs later in the paper.A key ingredient is the pair of sets F r (u) and G r (u) defined next.Björner and Wachs introduced these sets F r (u) and G r (u) for recursive atom orderings, but we find it convenient to define them more generally.Definition 2.16.Consider any rooted interval [u, 1] r in a finite bounded poset P .Let u − be the element of r covered by u and let r − be the root for [u − , 1] obtained by omitting u from r and otherwise preserving r.Let Λ be either a total order on the atoms of [u − , 1] r − or any richer structure, such as a chain-atom ordering, which specifies such a total order.Define F Λ r (u) to be the set of atoms a of [u, 1] r such that a > P a ′ for some atom a ′ in [u − , 1] r − that comes earlier than u in Λ. Define G Λ r (u) to be the set of all atoms of [u, 1] r that are not contained in F Λ r (u).Given any v ∈ P satisfying u < v, define F Λ r (u, v) to be the set of atoms a of [u, v] r such that a > P a ′ for some a ′ that covers u − and comes earlier than u in Λ| [u − ,v] r − .Define G Λ r (u, v) to be the set of atoms of [u, v] r that are not contained in F Λ r (u, v).Sometimes we will denote these sets simply by F r (u, v), G r (u, v), F r (u) and G r (u) when the choice of Λ is clear from context.
Remark 2.17.By definition, we have Proof sketch of how an RAO yields a CL-labeling: Given an RAO a 1 , . . ., a n for P , start by labeling each cover relation of the form 0 ⋖ a i with the integer i.By induction on the length of the longest saturated chain in P , one may assume that each rooted interval [a i , 1] has its own RAO that induces a CL-labeling for [a i , 1].The plan is to modify this CL-labeling for [a i , 1] for each i so that the modified labels may be taken together with the labels λ( 0, a i ) = i for i = 1, 2, . . ., n to give a CL-labeling λ for all of P .It will suffice to describe how to modify the labels on the cover relations upward from a i to the atoms of [a i , 1], then to apply this same label modification process inductively to handle the rooted intervals higher in the poset.
Letting x 1 , . . ., x u be our RAO for [a i , 1] which is guaranteed to exist by the definition of an RAO for P , Björner and Wachs made the key observation that there always exists some j ∈ {1, . . ., u} such that x 1 , . . ., x j ∈ F 0⋖a i (a i ) while x j+1 , . . ., x u ∈ G 0⋖a i (a i ).For example, one may label each cover relation a i ⋖ x i ′ for i ′ ≤ j with the label λ(a i , x i ′ ) = λ( 0, a i ) − (j − i ′ ) − 1, and one may label each cover relation a i ⋖ x i ′ for i ′ ≥ j + 1 with the label λ(a i , x i ′ ) = λ( 0, a i ) + (i ′ − j).The resulting labeling has the following two properties: (1) λ( 0, a i ) ≥ λ(a i , x i ′ ) if and only if In other words, the shifting of label values preserves the relative order of the labels on cover relations upward from a fixed element with the same fixed choice of root below, but at the same time it creates ascents and descents exactly where they are needed in order to have a CL-labeling.See Theorem 3.2 in [BW83] for further details of this proof.End of proof sketch.
We next review the notions of CC-shellability and EC-shellability from [Ko], doing so using the language of topological ascents and descents (defined next) that was introduced in [He03A].Let λ be an edge labeling on the cover relations of a poset P by elements of some poset Q.A topological ascent occurs in P at u ⋖ v ⋖ w whenever the ordered pair of labels (λ(u, v), λ(v, w)) is lexicographically earlier than all other ordered sequences of labels on all other saturated chains from u to w.If (λ(u, v), λ(v, w)) is not a topological ascent, then it is a topological descent.
For λ a chain-edge labeling, we define topological ascents and descents in the same way, but now with respect to a choice of root.That is, we have a topological ascent at u ⋖ v ⋖ w with respect to root r if the ordered pair (λ(u, v), λ(v, w)) is lexicographically smaller than all other label sequences on saturated chains from u to w with this same root r.We have a topological descent at u ⋖ v ⋖ w with respect to root r otherwise.
If c is a saturated chain from an element u to an element v consisting entirely of topological ascents (in either an edge labeling or a chain-edge labeling), then we say that c is a topologically ascending chain from u to v. If c is a saturated chain from u to v consisting entirely of topological descents, then it is said to be topologically descending.
The next definition is rephrased from how it appears in [Ko], using the language of topological ascents and descents, but is entirely equivalent to his definition.
Definition 2.18 ( [Ko]).An EC-labeling of a finite bounded poset P is an edge labeling λ : E(P ) → Q on the cover relations of P with labels belonging to a poset Q, subject to the requirements that (1) every interval [x, y] has a unique saturated chain consisting entirely of topological ascents, and (2) the label sequences for the saturated chains of P (and hence of each interval [x, y]) are all distinct with no label sequence being a prefix of any other label sequence.
If P admits an EC-labeling, then P is said to be EC-shellable.
Just as EL-shellability was generalized to CL-shellability, the notion of EC-shellability was likewise generalized by Kozlov from edge labelings to chain-edge labelings.His definition from [Ko] may be rephrased as follows.
Definition 2.19 ( [Ko]).A CC-labeling of a finite bounded poset P is a chain-edge labeling λ : E * (P ) → Q with labels belonging to a poset Q, subject to the requirements that (1) every rooted interval [u, v] r has a unique saturated chain consisting entirely of topological ascents, and (2) the label sequences for the saturated chains of [u, v] r are all distinct with no label sequence being a prefix of another label sequence.
If a finite bounded poset P admits a CC-labeling, then P is said to be CC-shellable.
Kozlov proved in Theorem 3.8 of [Ko] that the order complex of any finite bounded poset P admitting a CC-labeling is shellable, doing so by taking any linear extension of the lexicographic order on the label sequences for the maximal chains of P as the shelling order for the corresponding facets of ∆(P ).
Kozlov noted in [Ko] that any finite bounded poset that is CL-shellable is also CC-shellable, leaving the proof to the reader.We include a proof of this result as Proposition 2.20.Our proof below closely follows parts of the proof of Björner and Wachs that any recursive atom ordering induces a CL-labeling (see Theorem 3.2 in [BW83]).This proof is included because it introduces several additional important ideas that will be used in various ways in other proofs later in the paper.
Proposition 2.20.Any recursive atom ordering a 1 , . . ., a t of a finite bounded poset P gives rise to a CC-labeling of P .Thus, any finite bounded poset that is CL-shellable is CC-shellable.
Proof.Given a recursive atom ordering Λ, we start by describing the desired CC-labeling λ derived from Λ. Letting a 1 , . . ., a t denote the atom ordering of P given by Λ, we assign the label λ( 0, a i ) = i to each cover relation of the form 0 ⋖ a i .Now, for any u ∈ P and any root r leading up to u, consider any recursive atom ordering a ′ 1 , . . ., a ′ t ′ for [u, 1] r of the type guaranteed to exist recursively in the definition of RAO.Assign the labels λ r (u, a ′ i ) = i for i = 1, 2, . . ., t ′ .
By construction, this labeling has two of the requisite properties of a CC-labeling, namely the requirements that 1) no two saturated chains of [u, 1] r have the same label sequence and 2) no two saturated chains of [u, 1] r have the property that the label sequence of one is a prefix of the label sequence of the other.Now we verify that each rooted interval [u, v] r in P has a unique topologically ascending maximal chain and that this chain is lexicographically first.We do so by induction on the length of the longest chain in [u, v] r .This hypothesis clearly holds for intervals of length two, the base case.Let c which is given by u ⋖ u 1 ⋖ . . .⋖ u k = v be the lexicographically first maximal chain in [u, v] r .Because it is lexicographically first, this forces c to be topologically ascending.
Suppose that there is another topologically ascending maximal chain ).As c is lexicographically earlier than c ′ , we may conclude that λ(u, u 1 ) ≤ λ(u, u ′ 1 ).But our choice of chain-edge labeling based on an RAO ensures that λ(u, u 1 ) = λ(u, u ′ 1 ), implying λ(u, u 1 ) < λ(u, u ′ 1 ) from which we deduce that u 1 comes before u ′ 1 in the given RAO of [u, 1] r .By (ii) of Definition 2.14, there exists an atom a of [u, v] r and an element z ∈ P such that 1 and, by the induction hypothesis, not topologically ascending.This gives a contradiction to the claim that c ′ consists entirely of topological ascents.Thus, we confirm that the chain-edge labeling λ has a unique topologically ascending chain in [u, v] r , completing our proof that this is a CC-labeling.
Combining the above argument with the result of [BW83] that a finite bounded poset is CL-shellable if and only if it admits a recursive atom ordering shows that every CL-shellable poset is CC-shellable.

Generalized recursive atom ordering
In this section we introduce a generalization of the notion of recursive atom ordering and prove several fundamental properties of these generalized recursive atom orderings.
Remark 3.1.The generalized recursive atom orderings (GRAOs) we are about to introduce will be defined as orderings on the atoms of a finite poset P that are extendible to chainatom orderings of P meeting certain requirements.It will often be convenient to think of a generalized recursive atom ordering as a chain-atom ordering of this type.Now we are ready to give the main new definition of the paper: Definition 3.2.A finite bounded poset P admits a generalized recursive atom ordering (GRAO) 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 , . . .a t on the atoms of P satisfying: (i) (a) For 1 ≤ j ≤ t, [a j , 1] admits a GRAO.(b) For any atom a j and any x, w ∈ P satisfying a j ⋖ x ⋖ w, the following property holds when the chain-atom ordering given by the GRAO from (i)(a) is restricted to [a j , w]: either the first atom of [a j , w] is above an atom a i with i < j, or no atom of [a j , w] is above any atom a i with i < j. (ii) For any y ∈ P and any atoms a i , a j satisfying a i < y and a j < y with i < j, there exists an element z ∈ P with z ≤ y and an atom a k with k < j such that a j ⋖ z and a k < z.
Remark 3.3.When we say that not all recursive atom orderings are generalized recursive atom orderings and when we say that GRAOs are strictly more general than RAOs, we are regarding RAOs and GRAOs as being types of chain-atom orderings.
Remark 3.4.Conditions (i)(a) and (ii) in the definition of GRAO are exactly equivalent to corresponding statements within the definition of recursive atom ordering.However, condition (i)(b) above is considerably less restrictive than (i)(b) of the definition of RAO.
