IDEMPOTENTS IN THE ENDOMORPHISM ALGEBRA OF A FINITE LATTICE

SERGE BOUC AND JACQUES THÉVENAZ

Abstract. We give a direct construction of a specific central idempotent in the endomorphism algebra of a finite lattice T. This idempotent is associated with all possible sublattices of T which are totally ordered. A generalization is considered in a conjectural fashion and proved experimentally to hold in many cases.

Serge Bouc, CNRS-LAMFA, Université de Picardie - Jules Verne, 33, rue St Leu, F-80039 Amiens Cedex 1, France.

serge.bouc@u-picardie.fr

Corresponding author : Jacques Thévenaz, Institut de mathématiques, EPFL, Station 8, CH-1015 Lausanne, Switzerland.

jacques.thevenaz@epfl.ch

1. Introduction

Let T be a finite lattice and let E be the monoid of all join-morphisms from T to T. If k is a commutative ring, the k-algebra of this monoid is the set kE of all k-linear combinations of elements of E, with multiplication induced by bilinearity from the composition in E. This k-algebra is of interest from a combinatorial point of view because it reflects the structure of T in an algebraic fashion. Moreover, kE plays an important role in our work on correspondence functors [BT1, BT2, BT3, BT4, BT5], providing another motivation for the present work.

Let kE_{tot} be the two-sided ideal of kE spanned by the set E_{tot} of all joinendomorphisms whose image is totally ordered. We proved in [BT2] that kE_{tot} is generated by a central idempotent e_{tot}. In other words, there is a ring isomorphism

kE ≅ kE_{tot} × D ,

where D is a complementary two-sided ideal. We also proved that kE_{tot} is isomorphic to a product of matrix algebras over k, hence semi-simple when k is a field.

Unfortunately, the definition of e_{tot} relies on some rather cumbersome constructions. In the present paper, we give a new point of view for this idempotent. We define an element f_{tot} ∈ kE by means of an easy and explicit formula (see Definition 2.2), which has the advantage of allowing for computer calculations and which does not depend on our previous work. Our main result is the following, which is proved in Section 5.

Date: May 5, 2022.

2010 Mathematics Subject Classification. 06A05, 06A07, 06A11, 06A12, 06B05, 16S99.

Key words and phrases. Poset, lattice, total order, idempotent.

1.1. Theorem.

(a) f_{tot} is a central idempotent of kE.

(b) f_{tot} is the identity element of the ideal kE_{tot}.

This implies that f_{tot} coincides with the idempotent e_{tot} of our previous paper, but with a much easier definition. However, no improvement is made here concerning the semi-simplicity result mentioned above, which depends heavily on the methods of [BT2].

Pole lattices are generalizations of totally ordered lattices and are defined in Section 2. They play a role in our work on correspondence functors because they occur in the semi-simple part of the representation theory developed in [BT5]. Let kE_{pole} be the two-sided ideal of kE spanned by the set E_{pole} of all join-endomorphisms whose image is a pole lattice. We proved in [BT5] that kE_{pole} is generated by a central idempotent e_{pole}, so that there is a ring isomorphism

kE ≅ kE_{pole} × D′

for some two-sided ideal D′. Moreover, we proved that kE_{pole} is isomorphic to a product of matrix algebras over either k or the group algebra of a group of order 2, hence semi-simple when k is a field of characteristic different from 2.

In the present paper, we give also a new point of view for the idempotent e_{pole}, but in a conjectural fashion. We define an element f_{pole} ∈ kE by means of a formula which is analogous to the definition of f_{tot}. We then conjecture the following :

1.2. Conjecture.

(a) f_{pole} is a central idempotent of kE.

(b) f_{pole} is the identity element of the ideal kE_{pole}.

In other words, we conjecture that f_{pole} coincides with the idempotent e_{pole} of our previous paper. Since the formula for f_{pole} allows for computer calculations, we were able to verify that the conjecture holds in a very large number of cases.

The paper is organized as follows. Section 2 is introductory and contains the definition of f_{tot} and f_{pole}. The formula for f_{tot} is simplified in Sections 3 and 4, while the main theorem is proved in Section 5. Section 6 contains an alternative proof of the equality f_{tot} = e_{tot}, using the original approach of [BT2]. Finally Section 7 deals with the conjecture in the case of pole lattices.

2. Endomorphisms of finite lattices

We first recall some basic facts we need about finite lattices. For the rest of this paper, T denotes a finite lattice. Most constructions depend on T but we usually do not include T in the subsequent notation in order to keep it simple. We write ≤ for the partial order of T, ∨ for its join, ∧ for its meet, ˆ0 = ˆ0_{T} for its least element, and ˆ1 = ˆ1_{T} for its greatest element. Recall that an empty join is equal to ˆ0, while an empty meet is equal to ˆ1. If x ≤ y, we define the intervals

[x, y] = {z ∈ T | x ≤ z ≤ y} and ]x, y[ = {z ∈ T | x < z < y} .

Recall that, because T is finite, the meet is uniquely determined by the join thanks to the finite expression

x ∧ y = ⋁_{\begin{matrix}a∈T \\ a≤x,a≤y \end{matrix}} a .

In particular, any join-closed subposet S of T is a lattice in its own right, with a meet induced by its join, but this meet may not coincide with the meet of T. Note that S contains ˆ0 (the empty join), but not necessarily ˆ1. We emphasize that we shall actually only use the join-semilattice structure of T.

If T′ is another finite lattice, a join-morphism φ : T → T′ is a map such that, for any subset X ⊆ T, we have

φ( ⋁_{x∈X} x) = ⋁_{x∈X} φ(x) .

The case X = ∅ yields the property φ(ˆ0) = ˆ0. However, a join-morphism need not respect the meet, and in particular need not map ˆ1 to ˆ1. We note that, whenever φ : T → T′ is a join-morphism, its image φ(T) is a join-closed subposet of T′, hence a lattice. We let E be the set of all join-endomorphisms of T. This is a monoid for the composition of maps and we let kE be the k-algebra of this monoid, where k is a fixed commutative base ring.

Let F be a family of finite lattices, closed under the following two operations :

(a) If S ∈ F and S′ is a join-closed subposet of S, then S′ ∈ F.

(b) If S ∈ F and φ : S → S′ is a join-morphism, then φ(S) ∈ F.

In particular, F is closed under taking images of join-endomorphisms :

S ∈ F, α ∈ E ⟹ α(S) ∈ F .

We define E_{F} to be the subset of E consisting of all join-morphisms α : T → T such that the image α(T) belongs to F. We let kE_{F} be the k-linear span of E_{F} in kE.

2.1. Lemma. kE_{F} is a two-sided ideal of kE.

Proof : Let α ∈ E_{F} and φ ∈ E. Since φ(T) is a join-closed subposet of T, it is clear that αφ(T) is a join-closed subposet of α(T), hence belongs to F. Thus αφ ∈ E_{F}. On the other hand, the lattice α(T) belongs to F and is mapped by φ to φα(T) ∈ F. Therefore φα ∈ E_{F}. The result follows by considering k-linear combinations.

The monoid E is partially ordered by defining, for any α, β ∈ E,

α ≤ β ⟺ α(t) ≤ β(t) ∀t ∈ T .

It is actually also a lattice via the join

(α ∨ β)(t) = α(t) ∨ β(t) ,

and the induced meet. We are interested in the subposet {α ∈ E | α ≥ id} and we note that, whenever α ≥ id, the join-closed image α(T) contains ˆ1 because α(ˆ1) ≥ id(ˆ1) = ˆ1.

If F is a family of finite lattices as above, we define

H_{F} := {α ∈ E | α ≥ id, α(T) ∈ F}

and consider its Möbius function μ_{H_{F}} (β, α), viewing H_{F} as a full subposet of E. We then define the element

f_{F} = ∑_{α∈H_{F}} ∑_{β∈H_{F}} μ_{H_{F}} (β, α) α ∈ kE ,

which will be our main concern throughout this paper. Since μ_{H_{F}} (β, α) = 0 whenever β ≰ α, the second sum actually runs over β ∈ H_{F} such that β ≤ α. In that case, μ_{H_{F}} (β, α) is equal to the reduced Euler characteristic χ̃(]β, α[_{H_{F}} ) of the poset ]β, α[_{H_{F}} of all endomorphisms γ ∈ H_{F} such that β < γ < α. It is convenient to add to H_{F} an element −∞, smaller than all elements of H_{F}, so that

∑_{β∈H_{F}} μ_{H_{F}} (β, α) = −μ_{H_{F}} (−∞, α) .

