The canonical join complex

In this paper, we study the combinatorics of a certain minimal factorization of the elements in a finite lattice $L$ called the canonical join representation. The join $\bigvee A =w$ is the canonical join representation of $w$ if $A$ is the unique lowest subset of $L$ satisfying $\bigvee A=w$ (where"lowest"is made precise by comparing order ideals under containment). When each element in $L$ has a canonical join representation, we define the canonical join complex to be the abstract simplicial complex of subsets $A$ such that $\bigvee A$ is a canonical join representation. We characterize the class of finite lattices whose canonical join complex is flag, and show how the canonical join complex is related to the topology of $L$.


Introduction
In a finite lattice L, the canonical join representation of an element w is a certain unique minimal factorization of w in terms of the join operation. Specifically, the join-representation Ž A " w is the canonical join representation of w if the join Ž A is irredundant and the set A is taken as low as possible in the partial order on L. (See Section 3.1 for the precise definition.) There is an analogous factorization in terms of the meet operation called the canonical meet representation that is defined dually (replacing " Ž " with " Ź " and "lowest" with "highest" in the sentence above). The canonical join representation or canonical meet representation for a given element may not exist. See Figure 3 for two examples. If each element in L has a canonical join representation then L is join-semidistributive. We say that L is semidistributive if each element also has a canonical meet representation. (See Section 3.1 and Theorem 3.1 in particular for an equivalent definition.) When L is join-semidistributive, we define the canonical join complex to be the abstract simplicial complex whose faces are the subsets A Ă L such that the join Ž A is a canonical join representation. (Proposition 3.7 says that this is indeed a complex.) We define the canonical meet complex similarly. Recall that a simplicial complex is flag if it is the clique complex of its 1-skeleton, or equivalently, its minimal non-faces have size two. Our main result is: In other words, if each element in L admits a canonical join representation, then the canonical join complex for L is flag if and only if each element also admits a canonical meet representation. In light of Theorem 1.1, we define the canonical join graph for L to be the one-skeleton of its canonical join complex. Canonical join representations and the canonical join graph appear in many familiar guises. See Section 2 for connections to comparability graphs and noncrossing partitions.
It is not hard to find examples of finite join-semidistributive lattices whose canonical join complex is not flag. A key example is shown below in Figure 1 that each pair of atoms in the lattice in Figure 1 is a face in the canonical join complex. Since the join of all three atoms is redundant (because we can remove b and obtain the same join), the canonical complex is an empty triangle. Note that the bottom element0 of this lattice does not have a canonical meet representation: Both a^e and c^d are minimal, highest meet-representations for0. We will see below that the combinatorics of the canonical join complex (and canonical meet complex) are closely related to the topology of the its lattice.
Recall that the crosscut complex of L is the abstract simplicial complex whose faces are the subsets A 1 of atoms in L such that Ž A 1 ă1. A lattice is crosscutsimplicial if for each interval rx, ys the join of each proper subset of atoms in rx, ys is strictly less than y. Recall that the order complex of a finite poset P is homotopy equivalent to its crosscut complex ( [4,Theorem 10.8]). Therefore, if L is crosscut-simplicial then each interval rx, ys in L is either contractible or homotopy equivalent to a sphere with dimension two less than the number of atoms in rx, ys (see also [19,Theorem 3.7]). In particular, µpx, yq P t´1, 0, 1u.
Observe that the facets of the crosscut complex for the lattice L in Figure 1 are ta, bu and tb, cu. Therefore, L is not crosscut-simplicial. By contrast, Hersh and Mészáros recently showed that a large class of finite semidistributive latticesincluding the class of finite distributive lattices, the weak order on a finite Coxeter group, and the Tamari lattice ([19, Theorems 5.1, 5.3 and 5.5])-are crosscutsimplicial. Building on this work, McConville proved that if L is semidistributive, then it is crosscut-simplicial ( [22,Theorem 3.1]). When each element in L has a canonical join representation, we prove that the converse is true. Theorem 1.2. Suppose that L is a finite join-semidistributive lattice. The following are equivalent: (1) The canonical join complex for L is flag.
As an immediate corollary, we obtain the following topological obstruction to the flag-property of the canonical join complex. Corollary 1.3. Suppose that L is a finite join-semidistributive lattice and its canonical join complex is flag. Then: (1) Each interval rx, ys in L is either contractible or homotopy equivalent to S d´2 , where d is the number of atoms in rx, ys; (2) The Möbius function takes only the values t´1, 0, 1u on the intervals of L.
McConville showed in [22,Corollary 5.4] that if L is crosscut-simplicial then so is each of its lattice quotients. Because semidistributivity is preserved under taking sublattices and quotients when L is finite (see Section 4.1), we immediately obtain the following extension of McConville's result for finite join-semidistributive lattices.
Corollary 1.4. Suppose that L is a finite join-semidistributive lattice that is crosscutsimplicial. Then each sublattice and quotient lattice of L is also crosscut-simplicial. Theorem 1.1 is surprising in part because its proof does not explicitly use the canonical meet representation of the elements in L. Instead, we make use of local characterization of canonical join representations in terms of the cover relations, and a bijection κ from the join-irreducible to the meet-irreducible elements in L.
As an easy consequence of this approach, we obtain the following nice result: Corollary 1.5. Suppose that L is a finite semidistributive lattice. Then the bijection κ induces an isomorphism from canonical join complex to the canonical meet complex of L.
Using the isomorphism from Corollary 1.5, one obtains an operation on the canonical join complex that generalizes the operation of rowmotion (on the set of antichains in a poset) and the operation of Kreweras complementation (on the set of noncrossing partitions). See Remark 3.15.
The canonical join complex was first introduced in [26], in which Reading showed that it is flag for the special case of the weak order on the symmetric group (see Example 2.6). Recently, canonical join representations have played a role in the study of functorially finite torsion classes for the preprojective algebra of Dynkintype W , when W is a simply laced Weyl group (see for example [15,21]). In the forthcoming [11], the authors study the canonical join complex for any finite dimensional associative algebra Λ of finite representation type. Since the weak order on any finite Coxeter group W and the lattice of torsion classes for Λ of finite representation type are both examples of finite semidistributive lattices (see [7,Lemma 9] and [15,Theorem 4.5]), we obtain the following two applications of Theorem 1.1: Corollary 1.6. Suppose that W is a finite Coxeter group. Then the canonical join complex for the weak order on W is flag. Corollary 1.7. Suppose that Λ is an associative algebra of finite representation type, and torspΛq is its lattice of torsion classes ordered by containment. Then the canonical join complex for torspΛq is flag.

Motivation and Examples
Before we give the technical background for our main results, we describe several familiar examples in which the combinatorics of canonical join representations appear. We begin with an example from number theory and commutative algebra.
Example 2.1 (The divisibility poset). It is often useful to give a canonical factorization of the elements in a set of equipped with some algebraic structure. A familiar example from number theory and commutative algebra is the primary decomposition of ideals. The canonical join representation is the natural lattice-theoretic analogue. Indeed, when L is the the divisibility poset (whose elements are the positive integers ordered r ď s if and only if r|s), the canonical join representation of x P L coincides with the primary decomposition of the ideal generated by x: x " ł tp d : p is prime and p d is the largest power of p dividing xu.
Suppose that L is a finite lattice, such that each element in L admits a canonical join representation. One pleasant property of the canonical join representation (and its dual, the canonical meet representation) is that it "sees" the geometry the Hasse diagram for L. Suppose that w P L has the canonical join representation Ž A. We will shortly prove that the factors that appear in A are naturally in bijection with the elements covered by w. So, the down-degree of w is equal to the size of A. Specifically, we will prove the following proposition (see Lemma 3.3 and Proposition 3.4): w is a face in the canonical join complex for L. Then, for each element y that is covered by w there is a corresponding element j P A such that j _ y " w, and j is the unique minimal element in L with this property. The correspondence y Þ Ñ j is a bijection.