One of the main points of this relaxation of the notion of recursive atom ordering is that it gives considerably more flexibility than a recursive atom ordering in how we may order those atoms in a rooted interval [u, 1] r that are not the first atom of that interval.For this reason, the following rephrasing of condition (ii) also seems quite useful to note: Remark 3.5.Condition (ii) in Definition 3.2 is logically equivalent to the following statement: if a i < y and a j < y for i < j, then there exists z ∈ P such that a j ⋖ z ≤ y where a j is not the earliest atom in [ 0, z].
Example 3.6.The chain-atom ordering given on the left in Figure 3 is a generalized recursive atom ordering but is not a (traditional) recursive atom ordering.In particular, the atom of [a, 1] labeled 3 in the poset on the left causes condition (i)(b) of Definition 2.14 to fail.The chain-atom ordering given on the right is a recursive atom ordering.We highlight in color the place where the two chain-atom orderings differ.
Remark 3.7.Implicit in the definition of GRAO is that a GRAO for a finite bounded poset P induces a GRAO for [u, 1] r for each u ∈ P and each root r.Lemma 3.8 will show that it also induces a GRAO for each [u, v] r .This result will allow us henceforth to refer to the restriction of any GRAO for a finite bounded poset to the rooted interval [u, v] r as the GRAO for [u, v] r induced by the GRAO for P .
Lemma 3.8.Let P be a finite bounded poset with Γ a GRAO for P .Then for any u < v and any root r for [u, v], restricting the GRAO for [u, 1] Proof.Our proof will be by induction on the length of the longest saturated chain from u to 1.We can use maximum length 2 for our base case, obtaining this case by using the fact that every possible chain-atom ordering on a finite bounded poset with maximum chain length of 1 or 2 is a GRAO.We then obtain condition (i)(a) by induction as follows.Given any a i satisfying u⋖a i ≤ v, the length of the longest saturated chain from u to 1 is strictly larger than the length of the longest saturated chain from a i to 1. Thus, our inductive hypothesis gives us that the GRAO for [a i , 1] r∪a i will restrict to a GRAO for [a i , v] r∪a i .But it is immediate from the definition of GRAO that the properties of a GRAO given in conditions (i)(b) and (ii) from the definition of GRAO will restrict from [u, 1] r to [u, v] r .Thus, the GRAO for P restricts just as desired.
Next we show how the statement about cover relations in condition (i)(b) in the definition of GRAO can be strengthened to a corresponding statement about all order relations.Lemma 3.9.Let P be a finite bounded poset, and let Λ be a GRAO for P with atom ordering for [a j , v] with the following property: either (a) the first atom of [a j , v] is greater than some atom a i satisfying i < j or (b) no atom of [a j , v] is greater than any atom a i satisfying i < j.
Proof.By Lemma 3.8, Λ| [a j ,v] is guaranteed to be a GRAO.Let x be the first atom in [a j , v] with respect to Λ. Suppose that some other atom y of [a j , v] is greater than some atom a i of P with i < j.Since x and y are both less than v with x coming before y in Λ| [a j , 1] , Definition 3.2 (ii) guarantees the existence of elements w 1 and y 1 , where y 1 comes before y in Λ| [a j , 1] and where y ⋖ w 1 ≤ v and y 1 < w 1 .Since y covers a i with i < j, Definition 3.2 (i)(b) ensures that the first atom in [a j , w 1 ] must be greater than some atom a i ′ satisfying i ′ < j.We may take y 1 to be the first atom in [a j , w 1 ], i.e., we may replace y 1 by this first atom if it is not already this atom.If y 1 also equals x, then x is greater than some atom that comes before a j in Λ, namely a i ′ , as desired.
If, on the other hand, y 1 = x, then we repeat this process as many times as necessary to get the desired result that x is greater than some atom that comes before a j in Λ, doing so as follows.Since x and y 1 are both less than v, we may use Definition 3.2 (ii) to deduce the existence of y 2 and w 2 where y 2 comes before y 1 in Λ| [a j , 1] and where y 2 < w 2 and y 1 ⋖ w 2 ≤ v.We may take y 2 to be the first atom in [a j , w 2 ].But then Definition 3.2 (i)(b) guarantees that y 2 must be greater than some atom a i ′′ where i ′′ < j.If y 2 = x, this yields the desired result.If y 2 = x, then we repeat this process again obtaining elements y 3 and w 3 in place of y 2 and w 2 .
Continuing in this manner, we obtain a sequence y 1 , y 2 , y 3 , . . . of atoms of [a j , v] where for each i ≥ 2 we have that: (1) y i comes earlier in Λ| [a j ,v] than y i−1 , and (2) y i is greater than an atom of P that comes before a j in Λ.Note that (1) implies each y i is distinct, allowing us to deduce from finiteness of [a j , v] that the sequence y 1 , y 2 , y 3 , . . .must terminate at some y n .This terminal element y n must satisfy y n = x since we have shown that otherwise there would be an atom y n+1 of [a j , v] coming still earlier than y n .Thus we obtain the desired result that x is greater than some atom that comes before a j in Λ.
The sets F r (u, v) and G r (u, v) appearing in the next lemma will play a key role later in the paper in allowing us to transform any generalized recursive atom ordering into a recursive atom ordering.Our justification that this atom reordering process converts a GRAO to an RAO will rely heavily on the property of the sets F r (u, v) and G r (u, v) that we justify next.See Definition 2.16 for the definitions of the sets F r (u, v) and G r (u, v).
Lemma 3.10.Let Γ be a chain-atom ordering for a finite bounded poset P .Consider any two atoms a i , a i+1 of a rooted interval [u, 1] r that are consecutive atoms of [u, 1] r with respect to Γ. Suppose the pair of atoms a i , a i+1 has the further property for every rooted interval [u, v] r containing both a i and a i+1 that neither a i nor a i+1 is the first atom of [u, v] r with respect to Γ| [u,v]r .Let Λ be the chain-atom ordering on P obtained from Γ by switching the order of a i and Proof.Throughout this proof, let r be the chain 0 r by deleting u.We will first show how to handle all cases which do not satisfy both of the conditions u ⋖ u ′ and r Observe that if we do not have u = u ′ l and also have then swapping the order of the atoms a i and a i+1 in [u, 1] r has no impact on the ordering of the atoms of ) cannot be impacted by the swap of a i and a i+1 unless u ⋖ u ′ and r ′ = r ∪ {u ′ }.
We subdivide the task of handling those situations with u ⋖ u ′ and r ′ = r ∪ {u ′ } into three cases, namely: (a) u ′ = a i , (b) u ′ = a i+1 , and (c) u ′ ∈ {a i , a i+1 }.We start with the easiest of these cases, namely (c).Our choice of a i , a i+1 as consecutive atoms of [u, 1] r with respect to Γ implies that a i , a i+1 are consecutive atoms of This ensures for such [u, v ′ ] r we must either have both a i and a i+1 before the atom u ′ or both a i and a i+1 after u ′ with respect to Γ| [u,v ′ ]r .In either case, swapping the order of a i , a i+1 does not impact which atoms of [u, v ′ ] r come earlier than u ′ .Thus, this swap does not impact which atoms of If, on the other hand, we do not have both a i and a i+1 in [u, v ′ ] r , then swapping the order of a i and a i+1 in [u, 1] r leaves the ordering of the atoms of [u, v ′ ] r unchanged.In particular, the swap preserves which atoms of [u, v ′ ] r come earlier than u ′ and which come later than u ′ .Again this implies that the swap does not impact which atoms of ).Now we turn to the cases (a) and (b), the cases in which u ′ = a i and u ′ = a i+1 , respectively.We handle these two cases simultaneously because the latter parts of the arguments for these two cases are intrinsically intertwined with each other.If an interval [u, v ′ ] r does not contain both of the atoms a i and a i+1 , then swapping a i and a i+1 has no impact on the ordering of the atoms of [u, v ′ ] r , yielding the desired equalities in this case.For the remainder of cases (a) and (b), we therefore may assume the rooted interval [u, v ′ ] r under consideration contains both a i and a i+1 .Since swapping a i and a i+1 moves a i to after a i+1 while otherwise preserving the chain-atom ordering, every atom of [u, v ′ ] r that came earlier than a i in [u, v ′ ] r before the swap still comes earlier than a i after the swap.In other words, we deduce the set containment Then by definition of F Λ r∪a i+1 (a i+1 , v ′ ) the rooted interval [u, a ′ ] r has an earlier atom than a i+1 with respect to Γ| [u,a ′ ]r .Denote by a 1 the earliest atom of [u, a ′ ] r with respect to Γ.If a i ∈ [u, a ′ ] r , then both a i and a i+1 are atoms of [u, a ′ ] r ; since neither a i nor a i+1 is allowed to be the first atom of any rooted interval [u, w] r containing both of them, a 1 ∈ {a i , a i+1 } in this case.On the other hand, for a i ∈ [u, a ′ ] r we must have a 1 = a i since a 1 ∈ [u, a ′ ] r ; we may also deduce a 1 = a i+1 in this case from our assumption that a ′ ∈ F Γ r∪a i+1 (a i+1 , v ′ ).Thus, regardless of whether a i ∈ [u, a ′ ] r or not, swapping the order of a i and a i+1 in [u, 1] r does not change the fact that a 1 is the earliest atom of [u, a ′ ] r .This means that after the swap a ′ is still above the atom a 1 which still comes earlier than a i+1 , implying If we swap a i with a i+1 twice in succession, the second swap takes the chain-atom ordering Λ and produces from it the original chain-atom ordering Γ.The atom a i+1 comes earlier than a i in [u, 1] r with respect to Λ, but a i+1 and a i are still consecutive atoms of [u, 1] r with respect to Λ; also observe that a i+1 and a i still satisfy our requirement with respect to Λ that any rooted interval [u, v] r containing both a i+1 and a i must have neither a i+1 nor a i as its first atom.Thus, we may apply the argument from the prior paragraph to the chain-atom ordering Λ using the swap of a i+1 and a i in the rooted interval [u, 1] r which outputs the chain-atom ordering Γ.This yields the set containments Before proving the main result of the remainder of this section, Theorem 3.13, we prove a pair of lemmas that will provide two of the main pieces of the proof of Theorem 3.13.Lemma 3.11.Let Γ be a GRAO for a finite bounded poset P .Suppose a pair of consecutive atoms a i , a i+1 in [u, 1] r with chain-atom ordering Γ| [u, 1]r has the property for each rooted interval [u, w] r containing both a i and a i+1 that neither a i nor a i+1 is the earliest atom of [u, w] r with respect to Γ| [u,w]r .Then the chain-atom ordering Λ on P obtained from Γ by swapping the order of a i and a i+1 in [u, 1] r satisfies condition (i)(b) of a GRAO.