The definition of f_{F} then takes the following form :

2.2. Definition.

f_{F} = − ∑_{α∈H_{F}} μ_{H_{F}} (−∞, α) α .

We shall be first interested in F = tot, the family of totally ordered lattices, and the corresponding element f_{tot} ∈ kE. Our main goal is to prove that f_{tot} is a central idempotent of kE and that it generates the two-sided ideal kE_{tot}. To this end, we use a series of reductions of the above formula to a much simpler form. These reductions only work for the family of totally ordered lattices. They appear in Sections 3 and 4.

Later in Section 7, we shall also consider the family of pole lattices and the corresponding element f_{pole} ∈ kE. Recall from [BT5] that a pole lattice is a finite lattice P such that every x ∈ P is comparable to every element of P except at most one. In the latter case, if x and y are not comparable, then y is called the twin of x (and x is in turn the twin of y). In other words, x and y have the same height and we have

P = P_{<{x,y}} ⊔ {x, y} ⊔ P_{>{x,y}} .

Moreover,

P_{<{x,y}} = P_{<x} = P_{<y} = [ˆ0 , x ∧ y] and P_{>{x,y}} = P_{>x} = P_{>y} = [x ∨ y , ˆ1] .

Clearly, any totally ordered lattice is a pole lattice (with no twins). Some motivation for working with the family of pole lattices comes from our work on correspondence functors [BT5], because the part of this representation theory which is semi-simple is closely related to pole lattices.

To end this preliminary section, recall that the opposite partial order on the finite lattice T yields the opposite lattice T^{op}, swapping the role of ∨ and ∧, and with ˆ0_{T^{op}} = ˆ1_{T} and ˆ1_{T^{op}} = ˆ0_{T}. Associated with a join-morphism φ : T → T′, there is its opposite

φ^{op} : T′^{op} ⟶ T^{op} , φ^{op}(t′) = ⋁_{\begin{matrix}t∈T \\ φ(t)≤t′ \end{matrix}} t .

2.3. Lemma. Let φ : T → T′ be a join-morphism between two finite lattices.

(a) φ^{op} : T′^{op} → T^{op} is a join-morphism (that is, a meet-morphism T′ → T).

(b) (φ^{op})^{op} = φ.

(c) If t′ ∈ φ(T), then

φ^{op}(t′) = ⋁_{\begin{matrix}t∈T \\ φ(t)=t′ \end{matrix}} t = sup\{t ∈ T | φ(t) = t′\} ,

and in particular φ(φ^{op}(t′)) = t′.

(d) ∀t ∈ T and ∀t′ ∈ T′, we have

φ(t) ≤ t′ ⟺ t ≤ φ^{op}(t′) .

Proof : See Lemma 8.1 and Property 8.3 in [BT2].

3. Endomorphisms with a totally ordered image

We consider the family F = tot of all totally ordered lattices. Recall that a chain in T is a totally ordered subset of T. Recall also that

H_{tot} := {α ∈ E | α ≥ id , α(T) ∈ tot} .

Let σ ∈ H_{tot} and let A := σ(T) be its totally ordered image. Then A is a chain containing ˆ0 (because σ(ˆ0) = ˆ0). Moreover, since σ ≥ id, the chain A also contains ˆ1, hence

A = {ˆ0 = a_{0} < a_{1} < … < a_{n} = ˆ1}

for some n ≥ 0. We let Z_{n} be the set of all chains of size (n + 1) in T whose least element is ˆ0 and greatest element is ˆ1 and we set Z := ⋃_{n≥0} Z_{n}. Letting

b_{i} = σ^{op}(a_{i}) = sup{t ∈ T | σ(t) = a_{i}} ,

we obtain a chain B = {ˆ0 = b_{0} < b_{1} < … < b_{n} = ˆ1} ending with ˆ1 (because σ^{op}(ˆ1) = ˆ1) and starting with ˆ0 (because id ≤ σ, hence b_{0} ≤ σ(b_{0}) = ˆ0). In other words, A, B belong to Z_{n} and σ is characterized by the rule

(3.1)  σ(t) = a_{i} ⟺ t ≤ b_{i} and t ≰ b_{i−1} .

In particular, we have

(3.2)  b_{i} ≤ a_{i} for all 0 ≤ i ≤ n ,

because b_{i} = id(b_{i}) ≤ σ(b_{i}) = a_{i}. We now show the converse.

3.3. Lemma. Given two chains of the same length A, B ∈ Z_{n} and satisfying (3.2), the rule (3.1) defines a join-endomorphism σ ∈ H_{tot}.

Proof : The rule (3.1) defines a map σ : T → T whose image is A. Let s, t ∈ T. Since A = σ(T) is totally ordered, we can assume that σ(s) ≤ σ(t). We let a_{i} = σ(s) and a_{j} = σ(t), so that i ≤ j. In particular, b_{i} ≤ b_{j}. The rule (3.1) implies that s ≤ b_{i} and that t ≤ b_{j}, t ≰ b_{j−1}. It follows that s∨t ≤ b_{j} and s∨t ≰ b_{j−1}. Therefore σ(s ∨ t) = a_{j} = σ(t) = σ(s) ∨ σ(t). Since we also have σ(ˆ0) = ˆ0 by (3.1), this shows that σ is a join-endomorphism. Finally (3.1) and (3.2) imply that σ ≥ id. It follows that σ ∈ H_{tot}.

3.4. Corollary. The set H_{tot} is parametrized by the set of pairs (A, B) of chains A, B ∈ Z of the same length and satisfying (3.2).

We write σ_{B}^{A} for the endomorphism parametrized by A and B. We first show that the case A = B plays a special role.

3.5. Lemma. Let A, B ∈ Z_{n} satisfying (3.2) and let σ = σ_{B}^{A}. Then σ^{2} = σ if and only if A = B.

Proof : Suppose that σ^{2} = σ. Since σ(b_{i}) = a_{i}, we get σ(a_{i}) = σ^{2}(b_{i}) = σ(b_{i}) = a_{i}, hence

a_{i} ≤ sup{t ∈ T | σ(t) = a_{i}} = b_{i} ≤ a_{i} ,

using (3.2). Therefore a_{i} = b_{i}, hence A = B.

Suppose now that A = B. Since σ(b_{i}) = a_{i}, we get σ(a_{i}) = a_{i}. Now, for any t ∈ T, we have σ(t) = a_{i} for some i, hence σ^{2}(t) = σ(a_{i}) = a_{i}. This shows that σ^{2} = σ.

In the case A = B, we write simply σ_{A} := σ_{A}^{A}. By the rule (3.1), the map σ_{A} can be described directly by the following formula :

(3.6)  σ_{A}(t) = min{a ∈ A | t ≤ a}.

Because of its importance in what follows, we introduce another useful notation :

G_{tot} := {σ ∈ H_{tot} | σ^{2} = σ} , i.e. G_{tot} := {σ ∈ E | σ ≥ id , σ(T) ∈ tot , σ^{2} = σ} .

By Corollary 3.4 and Lemma 3.5, the set G_{tot} is parametrized by the set of chains A ∈ Z, noticing that any pair (A, A) obviously satisfies Condition (3.2). The endomorphism parametrized by A is σ_{A} and therefore

(3.7)  G_{tot} = {σ_{A} | A ∈ Z} .

In the definition of f_{tot} given by (2.2), the sum runs over all α ∈ H_{tot} and they have the form α = σ_{B}^{A} for some chains A, B ∈ Z satisfying (3.2). Our first main result asserts that many such terms have a zero coefficient in the expression of f_{tot}.

3.8. Theorem. Let A, B ∈ Z satisfying (3.2).

(a) If A = B, we have μ_{G_{tot}}(−∞, σ_{A}) = μ_{H_{tot}}(−∞, σ_{A}).

(b) If A ≠ B, we have μ_{H_{tot}}(−∞, σ_{B}^{A}) = 0.