With this proposition in mind, we consider the class of finite distributive lattices. Example 2.3 (Finite distributive lattices). Suppose that L is a finite distributive lattice. Recall that the fundamental theorem of finite distributive lattices (see for example [31,Theorem 3.4.1]) says that L is the lattice JpPq of order ideals of some finite poset P. Suppose that A is an antichain in P. We write I A for the order ideal generated by A (that is, the elements of A are the maximal elements of I A ). Dually, we write I A for the order ideal satisfying: A is the set of minimal elements in PzI A . Observe that the order ideals covered by I A are exactly of the form I Aztyu " I A ztyu, where y P A. Since I y is the smallest order ideal in JpPq containing y, it follows immediately from Proposition 2.2 that the canonical join representation of I A is Ť tI y : y P Au. (Dually, the canonical meet representation for the ideal I A is Ş tI y : y P Au.) It follows that the canonical join graph of JpPq is the incomparability graph of P.  Table 1] or the complement of any graph appearing in [34, Table 2].
As an immediate corollary we have the following characterization of the canonical join graphs for finite distributive lattices. Each finite distributive lattice is, in particular, semidistributive. By Theorem 1.1, we obtain a complete characterization of the canonical join complexes for finite distributive lattices.
Proposition 2.5. The graph G is the canonical join graph for a finite distributive lattice if and only if G does not contain, as an induced subgraph, the complement of any graph forbidden by Theorem 2.4.
Example 2.6 (The Symmetric group and noncrossing diagrams). Recently Reading gave an explicit combinatorial model for the canonical join complex of the weak order on the symmetric group S n in terms of certain noncrossing diagrams. A noncrossing diagram is a diagram consisting of n vertices arranged vertically, together with a collection of curves called arcs that must satisfy certain compatibility conditions. In particular, the arcs in a noncrossing diagram do not intersect in their interiors (see [26] for details). Each diagram is determined by its combinatorial data: the endpoints of its arcs, and on which side (either left or right) each arc passes the vertices in the diagram. For example, a we say that a diagram contains only left arcs if it has no arc that passes to the right of any vertex (see the leftmost noncrossing diagram in Figure 2). We say that two arcs are compatible if there is a noncrossing diagram that contains them. The following is a combination of [26, Corollary 3.4 and Corollary 3.6]. (In the statement of the Theorem, we take "a collection of arcs" to also mean collection of noncrossing diagrams, each containing a single arc.) Theorem 2.7. There is a bijection δ from the set of join-irreducible permutations in S n to the set of noncrossing diagrams on n vertices with a unique arc. Moreover, a collection of arcs E corresponds to a face in the canonical join complex for S n if and only if the arcs in E are pairwise compatible.
Example 2.8 (The Tamari lattice and noncrossing partitions). We conclude our list of examples by considering the Tamari lattice. The Tamari lattice T n is a finite semidistributive lattice (see for example [18,Theorem 3.5]), which can be realized as an ordering on the set of triangulations for a fixed convex pn`3q-gon P n . Recall that the rank n associahedron is a simple convex polytope, whose faces are in bijection with the collections of pairwise noncrossing diagonals of P n (see [13, Figure 3.5]). In particular, its vertices are parametrized by triangulations of P n in such a way that we obtain the Hasse diagram for T n as an orientation of its 1skeleton. Since the number of factors in a canonical join representation (called the canonical joinands) for w P T n is equal to the down-degree of w, we obtain the following result: Proposition 2.9. The f -vector for the canonical join complex of the Tamari lattice T n is equal to the the h-vector of the rank n associahedron. Specifically, the number of size-k faces in the canonical join complex is equal to the Narayana number N pn, kq " 1 n`1ˆn`1 k`1˙ˆn`1 k˙.
Indeed, the canonical join representation of w P T n is essentially a noncrossing partition. It is well-known that the Tamari lattice T n may be realized as the set of permutations avoiding the 231-pattern. It is a fact that a permutation avoids the 231-pattern if and only if its image under the bijection δ from Theorem 2.7 is a noncrossing arc diagram consisting of only left arcs. Rotating such a diagram by a quarter-turn gives the familiar representation of a noncrossing partition as a bump diagram. (See [26,Example 4.5] for details, and [28, Theorem 2.7] and the discussion following [28,Proposition 8.8] for a type-free discussion.)

Finite semidistributive lattices
3.1. Definitions. In this paper, we study only finite lattices. We write0 for the unique smallest element in L and1 for the unique largest element. A joinrepresentation of w is an expression Ž A which evaluates to w in L. At times we will also refer to the set A as a join-representation. We write cov Ó pwq for the set ty P L : wą yu. Similarly, we write cov Ò pwq for the set of upper covers of w. Recall that w is join-irreducible if w " Ž A implies that w P A. (In particular, the bottom element0 is not join-irreducible, because it is equal to the empty join.) Since L is finite, w is join-irreducible when cov Ó pwq has exactly one element. Meet-irreducible elements satisfy the dual condition. We write IrrpLq for the set of join-irreducible elements of L.
A join-representation Each irredundant join-representation is an antichain in L. We say that the subset A of L join-refines a subset B if for each element a in A, there exists some element b in B such that a ď b. Join-refinement defines a preorder on the subsets of L that is a partial order (corresponding to the containment of order ideals) when restricted to the set of antichains in L.
We write ijrpwq for the set of irredundant join-representations of w. The canonical join representation of w in L, when it exists, is the unique minimal element, in the sense of join-refinement, of ijrpwq. We write canpwq for the canonical join representation of w. An element j P canpwq is a canonical joinand for w. If A " canpwq, we say that A joins canonically . It follows immediately from the definition that each canonical joinand of w is join-irreducible. Moreover, the canonical join representation of each join-irreducible element j exists and is equal to tju. The canonical meet representation of w (when it exists) is defined dually.
In Figure 3, we give two examples in which the canonical join representation of 1 does not exist.  In the modular lattice on the left each pair of atoms is a lowest-possible, irredundant join-representation for the top element. Since there is no unique such join-representation, the canonical join representation for1 does not exist. Arguing dually, we see that the canonical meet representation for the bottom element0 does not exist either. In the lattice on right, each element has a canonical meet representation. However, both a_d and b_c are minimal elements of ijrp1q. Again, the canonical join representation of1 does not exist.
In the lattice on the right, we observe the following failure of the distributive law: both e _ a and e _ b are equal to the top element, but e _ pa^bq is equal to e. (A similar failure is easily verified among the atoms of the modular lattice.) We will see that correcting for precisely this kind of failure of distributivity guarantees the existence of canonical join representations when L is finite.
A lattice L is join-semidistributive if L satisfies the following implication for every x, y and z: L is meet-semidistributive if it satisfies the dual condition: (SD^) If x^y " x^z, then x^py _ zq " x^y A lattice is semidistributive if it is join-semidistributive and meet-semidistributive.
The following result, the finite case of [12,Theorem 2.24], says that this definition is equivalent to one given in the introduction. Assume that L is a finite join-semidistributive lattice, and let j P IrrpLq. We write j˚for the unique element covered by j, and Kpjq for the set of elements a P L such that a ě j˚and a ě j. When it exists, we write κpjq for the unique maximal element of Kpjq. It is immediate that κpjq is meet-irreducible. Below, we quote [12, Theorem 2.56]: Proposition 3.2. A finite lattice is meet-semidistributive if and only if κpjq exists for each join-irreducible element j.