Proof.We break the proof into the following three cases based on the nature of the element u ∈ P that is covered by a i and a i+1 : (1) u = 0 (2) u is an atom of P (3) u is some other element of P .
To show that Λ satisfies condition (i)(b), it suffices to prove for each u ∈ P the following property for every atom a j ∈ P and every v ∈ P such that there exists an x ∈ P with a j ⋖ x ⋖ v: either the first atom of [a j , v] 0⋖a j with respect to Λ is above an earlier atom of P than a j or no atom of [a j , v] 0⋖a j is above an earlier atom than a j .We sometimes speak below of checking "condition (i)(b) for the rooted interval [a j , v] 0⋖a j ", by which we mean that we are checking the condition above just for that fixed choice of a j and v.In the remainder of the proof, we suppress the notation indicating our choice of root for intervals [a j , v] where a j is an atom so as to simplify notation, since it is clear from context for such intervals that the root must be 0 ⋖ a j .
We start with the easiest case, namely case (3) where we have u = 0 and u also not an atom of P .In this case, swapping the order of the atoms a i and a i+1 in the rooted interval [u, 1] r cannot impact the ordering of the atoms of P .The swap also cannot impact the ordering on the atoms of [a j , v] for any atom a j of P and any v ∈ P .Thus, swapping the order of a i , a i+1 in [u, 1] r while otherwise leaving a chain-atom ordering unchanged cannot cannot whether (i)(b) is satisfied.Since P satisfies condition (i)(b) with respect to Γ, P therefore also satisfies condition (i)(b) with respect to Λ.This completes case (3).
Next we handle case (2), i.e., the case where u is an atom of P .First we check condition (i)(b) for each interval [a j , v] with a j = u where a j is an atom of P and v satisfies a j ⋖ x ⋖ v for some x ∈ P ; then we will separately handle the intervals with a j = u.Since swapping the order of a i and a i+1 in [u, 1] 0⋖u has no impact on the order of the atoms of P or on the order of the atoms of [a j , v] for a j = u, it follows from the fact that [a j , v] satisfied condition (i)(b) before the swap that it also satisfies condition (i)(b) after the swap.Next we prove condition (i)(b) for those intervals [a j , v] with a j = u such that there exists x with u ⋖ x ⋖ v, splitting this in two parts based on whether or not a i and a i+1 are both elements of [u, v] 0⋖u .First consider any such interval [u, v] that contains at most one of a i and a i+1 .Notice that [ 0, v] contains at most one of a i , a i+1 by virtue of our having u ⋖ a i and u ⋖ a i+1 with a i , a i+1 not both in [u, v].But then Γ| [ 0,v] has the property that swapping the order of a i , a i+1 in [u, 1] does not change the chain-atom ordering Γ| [ 0,v] which therefore is still a GRAO on [ 0, v] after the swap.Therefore, condition (i)(b) of a GRAO holds for Λ| [ 0,v] , implying that [u, v] with at most one of a i , a i+1 in [u, v] satisfies condition (i)(b) with respect to Λ. Next consider the intervals [u, v] 0⋖u containing both a i and a i+1 .Neither a i nor a i+1 may be the first atom with respect to Γ of any interval to which both belong, implying neither is the first atom in [u, v] with respect to Γ. Therefore, swapping the order of a i and a i+1 has no impact on whether the first atom of [u, v] is greater than an earlier atom of P than u.This swap also has no impact on whether there exists an atom of [u, v] that is greater than an earlier atom of P than u.Thus, we use that condition (i)(b) holds for each such interval [u, v] with respect to Γ to deduce that it also holds for each such interval with respect to Λ.
Finally, we turn to case (1), the case in which u = 0.Here we subdivide the task of checking condition (i)(b) for the pertinent rooted intervals [a j , v] based on whether we have (a) First consider for u = 0 any rooted interval [a j , v] of type (a).Since neither a i nor a i+1 is allowed to be the first atom with respect to Γ in any rooted interval containing both a i and a i+1 , there exists an atom a of [ 0, v] that comes earlier than both a i and a i+1 with respect to Γ. Since Γ is a GRAO, the ordered pairs of atoms (a, a i ) and (a, a i+1 ) each satisfy condition (ii) of a GRAO when using the chain-atom ordering Γ, in the following sense: for each [ 0, y] having a, a i ∈ [ 0, y] (resp.a, a i+1 ∈ [ 0, y]) there exists z ∈ [ 0, y] and an atom a l ′ coming earlier than a i (resp.a i+1 ) in Γ with a l ′ < z and a i ⋖ z (resp.a i+1 ⋖ z).Therefore, there exists an atom z of [a i , v] (resp.[a i+1 , v]) that is above an earlier atom a l ′ of [ 0, v] than a i (resp.a i+1 ) with respect to Γ.This together with the fact that Γ is a GRAO allows us to deduce that the earliest atom of [a i , v] (resp.[a i+1 , v]) is above an earlier atom than a i (resp.a i+1 ) with respect to Γ.But then the earliest atom b ′ of [a i , v] is still above an earlier atom than a i after swapping the order of a i and a i+1 since every atom that came before a i with respect to Γ still comes before a i after the swap.This confirms condition (i)(b) for [a j , v] when a j = a i .Next consider the possibility that a j = a i+1 .We have already shown above that the earliest atom b ′ of [a i+1 , v] is above an earlier atom than a i+1 with respect to Γ.What remains is to rule out the possibility that a i is the only earlier atom than a i+1 with respect to Γ that is below b ′ .But this would imply a i < b ′ and a i+1 < b ′ , which by our hypotheses implies that neither a i nor a i+1 is the earliest atom of [ 0, b ′ ].This gives a contradiction to our assumption that a i was the only earlier atom than a i+1 below b ′ , completing our treatment of intervals of type (a).
Continuing the u = 0 case, next we handle the intervals of type (b).That is, we consider intervals [a j , v] with a j = a i but a i+1 ≤ v (resp.a j = a i+1 but a i ≤ v) where a j ⋖ x ⋖ v for some x ∈ P .In each of these two cases, swapping the order of the atoms a i , a i+1 of P leaves Γ| [ 0,v] unchanged.Thus, we may use the fact that condition (i)(b) holds for Γ to deduce condition (i)(b) for intervals of type (b) with respect to Λ.
We now complete the u = 0 case by handling the intervals of type (c).That is, consider the intervals [a j , v] where a j ∈ {a i , a i+1 } such that there exists x ∈ P with a j ⋖ x ⋖ v. Since a i and a i+1 are consecutive atoms in Γ, a j must either come before both a i and a i+1 in Γ or come after both a i and a i+1 in Γ.In either event, the following three facts may easily be checked: that Γ satisfies condition (i)(b) due to Γ being a GRAO, that swapping a i and a i+1 preserves Γ| [a j ,v] , and that the set of atoms coming earlier than a j with respect to Γ equals the set of atoms coming earlier than a j in the chain-atom ordering obtained from Γ by swapping the order of a i and a i+1 .These facts combine to imply that [a j , v] satisfies condition (i)(b) after the swap, namely with respect to Λ. Lemma 3.12.Let Γ be a GRAO for a finite bounded poset P .Suppose a pair of consecutive atoms a i , a i+1 in the induced GRAO for [u, 1] r has the property that each rooted interval [u, w] r containing both a i and a i+1 has neither a i nor a i+1 as its earliest atom with respect to Γ| [u,w]r .Then the chain-atom ordering Λ obtained from Γ by swapping the order of a i and a i+1 in [u, 1] r satisfies condition (ii) from the definition of GRAO.
Proof.If u = 0, swapping the order of the atoms a i and a i+1 of [u, 1] r does not impact the ordering of the atoms of P .Since condition (ii) held before the swap, condition (ii) therefore also holds after the swap.Thus, we may assume u = 0 for the remainder of the proof.
For each interval [ 0, y] in P , we need to show the following: for any pair of atoms a l , a m of [ 0, y] with a l coming before a m with respect to Λ, there exists an atom a k coming earlier than a m in Λ and an element z ∈ [ 0, y] such that a k < z and a m ⋖ z.Within this proof we call what must be checked for any fixed choice of y ∈ P "condition (ii-int) for the interval [ 0, y] within P ".We do not call this condition (ii) because we wish to emphasize the fact which will be important to our proof that this condition on [ 0, y] is strictly easier to check than regarding [ 0, y] as a poset with a chain-atom ordering and checking condition (ii) from the definition of GRAO on that poset.
First we show for each y ∈ P having a i , a i+1 ∈ [ 0, y] that condition (ii-int) holds for [ 0, y] with respect to Λ. Consider any pair a l , a m of atoms of [ 0, y] where a l comes earlier than a m with respect to Λ. Since a m is not the first atom of [ 0, y] with respect to Λ, we claim that it also cannot be the first atom of [ 0, y] with respect to Γ; we confirm this claim by noting that the swap of a i , a i+1 converting Λ back to Γ does not impact which atom is first in [ 0, y] since neither a i nor a i+1 is allowed to be the first atom of [ 0, y] with respect to Γ which implies they also cannot be first with respect to Λ. Since Γ is a GRAO, [ 0, y] satisfies condition (ii-int) with respect to Γ. Since a m is not the first atom of [ 0, y] with respect to Γ, there exists an atom in [ 0, y] coming earlier than a m with respect to Γ.But then condition (ii) of a GRAO implies there exists an atom a k ∈ [ 0, y] and an element z ∈ [ 0, y] such that a k comes earlier than a m with respect to Γ, a k < z and a m ⋖ z.We will show that we may use this same z and a k after the swap of a i and a i+1 to demonstrate the existence of both an atom a ′ k ∈ [ 0, y] that comes earlier than a m with respect to Λ and an element z ′ ∈ [ 0, y] that covers a m and is greater than a ′ k .What needs to be checked in order to justify letting a ′ k = a k and z ′ = z is that swapping a i and a i+1 cannot cause a k to come after a m with respect to Λ.This could only potentially happen if a k = a i and a m = a i+1 .But then we could use the fact that a i , a i+1 ∈ [ 0, z] (due to having a k , a m ∈ [ 0, z]) to conclude that neither a i nor a i+1 could be the first atom of [ 0, z].This would imply the existence of some atom a ∈ [ 0, z] coming earlier than both a i and a i+1 .But then we could use this element a to serve as our desired atom a k coming earlier than a m , and we could use the same z as before the swap.This confirms condition (ii-int) for all intervals [ 0, y] having a i , a i+1 ∈ [ 0, y].