This will be proved in Section 4. It follows from (b) that the sum in the expression of f_{tot} (in Definition 2.2) only runs over endomorphisms σ_{A} where A ∈ Z. Therefore, using (a), we obtain

(3.9)  f_{tot} = − ∑_{A∈Z} μ_{H_{tot}}(−∞, σ_{A}) σ_{A} = − ∑_{A∈Z} μ_{G_{tot}}(−∞, σ_{A}) σ_{A} .

The set Z is partially ordered by inclusion. It has no greatest element (unless T is totally ordered) and we let ∞ be an additional element, larger than any A ∈ Z.

3.10. Lemma. Consider the map

θ : Z ⊔ {∞} ⟶ {−∞} ⊔ G_{tot} , θ(A) = σ_{A} and θ(∞) = −∞ .

(a) θ is an anti-isomorphism of posets : A ⊆ B ⟺ σ_{A} ≥ σ_{B}.

(b) For any A ∈ Z, we have μ_{G_{tot}}(−∞, σ_{A}) = μ_{Z}(A, ∞).

Proof : By Corollary 3.4 and Lemma 3.5, the map θ is a bijection. Let A, B ∈ Z and suppose first that A ⊆ B. Let t ∈ T and set a = σ_{A}(t), so in particular t ≤ a. Then a ∈ A, hence a ∈ B, and so σ_{B}(a) = a. Thus

σ_{B}(t) ≤ σ_{B}(a) = a = σ_{A}(t)

and therefore σ_{B} ≤ σ_{A}.

Assume now that σ_{B} ≤ σ_{A} and let a ∈ A. Then

a ≤ σ_{B}(a) ≤ σ_{A}(a) = a ,

hence σ_{B}(a) = a, which implies that a ∈ B. Therefore A ⊆ B.

Statement (b) is an immediate consequence of (a).

Together with (3.9), Lemma 3.10 yields the following corollary :

3.11. Corollary. f_{tot} = − ∑_{A∈Z} μ_{Z}(A, ∞) σ_{A}.

From now on, we shall work with this much simpler expression of f_{tot}. For later use, we show that the Möbius function of Z can be expressed in terms of the Möbius function of T.

3.12. Lemma. Let A = {ˆ0 = a_{0} < a_{1} < … < a_{n} = ˆ1} be an element of Z_{n}. Then

μ_{Z}(A, ∞) = (−1)^{n+1} ∏_{k=1}^{n} μ_{T} (a_{k−1}, a_{k}) ,

where μ_{T} (a_{k−1}, a_{k}) denotes the Möbius function of T.

Proof : For any poset X, let s_{i}(X) be the number of chains of cardinality i in X. For i = 0, there is the empty chain, so s_{0}(X) = 1. It is well-known that

μ_{Z}(A, ∞) = χ̃(]A, ∞[ ) = ∑_{i≥0}(−1)^{i−1}s_{i}( ]A, ∞[ ) .

The sign is (−1)^{i−1} because a chain of cardinality i is an (i−1)-simplex. Now if A′ is a chain with A ⊆ A′, then A′ is obtained from A by inserting a chain in each interval ]a_{k−1}, a_{k}[ independently. Therefore

s_{i}( ]A, ∞[ ) = ∑_{\begin{matrix}i_{1},...,i_{n}≥0 \\ i_{1}+...+i_{n}=i \end{matrix}} ∏_{k=1}^{n} s_{i_{k}}( ]a_{k−1}, a_{k}[ )

and it follows that

μ_{Z}(A, ∞) = ∑_{i_{1},...,i_{n}≥0}(−1)^{i_{1}+...+i_{n}−1} ∏_{k=1}^{n} s_{i_{k}}( ]a_{k−1}, a_{k}[ )

= (−1) ∏_{k=1}^{n} ( ∑_{i_{k}≥0}(−1)^{i_{k}}s_{i_{k}}( ]a_{k−1}, a_{k}[)

= (−1)^{n+1} ∏_{k=1}^{n} ( ∑_{i_{k}≥0}(−1)^{i_{k}−1}s_{i_{k}}( ]a_{k−1}, a_{k}[)

= (−1)^{n+1} ∏_{k=1}^{n} μ_{T} (a_{k−1}, a_{k}) ,

as was to be shown.

3.13. Remark. In the expression of f_{tot} in Corollary 3.11, the sum could be restricted to all A ∈ Z such that the lattice [a_{k−1}, a_{k}] is complemented for each k = 1, …, n, where A = {ˆ0 = a_{0} < a_{1} < … < a_{n} = ˆ1}. This is because

μ_{Z}(a, ∞) = (−1)^{n+1} ∏_{k=1}^{n} μ_{T} (a_{k−1}, a_{k})

by Lemma 3.12 and μ_{T} (a_{k−1}, a_{k}) = 0 whenever the lattice [a_{k−1}, a_{k}] is not complemented, by Crapo’s formula (see Exercice 92 of Chapter 3 in [St]).

4. Proof of Theorem 3.8

Throughout this section, we simplify notation and define H = H_{tot} and G = G_{tot}. By Corollary 3.4, the elements of H have the form σ_{B}^{A} where A, B ∈ Z satisfy (3.2). By (3.7), the elements of G have the form σ_{A} := σ_{A}^{A} where A ∈ Z. For any τ ∈ H, we also write G_{≤τ} := {σ ∈ G | σ ≤ τ} and G_{<τ} := {σ ∈ G | σ < τ}. We shall use some of Quillen’s basic results on homotopy of posets from [Qu].

4.1. Theorem. For any λ ∈ H − G, the poset G_{≤λ} is contractible.

Before proving Theorem 4.1, we first show that Theorem 3.8 follows easily from Theorem 4.1.

4.2. Lemma. Let τ ∈ H.

(a) The inclusion map G_{<τ} ↪ H_{<τ} is a homotopy equivalence.

(b) If τ ∈ G, we have μ_{G}(−∞, τ) = μ_{H}(−∞, τ).

(c) If τ ∉ G, we have μ_{H}(−∞, τ) = 0.

Proof : (a) For any σ ∈ H_{<τ}, the poset G_{≤σ} is contractible. This is obvious if σ ∈ G because G_{≤σ} is a cone on σ, while this follows from Theorem 4.1 if σ ∉ G. Thus Proposition 1.6 in [Qu] applies and the inclusion map G_{<τ} ↪ H_{<τ} is a homotopy equivalence.

(b) This follows from (a) by taking reduced Euler characteristics.

(c) We have G_{<τ} = G_{≤τ} because τ ∉ G. Thus G_{<τ} is contractible by Theorem 4.1 and so is H_{<τ} by the homotopy equivalence of part (a). Therefore the reduced Euler characteristic of H_{<τ} is zero, in other words the Möbius function in (c) vanishes.

Proof of Theorem 3.8 : By Corollary 3.4, any τ ∈ H has the form τ = σ_{B}^{A} for A, B ∈ Z satisfying (3.2). Moreover, τ ∈ G if and only if A = B, by Lemma 3.5. Now parts (a) and (b) of Theorem 3.8 are exactly parts (b) and (c) of Lemma 4.2, written differently.

Proof of Theorem 4.1 : Using Corollary 3.4 and Lemma 3.5, we fix λ = σ_{B}^{A} with A ≠ B and we write

A = {ˆ0 = a_{0} < a_{1} < … < a_{n} = ˆ1} and B = {ˆ0 = b_{0} < b_{1} < … < b_{n} = ˆ1} .

For any 1 ≤ k ≤ n, replace {b_{k}, …, b_{n}} by {a_{k}, …, a_{n}} and define

B_{k} = {ˆ0 = b_{0} < … < b_{k−1} < a_{k} < … < a_{n} = ˆ1} .

Define λ_{k} = σ_{B_{k}}^{A} and notice that λ_{n} = σ_{B}^{A} = λ and λ_{1} = σ_{A}^{A} = σ_{A} ∈ G. Using (3.1), it is elementary to check that

(4.3)  λ_{k} ≤ λ_{k+1} , ∀ 1 ≤ k ≤ n − 1 .

Our strategy is to pass from G_{≤λ_{n}} to the single point {λ_{1}} by a sequence of homotopy equivalences.

We first need to characterize the elements of G_{≤λ_{k}}.