Suppose that w P L. For each y P cov Ó pwq, there is some element j P canpwq such that y _ j " w (because there is some element j P canpwq such that j ď y). For this j, the set canpwq join-refines tj, yu. Because canpwq is an antichain, each j 1 P canpwqztju satisfies j 1 ď y. Therefore, j is the unique canonical joinand of w such that y _ j " w. We define a map η : cov Ó pwq Ñ canpwq which sends y to the unique canonical joinand j such that y _ j " w. Lemma 3.3. Suppose that L is a finite join-semidistributive lattice, and w P L. Then the map η : cov Ó pwq Ñ canpwq is a bijection such that y ě Ž canpwqztηpyqu and y P Kpηpyqq for each y P cov Ó pwq.
Proof. Suppose there exist distinct y and y 1 in cov Ó pwq satisfying ηpyq " ηpy 1 q. Then, y _ y 1 " w, and canpwq does not join-refine ty, y 1 u (because ηpyq is below neither y nor y 1 ). We have a contradiction, because canpwq is the unique minimal element (in join-refinement) of ijrpwq. By this contradiction, we conclude that η is injective. Suppose that j P canpwq. Since Ž canpwq is irredundant, Ž pcanpwqztjuq ă w. Thus, there is some y P cov Ó pwq such that y ě Ž pcanpwqztjuq. If y ě j then y " w, and that is absurd. We conclude that j " ηpyq, and that η is a bijection.
We have already argued, in the paragraph above the statement of the proposition, that y ě Ž canpwqztηpyqu. To complete the proof, suppose that y _ ηpyq˚" w. Since, canpwq does not join-refine ty, ηpyq˚u (because ηpyq ď ηpyq˚and ηpyq ď y), we obtain a contradiction as above. We conclude that y _ ηpyq˚ă w. Since y is covered by w, we have y _ ηpyq˚" y. Thus, y P Kpηpyqq, for each y P cov Ó pwq.
As a consequence we obtain a proof of Proposition 2.2, which we restate here with the notation from of Lemma 3.3.
Proposition 3.4. Suppose that L is a finite join-semidistributive lattice, and y is covered by w in L. Then, ηpyq is the unique minimal element of L such that ηpyq _ y " w.
Proof. Suppose that x P L has x _ y " w. Since canpwq join-refines tx, yu and ηpyq and y are incomparable, we conclude that ηpyq ď x.
In fact, the previous proposition characterizes of finite join-semidistributive lattices. (Similar constructions exist; for example, see the proof of [1, Theorem 3-1.4].) Because the proof is similar to the proof of Lemma 3.3, we leave the details to the reader.
Proposition 3.5. Suppose that L is a finite lattice. The following conditions are equivalent: (1) For each w P L, there is a unique minimal element ηpyq P L satisfying y _ ηpyq " w, for each y P cov Ó pwq.
Suppose that L is a finite join-semidistributive lattice, j P IrrpLq and F is a face of the canonical join complex for L. The following will be useful for determining when F Y tju is also a face.
Lemma 3.6. Suppose that L is a finite join-semidistributive lattice and j P IrrpLq. Then j is a canonical joinand of y _ j, for each y P Kpjq. In particular, j is a canonical joinand of Proof. If y " j˚, then the first statement is obvious (because tju is the canonical join representation), so we assume that y and j are incomparable. We write w for the join j _ y, and we write A " tj 1 P canpwq : j 1 ď ju and A 1 " tj 1 P canpwq : j 1 ď yu. Because canpwq join-refines tj, yu, we have A Y A 1 " canpwq. Also, the set A is not empty because the join y _ j is irredundant. We want to show that A " tju. Since j is join-irreducible, it is enough to show that j " then j˚_ y " j _ y, and that is impossible because y P Kpjq. We conclude that j is a canonical joinand of y _ j. If The remaining direction of the second statement is straightforward to verify.
We close this subsection by quoting the following easy proposition (for example see [26,Proposition 2.2]), which says that the canonical join complex is indeed a simplicial complex.
Proposition 3.7. Suppose L is a finite lattice, and the join Ž A is a canonical join representation in L. Then each proper subset of A also joins canonically.
3.2. The flag property. In this section we prove Theorem 1.1. We begin by presenting the key arguments in one direction the proof: If L is a finite semidistributive lattice, then its canonical join complex is flag. Most of the work is done in the following two lemmas.
Lemma 3.8. Suppose that L is a finite semidistributive lattice, and F is a subset of IrrpLq such that |F | ě 3 and each proper subset of F is a face in the canonical join complex for L. Then the joins Ž pF ztjuq and Ž pF ztj 1 uq are incomparable for each distinct j and j 1 in F .
Proof. Without loss of generality we assume that Ž F "1. Suppose there exists distinct j, j 1 P F such that Ž pF ztjuq ě Ž pF ztj 1 uq. On the one hand, we have Ž pF ztjuq _ Ž pF ztj 1 uq " Ž F "1. On the other hand, Ž pF ztjuq _ Ž pF ztj 1 uq is equal to Ž pF ztjuq. Thus, Ž pF ztjuq "1. Since F has at least three elements, there exists j 2 P F ztj, j 1 u. We write w 1 for Ž pF ztj 1 uq and w 2 for Ž pF ztj 2 uq. Because both F ztj 1 u and F ztj 2 u are faces in the canonical join complex, j is a canonical joinand for both w 1 and w 2 . Lemma 3.3 implies that there exists y 1 P cov Ó pw 1 q and y 2 P cov Ó pw 2 q such that y 1 , y 2 P Kpjq. Moreover, y 1 ě Ž pF ztj, j 1 uq and similarly y 2 ě Ž pF ztj, j 2 uq. So, we have: Since Ž pF ztjuq "1 we conclude that Ž Kpjq "1, contradicting Proposition 3.2.
Lemma 3.9. Suppose that L is a finite join-semidistributive lattice, and F is a subset of IrrpLq satisfying the following conditions: First, |F | ě 3; second, each proper subset of F is a face in the canonical join complex for L; third, Ž F is irredundant; fourth F is not a face of the canonical join complex. Then there exists j P F such that κpjq does not exist.
Proof. Without loss of generality, we assume that Ž F "1. Since the join Ž F is irredundant, there exists some j P F such that j R canp1q. Lemma 3.6 implies that j˚_ Ž pF ztjuq "1. Let j 1 and j 2 be distinct elements in F ztju. As in the proof of Lemma 3.8, let y 1 and y 2 be the unique elements covered by Ž F ztj 1 u and Ž F ztj 2 u, respectively, with y 1 _ j " Ž F ztj 1 u and y 2 _ j " Ž F ztj 2 u. By Lemma 3.3, y 1 and y 2 are both members of Kpjq, so that y 1 , y 2 ě j˚. Also y 1 ě Ž pF ztj, j 1 uq and y 2 ě Ž pF ztj, j 2 uq. Therefore, y 1 _ y 2 ě j˚_ Ž pF ztjuq "1. The statement follows.
Proof of one direction of Theorem 1.1. We show that if L is semidistributive, then its canonical join complex is flag. Suppose that F Ă IrrpLq such that |F | ě 3 and each proper subset of F is a face of the canonical join complex. By Lemma 3.9, it is enough to show that Ž F is irredundant. Without loss of generality, assume that Ž F "1. Lemma 3.8 says that for each distinct j and j 1 in F , the joins Ž pF ztjuq and Ž pF ztj 1 uq are incomparable. So, for any distinct j and j 1 in F , we have Ž pF ztjuq ă Ž pF ztjuq _ Ž pF ztj 1 uq which is1. We conclude that Ž F is irredundant, and thus a face of the canonical join complex.