Finally, we verify that condition (ii-int) holds with respect to Λ for all intervals [ 0, y] such that a i , a i+1 are not both elements of [ 0, y].Again, consider any pair of atoms a l , a m ∈ [ 0, y] such that a l comes earlier than a m with respect to Γ| [ 0,y] .Since Γ is a GRAO, condition (ii) of a GRAO ensures that there exists z ∈ [ 0, y] and an atom a k of [ 0, y] which comes earlier than a m with respect to Λ such that we also have a k < z and a m ⋖ z.Since at most one of the elements a i , a i+1 is an atom of [ 0, y] in this case, the swap of a i , a i+1 cannot change the relative order of the atoms of [ 0, y].Thus, we may use this same a k and z after the swap of a i and a i+1 to demonstrate the existence of an element z ∈ [ 0, y] and an atom a k coming earlier than a m such that a k < z and a m ⋖ z.This confirms condition (ii-int) with respect to Λ for all intervals [ 0, y] not having both a i and a i+1 as elements of [ 0, y].
While the statement of the next result may seem somewhat technical, it captures in a precise and seemingly useful way how certain types of localized moves are guaranteed to transform a GRAO into a new GRAO.Thus, this theorem pins down a certain type of flexibility one has in choosing a chain-atom ordering that will be a GRAO.We will use this result in our proof later in the paper that any GRAO may be transformed into an RAO.Theorem 3.13.Let Γ be a GRAO for a finite bounded poset P .Suppose a pair of consecutive atoms a i , a i+1 in the GRAO for [u, 1] r induced by Γ has the property that each rooted interval [u, w] r containing both a i and a i+1 has neither a i nor a i+1 as its earliest atom with respect to Γ. Then the chain-atom ordering Λ obtained from Γ by swapping the order of a i and a i+1 in [u, 1] r is itself a GRAO for P .
Proof.Our proof is by induction on the length of the longest saturated chain in P .It will suffice to show that each of the requirements of a GRAO is preserved under swapping the order of two consecutive atoms a i and a i+1 in a rooted interval [u, 1] r , provided that neither a i nor a i+1 is the earliest atom of any rooted interval [u, w] r that contains both a i and a i+1 .
To show that condition (i)(a) in the definition of GRAO still holds after the swap, it suffices to show for each of the atoms a in P that Λ| [a, 1] is a GRAO.First consider the case with u ∈ [a, 1].One may easily observe in this case that Λ| [a, 1] is the same chain-atom ordering one obtains by restricting Γ to [a, 1] and then performing the swap on this restricted GRAO.Since Γ| [a, 1] is a GRAO, our inductive hypothesis allows us to deduce that Λ| [a, 1] is also a GRAO.Next consider the case with u ∈ [a, 1].Then the swap of a i , a i+1 leaves Γ| [a, 1] unchanged; since this chain-atom ordering on [a, 1] is a GRAO before the swap, it remains a GRAO after the swap.This proves condition (i)(a) for Λ.
Since Γ is a GRAO, we may apply Lemma 3.11 to deduce that Λ also satisfies condition (i)(b) from the definition for GRAO.Likewise we may use that Γ is a GRAO and apply Lemma 3.12 to deduce that condition (ii) from the definition of GRAO holds for Λ.

Equivalence of admitting a generalized recursive atom ordering to admitting a recursive atom ordering
In this section we prove that a finite bounded poset admits a generalized recursive atom ordering (see Definition 3.2) if and only if it admits a recursive atom ordering.We accomplish the more challenging half of this result constructively by a procedure that transforms any generalized recursive atom ordering into a recursive atom ordering, a process we call "atom reordering."Along the way, we will develop several properties of such reorderings.
First we carry out the other much easier direction of the result.
Lemma 4.1.Every recursive atom ordering is a generalized recursive atom ordering.
Proof.We first verify that conditions (i)(a) and (i)(b) of Definition 3.2 hold for any given RAO, doing so by way of a proof by induction on the length l of the longest saturated chain of P .Any ordering of the atoms of a finite bounded poset whose longest saturated chain is of length 1 or 2 is a generalized recursive atom ordering, giving the base case.Let a 1 , a 2 , . . ., a t be an RAO for a finite bounded poset P having l > 2. By induction, we may assume for each atom a j ∈ P that the RAO on [a j , 1] guaranteed to exist by condition (i)(a) of the definition of RAO is a GRAO, giving (i)(a) from the definition of GRAO.Now we turn to condition (i)(b) of Definition 3.2.By the definition of RAO, each atom in F 0⋖a j (a j ) (see Definition 2.16) must come earlier in our given RAO for [a j , 1] than every atom in G 0⋖a j (a j ).This implies that if any atom of [a j , 1] is greater than a i for some i < j, then the first atom of [a j , 1] is greater than a i ′ for some i ′ < j.Once we check that any RAO for [a j , 1] restricts to an RAO for [a j , w] for each w > a j in P , we will likewise be able to deduce the following implication: if any atom of [a j , w] is greater than a i for some i < j, then the first atom of [a j , w] is greater than a i ′ for some i ′ < j.Now we verify the desired claim about restriction of an RAO.Given an RAO, we first apply the construction of Bjöner and Wachs from [BW83] that produces a CL-labeling from any RAO.Then we note that this CLlabeling restricts to a CL-labeling for the interval [a j , w].Finally we apply the construction of Björner and Wachs from [BW83] which produces an RAO from any CL-labeling to get an RAO for [a j , w].It is easy to see that the chain-atom ordering obtained this way is exactly the restriction of our given RAO to [a j , w].Having verified this claim, we have completed the confirmation of condition (i)(b) of Definition 3.2.
Condition (ii) in the definition of GRAO is the same as condition (ii) in the definition of RAO, hence is guaranteed to hold for any RAO.
Remark 4.2.It is not true that every generalized recursive atom ordering is a recursive atom ordering.See Figure 3 for an example illustrating this.
Next, in Algorithm 4.3, we describe the atom reordering process that will allow us to transform any generalized recursive atom ordering into a recursive atom ordering.This algorithm takes as its input any chain-atom ordering Λ, and it outputs a chain-atom ordering that we denote by Λ re .One may think of this superscript re as shorthand for "reordered." The atom reordering process is designed to output a chain-atom ordering that will satisfy condition (i)(b) from the definition of recursive atom ordering.Moreover, it is set up to do so in such a way that when applied to a GRAO, it preserves useful structure that is present in a GRAO, including preserving condition (ii) from the definition of GRAO.
Broadly, the algorithm starts at the bottom of the poset P and works its way to the top, reordering the atoms of each rooted interval in a way that takes into account the reordering that has already occurred lower in the poset.Each of these reordering steps moves those atoms of a rooted interval [u, 1] r that are above an earlier atom than u in the rooted interval [u − , 1] r − ahead of those that are not, otherwise preserving the ordering on atoms.The algorithm progressively builds up a chain-atom ordering Λ pr by defining Λ pr ([u, 1] r ) for more and more choices of u ∈ P and of root r for [u, 1].The superscript pr in Λ pr is shorthand for "partially reordered".Once the algorithm has made Λ pr into an entire chain-atom ordering, it outputs this chain-atom ordering and calls it Λ re .Readers may find it helpful to refer to Definition 2.16 for an explanation of the notations F Λ pr r (u, v) and G Λ pr r (u, v) as they read Algorithm 4.3.
It is sometimes necessary (e.g. in the proof of Lemma 4.9) to take a different viewpoint on this algorithm, keeping track of more data at the intermediate stages in the algorithm in a way that does not impact the output of the algorithm or the essentials of how the algorithm proceeds.In this enriched version of the algorithm, Λ pr will denote an entire chain-atom ordering at each step of the atom reordering process.This is accomplished by initializing Λ pr to equal Λ and then otherwise leaving the algorithm unchanged.The effect is that progressively more and more of the values Λ pr ([u, 1] r ) are re-set from what they equal in Λ to what they will equal in Λ re .This allows one to think of Λ as evolving into Λ re over the course of the algorithm.Our upcoming proofs of various properties of the atom reordering process will all hold regardless of which of these two viewpoints one takes, namely regarding Λ pr as growing or as evolving, as one may easily check by noting how these two versions of the algorithm really only differ in terms of notation, not in substance.
The reason we take the former viewpoint within Algorithm 4.3 itself is so that the sets F Λ pr r (u, v) and G Λ pr r (u, v) will be defined in an unambiguous way when they are used later (e.g.within the proof of Lemma 4.7).When we speak of the atom reordering process transforming Λ into Λ re , both in the introduction of the paper and in the proof of Lemma 4.9, we are taking the latter viewpoint.Algorithm 4.3: The atom reordering process applied to Λ which outputs Λ re .
input: A finite bounded poset P equipped with a chain-atom ordering Λ output: A chain-atom ordering of P , denoted Λ re , also called the atom reordering of P .begin (1) Choose any linear extension u 0 , u 1 , u 2 , . . ., u n of P ; (2) Set Λ pr ([ 0, 1]) := Λ([ 0, 1]), thereby determining Λ pr ([u 0 , 1]); (3) for i = 1 to n do (a) Choose an ordering of the roots r 1 , r 2 . . .r t i for [u i , 1]; (b) for j = 1 to t i do i.Calculate F Λ pr r j (u i ) and G Λ pr r j (u i ); ii.Order the elements of F Λ pr r j (u i ) in the same relative order as in Λ([u i , 1] r j ); iii.Order the elements of G Λ pr r j (u i ) in the same relative order as in Λ([u i , 1] r j ); iv.Determine Λ pr ([u i , 1] r j ) as follows: put all elements of F Λ pr r j (u i ) in the order from step ii, followed by all elements of G Λ pr r j (u i ) in the order from step iii; v. Increase j by 1; end (c) Increase i by 1; end (4) Set Λ re to be the chain-atom ordering that has Λ re ([u i , 1] r j ) = Λ pr ([u i , 1] r j ) for every u i ∈ P and every choice of root r j for [u i , 1]; end Example 4.4.Figure 3 shows a poset P with a GRAO on the left and a RAO on the right.The RAO on the right is obtained by applying the atom reordering process described in Algorithm 4.3 to the GRAO on the left.In this case, the atom reordering process changes the order of the atoms above the element a, in particular swapping the 2nd and 3rd atoms above a (highlighted in red).Note that this example was chosen to have the further property in both the ordering on the left and the reordering on the right that the atom ordering for any interval [u, 1] r is independent of choice of root r.This makes the figures more understandable, but is a very special case.