4.4. Lemma. Let C ∈ Z and 1 ≤ k ≤ n.

σ_{C} ∈ G_{≤λ_{k}} ⟺ \{ \begin{matrix}if 1 ≤ i ≤ k − 1 and t ≤ b_{i}, then σ_{C}(t) ≤ a_{i}. \\ a_{k}, …, a_{n} ∈ C. \end{matrix}

Proof : Suppose that σ_{C} ≤ λ_{k}. If t ≤ b_{i} with i ≤ k − 1, then σ_{C}(t) ≤ λ_{k}(t) ≤ λ_{k}(b_{i}) = a_{i}. If i ≥ k, then since id ≤ σ_{C}, we have a_{i} ≤ σ_{C}(a_{i}) ≤ λ_{k}(a_{i}) = a_{i}, hence σ_{C}(a_{i}) = a_{i}, that is, a_{i} ∈ C.

Suppose conversely that the right hand side conditions hold. If 1 ≤ i ≤ k − 1, t ≤ b_{i}, and t ≰ b_{i−1}, then σ_{C}(t) ≤ a_{i} = λ_{k}(t) by applying the rule (3.1) to λ_{k}. If k ≤ i ≤ n, t ≤ a_{i}, t ≰ a_{i−1} (respectively t ≰ b_{k−1} in case i = k), then a_{i} ∈ C and therefore σ_{C}(t) ≤ σ_{C}(a_{i}) = a_{i} = λ_{k}(t), by applying again the rule (3.1) to λ_{k}.

For any chain C satisfying a_{k}, …, a_{n} ∈ C and for any k ≤ i ≤ n, we let

C_{i} := {c ∈ C | c < a_{i}, c ≰ a_{i−1}} and C_{≰a_{k−1}} := {c ∈ C | c ≰ a_{k−1}} .

It should be noticed that such a chain C may be (much) longer than A, because C_{i} = C ∩]a_{i−1}, a_{i}[ if i ≥ k + 1 and this may be nonempty. For any 1 ≤ k ≤ n, define now

G′_{≤λ_{k}} = {σ_{C} ∈ G_{≤λ_{k}} | C_{≰a_{k−1}} = {a_{k}, …, a_{n}} } ,

In other words, σ_{C} ∈ G′_{≤λ_{k}} if and only if C_{i} = ∅ for any k ≤ i ≤ n. It follows that, whenever σ_{C} ∈ G′_{≤λ_{k}}, we obtain

k ≤ i ≤ n, t ≤ a_{i}, t ≰ a_{i−1} ⟹ σ_{C}(t) = a_{i} ,

(whereas (3.6) only implies an inequality σ_{C}(t) ≤ a_{i} when |C_{i}| ≥ 2). Clearly, (4.3) implies that G_{≤λ_{k}} ⊆ G_{≤λ_{k+1}} and G′_{≤λ_{k}} ⊆ G′_{≤λ_{k+1}} for any k < n. It follows that we have inclusions

G′_{≤λ_{k}} ⊆ G_{≤λ_{k}} ∩ G′_{≤λ_{k+1}} ⊆ G_{≤λ_{k}} .

Now define

ρ_{n} : G_{≤λ_{n}} ⟶ G′_{≤λ_{n}} , ρ_{n}(σ_{C}) = σ_{C−C_{n}} ,

and similarly, for 1 ≤ k ≤ n − 1,

ρ_{k} : G_{≤λ_{k}} ∩ G′_{≤λ_{k+1}} ⟶ G′_{≤λ_{k}} , ρ_{k}(σ_{C}) = σ_{C−C_{k}} .

4.5. Lemma. ρ_{k} is a map of posets and is a homotopy equivalence.

Proof : If σ_{C} ∈ G_{≤λ_{k}} ∩ G′_{≤λ_{k+1}} (respectively σ_{C} ∈ G_{≤λ_{n}} if k = n), then C_{i} = ∅ for any i > k and therefore (C − C_{k})_{i} = ∅ for any i ≥ k. It follows that σ_{C−C_{k}} ∈ G′_{≤λ_{k}} and so the map ρ_{k} is well-defined. It is order-preserving because, by Lemma 3.10,

σ_{C} ≥ σ_{C′} ⇒ C ⊆ C′ ⇒ C − C_{k} ⊆ C′ − C′_{k} ⇒ σ_{C−C_{k}} ≥ σ_{C′−C′_{k}} ,

using the fact that C′_{k} ∩ C = C_{k}.

Since C −C_{k} ⊆ C, we have ρ_{k}(σ_{C}) ≥ σ_{C}. Therefore, if j : G′_{≤λ_{k}} → G_{≤λ_{k}} ∩G′_{≤λ_{k+1}} (respectively j : G′_{≤λ_{k}} → G_{≤λ_{k}} if k = n) denotes the inclusion map, then ρ_{k}j = id and jρ_{k} ≥ id. By Property 1.3 in [Qu], jρ_{k} is homotopic to id and it follows that ρ_{k} is a homotopy equivalence.

Now for 2 ≤ k ≤ n, define

π_{k} : G′_{≤λ_{k}} ⟶ G_{≤λ_{k−1}} ∩ G′_{≤λ_{k}} , π_{k}(σ_{C}) = σ_{C∪{a_{k−1}}} .

4.6. Lemma. π_{k} is a map of posets and is a homotopy equivalence.

Proof : Let σ_{C} ∈ G′_{≤λ_{k}}, so that C_{≰a_{k−1}} = {a_{k}, …, a_{n}}. We first prove that C ∪ {a_{k−1}} is totally ordered (i.e. a chain in Z), so that σ_{C∪{a_{k−1}}} makes sense. We must show that any c ∈ C is comparable to a_{k−1}. This is clear if c ≤ a_{k−1}, so we may assume that c ≰ a_{k−1}. Then c = a_{i} for some i ≥ k by the definition of G′_{≤λ_{k}}, hence c = a_{i} > a_{k−1}.

Now we get a_{k−1}, a_{k}, …, a_{n} ∈ C∪{a_{k−1}} and Lemma 4.4 implies that σ_{C∪{a_{k−1}}} ∈ G_{≤λ_{k−1}}. It follows that σ_{C∪{a_{k−1}}} ∈ G_{≤λ_{k−1}} ∩ G′_{≤λ_{k}} and the map π_{k} is well-defined. It is order-preserving, by using Lemma 3.10.

Since C ∪ {a_{k−1}} ⊇ C, we have π_{k}(σ_{C}) ≤ σ_{C}. Therefore, if j : G_{≤λ_{k−1}} ∩ G′_{≤λ_{k}} → G′_{≤λ_{k}} denotes the inclusion map, then π_{k}j = id and jπ_{k} ≤ id. By Property 1.3 in [Qu], jπ_{k} is homotopic to id and it follows that π_{k} is a homotopy equivalence, as was to be shown.

Now the sequence of homotopy equivalences ρ_{n}, π_{n}, ρ_{n−1}, π_{n−1}, …, ρ_{2}, π_{2}, ρ_{1} shows that G_{≤λ_{n}} is homotopy equivalent to G′_{≤λ_{1}}. But for any σ_{C} ∈ G′_{≤λ_{1}}, we have

C = {ˆ0 = a_{0} < a_{1} < … < a_{n} = ˆ1} = A ,

so σ_{C} = σ_{A} = λ_{1}. Thus G′_{≤λ_{1}} is a singleton, hence contractible, and therefore G_{≤λ_{n}} is contractible as well. This completes the proof of Theorem 4.1.

5. The main theorem

In this section, we prove the main properties of the element f_{tot}, establishing the connection between f_{tot} and the two-sided ideal kE_{tot} of the k-algebra kE. In order to simplify notation throughout this section, we write μ instead of μ_{Z} and σ_{A} instead of σ_{A}^{A}.

5.1. Theorem. Let f_{tot} be as in Corollary 3.11.

(a) f_{tot} is a central idempotent of kE.

(b) f_{tot} is the identity element of the ideal kE_{tot}.

The proof depends on a main lemma.