We now turn to the other direction of Theorem 1.1. In the following lemmas we will assume that L is a finite join-semidistributive lattice which fails SD^. By Proposition 3.2, there is some j P IrrpLq such that κpjq does not exist. Our goal is to construct a set A Ă IrrpLq satisfying: (1) A Y tju is not a face in the canonical join complex for L and (2) each pair of elements in A Y tju is a face in the canonical join complex. The essential idea is that among all Y Ă IrrpLq satisfying (1), a set A chosen as low as possible in L will also satisfy (2). For us, "as low as possible in L" means that A is chosen to be minimal in join-refinement. The argument is somewhat delicate because join-refinement is a preorder, not a partial order, on subsets of L. So, we must take extra care to compare only antichains Y Ă IrrpLq satisfying (1). To further emphasize this point, we write A ! B when A join-refines B, for antichains A and B. We write A j for the collection of antichains Y Ď Lztju satisfying Y Y tju is an antichain. Lemma 3.10. Suppose that L is a finite join-semidistributive lattice and j is in IrrpLq such that κpjq does not exist. Let X denote the set of j 1 P IrrpLqztju such that j 1 _ j is a canonical join representation. Then: If a is not less than y, then y _ a " Ž X _ j. Proposition 3.4 says that j is the unique minimal element of L whose join with y is equal to Ž X _ j. Therefore, j ď a, contradicting the fact that a P Kpjq. We conclude that a ď y. We have proved that y " κpjq, contradicting our hypothesis. Thus, Ž X _ j " Ž X _ j˚. For the second statement, observe that if X is empty, then Lemma 3.6 implies that Kpjq " tj˚u, contradicting the assumption that κpjq does not exist. We conclude that X is nonempty. Since the antichain of maximal elements Y Ď X satisfies Ž Y " Ž X, we have the desired result.
there is a nonempty minimal (in join-refinement) antichain. In particular, there is an antichain that is minimal with this property among the antichains of the set X " tj 1 P IrrpLqztju : j 1 _ j is a canonical join representationu. For this antichain A, we have that A Y tju is not a face of the canonical join complex, while ta, ju is a face, for each a P A. The next two lemmas are key in showing that ta, a 1 u is a face in the canonical join complex for pair a, a 1 P A.
Before we begin, we point out two easy observations about the join-refinement relation. Lemma 3.11. Suppose that L is a finite join-semidistributive lattice, and j is in IrrpLq such that κpjq does not exist. Among all nonempty antichains Y in A j such that Ž Y _ j " Ž Y _ j˚, let B be minimal in join-refinement. Then the join Ž pBztbuq _ j is a canonical join representation, for each b P B.
Proof. Lemma 3.10 implies that such an antichain B exists. Observe that Bztbu ! B, by (JR1). We conclude that Ž pBztbuq _ j˚ă Ž pBztbuq _ j. Lemma 3.6 says that j is a canonical joinand of Ž pBztbuq _ j and not a canonical joinand of Ž B _ j. Thus, Let C Y j be the canonical join representation of Ž pBztbuq _ j. If we have Ž pC Y tbuq _ j˚ă Ž pC Y tbuq _ j then Lemma 3.6 says that j is a canonical joinand of Ž pC Y tbuq _ j " Ž B _ j. That is a contradiction. Therefore, Ž pC Y tbuq _ j " Ž pC Y tbuq _ j˚. We claim that C Y tbu " B. Since C Y tju is the canonical join representation for Ž pBztbuq _ j, we have C Y tju ! pB Y tjuqztbu. By (JR2), we have C ! Bztbu. If C Y tbu is an antichain, then applying (JR2) again, we get C Y tbu ! B. By minimality of B we conclude that C Y tbu " B, as desired.
So, we assume that C Y tbu is not an antichain. By (3.1) we have Therefore, there exists no c P C with b ď c. Let C 1 be the set of all c P C with c ă b. We make three easy observations: First, pCzC 1 q Y tbu is member of A j . Second, applying (JR1) to the relation C ! Bztbu, we have that CzC 1 ! Bztbu. By (JR2), we conclude that pCzC 1 q Y tbu ! B. Third, we have: Therefore, by the minimality of B, we have B " pCzC 1 q Y tbu. Since C ! Bztbu, we have that C join-refines its proper subset CzC 1 . That is a contradiction (because C is an antichain). Thus, C 1 is empty. We have proved the desired result.
Lemma 3.12. Suppose that L is a finite join-semidistributive lattice and j P IrrpLq such that κpjq does not exist. Let X be the set of j 1 P IrrpLqztju such that j 1 _ j is a canonical join representation. Let A be nonempty and minimal in join-refinement among all antichains Y Ď X such that Then A is minimal among all elements in A j , in join refinement, with this property.
Proof. Lemma 3.10 implies that such an antichain A exists. Suppose that B P A j satisfies Ž B _ j " Ž B _ j˚, and B ! A. Without loss of generality, assume that B is minimal in join-refinement with this property. If B has two or more elements, then Lemma 3.11 implies that B Ă X. Therefore, B " A. Thus we can assume that B " tbu. Since B join-refines A, there is some a P A such that b ď a. Write w for the element a _ j. Since a _ j is the canonical join representation of w, Lemma 3.3 implies that cov Ó pwq has precisely two elements, y and y 1 . Let ηpyq " j and ηpy 1 q " a, so that y P Kpjq and y ě a. Thus, we have b ď a ď y. On the one hand, pb _ jq _ y " pb _ j˚q _ y " y. On the other hand, b _ pj _ yq " b _ w " w. By this contradiction, we have proved the result.
As in the previous lemma, let A be minimal (in join-refinement) among all of the antichains Y Ď X with the property that In the next lemma, we show that each pair ta, a 1 u in A is canonical join representation.
Lemma 3.13. Suppose that L is a finite join-semidistributive lattice and j P IrrpLq such that κpjq does not exist. Let X be the set of j 1 P IrrpLqztju such that j 1 _ j is a canonical join representation. Let A be nonempty and minimal in join-refinement among all antichains Y Ď X such that Then each pair of elements in A is a face in the canonical join complex.
Proof. Lemma 3.12 says that A is minimal (in join-refinement) in A j among all B P A j with the property that Ž B _ j " Ž B _ j˚. So, Lemma 3.11 says that for each a P A, the join Ž pAztauq _ j is a canonical join representation. If A has three or more elements, then each pair of elements joins canonically by Proposition 3.7. Assume that A has two elements, a 1 and a 2 . Minimality of A (in join-refinement) implies that the join a 1 _ a 2 is irredundant. We will argue that a 1 is a canonical joinand of a 1 _ a 2 , and complete the proof by symmetry.
Assume that pa 1 q˚_a 2 " a 1 _a 2 . We observe that pa 1 q˚_a 2 _j " pa 1 q˚_a 2 _j˚. Since A Ď X, we have that both ta 1 , ju and ta 2 , ju are faces in the canonical join complex. If pa 1 q˚ă j, then we have a 2 _ j " a 2 _ j˚, contradicting Lemma 3.6. Also, j ď pa 1 q˚because a 1 is not comparable to j. So, we have tpa 1 q˚, a 2 u P A j with pa 1 q˚_ a 2 _ j " pa 1 q˚_ a 2 _ j˚, and tpa 1 q˚, a 2 u join-refines ta 1 , a 2 u. But this contradicts Lemma 3.12 which says that A is minimal in A j . By this contradiction, we conclude that pa 1 q˚_ a 2 ă a 1 _ a 2 . Lemma 3.6 says that a 1 is a canonical joinand of a 1 _ a 2 .
Finally, we complete the proof of the main result.
Proof of the remaining direction of Theorem 1.1. We show that if L is a finite joinsemidistributive lattice and the canonical join complex for L is flag, then L is semidistributive. By Proposition 3.2, it is enough to show that for each j P IrrpLq the element κpjq exists.