We next deduce several fundamental properties of atom reorderings.Proof.Let r = 0 ⋖ t 1 ⋖ t 2 . . .⋖ u be a root for the interval [u, v] in P .Let r − be the root obtained by eliminating u from r, and let u − be the highest element of r − .Recall from Definition 2.16 that F Λ re r (u, v) refers to the set of atoms of [u, v] that cover an earlier atom of Our main task will be to prove that , and we call this Claim (I).Notice that Claim (I) would imply that the union of sets But this last set containment would actually be an equality of sets by virtue of both of the sets in the containment equalling the set of all atoms in [u, v].This set equality would imply that each of the component set containments would also be a set equality, which would complete the proof.
What remains is to prove Claim (I).We do so by induction on the length of the longest saturated chain from 0 to u.The point is that a ∈ F Λ re r (u, 1) (resp.a ∈ G Λ re r (u, 1)) implies that there exists (resp.does not exist) an atom a ′ of [u − , 1] r − with a ′ < a in P such that a ′ is earlier than u in Λ re | [u − , 1] r − .But any such a ′ is also an atom of [u − , v] r − since u − ⋖ a ′ < a ≤ v in P .Moreover, our inductive hypothesis ensures that a ′ is an earlier atom than u in [u − , v] r − regardless of whether we are reordering within P or within [ 0, v].This is exactly what is needed to show that a ∈ F Λ re r (u, v).On the other hand, when such a ′ does not exist within P , this implies that no such a ′ exists within [ 0, v] since [ 0, v] is a subset of P , yielding the desired claim that G Λ re r (u, 1) ∩ [u, v] r ⊆ G Λ re r (u, v).In light of Proposition 4.5, we may henceforth speak interchangeably of the restriction to [u, v] r of the atom reordering Λ re of a chain-atom ordering Λ of a finite bounded poset P and of the restriction to [u, v] r of the atom reordering of Λ| [ 0,v] for any given u < v in P and any root r for [u, v].
Next we give a variation on condition (i)(b) from the definition of GRAO that will be useful in upcoming inductive arguments.
Proposition 4.6.Let P be a finite bounded poset with a GRAO.Consider t, u, v ∈ P such that t ⋖ u < v and consider a choice of root r for [t, v].Then either the first atom of [u, v] r∪u in the GRAO is above an earlier atom of [t, v] r than u or else u is the first atom of [t, v] r .
Proof.Lemma 3.9 immediately implies that either the first atom of [u, v] r∪u is above an earlier atom of [t, v] r than u or no atom of [u, v] r∪u is above an earlier atom of [t, v] r than u.It suffices to consider the case where no atom of [u, v] r∪u is above an earlier atom than u in the GRAO of [t, v] r , and to show in this case that u is the first atom in the GRAO of [t, v] r .Suppose by way of contradiction that u ′ is the first atom in the GRAO of [t, v] r for some u ′ = u.Since both u ′ and u are below v, Definition 3.2, part (ii), implies there must exist some atom u ′′ of [t, v] r that comes before u in the GRAO and some element x ′ such that u⋖x ′ and u ′′ < x ′ < v.The existence of such x ′ contradicts the fact that no atom of [u, v] r∪u is above an atom earlier than u in the GRAO of [t, v] r .This completes the proof.
Next is a result that plays an important role in explaining (and in justifying) the fact that our atom reordering process transforms any GRAO into an RAO.
Lemma 4.7.Let P be a finite, bounded poset with Λ a GRAO for P .Then for any u < v in P and any root r for [u, v], the first atom of [u, v] r in Λ is the first atom of [u, v] r in Λ re , namely in the atom reordering of Λ.
Proof.Proposition 3.8 ensures that Λ restricts to a GRAO for [u, v] r .Proposition 4.5 allows us to speak interchangeably about the atom reordering of [ 0, v] restricted to [u, v] r and the atom reordering of P restricted to [u, v] r .Let r = 0 ⋖ t 1 ⋖ t 2 . . .⋖ t n ⋖ u be a root for the interval [u, v] in P and let r − denote the root for [t n , v] obtained from r by removing u.Recall from Definition 2.16 that F Λ r (u, v) refers to the set of atoms of [u, v] r that cover an earlier atom of ).We will prove that the first atom of Λ([u, v] r ) is the first atom of Λ re ([u, v] r ) for all u ∈ P and all choices of root r.We will do so by induction on the length l of the root r from 0 to u.For the base case, let u be an atom of [ 0, v].Let x be the first atom in Λ ([u, v] ) since u is an atom of P and the atom reordering process does not change the order of the elements covering 0. Since we put the elements of F Λ pr r (u, v) before all other atoms of [u, v] r during the atom reordering of Λ| [ 0,v] , x remains first among the atoms of [u, v] u, v).By Proposition 4.6, u is the first atom in Λ ([ 0, v]).Since the atom reordering process does not change the order of the elements covering 0, u is also the first atom in Λ re ([ 0, v]).The fact that u is first in Λ re ([ 0, v]) implies that every atom in [u, v] r must be in G Λ pr r (u, v).Since the atom reordering process preserves the relative ordering of the elements of G Λ pr r (u, v) given by Λ([u, v] r ), the atom reordering process preserves the ordering of all atoms of [u, v] r .Thus, x remains first in Λ re ([u, v] r ).This completes the base case.
For the inductive step, assume that the first atom in Λ([u, v] r ) is also the first atom in Λ re ([u, v] r ) for any u < v and any root r of length l, where l ≤ n.We will prove that the first atom in Λ ([u, v] r ) is also the first atom in Λ re ([u, v] r ) for any u < v and any root r of length n + 1.Let x be the first atom in Λ ([u, v] r ) where r is a root of length n + 1. Suppose by way of contradiction that x is not first in Λ re ([u, v] r ).This implies x is moved to a later position by the atom reordering process applied to Λ, hence it implies x ∈ G Λ pr r (u, v).Since x is the first atom in Λ([u, v] r ), Lemma 3.9 gives us that either (1) x is greater than an atom that comes before u in Λ([t n , v] r − ) or (2) no atom of [u, v] r is greater than an atom that comes before u in Λ([t n , v] r − ).We consider these two cases separately.
Case 1: This is the case in which x is above an earlier atom than u in Λ([t n , v] r − ).See Figure 4 for an illustration of this case.Let u 1 be the first atom in Λ([t n , x] r − ).By our assumption 0 Case 1 in the proof of Lemma 4.7.about x, note that u 1 = u; this implies that u 1 comes earlier than u in Λ([t n , v] r − ).Since we already showed x ∈ G Λ pr r (u, v), u 1 must come later than u in Λ re ([t n , v] r − ).Because of Proposition 4.5, u 1 must come later than u in Λ re ([t n , x] r − ) as well.In particular, u 1 is not first in Λ re ([t n , x] r − ).But the root r − has length n, which contradicts our inductive hypothesis that requires u 1 to remain the earliest atom in Λ re ([t n , x] r − ).Thus, we rule out this case.
Case 2: This is the case in which no atom of [u, v] r is above an atom that is earlier than u in Λ([t n , v] r − ).See Figure 5 for an illustration of this case.By Proposition 4.6, u must then be first in Λ([t n , v] r − ).Since r − has length n, our inductive hypothesis ensures that u must also be first in Λ re ([t n , v] r − ).This means that all atoms of [u, v] r are in G Λ pr r (u, v).This implies that the ordering of all atoms of [u, v] r is preserved by the atom reordering process.In particular, x remains first among the atoms of [u, v] r , contradicting our assumption that x is not the first atom of [u, v] Next is a lemma that sheds light on what the reordering process does to each rooted interval [u, 1] r in a finite bounded poset P endowed with a chain-atom ordering.This will be helpful for proving that applying the atom reordering process to a GRAO yields a chain-atom ordering that is still a GRAO.
Lemma 4.8.Let Σ be a chain-atom ordering on a finite bounded poset P .Given any u ∈ P and any root r for [u, 1], consider the chain-atom ordering Σ re | [u, 1]r on [u, 1] r that is obtained by applying the atom reordering process to Σ and then restricting the resulting chain-atom ordering for P to [u, 1] r .transpositions in which each of these adjacent transpositions fixes which atom of [u, v] r is first for every v > u in P .c) If Σ is a GRAO for P , then the chain-atom ordering for P obtained from Σ by applying the permutation π from (a) to permute the order of the atoms of [u, 1] r while otherwise leaving Σ unchanged is a GRAO for P .
Proof.First we prove (a).Applying the reordering process to all of P , the point will be to observe what happens to [u, 1] r .The earliest step in which the chain-atom ordering for [u, 1] r gets modified by the reordering process is the step where the atoms of [u, 1] r get permuted by moving the elements of F Σ r (u, 1) ahead of the elements of G Σ r (u, 1), preserving the order of the elements within F Σ r (u, 1) and within G Σ r (u, 1).This gives the desired permutation π on the ordering of the atoms of [u, 1] r .Now one may observe that the subsequent reordering of the atoms of each rooted interval [u ′ , 1] r ′ for u ′ ∈ (u, 1] and each root r ′ containing r depends only on π and on the reordering of [u, 1] r that takes place prior to reaching u ′ and r ′ .More precisely, one may confirm by induction on the length of the longest saturated chain from u to u ′ that the reordering of the atoms of [u ′ , 1] r ′ accomplished within the course of reordering P gives the same atom ordering for [u ′ , 1] r ′ as we get by first restricting Σ to [u, 1] r , then applying the permutation π to the atoms of [u, 1] r , and finally applying the reordering process just to the resulting chain-atom ordering on [u, 1] r (i.e.not working our way up from 0).This completes the proof of (a) Next we prove (b).Lemma 4.7 implies that the permutation π from (a) has the property for each v > u and each root r that the earliest atom of [u, v] r before applying π is still the earliest atom of [u, v] r after applying π; equivalently, the permutation that π induces on the atoms of [u, v] r fixes the first atom of [u, v] r .Let s 1 • • • s k be any reduced expression for π, i.e. any expression for π as a product of adjacent transpositions (i, i + 1) with k as small as possible.The natural correspondence between reduced expressions for any fixed π ∈ S n and saturated chains from e to π in left weak order for S n (which is discussed extensively e.g. in [BB]) may be combined with Corollary 3.1.4in [BB] to give us the well-known fact that for any 1 ≤ l < m ≤ n satisfying π(l) < π(m), we also must have We will apply this property of reduced expressions to certain pairs of atoms a, a ′ ∈ [u, v] r such that a comes before a ′ both in the ordering A on the atoms of [u, 1] r given by Σ and in the ordering π(A) on the atoms of [u, 1] r obtained by applying the permutation π to A. For any two atoms a, a ′ of an interval [u, v] r such that a is the first atom of [u, v] r both in the ordering for the atoms of [u, v] r inherited from A and in the ordering for the atoms of [u, v] r inherited from π(A), a comes before a ′ both in A and in π(A).Thus, our above property of reduced expressions implies that a must come before a ′ in s j • • • s k (A) for each j ≤ k.This implies that a must be the first atom in the ordering on the atoms of [u, v] r that is inherited from the ordering s j • • • s k (A) on the atoms of [u, 1] r for each j ≤ k.Thus, each adjacent transposition in s 1 . . .s k applied to A from right to left preserves which atom is first in every rooted interval of P .