5.2. Lemma. f_{tot} ψ = ψ, for all ψ ∈ E_{tot}.

Proof : By Corollary 3.11, recall that f_{tot} = − ∑_{A∈Z} μ(A, ∞) σ_{A}. Let X = Im(ψ),

which is totally ordered because ψ ∈ E_{tot}. We decompose ψ as the composite of a surjective join-morphism T → X followed by the inclusion map i_{X} : X → T. It suffices to prove that f_{tot} i_{X} = i_{X} for any chain X in T starting with ˆ0. Note that f_{tot} i_{X} is a k-linear combination of maps from X to T, so all the following computations must take place in the k-linear span of all maps X → T. Now we have

(5.3)  f_{tot} i_{X} = − ∑_{A∈Z} μ(A, ∞) σ_{A}i_{X} = ∑_{φ:X→T} ( − ∑_{\begin{matrix}A∈Z \\ σ_{A}i_{X}=φ \end{matrix}} μ(A, ∞)) φ .

By the definition of σ_{A}, the equation σ_{A}i_{X} = φ means that, for any x ∈ X, the element φ(x) is the least element of A such that φ(x) ≥ x. In other words, the condition σ_{A}i_{X} = φ is equivalent to

(5.4)  [x, φ(x)] ∩ A = {φ(x)} , ∀ x ∈ X .

Any map φ : X → T appearing in the sum (5.3) must satisfy the following 3 conditions :

(a) φ is order-preserving and φ(ˆ0) = ˆ0 (that is, φ is a join-morphism).

(b) φ(x) ≥ x, for all x ∈ X.

(c) If x, y ∈ X satisfy x ≤ y ≤ φ(x), then φ(y) = φ(x).

In order to prove this, we note that the coefficient of φ in (5.3) is nonzero only if there exists at least one A ∈ Z such that σ_{A}i_{X} = φ. Condition (a) follows from the fact that both i_{X} and σ_{A} are order-preserving and map ˆ0 to ˆ0, hence φ = σ_{A}i_{X} has the same properties. Condition (b) is clear because σ_{A} ≥ id. For condition (c), note that the only element of [x, φ(x)] ∩ A is φ(x), so the definition of σ_{A} yields σ_{A}(y) = φ(x), that is, φ(y) = φ(x).

Now we prove that, if φ satisfies (b) and (c), then (5.4) is equivalent to

(5.5)  ( ⋃_{x∈X}[x, φ(x)]) ∩ A = φ(X) .

It is clear that (5.4) implies (5.5). Assume now (5.5) and let x ∈ X. Notice that [x, φ(x)] ∩ A is nonempty because φ(x) ∈ φ(X), hence φ(x) ∈ A by (5.5), and so φ(x) ∈ [x, φ(x)] ∩ A. For any b ∈ [x, φ(x)] ∩ A, we have b ∈ φ(X) by (5.5), that is, b = φ(y) for some y ∈ X. Since X is totally ordered, we have either x ≤ y ≤ φ(y) = b ≤ φ(x), hence φ(y) = φ(x) by (c), or y ≤ x ≤ b = φ(y), hence φ(x) = φ(y) by (c) again. Therefore b = φ(x), showing that [x, φ(x)]∩A = {φ(x)}. This proves that (5.5) implies (5.4).

We now fix a map φ : X → T satisfying (a), (b), (c), and we set

C = ⋃_{x∈X}[x, φ(x)]  and  D = φ(X) ,

so that (5.5) becomes C ∩ A = D. Since φ is order-preserving and φ(ˆ0) = ˆ0, the set D is a chain in T starting with ˆ0. Let \overline{C} = C ∪ \{ˆ1\} and \overline{D} = D ∪ \{ˆ1\}, so that \overline{D} ∈ Z. Clearly, ˆ1 ∉ C if and only if ˆ1 ∉ D, and therefore the condition C ∩ A = D is equivalent to \overline{C} ∩ A = \overline{D}. It follows that the coefficient of φ in (5.3) is equal to

− ∑_{\begin{matrix}A∈Z \\ σ_{A}i_{X}=φ \end{matrix}} μ(A, ∞) = − ∑_{\begin{matrix}A∈Z \\ C∩A=D \end{matrix}} μ(A, ∞) = − ∑_{\begin{matrix}A∈Z \\ \overline{C}∩A=\overline{D} \end{matrix}} μ(A, ∞) ,

because the condition σ_{A}i_{X} = φ is equivalent to (5.5) by the discussion above. By the defining property of the Möbius function, we obtain

− ∑_{\begin{matrix}A∈Z \\ \overline{C}∩A=\overline{D} \end{matrix}} μ(A, ∞) = ∑_{\begin{matrix}A,B∈Z \\ A⊆B \\ \overline{C}∩A=\overline{D} \end{matrix}} μ(A, B) = ∑_{\begin{matrix}B∈Z \\ \overline{D}⊆B \end{matrix}} ∑_{\begin{matrix}A∈Z \\ A⊆B \\ \overline{C}∩A=\overline{D} \end{matrix}} μ(A, B) .

For a fixed B ∈ Z, the chain A runs over the interval [\overline{D}, B] with the additional condition \overline{C} ∩ A = \overline{D}, which can also be written (\overline{C} ∩ B) ∩ A = \overline{D} because A ⊆ B. By a well-known property of the Möbius function (Corollary 3.9.3 in [St]), the corresponding sum

∑_{\begin{matrix}A∈[\overline{D},B] \\ (\overline{C}∩B)∩A=\overline{D} \end{matrix}} μ(A, B)

is zero, provided the fixed element \overline{C} ∩ B is not equal to the top element B. If otherwise \overline{C} ∩ B = B, then B ⊆ \overline{C} and A = \overline{C} ∩ A = \overline{D}, so that the sum over A has the single term μ(\overline{D}, B) for A = \overline{D}.

Going back to the coefficient of φ in (5.3), we obtain

− ∑_{\begin{matrix}A∈Z \\ σ_{A}i_{X}=φ \end{matrix}} μ(A, ∞) = ∑_{\begin{matrix}B∈Z \\ \overline{D}⊆B⊆\overline{C} \end{matrix}} μ(\overline{D}, B) = ∑_{\begin{matrix}B∈Z\overline{C} \\ \overline{D}⊆B \end{matrix}} μ(\overline{D}, B) = −μ(\overline{D}, ∞)

where the latter symbol ∞ denotes a top element added to the poset Z\overline{C} (consisting of all chains in \overline{C} having least element ˆ0 and greatest element ˆ1).

Recall that φ ≥ i_{X} by condition (b). We now assume that φ > i_{X} and we want to prove that μ(\overline{D}, ∞) = 0. Let y ∈ X be minimal such that φ(y) > y. We claim that, for any B ∈ Z\overline{C}, the union B ∪ \{y\} is totally ordered. We have to prove that any b ∈ B is comparable with y. Since b ∈ \overline{C}, either b = ˆ1 and then we are done because y ≤ ˆ1, or there exists x ∈ X such that b ∈ [x, φ(x)]. If y ≤ x, then y ≤ b and we are done again. We can assume now that y ≰ x, hence x < y because X is totally ordered. By minimality of y, we must have φ(x) = x, hence [x, φ(x)] = {x} and b = x. It follows that b < y. This completes the proof that B ∪ {y} is totally ordered.

We claim now that y does not belong to D = φ(X). Otherwise y = φ(z) for some z ∈ X. If we had z = y, we would obtain φ(y) = y, contrary to the choice of y.

It follows that the relation z ≤ φ(z) = y must be a strict inequality z < φ(z). This contradicts the minimality of y and proves the claim. Moreover, y ∉ \overline{D} because y < φ(y), hence y ≠ ˆ1. Consequently, the poset ]\overline{D}, ∞[ is conically contractible (see 1.5 in [Qu]) via the contraction

B ≤ B ∪ \{y\} ≥ \overline{D} ∪ \{y\}

and it follows that μ(\overline{D}, ∞) = 0.

This shows that the coefficient of φ in (5.3) is zero whenever φ > i_{X}. Therefore we are left with a single term for φ = i_{X}, namely

f_{tot} i_{X} = −μ(\overline{D}, ∞) i_{X} .

But for φ = i_{X}, we have

C = ⋃_{x∈X}[x, φ(x)] = ⋃_{x∈X}{x} = X = i_{X}(X) = D ,

and consequently the only chain in \overline{C} containing \overline{D} is \overline{D} itself. In other words ]\overline{D}, ∞[= ∅ and μ(\overline{D}, ∞) = −1. The required equality f_{tot} i_{X} = i_{X} follows and this completes the proof of Lemma 5.2.