Suppose j P IrrpLq and κpjq does not exist. As above, let X be the set of j 1 P IrrpLqztju such that j 1 _ j is a canonical join representation. Among all nonempty antichains of X, choose A to be minimal in join-refinement with the property that Ž A _ j " Ž A _ j˚. Lemma 3.10 implies that such an antichain A exists. Lemma 3.6 implies that A Y tju is not face of the canonical join complex. Since A Ď X, we have that ta, ju is a face of the canonical join complex, for each a P A. In particular, A has at least two elements. Finally, Lemma 3.13 says that ta, a 1 u is face in the canonical join complex, for each pair a, a 1 P A. We have reached a contradiction to our hypothesis that the canonical join complex is flag. By this contradiction, we conclude that L is semidistributive.
Suppose that m is meet-irreducible and write m˚for the unique element covering m. When it exists, let κ˚pmq be the unique smallest element j P L with j ď må nd j ď m. It is immediate that κ˚pmq is join-irreducible. Proposition 3.2, applied to the dual lattice, says that L is meet-semidistributive if and only if κ˚pmq exists for each meet-irreducible element m. In fact, L is semidistributive if and only if κ is a bijection, with inverse map κ˚; this is the finite case of [12, Corollary 2.55]. Applying the dual argument for the canonical meet complex, we immediately obtain the following result. (Recall that Theorem 3.1 says that each element in L has a canonical meet representation if and only if L is meet-semidistributive.) Corollary 3.14. Suppose that L is a finite meet-semidistributive lattice. Then, the canonical meet complex for L is flag if and only if L is semidistributive.
Next, we prove Corollary 1.5 by showing that the bijection κ taking a joinirreducible element j to κpjq induces an isomorphism from the canonical join complex of L to the canonical meet complex of L.
Proof of Corollary 1.5. Corollary 3.14 says that the canonical meet complex of L is flag, so it is enough to show that κ bijectively maps edges of the canonical join complex to edges of the canonical meet complex. Suppose that tj 1 , j 2 u is a face of the canonical join complex, and write m 1 for κpj 1 q and m 2 for κpj 2 q. Suppose that m 1^m2 " pm 1 q˚^m 2 . Lemma 3.3 implies that there exists some y P cov Ó pj 1 _ j 2 q satisfying : j 1 ď y ď κpj 2 q (see Figure 4 for an illustration). Since j 1 ď pm 1 q˚, we conclude that j 1 ď pm 1 q˚^m 2 " m 1^m2 . We see that j 1 ď m 1 and that is a contradiction. Therefore, pm 1 q˚^m 2 ą m 1^m2 . By the dual statement of Lemma 3.6, we conclude that m 1 is a canonical meetand of m 1^m2 , and by symmetry m 2 is also a canonical meetand of m 1^m2 . The dual argument establishes the desired isomorphism.
We close this section by relating Corollary 1.5 to Example 2.3 and Example 2.8, from Section 2.
Remark 3.15. Suppose that F is a face of the canonical join complex for a finite semidistributive lattice L. Corollary 1.5 says that Ž κpF q is a canonical meet representation. By taking the canonical join representation of Ž κpF q, we can view the map κ as an operation on the canonical join complex. Similarly, we can view κ˚as an action on the canonical meet complex.
The main premise of [2] is that the action of Kreweras complementation on the set of noncrossing partitions and the action of Panyshev complementation on the set of nonnesting partitions (that is, the set of antichains in the root poset for a finite cystrallographic root system) coincide. Indeed, both maps are an instance of the operation of κ (or κ˚) on the canonical join complex (or canonical meet complex). On the one hand, the action of κ on the canonical join complex for the Tamari lattice coincides with Kreweras complementation (recall from Example 2.8 that canonical join representations in the Tamari lattice are essentially noncrossing partitions). On the other hand, Panyshev complementation is a special case of an operation on the set of antichains in a finite poset P called rowmotion, as we now explain. When A is an antichain in P, we write RowpAq for the antichain tx P P : x is minimal among elements not in I A u. (Our notation is based on [33]. See also [3,5,6,10,14,24].) So, we have I A " I RowpAq . It follows immediately from the definition of κ˚that κ˚pI y q Þ Ñ I y . We obtain the following result. Proposition 3.16. Suppose that P is a finite poset, and A is an antichain in P. Then the map κ˚, acting on faces of the canonical meet complex of JpPq, sends the order ideal I A to the order ideal I RowpAq .
3.3. Crosscut-simplicial lattices. In this section, we prove Corollary 1.2. Recall that one direction of the proof was given as [22,Theorem 3.1]. Because it is easy, we give an alternative argument below. Write A for the set of atoms in L. When L is a finite semidistributive lattice every join of two atoms is a canonical join representation. In particular, Theorem 1.1 implies that each distinct subset of atoms gives rise to a distinct element in L. Thus the crosscut complex for L is either the boundary of the simplex on A or equal to the simplex on A, depending on whether Ž A "1 or Ž A ă1. Since each interval in L inherits semidistributivity, it follows that L is crosscut-simplicial.
Before we proceed with the proof of the converse, we point out that the joinsemidistributivity hypothesis in Corollary 1.2 is crucial. (For example, consider the crosscut-simplicial lattice shown in Figure 5. This lattice fails both SD _ and SD^.) Join-semidistributivity gives us a powerful restriction: A finite join-semidistributive lattice L fails SD^if and only if L contains the lattice shown in Figure 1 as a sublattice ([12, Theorem 5.56]).
We now begin our proof. The following lemmas will be useful; the first is [25, Lemma 9-2.5]. Lemma 3.17. Suppose that L is a finite lattice satisfying the following property: If x, y, and z are elements of L with x^y " x^z and if y and z cover a common element, then x^py _ zq " x^y. Then, L is meet-semidistributive. Lemma 3.18. Suppose that L is a finite join-semidistributive lattice that is not meet-semidistributive. Then there exists x, y, and z such that y _ z ą x and x, y, and z cover a common element.
Proof. We prove the proposition by induction on the size of L. As mentioned above, L contains the lattice shown in Figure 1 as sublattice, and this proves the base case. By Lemma 3.17, we can assume that there exist x, y, and z in L such that x^y " x^z, x^py _ zq ‰ x^y, and cov Ó pyq X cov Ó pzq is not empty. We choose such a triple so that the set tx, y, zu is minimal in join-refinement, among all such triples. Write a for the element in cov Ó pyq X cov Ó pzq (if there is more than one element in cov Ó pyq X cov Ó pzq, then y^z does not exist). If x also covers a, then we are done (because if xą a and y _ z ą x, then py _ zq^x " a, and that contradicts our assumption that tx, y, zu fail SD^). So we assume that x does not cover a.
We first prove that x ă y _ z (see Figure 6 for an illustration). We write w for x^py _ zq. Since x^y " x^z, we have x^y ă w (because x, y and z fail SD^, the inequality is strict). On the one hand w^px^yq " x^y. On the other hand, x ě w, so px^wq^y " w^y. By symmetry, w^z " x^z. Therefore, w^y " w^z. Observe that w ‰ y^w (otherwise w ď y^x, and that is absurd). Since, w^py _ zq " w we have tw, y, zu fails SD^. Since tw, y, zu join-refines tx, y, zu, minimality of tx, y, zu implies that w " x. We have proved the claim that y _ z ą x. By induction, we may assume that y _ z "1.
Next, we claim that x _ y and x _ z are incomparable. By way of contradiction assume that x _ z ě x _ y, so we have x _ z ě x, y, z. Therefore, z _ x " z _ y. Since z _ px^yq " z and L is join-semidistributive, we have z "1, contradicting the fact that x^z ‰ x^py _ zq. We have proved the claim that x _ y and x _ z are incomparable.