Finally we prove (c).Part (b) tells us that each of the adjacent transpositions in s 1 • • • s k has the property when it is applied in turn to A that it preserves which atom is first in each [u, v] r .This enables us to use Theorem 3.13 repeatedly, once for each of the adjacent transpositions in s 1 • • • s k , to deduce that the chain-atom ordering obtained from Σ by reordering the atoms of [u, 1] r by π is a GRAO for P .
We are now prepared to prove that applying the atom reordering process to a GRAO yields a GRAO and then to go on from there to prove that the resulting GRAO is in fact an RAO.
Lemma 4.9.The atom reordering process applied to a GRAO for a finite bounded poset P yields a GRAO for P .
Proof.Throughout this proof, we will work with the more enriched variation on the atom reordering process discussed just prior to Algorithm 4.3.That is, we regard Λ pr at each step in the process as being a full chain-atom ordering, doing so as follows.We initialize Λ pr to equal Λ, and then otherwise run the algorithm just as in Algorithm 4.3.In other words, at each of the steps of the algorithm in which Λ pr ([u, 1] r ) has not yet been defined for a given u ∈ P and a given choice of root r in the usual algorithm, we set Λ pr ([u, 1] r ) equal to Λ([u, 1] r ) in our modified atom reordering process, otherwise leaving the process entirely unchanged.Since this is merely an enrichment of the data contained in Λ pr at the intermediate stages of the algorithm, one may easily see that this does not impact the output of the algorithm or the applicability of the proofs of Lemmas 4.5, 4.7 and 4.8 to this enriched version of the atom reordering process.With these conventions, Λ pr progressively evolves from being the chain-atom ordering Λ at the start of the algorithm to equalling the chain-atom ordering Λ re at the end of the algorithm, as will be necessary for the argument below to apply.
By Lemma 4.7, the atom reordering process for Λ takes [u, 1] r for each u ∈ P and each root r for [u, 1] and permutes the atoms a 1 , . . ., a m of [u, 1] r in a way that preserves which atom comes first in every rooted interval [u, v] r for v > u with our fixed root r.Thus, we may apply Lemma 4.8, part (b), to deduce that this permutation on the atoms of [u, 1] r is expressible as a reduced expression comprised of a series of adjacent transpositions, each of which preserves which atom comes first in [u, v] r for each v > u.But this means that we may express this enriched version of the atom reordering process as a series of such steps by considering each u and each r in turn, applying such a series of adjacent transpositions as in Lemma 4.8, part (b), for each u and each r.If we start with a GRAO, then Lemma 3.13 tells us that after each of these steps, i.e. after each of these adjacent transpositions, we still have a GRAO.In particular, the end-result of this enriched version of the atom reordering process applied to a GRAO for P is a GRAO for P .Since by design this enriched version of the atom reordering process has the same output Λ re as the atom reordering process given in Algorithm 4.3, this completes the proof.
Before getting to our main result of this section, we mention a consequence of what we have just proven that could give useful insight into the atom reordering process.
Corollary 4.10.The reordering process applied to a GRAO of a finite bounded poset P preserves the sets F r (u, v) and G r (u, v) for each u < v in P and each root r for [u, v].
Proof.This follows from Lemma 3.10 and the proof of Lemma 4.9.Now we are ready to prove the main result of this section.
Theorem 4.11.A finite bounded poset admits a generalized recursive atom ordering (GRAO) if and only if it admits a recursive atom ordering (RAO).Moreover, every RAO is a GRAO while every GRAO may be transformed into an RAO by the atom reordering process.
Proof.By Lemma 4.1, any RAO is a GRAO.What remains is to prove that applying the reodering process from Algorithm 4.3 to any GRAO for a finite bounded poset P yields an RAO.We will do this by induction on the length l of the longest saturated chain in P .For l = 2, any ordering of the atoms of P is a recursive atom ordering, giving the base case for our proof by induction.Now assuming the result for l ≤ n, it will suffice to show that this implies the result for l = n + 1.Let Λ be a GRAO for P , regarded as a chain-atom ordering.Let a 1 , a 2 , . . ., a q be the ordering of the atoms of P with l = n+1 in Λ.In this case, for each atom a j ∈ P , we have that [a j , 1] is an interval whose longest saturated chain is of length at most n.By condition (i)(a) in the definition of GRAO, Λ| [a j , 1] is a GRAO for [a j , 1].By Lemma 4.8, parts (a) and (b), applying the atom reordering process to Λ and then restricting the resulting chainatom ordering Λ re to [a j , 1] yields the same chain-atom ordering for [a j , 1] that we get by instead permuting the atoms of [a j , 1] in a way that yields a new GRAO for [a j , 1] that is denoted by π(Λ| [a j , 1] ) and then applying the atom reordering process directly to π(Λ| [a j , 1] ).By our inductive hypothesis, the result of this latter series of operations is an RAO for [a j , 1].This implies that condition (i)(a) of Definition 2.14 holds for Λ re .By definition of the atom reordering process, those atoms of [a j , 1] that are above an earlier atom of P than a j all come before all of the other atoms of [a j , 1] in Λ re , so condition (i)(b) of Definition 2.14 is also satisfied for Λ re .
What remains is to prove that condition (ii) of Definition 2.14 holds for Λ re .Here we may use Lemma 4.9 which tells us that applying the atom reordering process to Λ yields a GRAO for P .Therefore Λ re satisfies condition (ii) in the definition of GRAO.But condition (ii) in the definition of GRAO is the same as condition (ii) in the definition of RAO, albeit phrased slightly differently, completing our proof.

Self-consistency, the UE property and TCL-shellability
In this section, we introduce a condition that a CC-labeling may have called self-consistency.We also introduce a fairly natural and readily checkable property called the UE property that will imply self-consistency.We prove that all CL-labelings have the UE property.
These notions are introduced in preparation for a result later in the paper where we will prove that a finite bounded poset is CL-shellable if and only if it is CC-shellable by way of a self-consistent CC-labeling.To help us further clarify the relationship between the established notions of CC-shellability and CL-shellability, we also introduce a variation on CC-shellability that we call TCL-shellability.
Definition 5.1.Consider a chain-edge labeling λ such that each rooted interval has a unique lexicographically earliest saturated chain.We define such λ to be self-consistent if for any rooted interval [u, v] r we have the following condition: if a is the atom in the lexicographically first saturated chain of [u, v] r and b = a is also an atom of [u, v] r , then for any [u, v ′ ] r containing a and b, all saturated chains of [u, v ′ ] r containing b come lexicographically later than all saturated chains of [u, v ′ ] r containing a.If a chain-edge labeling is not self-consistent, then it is said to be self-inconsistent.
Next we introduce a property that will imply self-consistency that is more readily checkable.
Definition 5.2.A chain-edge labeling λ of a finite bounded poset P has the unique earliest (UE) property if for each rooted interval [u, v] r in P , the smallest label occurring on any cover relation upward from u only occurs on one such cover relation.
Lemma 5.3.If a CC-labeling has the UE property, then it is self-consistent.
Proof.Let a be the unique atom of [u, v] r for which λ(u, a) is smallest among all labels upward from u.Consider any other atom The label sequences on saturated chains of [u, v ′ ] r containing a must be lexicographically smaller than those containing b by virtue of λ(u, a) being smaller than λ(u, b) with respect to root r.
Corollary 5.4.Any recursive atom ordering for a finite bounded poset gives rise to a CClabeling with the UE property, hence to a self-consistent CC-labeling.
Proof.Simply observe that the CC-labeling constructed in the proof of Proposition 2.20 has the UE property, then apply Lemma 5.3.
The next result gives some evidence that the UE property is not an unduly burdensome condition to impose.Proof.Consider a pair of elements u < v in a finite bounded poset P with a CL-labeling λ, and consider any root r of [u, v] from 0 to u. Suppose that there are distinct atoms a, a ′ ∈ [u, v] such that λ r (u, a) = λ r (u, a ′ ).Further suppose λ r (u, a) ≤ λ r (u, a ′′ ) for all other atoms a ′′ ∈ [u, v] r .We may choose a to belong to the lexicographically earliest saturated chain M from u to v in [u, v] r since this saturated chain will begin with the smallest possible first label.
⋖v be the lexicographically first saturated chain of [u, v] r that contains a ′ .Denote by M ′ this latter saturated chain from u to v. We must have λ r (u, a) < λ r∪a (a, x 2 ) since M is an ascending chain.But we also have λ r∪a (a, x 2 ) ≤ λ r∪a ′ (a ′ , y 2 ) since the label sequences for M and M ′ both start with the same label and M has a lexicographically smaller label sequence than M ′ .Thus, we have implying that M ′ has an ascent at a ′ .But M ′ is also ascending from a ′ to v since M ′ is lexicographically first in [a ′ , v] r∪a ′ .Thus, M and M ′ are both ascending chains on [u, v] r , giving a contradiction.
Example 5.6.Figure 6 shows a poset P and three different CC-labelings (all of which are in fact EC-labelings).The leftmost CC-labeling is self-inconsistent because both [ 0, x] and [ 0, y] contain a and b as atoms, but a is first in [ 0, x] while b is first in [ 0, y].Given that this labeling is self-inconsistent, it also cannot have the UE property.The CC-labeling in the middle of Figure 6 is self-consistent but does not have the UE property because both cover relations up from 0 have label 1.The CC-labeling on the right has the UE property and is self-consistent.