Proof of Theorem 5.1 : We first show that f_{tot} is idempotent. Lemma 5.2 implies that f_{tot} σ_{A} = σ_{A} for any A ∈ Z, hence

(f_{tot})^{2} = f_{tot}( − ∑_{A∈Z} μ(A, ∞) σ_{A}) = − ∑_{A∈Z} μ(A, ∞) σ_{A} = f_{tot} ,

as required.

Next we want to pass to opposite lattices and use Lemma 5.2 for T^{op}. To this end, we include T in the notation, so we write ≤_{T} for ≤ and we define f_{tot}(T) = f_{tot}, E_{tot}(T) = E_{tot}, and Z(T) = Z. Since σ_{A} is an endomorphism of T, σ_{A}^{op} is an endomorphism of T^{op} and we claim that σ_{A}^{op} = σ_{A^{op}}, where A^{op} is equal to A, with its opposite order inside T^{op}. In order to prove the claim, we let s, t ∈ T. Clearly σ_{A}(t) ≤_{T} s if and only if s ≤_{T^{op}} σ_{A}(t). Since σ_{A}(t) ∈ A and since σ_{A^{op}}(s) is the smallest element of A^{op} containing s in T^{op}, by (3.6), we get

s ≤_{T^{op}} σ_{A}(t) ⟺ σ_{A^{op}}(s) ≤_{T^{op}} σ_{A}(t) ⟺ σ_{A}(t) ≤_{T} σ_{A^{op}}(s) ⟺ t ≤_{T} σ_{A^{op}}(s) ,

where the latter equivalence comes from the fact that σ_{A^{op}}(s) ∈ A and σ_{A}(t) is the smallest element of A containing t in T, by (3.6) again. Now the definition of σ_{A}^{op} yields

σ_{A}^{op}(s) = ⋁_{\begin{matrix}t∈T \\ σ_{A}(t)≤_{T} s \end{matrix}} t = ⋁_{\begin{matrix}t∈T \\ t≤_{T} σ_{A^{op}} (s) \end{matrix}} t

by the above, and the latter element is obviously equal to σ_{A^{op}}(s). Therefore σ_{A}^{op} = σ_{A^{op}}, proving the claim.

Now the opposite of our idempotent is

f_{tot}(T)^{op} = − ∑_{A∈Z(T)} μ(A, ∞) σ_{A}^{op} = − ∑_{A∈Z(T)} μ(A, ∞) σ_{A^{op}} .

Using the fact that the passage to opposite chains preserves the inclusion relation between chains, we obtain

f_{tot}(T)^{op} = − ∑_{A^{op}∈Z(T^{op})}μ(A^{op}, ∞) σ_{A^{op}} ,

that is, f_{tot}(T)^{op} = f_{tot}(T^{op}).

Lemma 5.2 for the lattice T^{op} asserts that f_{tot}(T^{op}) ρ = ρ for every ρ ∈ E_{tot}(T^{op}). Since ρ = ψ^{op} for some ψ ∈ E_{tot}(T) (namely ψ = ρ^{op}), we get f_{tot}(T)^{op} ψ^{op} = ψ^{op}, that is, ψ f_{tot}(T) = ψ, for all ψ ∈ E_{tot}. Together with Lemma 5.2 for T, we obtain

(5.6)  ψ f_{tot} = ψ = f_{tot} ψ , ∀ψ ∈ E_{tot} ,

showing that f_{tot} is the identity element of the two-sided ideal kE_{tot}.

To prove that f_{tot} is central in kE, let φ ∈ E. Applying (5.6) to both f_{tot} φ and φ f_{tot} (which belong to kE_{tot}), we see that

f_{tot} φ = f_{tot} φ f_{tot} = φ f_{tot} ,

as required. This completes the proof of Theorem 5.1.

5.7. Corollary. There is a two-sided ideal D of kE and a ring isomorphism

kE ≅ kE_{tot} × D ,

(where kE_{tot} is identified with kE_{tot} × {0} and D with {0} × D, as usual).

Proof : By Theorem 5.1, the central idempotent, f_{tot} generates kE_{tot} and therefore the central idempotent id −f_{tot} generates a complementary two-sided ideal D.

Corollary 5.7 was first proved as Theorem 10.8 in [BT2], by means of a completely different approach which produces a central idempotent e_{tot} generating the twosided ideal kE_{tot}.

5.8. Corollary. e_{tot} = f_{tot}.

Proof : By Theorem 5.1, f_{tot} is the identity element of the ideal kE_{tot}. By Theorem 10.8 in [BT2], e_{tot} is also the identity element of kE_{tot}. Therefore they must be equal.

The formula for e_{tot} given in Theorem 10.8 of [BT2] comes from rather elaborate constructions, which we revisit in Section 6 below.

6. The original approach to the idempotent

In this section, we use the original approach of [BT2] and we consider the idempotent e_{tot}, defined in Theorem 10.8 of [BT2] by an explicit, but complicated, formula. Our goal is to give a second proof of Corollary 5.8, namely e_{tot} = f_{tot}, using the formula for f_{tot} obtained in Corollary 3.11. In other words, we want to prove the following :

(6.1)  e_{tot} = − ∑_{A∈Z} μ(A, ∞) σ_{A} ,

where we use again the simplified notation of Section 5. We will start from the expression of e_{tot} given in [BT2]. By using rather delicate arguments, we will show that this expression can be greatly simplified and finally reduces to the right hand side of (6.1).

This second proof of (6.1) is slightly shorter than the proof of Corollary 5.8 obtained by using the developments of Sections 4 and 5. However, it has the disadvantage of being quite technical, as it uses the cumbersome constructions of [BT2]. Moreover, this proof does not seem to provide useful ideas for proving Conjecture 7.1 below about pole lattices.

We first need to define the notation. For any n ∈ ℕ, we write \underline{n} = \{0, 1, …, n\}, a totally ordered lattice with ˆ0\underline{n} = 0 and ˆ1\underline{n} = n. We use the set P_{n} of all chains B = {b_{0} < b_{1} < … < b_{n} = ˆ1} in T whose greatest element is ˆ1. The set P_{n} parametrizes the set of surjective join-morphisms π : T → \underline{n} via the rule

b_{i} = π^{op}(i) = sup{t ∈ T | π(t) = i} .

Instead of \underline{n}, it will be convenient to use a totally ordered lattice P of cardinality n + 1, that is, a lattice isomorphic to \underline{n}, and to define r(p) = sup\{q ∈ P | q < p\}, for any p ∈ P − {ˆ0}. With this notation, a surjective join-morphism π : T → P corresponds to a chain B = {b_{p} | p ∈ P } defined by

b_{p} = π^{op}(p) = sup{t ∈ T | π(t) = p} ,

and satisfying b_{p} < b_{q} whenever p < q. We write π^{B} : T → P for the surjective join-morphism corresponding to the chain B ∈ P_{n}. Then π^{B}(t) = ˆ0 if t ≤ b_{0} and otherwise we recall the rule

π^{B}(t) = p if t ≤ b_{p} and t ≰ b_{r(p)} .

For any given B ∈ P_{n}, we choose an element a_{p} ∈ [b_{r(p)}, b_{p}] for each p ∈ P −{ˆ0}. This defines a family A = (a_{p})_{p∈P −{ˆ0}} of elements of T. We let J_{B} be the set of all families A = (a_{p})_{p∈P −{ˆ0}} of elements of T such that a_{p} ∈ [b_{r(p)}, b_{p}] for every p ∈ P − {ˆ0}. If A ∈ J_{B}, we also set aˆ0 = ˆ0 and we define

j_{A}^{B} : P ⟶ T , j_{A}^{B}(p) = a_{p} .

Clearly j_{A}^{B} is order-preserving (because if p < q in P, then p ≤ r(q), hence a_{p} ≤ b_{p} ≤ b_{r(q)} ≤ a_{q}), and it also maps ˆ0 to ˆ0. Therefore j_{A}^{B} is a join-morphism.