Suppose that ty, zu " cov Ò paq. Then, either y ď a _ x or z ď a _ x, but not both. (Indeed, if x _ a ě y, z then x _ a "1, so x _ a " x _ y " x _ z. This contradicts the fact that x _ y and x _ z are incomparable.) If y ă x _ a then y ă x _ a ď x _ z. Thus we have x _ y ď x _ z, contradicting the fact that x _ y and x _ z are incomparable. We conclude that there is some w 1 P cov Ò paqzty, zu with w 1 ď a _ x. The triple tw 1 , y, zu satisfies the statement of the proposition. Proof of Theorem 1.2. We prove that if L is join-semidistributive and crosscutsimplicial then it is semidistributive. Assume that L is fails SD^. Lemma 3.18 says that there exists x, y and z covering a common element a P L such that y _ z ą x. In particular, the interval ra, y _ zs is not crosscut-simplicial because ty, zu is not a face in the crosscut complex. That is a contradiction. Therefore, L is a finite semidistributive lattice, and the statement follows from Theorem 1.1.

Lattice-theoretic constructions
4.1. Sublattices and quotient lattices. A map φ : L Ñ L 1 between lattices L and L 1 is a lattice homomorphism if φ respects the meet and join operations. The image of φ is a sublattice of L 1 and a lattice quotient of L. It is immediate that each sublattice of a semidistributive lattice is also semidistributive. When L is finite, the image φpLq also inherits semidistributivity (see [25,). (Outside of the finite case, it is not generally true that if L is semidistributive, then φpLq is semidistributive; similarly for meet and join-semidistributivity.) We obtain the following result as an immediate corollary of Theorem 1.1.
Corollary 4.1. Suppose that L is a finite join-semidistributive lattice whose canonical join complex is flag. Then, the canonical join complex for each sublattice and quotient lattice of L is also flag.
An equivalence relation Θ on L is a lattice congruence if Θ satisfies the following: if x " Θ y, then x _ t " Θ y _ t and x^t " Θ y^t for each x, y, and t in L (see [16,Lemma 8]). It is immediate that the fibers of a lattice homomorphism φ constitute a lattice congruence of L. Conversely, each lattice congruence also gives rise to a lattice quotient (see [16,Theorem 11]).
When L is finite, Θ is lattice congruence if and only if it satisfies the following: Each class is an interval; the map π Θ Ó sending x P L to the smallest element in its Θ-class is order preserving; the map π Ò Θ sending x P L to the largest element in its Θ-class is order preserving. Both π Θ Ó and π Ò Θ are lattice homomorphisms onto their images such that π Θ Ó pLq and π Ò Θ pLq are isomorphic lattice quotients of L. The lattice quotient π Θ Ó pLq is a sub-join-semilattice of L, but not generally a sublattice of L. Similarly, π Ò Θ pLq is a sub-meet-semilattice of L. Below we quote [27,Proposition 6.3]. In the proposition, a join-irreducible element j P L is contracted by the congruence Θ if j is congruent to the unique element that it covers. Proposition 4.2. Suppose that L is a finite join-semidistributive lattice and Θ is a lattice congruence on L with associated projection map π Θ Ó . Then, the element x belongs to π Θ Ó pLq if and only if no canonical joinand of x is contracted by Θ.
Suppose that x P π Θ Ó pLq. Since π Θ Ó is a sub-join-semilattice of L, the canonical join representation of x taken in the lattice quotient π Θ Ó pLq is equal to the canonical join representation taken in L.

Corollary 4.3.
Suppose that L is a finite join-semidistributive lattice with canonical join complex ∆, and Θ is a lattice congruence of L. Then, the canonical join complex of π Θ Ó pLq is the induced subcomplex of ∆ supported on the set of joinirreducible elements not contracted by Θ.
Remark 4.4. The canonical join complex of a sublattice L 1 of L need not be an induced subcomplex of ∆. In fact, the sets IrrpL 1 q and IrrpLq may be disjoint. For example, consider the canonical join complex of the sublattice t0,1u in the boolean lattice B n , where n ą 1.
Remark 4.5. In general, not every induced subcomplex of ∆ is the canonical join complex for a lattice quotient of L. Each lattice congruence is determined by the set of join-irreducible elements that it contracts. But, a given collection of joinirreducible elements may not correspond to a lattice congruence. For j and j 1 in IrrpLq, we say that j forces j 1 if every congruence that contracts j also contracts j 1 . In N 5 pictured in Figure 7 both a and b force c. So, for example, there is no quotient of N 5 whose canonical join complex is the subcomplex induced by tb, cu.

4.2.
Products and sums. In the following easy propositions, we construct new semidistributive lattices from old ones, and give the corresponding construction for the canonical join complex.
Proposition 4.6. Suppose that L 1 and L 2 are finite, join-semidistributive lattices with corresponding canonical join complex ∆ i for i " 1, 2. Then the canonical complex for L 1ˆL2 is the join ∆ 1˚∆2 .
The ordinal sum of lattices L 1 and L 2 written L 1 ' L 2 is the lattice whose set of elements is the disjoint union L 1 Z L 2 , ordered as follows: x ď y if and only if x ď y in L i , for i " 1, 2, or x P L 1 and y P L 2 .
Proposition 4.7. Suppose that L 1 and L 2 are finite, join-semidistributive lattices with corresponding canonical join complex ∆ i , for i " 1, 2. Then the canonical join complex of L 1 ' L 2 is equal to the disjoint union ∆ 1 Z ∆ 2 Z tvu, in which the vertex v corresponds to the minimal element of L 2 .
We define the wedge sum L 1 ą L 2 to be the lattice quotient of the ordinal sum L 1 ' L 2 in which the minimal element of L 2 is identified with the maximal element of L 1 . (Our nonstandard terminology is inspired by the wedge sum of topological spaces.) Proposition 4.8. Suppose that L 1 and L 2 are finite, join-semidistributive lattices with corresponding canonical join complex ∆ i , for i " 1, 2. Then the canonical join complex of L 1 ą L 2 is equal to the disjoint union ∆ 1 Z ∆ 2 .

Day's doubling construction.
A subset C of L is order-convex if for each x, y P C with x ď y, we have that the interval px, yq belongs to C. Suppose that C Ď L is order convex, and let 2 be the two element chain 0 ă 1. We write X for the set of elements x P L such that x ě c for some c P C. Define LrCs to be the following induced subposet of Lˆ2: We say that LrCs is obtained by doubling L with respect to C. This procedure, due to Day [9], is defined more generally for all posets. If L is a lattice, then LrCs is a lattice and the map π C : LrCs Ñ L given by px, q Þ Ñ x is a surjective lattice homomorphism (see [9] or [22, Lemma 6.1]). In the next proposition, we show that when C is an interval in L, doubling L with respect to C also preserves semidistributivity.
Proposition 4.9. Suppose that L is a finite semidistributive lattice, I " ra, bs is an interval in L, and write E for the edge set of the canonical join graph for L. Then LrIs is semidistributive, and the canonical join graph for LrIs has edge set E 1 Z ttpj, 0q, pa, 1qu : j P canpwq for w P I and j ď au , where E 1 is the set of pairs tpj, q, pj 1 , 1 qu, such that tj, j 1 u P E, and pj, q and pj 1 , 1 q are the minimal elements of the fibers π´1 I pjq and π´1 I pj 1 q, respectively.
In the proof below we check that LrIs satisfies (1) from Proposition 3.5 (and the obvious dual argument gives meet-semidistributivity). One can also verify semidistributivity directly for LrIs using [22,Lemma 6.1]. Our approach has the advantage of giving the canonical join representation of each element of LrIs. In either case, the argument is tedious but, at least, elementary.