It seems plausible that many of the most interesting and the most natural examples of CClabelings will have the UE property.Section 7 shows that an EC-labeling from [HK] for the dual poset to the uncrossing order has the UE property, allowing us to invoke the upcoming Corollaries 6.5 and 6.6 to deduce dual CL-shellability for the uncrossing order.Section 8.1 shows how readily checkable the UE property is for numerous well-known EL-labelings and CL-labelings.5.1.TCL-shellability: a variation on CC-shellability.One may replace Kozlov's requirements of a CC-labeling (a) that saturated chains have distinct label sequences and (b) that no label sequence be a prefix of any other label sequence by simply requiring each rooted interval to have a unique lexicographically earliest saturated chain.With just this requirement, the proof of Björner and Wachs that a CL-labeling yields a shelling of the order complex will carry over using topological descents in place of descents and using topological ascents in place of ascents.In particular, the UE property suffices in place of these two aforementioned conditions of Kozlov to guarantee we get a shelling.
This yields a class of poset chain-edge labelings inducing shellings that includes all CLlabelings, whether or not the labels upward from the lowest element u in any rooted interval [u, v] r are all distinct from each other.In justifying his assertion that CC-shellability is the most general possible version of lexicographic shellability, Kozlov notes in [Ko] that any CL-labeling may be modified into a CC-labeling, implying that every CL-shellable poset is CC-shellable.
Definition 5.7.A TCL-labeling of a finite bounded poset P is a chain-edge labeling λ for P such that (i) each rooted interval [u, v] r has a unique saturated chain from u to v with lexicographically smallest label sequence, and (ii) every other saturated chain from u to v in [u, v] r has at least one topological descent.If a poset has a TCL-labeling, then it is TCL-shellable.
Theorem 5.8.Every CC-labeling is a TCL-labeling.However, not every TCL-labeling is a CC-labeling.
Proof.First notice that the requirement of a CC-labeling that the saturated chains in any rooted interval have distinct label sequences in particular implies the uniqueness of the lexicographically earliest label sequence on each rooted interval.Since the saturated chain from u to v with the lexicographically earliest label sequence on a rooted interval [u, v] r cannot have any topological descents, condition (1) in the definition of CC-labeling implies that every other saturated chain of [u, v] r must have a topological descent.This shows that every CC-labeling is a TCL-labeling.
Remark 5.9.It may very well be true that a finite bounded poset is CC-shellable if and only if it admits a TCL-labeling.The labeling given in the second half of the proof of Theorem 5.8 can easily be modified into a CC-labeling, and indeed it remains open whether these two notions of shellability are equivalent in terms of which posets admit such shellings.Likewise, as far as we know, it is open whether CC-shellability is equivalent to CL-shellability, though this seems to be quite a tricky question.
Theorem 5.10.Any recursive atom ordering of a finite bounded poset P induces a TCLlabeling for P that is self-consistent.
Proof.One may simply check that the CC-labeling constructed in the proof of Theorem 2.20 is in fact self-consistent (by virtue of having the UE property) and is a TCL-labeling.One may also invoke Theorem 5.8 for the latter claim.
Theorem 5.11.If a finite bounded poset P admits a TCL-labeling, then this induces one or more lexicographic shellings for ∆(P ).Specifically, any linear order on the maximal chains of P that is a linear extension of the partial order obtained by ordering maximal chains lexicographically is a shelling order.
Proof.The proof of Björner and Wachs in [BW82] that any CL-labeling of a finite bounded poset P induces a lexicographic shelling for ∆(P ) (for each choice of linear extension of the lexicographic order on saturated chains) still applies unchanged for TCL-labelings when we simply replace ascents by topological ascents, descents by topological descents, ascending chains by topologically ascending chains, and descending chains by topologically descending chains throughout the proof.

Equivalence of CL-shellability to self-consistent CC-shellability
In this section, we will prove that several different conditions on a finite bounded poset are equivalent to CL-shellability.Theorem 6.1.If a finite bounded poset P admits a generalized recursive atom ordering, then P admits a self-consistent CC-labeling.
Proof.We proved in Theorem 4.11 that any finite bounded poset P with a generalized recursive atom ordering also admits a recursive atom ordering.We showed in Corollary 5.4 that any such P admits a self-consistent CC-labeling, completing the proof.Remark 6.2.One may alternatively prove Theorem 6.1 by using any generalized recursive atom ordering for a finite bounded poset P to construct a self-consistent CC-labeling for P that is compatible (in the sense of Definition 8.5) with the GRAO without needing to use the atom reordering process at all.The construction of such a CC-labeling is carried out as follows.For each u ∈ P and each root r for [u, 1], the GRAO gives an ordering a 1 , . . ., a t(u,r) of the atoms of the rooted interval [u, 1] r .Assign the label i to the cover relation u ⋖ a i for this choice of root r.One may prove that this produces a UE (and thus self-consistent) CClabeling for P .It is worth noting that this is typically a different self-consistent CC-labeling for P than the one produced as in the proof above of Theorem 6.1, as we do not apply the atom reordering process to P in this more direct approach.See the proof of Theorem 8.6 for all of the details of this approach.Theorem 6.3.If a finite bounded poset P admits a self-consistent TCL-labeling, then P admits a generalized recursive atom ordering.
Proof.Let λ be a self-consistent TCL-labeling for P .By definition, λ restricts to a TCLlabeling for [u, 1] r for each u ∈ P and each root r.Consider the following chain-atom ordering of P that is compatible with λ in the sense that each rooted interval [u, v] r will have its earliest atom in the chain-atom ordering belonging to the lexicographically earliest saturated chain of [u, v] r according to λ.For any u ∈ P and any choice of root r, define the first atom of [u, 1] r in our chain-atom ordering to be the unique atom belonging to the lexicographically first saturated chain of [u, 1] r with respect to λ. Denote this atom as a 1 .Now assuming we have already chosen atoms a 1 , a 2 , . . ., a i of [u, 1] r for some i ≥ 1, choose a i+1 as follows.Among the saturated chains of [u, 1] r that do not include any of the atoms a 1 , a 2 , . . ., a i , choose one that is as small as possible in lexicographic order.There may be more than one choice, but pick any such saturated chain.Let a i+1 be the atom of [u, 1] r belonging to this chosen saturated chain.Continuing in this manner, we specify a total order Ω = a 1 , . . ., a t on the atoms of [u, 1] r .By construction, Ω is compatible with λ by virtue of λ being self-consistent.
We will prove that Ω is a generalized recursive atom ordering (GRAO), doing so with a proof by induction on the length of the longest saturated chain of P .For P of length 2, our base case, any ordering on the atoms of P is a GRAO.Let P be a finite bounded poset whose longest saturated chain is of length l ≥ 3. We assume by induction that the ordering Ω on the atoms of [a j , 1] is a GRAO for [a j , 1], directly yielding condition (i)(a) of Definition 3.2.
Next we prove that this GRAO on [a j , 1] satisfies condition (i)(b) of Definition 3.2.That is, we verify the following assertion for any w covering an element x satisfying a j ⋖ x: if there exists an atom x ′ of [a j , w] such that x ′ is greater than a i for some atom a i with i < j, then the first atom x 1 of [a j , w] is greater than an atom a i ′ with i ′ < j.Suppose this assertion is false.By definition of Ω and by its compatibility with λ, x 1 must be in the saturated chain of [a j , 1] with the lexicographically smallest label sequence among those saturated chains of [a j , 1] that contain w.Since λ is self-consistent, x 1 is also in the saturated chain of [a j , w] having the lexicographically smallest label sequence.Let a j ⋖x 1 ⋖u 1 ⋖u 2 ⋖. ..⋖u n ⋖w be this lexicographically first saturated chain in [a j , w] according to λ.Since λ is a TCL-labeling, all pairs of consecutive cover relations in a j ⋖ x 1 ⋖ u 1 ⋖ . . .⋖ u n ⋖ w must be topological ascents.But our assumption that x 1 is not above an earlier atom than a j implies that a j is the first atom of [ 0, x 1 ].Since the GRAO denoted by Ω and the TCL-labeling λ are compatible, this implies that 0 ⋖ a j ⋖ x 1 is a topological ascent with respect to λ.Thus, we have shown that w is a topologically ascending saturated chain from 0 to w with respect to the labeling λ.
We now show how our assumption that x 1 is not above an earlier atom than a j while there is some atom x ′ for [a j , w] that is above an earlier atom will imply the existence of another topologically ascending chain for [ 0, w] besides M. Let a w be the first atom of [ 0, w] with respect to the atom ordering Ω.Since a i < x ′ for some i < j, we have a i < w and hence have a w = a j .Since λ is a TCL-labeling compatible with Ω, (λ( 0, a w ), λ(a w , y)) is a topological ascent for any y such that a w ⋖ y < w.Let c be the unique topologically ascending chain in [a w , w].Then the saturated chain M ′ for [ 0, w] obtained by appending 0 to c is a topologically ascending chain in [ 0, w].Since M ′ = M, this contradicts M being the unique topologically ascending chain from 0 to w, completing our proof of condition (i)(b) of Definition 3.2.Now let us check that condition (ii) of Definition 3.2 holds for Ω.Consider any atoms a i , a j with i < j satisfying a i < y and a j < y for some element y.Let M be the lexicographically first saturated chain in [a j , y] and let a j ⋖x be the lowest cover relation in M. Let a y be the first atom of [ 0, y] with respect to the atom ordering Ω.Since a i comes before a j in Ω and a i < y, we must have a j = a y .Let M ′ be the lexicographically first saturated chain in [a y , y] and let a y ⋖ z be the lowest cover relation in M ′ .Since λ is a TCL-labeling compatible with Ω, a y and z are both in the lexicographically first saturated chain of [ 0, y].Thus, (λ( 0, a y ), λ(a y , z)) is a topological ascent, implying M ′ is a topologically ascending chain from 0 to y.Since M is topologically ascending from a j to y but is not the unique topologically ascending chain in [ 0, y], M must have a topological descent at (λ( 0, a j ), λ(a j , x)).This means there must exist 0 ⋖ a k ⋖ u such that u ≤ x and 0 ⋖ a k ⋖ u belongs to the lexicographically first saturated chain of [ 0, x].Since λ is self-consistent, a k is in the lexicographically first saturated chain of P that contains an atom of [ 0, x].As a j and a k are both atoms of [ 0, x], we conclude that a k comes before a j in Ω.This confirms that condition (ii) of Definition 3.2 holds for Ω.Theorem 6.4.Let P be a finite, bounded poset.Then the following are equivalent: (1) P admits a recursive atom ordering (2) P admits a generalized recursive atom ordering (3) P admits a CL-labeling (4) P admits a CL-labeling with the UE property (5) P admits a self-consistent CC-labeling.( 6) P admits a CC-labeling with the UE property (7) P admits a self-consistent TCL-labeling (8) P admits a TCL-labeling with the UE property Moreover, all of these implications are proven constructively.That is, for each implication, either a construction is given showing how to construct the latter type of object from the former or else the former type of object is proven also to be the latter type of object.