Now let B^{−} = {b_{r(p)} | p ∈ P − {ˆ0}} and for any A ∈ J_{B}, write

μ(B^{−}, A) = ∏_{p∈P −{ˆ0}} μ(b_{r(p)}, a_{p}) ,

where μ(b_{r(p)}, a_{p}) denotes the Möbius function for the lattice T. Now we allow the family A to vary (i.e. a_{p} varies in [b_{r(p)}, b_{p}] for each p ≠ ˆ0) and we define

j^{B} = (−1)^{n} ∑_{A∈J_{B}} μ(B^{−}, A) j_{A}^{B} ∈ Hom_{kL}(P, T) .

By Proposition 10.2 of [BT2], f_{B} = j^{B}π^{B} is an idempotent in E and when n ≥ 0 varies and B ∈ P_{n} varies, the idempotents f_{B} are pairwise orthogonal (Corollary 10.5 of [BT2]). The idempotent e_{tot} is then defined by the formula

e_{tot} = ∑_{n=0}^{N} ∑_{B∈P_{n}} f_{B} ,

see Theorem 10.8 of [BT2]. For each B ∈ P_{n}, the idempotent f_{B} is a linear combination of join-morphisms j_{A}^{B}π^{B}. We are going to prove that most of these join-morphisms cancel pairwise in the sum

(6.2)  e_{tot} = ∑_{n=0}^{N} ∑_{B∈P_{n}}(−1)^{n} ∑_{A∈J_{B}} μ(B^{−}, A) j_{A}^{B}π^{B} .

More precisely, we consider all triples {(n, B, A) | n ∈ ℕ, B ∈ P_{n}, A ∈ J_{B}} such that a_{x} < b_{x} for some x ∈ P. For such a triple, we let p ∈ P be minimal with respect to the condition a_{p} < b_{p}. Since a_{p} ∈ [b_{r(p)}, b_{p}] for p ≠ ˆ0, we can either have b_{r(p)} = a_{p} < b_{p} or b_{r(p)} < a_{p} < b_{p}. The case p = ˆ0 is special because we always have aˆ0 = ˆ0. It follows that p must satisfy one of the following 4 cases :

A1. p ≠ ˆ0, r(p) ≠ ˆ0, b_{r(p)} = a_{p} < b_{p}, and a_{x} = b_{x} for any x < p.

A2. p ≠ ˆ0, r(p) = ˆ0, b_{r(p)} = a_{p} < b_{p}, and ˆ0 = aˆ0 = bˆ0.

B1. p ≠ ˆ0, b_{r(p)} < a_{p} < b_{p}, and a_{x} = b_{x} for any x < p.

B2. p = ˆ0 and ˆ0 = aˆ0 < bˆ0.

We now examine each of these cases successively.

Case A1. Suppose we are in Case A1. Define

P̃ = P − {p} ,  b̃_{q} = b_{q} ∀ q ∈ P̃ − {r(p)} ,  b̃_{r(p)} = b_{p} .

This defines a chain B̃ in T and a surjective join-morphism πB̃ : T → P̃, satisfying in particular πB̃(b_{p}) = r(p). Let Ã ∈ JB̃ be the family defined by

ã_{q} = a_{q} ∀ q ∈ P̃ − {r(p)} ,  ã_{r(p)} = b_{r(p)} ,

and let jB̃ ; Ã : P̃ → T be the corresponding join-morphism. Then we obtain

b_{r(r(p))} < b_{r(p)} < b_{p} ,  that is,  b̃_{r(r(p))} < ã_{r(p)} < b̃_{r(p)} ,

so that P̃ and its element r(p) are in Case B1 (because r(p) ≠ ˆ0 by assumption A1). Moreover, j_{A}^{B}π^{B} = jB̃ ; ÃπB̃. This is easy to check on most elements of T, the only nontrivial case being

j_{A}^{B}π^{B}(b_{p}) = j_{A}^{B}(p) = a_{p} = b_{r(p)} = ã_{r(p)} = jB̃ ; Ã(r(p)) = jB̃ ; ÃπB̃(b_{p}) .

Finally, since μ(b_{r(p)}, a_{p}) = μ(b_{r(p)}, b_{r(p)}) = 1, the coefficient of j_{A}^{B}π^{B} is equal to

(−1)^{n}μ(B^{−}, A) = (−1)^{n} ∏_{x∈P −{ˆ0}} μ(b_{r(x)}, a_{x})

= (−1)^{n} ∏_{x∈P −{ˆ0,p}} μ(b_{r(x)}, a_{x})

= (−1)^{n} ∏_{x∈P̃−{ˆ0}} μ(b̃_{r(x)}, ã_{x})

= −(−1)^{n−1}μ(B̃^{−}, Ã) ,

using the fact that, for x = r(p), we have ã_{r(p)} = b_{r(p)} and also a_{r(p)} = b_{r(p)} by minimality of the choice of p. This shows that

(−1)^{n}μ(B^{−}, A) j_{A}^{B}π^{B}  and  (−1)^{n−1}μ(B̃^{−}, Ã) jB̃ ; ÃπB̃

cancel in the sum (6.2). Thus any Case A1 cancels with some Case B1.

Case B1. Suppose we are in Case B1. Define

P̂ = P_{<p} ⊔ {s} ⊔ P_{≥p} ,

with the total order defined by x < s for all x ∈ P_{<p} and s < x for all x ∈ P_{≥p}, so that r(p) = s. Moreover, define

b̂_{q} = b_{q} ∀ q ∈ P̂ − {s} ,  b̂_{s} = a_{p} .

This defines a chain B̂ in T and a surjective join-morphism πB̂ : T → P̂, satisfying in particular πB̂(a_{p}) = s and πB̂(b_{p}) = p. Finally, let Â ∈ JB̂ be the family defined by

â_{q} = a_{q} ∀ q ∈ P̂ − {s} ,  â_{s} = a_{p} ,

and let jB̂ ; Â : P̂ → T be the corresponding join-morphism. Then we obtain

a_{p} < b_{p} ,  that is,  b̂_{r(p)} = â_{p} < b̂_{p} ,

so that P̂ and its element p are in Case A1 (because r(p) = s ≠ ˆ0). Applying the procedure described in Case A1, we note that P̂ − {p} is isomorphic to P and it follows easily that we recover the Case B1 we started with. Thus every Case B1 has been canceled with a corresponding Case A1.

Case A2. Suppose we are in Case A2. Since r(p) = ˆ0, p is the least element of P − {ˆ0}. Define

P̃ = P − {p} ,  b̃_{q} = b_{q} ∀ q ∈ P̃ − {ˆ0} ,  b̃ˆ0 = b_{p} .

This defines a chain B̃ in T and a surjective join-morphism πB̃ : T → P̃, satisfying in particular πB̃(b_{p}) = ˆ0. Let Ã ∈ JB̃ be the family defined by

ã_{q} = a_{q} ∀ q ∈ P̃ − {ˆ0} ,  ãˆ0 = bˆ0 ,

and let jB̃ ; Ã : P̃ → T be the corresponding join-morphism. We have ˆ0 = bˆ0 by minimality of p and we obtain

ˆ0 = bˆ0 < b_{p} ,  that is,  ˆ0 = ãˆ0 < b̃ˆ0 ,

so that P̃ and its element ˆ0 are in Case B2. The argument for the Möbius function holds in the same way as in Case A1 and it follows that any Case A2 cancels with some Case B2 in the sum (6.2).

Case B2. Suppose we are in Case B2. Define

P̂ = {s} ⊔ P ,

with the total order defined by s < x for all x ∈ P, so that r(p) = s = ˆ0P̂. Moreover, define

b̂_{q} = b_{q} ∀ q ∈ P ,  b̂_{s} = ˆ0 .

This defines a chain B̂ in T and a surjective join-morphism πB̂ : T → P̂, satisfying in particular πB̂(b_{p}) = p. Finally, let Â ∈ JB̂ be the family defined by

â_{q} = a_{q} ∀ q ∈ P ,  â_{p} = ˆ0 ,

and let jB̂ ; Â : P̂ → T be the corresponding join-morphism. Then we obtain

ˆ0 < b_{p} ,  that is,  ˆ0 = âˆ0 = b̂ˆ0 = â_{p} < b̂_{p} ,

so that P̂ and its element p are in Case A2. Applying the procedure described in Case A2, we note that P̂ − {p} is isomorphic to P and it follows easily that we recover the Case B2 we started with. Thus every Case B2 has been canceled with a corresponding Case A2.