Proof. Suppose that pw, q is not in Iˆ1, where " 0, 1. Observe that the map π I : py, 1 q Þ Ñ y is a bijection from cov Ó ppw, qq to cov Ó pwq. For each y P cov Ó pwq, write ηpyq for the unique minimal element of L satisfying y _ ηpyq " w, and py, 1 q for the corresponding element in cov Ó ppw, qq. Let pηpyq, 2 q be the minimal element of the fiber π´1 I pηpyqq in LrIs. We claim that pηpyq, 2 q _ py, 1 q " pw, q. If " 0, the claim is immediate, and if " 1 then the claim follows from the fact that pw, 0q R LrIs. It is straightforward, using the surjection π I , to check that pηpxq, 2 q is the unique minimal element of LrCs whose join with px, 1 q is equal to py, q.
Suppose that pw, 1q P Iˆ1. If w " a, it is immediate that pw, 1q satisfies condition (1) of Proposition 3.5. So we assume that w ą a. Observe that the lower covers of pw, 1q are py, 1q such that y P cov Ó pyq X I and pw, 0q. For each y P cov Ó pwq X I, we claim that the set tηpyq : y P cov Ó pwq X Iu is precisely the set of canonical joinands of w that are not weakly below a. If y P cov Ó pwqzI, then y _ a " w. By minimality of ηpyq, we conclude that ηpyq ď a. If y P cov Ó pwq X I and ηpyq ď a, then ηpyq _ y " y, which is a contradiction. The claim follows. As above, it is straightforward to check that pηpyq, 0q is the unique minimal element in LrIs whose join with py, 1q is equal to pw, 1q, for each y P cov Ó pwq X I.
Below we gather some useful facts that follow immediately from the proof of Proposition 4.9.
Proposition 4.10. Suppose that L is a finite semidistributive lattice, I " ra, bs is an interval in L, and j P IrrpLq such that j ‰ a. For each w P L and , 1 P t0, 1u the following statements hold: (1) If pj, q is a canonical joinand of pw, 1 q in LrIs, then j is a canonical joinand of w. (2) If pj, q is a canonical joinand of pw, 1 q P Iˆ2 then " 0.
(3) If pj, q is a canonical joinand of pw, 0q P Iˆ0 and j ď a, then pj, q is also a canonical joinand of pw, 1q. (4) pw, 1 q has pa, 1q as canonical joinand if and only if pw, 1 q P Iˆ1.
A lattice is congruence uniform if it is obtained from the one element lattice by a finite sequence of doublings of intervals. Suppose that L is a finite congruence uniform lattice. Proposition 4.9 says that after each iteration of the doubling procedure, the resulting lattice has exactly one additional join-irreducible element, namely pa, 1q, where a is the smallest element of the interval that is doubled. Thus the canonical join graph of each congruence uniform lattice L has a natural labeling, in which the vertex labeled i is the join-irreducible element that is added in the i th step of the doubling sequence for L.
Remark 4.11. Non-isomorphic congruence uniform lattices may have the same labeled canonical join graphs. For example, doubling the boolean lattice B 2 with respect to any singleton interval I " txu, for x P B 2 , results in the labeled canonical join graph depicted in Figure 8 below. When x is equal to0 or1, we obtain the ordinal sums B 0 ' B 2 and B 2 ' B 0 , respectively. When x is either join-irreducible element of B 2 , the resulting lattice is isomorphic to N 5 from Figure 7.
We conclude this subsection with some examples of labeled and unlabeled graphs that can realized as the canonical join graph for some congruence uniform lattice.
Example 4.12 (Complete graphs). In our first example we consider the complete graph K n on n vertices, which can be realized as the canonical join graph for the boolean lattice B n . In fact, the boolean lattice is the only lattice whose canonical join graph is K n .  Figure 8. The canonical labeled join graph of three nonisomorphic congruence uniform lattices.
Proposition 4.13. Supposed that L is a finite semidistributive lattice with canonical join graph equal to the complete graph K n . Then, L is isomorphic to B n .
Proof. Write x S for the element with canonical join representation Ž ptj i : i P Su, where S Ď rns " t1, 2 . . . , nu. Suppose that x S ď x S 1 for some S 1 Ď rns, and there exists k P S that is not in S 1 . Since j k _ Ž ptj i : i P S 1 u is a canonical join representation, in particular this join is irredundant. So, j k ď Ž ptj i : i P S 1 u " x S 1 , and that is a contradiction. Therefore, the map x S Þ Ñ S is order preserving. It is immediate that the inverse map is order preserving.
Example 4.14 (Chordal graphs). Similar to the construction of the complete graph (as a labeled canonical join graph), one can construct certain chordal graphs as the canonical join graph for a congruence uniform lattice. In the construction, each doubling with respect to some interval I has Iˆ2 isomorphic to a boolean lattice.
Suppose that G is a graph. The closed neighborhood N rvs is the subgraph of G induced by the set of vertices v 1 adjacent to v, together with v. The open neighborhood N pvq is the subgraph induced by the set of vertices v 1 adjacent to v. A perfect elimination ordering for G is linear ordering v 1 ă v 2 ă¨¨¨ă v n of the vertices of G such that for each i " 1, 2, . . . , n, the intersection of N rv i s with the set tv i , v i`1 , . . . , v n u is a clique in G. Recall that a graph G is chordal if and only if it has a perfect elimination ordering. Proposition 4.15. Suppose that G is a labeled graph such that L " v n ă v n´1 ă . . . ă v 1 is a perfect elimination ordering. If N pv i`1 q Ď N pv i q for each i P rn´1s, then there exists a congruence uniform lattice L such that G is its labeled canonical join graph.
Proof. We prove the statement by induction on n. There exists a congruence uniform lattice L 1 whose labeled canonical join graph is the subgraph induced by the first n´1 vertices. In particular, L 1 is isomorphic to L 2 rIs where L 2 is congruence uniform, I " ra, bs is an interval in L 2 , and the vertex v n´1 corresponds to the join-irreducible element pa, 1q in L 1 .
We give the argument for the case when that v n and v n´1 are neighbors. The proof is similar when v n R N pv n´1 q. We write tv i1 , . . . , v i k u for the set of vertices N pv n qztv n´1 u, and j i1 , . . . , j i k for the corresponding join-irreducible elements of L 1 . Since L is a perfect elimination order, the vertices tv i1 , . . . , v i k , v n´1 u form a clique in the subgraph induced by V ztv n u. By Theorem 1.1, the join pa, 1q_ Ž tj i1 , . . . , j i k u is a canonical join representation for some element py, 1q in L 1 .
Consider the interval I 1 " rpa, 0q, py, 1qs. It is straightforward (with Proposition 4.9) to verify that the new join-irreducible element in L 1 rI 1 s joins canonically with each element in tj i1 , . . . , j i k , pa, 1qu. Suppose that pw, 1 q P I 1 and j P L 2 such that pj, 1 q is a canonical joinand of pw, q with pj, 1 q ď pa, 0q. We claim that pj, 1 q corresponds to a vertex in the set tv i1 , . . . , v i k , v n´1 u. The claim is obvious if pj, 1 q " pa, 1q, so we assume that j ‰ a.
First we show that pj, 1 q is adjacent to pa, 1q in the canonical join graph for L 1 . The last item of Proposition 4.10 implies that y P ra, bs. Therefore, w is also in ra, bs. The first item of Proposition 4.10 says that j is a canonical joinand of w (in L 2 ), and the second item says that 1 " 0. Therefore, j ď a. Proposition 4.9 implies that pj, 0q is adjacent to pa, 1q in the canonical join graph of L 1 , as desired.