Proof.The results we cite below for the various implications are all proven constructively.
Next we confirm the equivalence of (2) and (5).Theorem 5.8 followed by Theorem 6.3 gives one direction.The converse is given by Theorem 6.1.Now we prove the equivalence of ( 2) and ( 6).If λ is a CC-labeling with the UE property, then Lemma 5.3 implies λ is a self-consistent CC-labeling.This gives (5) which was already shown to imply (2).For the converse, Remark 6.2 explains how to construct a CC-labeling with the UE property from a GRAO; the justification that this indeed yields a CC-labeling may be found within the proof of Theorem 8.6.It is self-evident from how it is constructed that this CC-labeling has the UE property.
Next we verify the equivalence of ( 2) and ( 7).Theorem 6.1 followed by Theorem 5.8 gives the forward direction.The reverse direction is proven in Theorem 6.3.
Finally we prove the equivalence of ( 7) and (8).To show that (8) implies ( 7), we use the result from Lemma 5.3 that the UE property implies self-consistency.For the other direction, first use the result already proven above that (7) implies (4).Then use the fact that every CL-labeling is a TCL-labeling (by virtue of how these are defined) to get that (4) implies (8).Combining these implications shows that (7) implies (8).
We conclude this section with two simple consequences of the above results.They are included because they will both be used in Section 7 in our proof that the uncrossing order is dual CL-shellable.Corollary 6.5.Let P be a finite bounded poset with a self-consistent EC-labeling λ.Then λ may be transformed algorithmically into a CL-labeling for P .
Proof.This is immediate from Theorem 6.4, using the fact that every EC-labeling is a CClabeling.
Corollary 6.6.A finite bounded poset P admits a generalized recursive coatom ordering if and only if it admits a recursive coatom ordering.
Proof.Simply apply Theorem 4.11 to P * .

Dual CL-shellability of the uncrossing order
As an application of our earlier results, in this section we prove that the uncrossing order P n is dual CL-shellable.Interest in the family of uncrossing posets stems from their role as the face posets of naturally arising stratified spaces of planar electrical networks (see e.g.[La14a]).In [La14a], Lam proved P n to be Eulerian and conjectured that the dual to the uncrossing order is lexicographically shellable.Indeed, the uncrossing order was proven dual EC-shellable in [HK].Our results clarifying the relationship between CL-shellability and CC-shellability (as well as EC-shellability) will allow us to deduce dual CL-shellability of P n from the dual EC-shelling of [HK] for P n once we prove that the EC-labeling from [HK] has the UE property.
Let us begin by recalling the definition of P n .Figure 7 depicts two cover relations in P 3 while Figure 8 shows all of P 3 .These examples may be helpful for understanding both the definition of P n and how its cover relations are labeled.The poset P n has a unique minimal element 0. The other elements of P n are the various complete matchings on a set of 2n vertices (called boundary nodes) positioned around a circle and labeled clockwise 1, 2, . . ., 2n.By convention, we will typically mark the node labeled 1 (by making it larger than the other nodes) and leave the other nodes unlabeled in our examples.These complete matchings giving rise to the elements of P n \ { 0} may be represented by collections of n strands, with each strand connecting a matched pair of boundary nodes.P n is graded with the rank of any element other than 0 being one more than the minimal number of strand crossings needed to represent its strand diagram in the plane.One of these strand diagrams u will be covered by another strand diagram v whenever the rank of u is one less than the rank of v and u is obtainable from v by uncrossing a single pair of strands.Thus, a strand diagram u will be less than a strand diagram v in P n whenever u may be obtained from v by a series of steps, with each step uncrossing a pair of strands.The elements of P n covering 0 are exactly those strand diagrams with no two strands crossing each other; thus, for C n the n-th Catalan number, there are exactly C n such elements covering 0.
Next we describe an encoding of the elements of P n \ { 0} that was introduced in [HK].This will be used to define the dual EC-labeling for P n from [HK].Again, Figures 7 and 8 may help with parsing this discussion by illustrating the case with n = 3. Start by choosing one of the 2n boundary nodes to treat as a basepoint in each of the strand diagrams of P n .In Figures 7 and 8, this base node is enlarged and labeled with a 1.We associate to each strand diagram a sequence of 2n numbers as follows.First we assign the number 1, 2, . . ., n to the n strands by numbering the strands in the order they are first encountered as we proceed clockwise about the circle starting at our basepoint.Next, we label each strand endpoint with the number that has been assigned to its strand.Finally, we proceed clockwise about the circle from the basepoint reading off the sequence of numbers assigned to the 2n nodes.This procedure assigns to each strand diagram a sequence of length 2n comprised of two copies of each of the numbers from 1 to n.Among such sequences, the ones that arise this way are exactly those in which the first copy of i occurs earlier than the first copy of j for each i < j, and this map from P n \ { 0} to the set of such sequences (introduced in [HK]) is a bijection.
As an example, Figure 7 shows the strand diagram giving rise to the sequence 123312 as well as the two strand diagrams it covers that are obtained by uncrossing the pair of strands in the two different possible ways.These two elements it covers give rise to the sequences 112233 and 123321, with the former shown on the left and the latter shown on the right in Figure 7. See Figure 8 for all of the poset P 3 with the strand diagrams each labeled numerically so that the associated sequence may be read off by proceeding clockwise from the basepoint.
Uncrossing the pair of strands labeled i and j for i < j in a strand diagram will change a sequence having the subsequence ijij into a sequence having either the subsequence iijj or the subsequence ijji (with some of these letters possibly appearing in new positions in the sequence), depending on which of the two possible ways the uncrossing is done.This is discussed in more depth in [HK], but these further details are not critical to our upcoming proof that the dual EC-labeling of [HK] (which is also a dual CC-labeling) may be transformed into a dual CL-labeling.
With this encoding in hand, we next describe the dual EC-labeling λ for P n from [HK].Given a cover relation u ⋖ v which replaces subsequence ijij in v with ijji in u, we assign the label λ(u, v) = (i, j).Given a cover relation u ⋖ v which replaces ijij in v with iijj in u, we assign the label λ(u, v) = (j, i).Label each cover relation which proceeds downward from a strand diagram not having any crossings to the element 0 with the label L. Totally order the label set as follows: • For i < j we have (i, j) < L < (j, i).
• We make the label (i 1 , j 1 ) smaller than the label (i 2 , j 2 ) for i 1 < j 1 and i 2 < j 2 if and only if we either have i 1 < i 2 or we have i 1 = i 2 and j 1 < j 2 .In other words, we order these labels lexicographically.• We make the label (j 1 , i 1 ) smaller than the label (j 2 , i 2 ) for i 1 < j 1 and i 2 < j 2 if and only if we either have j 1 > j 2 or we have j 1 = j 2 and i 1 > i 2 .In other words, we order these labels in the reverse order to lexicographic order.It is proven in [HK] that this edge labeling λ is a dual EC-labeling for the uncrossing order P n (i.e. with label sequences proceeding from top to bottom in P n ).
Proof.By Corollary 6.5, it suffices to show that the dual EC-labeling λ for P n given in [HK] is self-consistent.To this end, we will show that the labels on the cover relations downward from a given element v ∈ P n are all distinct.This will immediately imply that the resulting edge labeling for P * n has the UE property.Lemma 5.3 shows that the UE property implies that the labeling is self-consistent.
Given any fixed element v = 0 in P n and given a label K on an edge down from v, we may determine the unique u for which λ(u, v) = K.This holds because we can read from the label K the names of the strands being uncrossed as well as which way they are being uncrossed.As these two pieces of information are the only data used to determine u from v when u ⋖ v, this means the labels on the cover relations downward from v uniquely determine each u and thus must be distinct.This implies this dual EC-labeling for P n has the UE property.

Further results, remarks, and open questions
We conclude with some further connections between our work and results in the literature, some open questions, and assorted other remarks.8.1.Classical EL-labelings and CL-labelings with the UE property.In light of the role of self-consistency in constructing a GRAO or a CL-labeling from a CC-labeling and also given the fact that self-consistency is implied by the UE property (see Lemma 5.3), it is natural to ask how readily checkable the UE property is.We already proved that every CL-labeling (and hence every EL-labeling) is self-consistent in Lemma 5.5.We now show how easy it is to verify the UE property directly for many of the classic examples of EL-labelings and CL-labelings.This together with the proof in Section 7 of the UE property for the EClabeling for dual uncrossing orders from [HK] may give readers some sense of how the UE property may well be readily verifiable for many of the sorts of CC-labelings and EC-labelings that people are likely to construct for posets of interest in the future.
Proposition 8.1.Let L be a geometric lattice and let a 1 , . . ., a t be any total order on the atoms of L. For each x ∈ L, denote by A(x) the set of atoms a ∈ L satisfying a ≤ x.Then the EL-labeling λ for geometric lattices which labels each cover relation u ⋖v with the smallest atom in A(v) \ A(u) satisfies the UE property.
Proof.Notice for each u ∈ L and any cover relation u⋖v with edge label a i that we must have v = u ∨ a i .In particular, v may be recovered from u and λ(u, v), implying that the labels on the cover relations upward from u are all distinct from each other.Thus, the smallest such label can only occur once, as needed.
Proposition 8.2.Let L be a finite, distributive lattice.Consider the EL-labeling obtained by first interpreting L as the poset J(P ) of order ideals in P ordered by containment and then labeling the cover relation I ⋖ I ∪ {x} with the label x, using any linear extension on P to order the set of edge labels.Then this EL-labeling λ satisfies the UE property.
Proof.Again the proof is based on observing for the cover relations upward from u that any element v ∈ L covering u may be determined from knowing u and λ(u, v).This implies that the edge labels on cover relations upward from a fixed element u are all distinct.Since the smallest such label only occurs once, this implies the UE property.

Figure 2 .
Figure 2. Poset with GRAO that is not RAO (left) and with RAO (right), with edge labels indicating the atom ordering.

Figure 3 .
Figure 3. GRAO that is not RAO (left) and RAO for same poset (right).

Figure 5 .
Figure 5. Case 2 in the proof of Lemma 4.7.

Figure 7 .
Figure 7. Cover relations uncrossing strands in two different ways in P 3 .