Applying the cancelations described above, we can now eliminate all the joinmorphisms j_{A}^{B}π^{B} corresponding to a triple (n ∈ ℕ, B ∈ P_{n}, A ∈ J_{B}) satisfying a_{x} < b_{x} for some x ∈ P. We are left with the triples satisfying a_{x} = b_{x} for all x ∈ P. In such a case, we have bˆ0 = ˆ0, that is, ˆ0 ∈ B, hence B ∈ Z_{n}, where Z_{n} is the set of chains described at the beginning of Section 3. Thus the second sum in (6.2) runs over B ∈ Z_{n}. Moreover, A = B − {ˆ0} =: B^{+} and so the third sum in (6.2) reduces to a single term.

Now we have j_{B^{+}}^{B}π^{B}(t) = b_{p} if π^{B}(t) = p, that is, if t ≤ b_{p} but t ≰ b_{r(p)}. In other words,

j_{B^{+}}^{B}π^{B}(t) = min{b ∈ B | t ≤ b}

and this is exactly the definition of the endomorphism σ_{B} considered in Section 3, see the formula (3.6). Thus

j_{B^{+}}^{B}π^{B} = σ_{B} ∀ B ∈ Z_{n} .

Moreover, the coefficient of j_{B^{+}}^{B}π^{B} in the expression for e_{tot} is the Möbius function

(−1)^{n}μ(B^{−}, B^{+}) = (−1)^{n} ∏_{p∈P −{ˆ0}} μ(b_{r(p)}, b_{p}) = −μ_{Z}(B, ∞) ,

by Lemma 3.12, where the latter Möbius function is the Möbius function of the poset Z ⊔ {∞}. It follows that the expression for e_{tot} given in (6.2) reduces to

e_{tot} = ∑_{n=0}^{N} ∑_{B∈Z_{n}}(−1) μ_{Z}(B, ∞) σ_{B} = − ∑_{B∈Z} μ_{Z}(B, ∞) σ_{B} .

This completes the proof of (6.1).

6.3. Remark. It is proved in Theorem 10.6 of [BT2] that the two-sided ideal kE_{tot} is isomorphic to a product of matrix algebras

kE_{tot} ≅ ∏_{n=0}^{N} M_{|Z_{n}|}(k) ,

where N is the maximal length of a chain in T. It should be noticed that the new approach to the generating idempotent of kE_{tot} explained in the present paper does not simplify in any way the proof of this result. In particular, if T is totally ordered, then

kE = kE_{tot} ≅ ∏_{n=0}^{N} M_{|Z_{n}|}(k) ,

and this is a semi-simple algebra whenever k is a field. As noticed in Remark 11.3 of [BT2], this result is similar, but not equivalent, to a theorem proved in [FHH] about the planar rook algebra.

7. The case of pole lattices

Pole lattices are generalizations of totally ordered lattices and are defined in Section 2. They play a role in our work [BT5] on correspondence functors because they have a property of semi-simplicity analogous to the results of Remark 6.3.

We consider now the endomorphism f_{pole} obtained by using the family of pole lattices instead of totally ordered lattices. Recall from Definition 2.2 that

f_{pole} := − ∑_{α∈H_{pole}} μ_{H_{pole}}(−∞, α) α ,

where H_{pole} = {α ∈ E | α ≥ id, α(T) ∈ pole}. Recall also that kE_{pole} denotes the two-sided ideal generated by all endomorphisms α ∈ E whose image α(T) is a pole lattice. In analogy with Theorem 5.1, we conjecture the following :

7.1. Conjecture. Let f_{pole} be as as above.

(a) f_{pole} is a central idempotent of kE.

(b) f_{pole} is the identity element of the ideal kE_{pole}.

By Theorem 8.3 of [BT5], we know that kE = kE_{pole} ⊕ D′ for some two-sided ideal D′. The projection of id onto the first factor is an idempotent e_{pole} satisfying the two properties of the Conjecture, but constructed in a very different way. Thus Conjecture 7.1 asserts in fact that f_{pole} = e_{pole}.

By using the computer system [GAP4], we can prove that Conjecture 7.1 holds in very many cases. Actually, the idea of the formula in Definition 2.2 slowly emerged in 2018 from complicated calculations using already the GAP software [GAP4], based on existing lists of finite posets (either from a personal database or from [Mc]).

In order to verify the conjecture, we consider a finite partially ordered set E and the lattices admitting E as subposet of join-irreducible elements. As there are in general too many such lattices for any given E, we deal only with two of them. The largest one is the distributive lattice D(E) of all lower subsets of E. There is also a smallest such lattice K(E), which is not hard to construct as the image of a suitable closure operator in the lattice D(E).

We have checked Conjecture 7.1 for D(E) and |E| ≤ 5, and for K(E) and |E| ≤ 8. In the case |E| = 8, two weeks were needed for running the program for K(E). In the next case, there are so many posets E of cardinal 9 that running the program for K(E) and |E| = 9 would require too much time.

7.2. Remark. In the case of totally ordered lattices, we were able to replace H_{tot} by the subposet G_{tot} of all α ∈ H_{tot} satisfying the additional condition α^{2} = α. This was achieved by means of a homotopy equivalence G_{tot,<τ} ↪ H_{tot,<τ} (see Lemma 4.2). No such reduction is possible in the case of pole lattices, because in the sum defining f_{pole} it may happen that some terms α are not idempotent (i.e. α^{2} ≠ α). The first instance of this occurs with the lattice

[figure: ˆ1; v; w; u; ˆ0]

If [x, y, z] denotes the endomorphism of this lattice such that u ↦ x, v ↦ y, w ↦ z, we obtain

f_{pole} = [v, ˆ1, ˆ1] − [u, ˆ1, ˆ1] − [v, v, ˆ1] − [v, ˆ1, w] + [u, v, ˆ1] + [u, ˆ1, w] + [v, v, w]

and neither [v, ˆ1, ˆ1] nor [v, ˆ1, w] are idempotent.

On the other hand, we easily get

f_{tot} = −[ˆ1, ˆ1, ˆ1] + [u, v, ˆ1] + [ˆ1, ˆ1, w] .

7.3. Remark. One may wonder if the definition of f_{F} for other families F gives rise to a central idempotent generating the ideal kE_{F}. Despite various attempts, we could not find any sufficient evidence for a result of this kind. There is a positive answer in many small cases, but also a negative one in some easy cases, such as the family tot(r) of all totally ordered lattices of size ≤ r. The two-sided ideal kE_{tot(r)} is generated by a central idempotent, but its expression may involve endomorphisms α of T which do not satisfy the condition α ≥ id. This is obvious when r = 1 since the constant map cˆ0 onto ˆ0 is the only join-endomorphism of T whose image is totally ordered of size 1 (so that kE_{tot(1)} has rank one), but cˆ0 ≱ id (whenever |T | ≥ 2).

References

[BT1] S. Bouc, J. Thévenaz. Correspondence functors and finiteness conditions, J. Algebra 495 (2018), 150–198.

[BT2] S. Bouc, J. Thévenaz. Correspondence functors and lattices, J. Algebra 518 (2019), 453–518.

[BT3] S. Bouc, J. Thévenaz. The algebra of Boolean matrices, correspondence functors, and simplicity, J. Comb. Algebra 4 (2020), 215–267.

[BT4] S. Bouc, J. Thévenaz. Tensor product of correspondence functors, J. Algebra 558 (2020), 146–175.

[BT5] S. Bouc, J. Thévenaz. Simple and projective correspondence functors, Representation Theory 25 (2021), 224–264.

[FHH] D. Flath, T. Halverson, K. Herbig. The planar rook algebra and Pascal’s triangle, Enseign. Math. 55 (2009), no. 1-2, 77–92.

[GAP4] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.11.0, 2020.

 http://www.gap-system.org

[Mc] B. D. McKay. Digraphs.

 http://users.cecs.anu.edu.au/bdm/data/digraphs.html

[Qu] D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. Math. 28 (1978), 101–128.

[St] R. P. Stanley. Enumerative Combinatorics, Vol. I, Second edition, Cambridge studies in advanced mathematics 49, Cambridge University Press, 2012.