Finally, we show that pj, 1 q " pj, 0q belongs to the subset tj i1 , . . . , j i k , pa, 1qu of neighbors of pa, 1q. The third and fourth items of Proposition 4.10 say that pw, 1q has pj, 0q and pa, 1q as canonical joinands. Since pw, q is in I 1 , so is pw, 1q. In particular, pj, 0q _ pa, 1q P I 1 . Since I 1 is a sublattice of L 1 , we have that the join rpa, 1q _ pj, 0qs _ rpa, 1q _ Ž tj i1 , . . . , j i k us also belongs to I 1 . Therefore, Ž ptj i1 , . . . , j i k , pa, 1q, pj, 0quq " py, 1q. Because L is a perfect elimination ordering (and pj, 0q corresponds to a vertex v l with l ă n´1), the set tj i1 , . . . , j i k , pj, 0q, pa, 1qu is a face of the canonical join complex for L 1 . Therefore, pj, 0q is a canonical joinand of py, 1q, as desired. The statement of the proposition now follows immediately from Proposition 4.9.
The same argument, replacing the interval rpa, 0q, py, 1qs with rpa, 1q, py, 1qs, proves the case in which v n and v n´1 are not adjacent.
Example 4.16 (Cycle graphs). For each positive integer n, we claim that there is a finite congruence uniform lattice whose canonical join graph is isomorphic to the unlabeled cycle graph C n on n vertices. We provide an illustration with examples for n " 5, 6, 7. Leftmost in Figure 9 is the Hasse diagram for a distributive lattice L, and rightmost is the Hasse diagram obtained by doubling the interval ra, es in L. (The middle Hasse diagram, which is isomorphic to the leftmost Hasse diagram, serves only to make the doubling as clear as possible.) Each distributive lattice is in particular congruence uniform, so the rightmost lattice is congruence uniform, as desired. It is an easy exercise to verify that the canonical join graph for this right-most lattice is isomorphic to C 5 .
The analogous construction is given in Figure 10 for n " 6 and 7. In these cases, the lattice L being doubled is not distributive. Because it is easy to check that L is congruence uniform, we leave the details to the reader. (Note that C n , for n ě 5 is among the minimal graphs excluded by Theorem 2.4, and so does not appear as the canonical join graph for a distributive lattice.)

Discussion and open problems
The discussion in Section 4 does not constitute a complete list of lattice theoretic operations which preserve (join)-semidistributivity. For example, the derived lattice CpLq discussed in [29], the box product defined in [17] (see also, [35,Corollary 8.2]), and the lattice of multichains from [20] all preserve (join)-semidistributivity.
Because it is relatively easy, we will discuss this last operation in a small example. Recall that an m-multichain in a poset P is a collection of m elements satisfying x 1 ď x 2 ď . . . ď x m . We write an m-multichain as a tuple px 1 , . . . , x m q or more compactly as a vector x. We write the set of all m-multichains, partially ordered component-wise, as P rms . When P is a lattice, then P rms is a sublattice of the m-fold direct product of P m (see [20,Theorem 2.4]). It follows immediately that  Figure 10. Doubling the interval ra, es in the leftmost congruence uniform lattice yields the left-middle lattice, whose canonical join graph is isomorphic to C 6 . Doubling the interval ra, es in the rightmiddle lattice yields the rightmost lattice, whose canonical join graph is isomorphic to C 7 .
if L satisfies SD _ or SD^then L rms also does, for each m P N (see also, [20,Proposition 2.10]). In the proposition below, pjq k is the element p0, . . . ,0, j, . . . , jq, where k is the left-most coordinate that is equal to j.
Proposition 5.1. Suppose that L is a finite lattice. Then, IrrpL rms q is equal to the set tpjq k : j P IrrpLqu, where k P rms.
Proof. We first show that pjq k is join-irreducible when j P IrrpLq. Suppose that w _ v " pjq k . We have w i _ v i " j, for each i ě k. Since j is join-irreducible, we may assume that w k " j. Since w is a multichain, we have that j ď w i for each i ě k. Thus, w " pjq k , as desired. Next, suppose that w P IrrpL rms q. Let w k be the first nonzero entry in w, and assume that w k R IrrpLq so that there exist a and b in Lztw k u with w k " a _ b. Then w " p0, . . . ,0, a, w k`1 , . . . , w m q _ p0, . . . ,0, b, w k`1 , . . . , w m q in L rms . By this contradiction, we conclude that w k P IrrpLq. Next, suppose that w i ‰ w k , for some i ą k. Since w k ă w i , there is an element y P cov Ó pw i q such that w k ď y. We have the following nontrivial join-representation of w: w " p0, . . . ,0, w k , . . . , y, w i`1 , . . . , w m q _ p0, . . . ,0, y 2 , w k`1 , . . . , w i , . . . , w m q, where y 2 P cov Ó pw k q. Therefore w i " w k , and the proposition follows.
Example 5.2. Let L be the weak order on the symmetric group S 3 , and consider L r2s . The lattice L and L r2s are shown in Figure 11, and the corresponding canonical join complexes are shown in Figure 12. Observe that if j _ j 1 is a canonical join representation in L then both p0, jq _ p0, j 1 q and pj, jq _ pj 1 , j 1 q are canonical join representations in L r2s . This accounts for the edges tp0, aq, p0, bqu and tpa, aq, pb, bqu in the complex for L r2s .
To see how we obtain the remaining edges in Figure 12, consider the canonical join representation of pd,1q. Observe that cov Ó ppd,1qq " tpd, dq, pb,1qu. It is easily checked that pd, dq is the smallest element in L r2s whose join with pb,1q is equal to pd,1q. Similarly, p0, aq is the smallest element whose join with pd, dq is equal to pd,1q. Therefore, the canonical join representation for pd,1q " pd, dq _ p0, aq. The canonical join representations of the remaining elements in L r2s are computed similarly. This example is emblematic of the general construction, as can be seen in the next proposition which describes the canonical join graph for L rms . We leave the details of proof to the reader. Proposition 5.3. Suppose that L is a finite semidistributive lattice with joinirreducible elements j and j 1 .
(1) If i ă k then tpjq i , pj 1 q k u is a face in the canonical join complex for L rms if and only if j 1 is a canonical joinand of j _ j 1 in L. (2) If i " k, then tpjq i , pj 1 q k u is a face in the canonical join complex for L rms if and only if tj, j 1 u is a face in the canonical join complex for L.
Note that the operation on the canonical join complex corresponding to L Þ Ñ L rms depends on the lattice L (not just the canonical join complex for L).
Question 5.4. What lattice theoretic operations (preserving join-semidistributivity) correspond to geometric operations on the canonical join complex that are independent of L?
Alternatively, it would be interesting to know which geometric operations (on the class of finite simplicial complexes) have a corresponding lattice theoretic analogue. We point out that conspicuously absent from the discussion in Section 4 is closure under taking induced subcomplexes (see Remark 4.5).
Question 5.5. Let C be the class of simplicial complexes that can be realized as the canonical join complex for some finite semidistributive lattice. Is C closed under taking induced subcomplexes?
Say that G n is the set of labeled graphs that can be realized the (labeled) canonical join graph for a congruence uniform lattice with n join-irreducible elements, and G is the union Ť nPN G n . Using Stembridge's poset Maple package ( [32]) and Proposition 4.9, we have counted the number of elements of G n for n ď 6. While our computations indicate that not every labeled graph appears, they also suggest that G is closed under subgraphs (so that the corresponding class of simplicial complexes is closed under taking subcomplexes). We close the paper by asking two related questions: Question 5.6. Which labeled graphs can be realized as the labeled canonical join graph for some congruence uniform lattice? Question 5.7. Suppose that G is the canonical join graph for a congruence uniform lattice L. What data, in addition to G, is necessary in order to determine L up to isomorphism?

Acknowledgements
The author thanks Patricia Hersh (who suggested the connection to the crosscut complex), Thomas McConville, Nathan Reading and Victor Reiner for many helpful suggestions.