Oriented Flip Graphs and Noncrossing Tree Partitions

In this paper, we study the lattice properties of posets of torsion pairs in the module category of a family of representation-finite gentle algebras called tiling algebras, introduced by Coelho Simoes and Parsons. We present a combinatorial model for torsion pairs using polyogonal subdivisions of a convex polygon. We use this model and the lattice theory to classify 2-term simple-minded collections in the bounded derived category of the corresponding tiling algebra. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver. Our model is developed using the dual tree of a polygonal subdivision. Given such a tree, we introduce a simplicial complex of noncrossing geodesics supported by the tree which we call the noncrossing complex. The facets of the noncrossing complex may be given the structure of an oriented flip graph. Special cases of the oriented flip graphs that may be expressed in this way include the Tamari order, type A Cambrian orders, oriented exchange graphs for quivers mutation-equivalent to a path quiver. We prove that the oriented flip graph of any noncrossing complex is a polygonal, congruence-uniform lattice. To do so, we express the oriented flip graph as a lattice quotient of a lattice of biclosed sets. The facets of the noncrossing complex have an alternate ordering known as the shard intersection order. We prove that this shard intersection order is isomorphic to a lattice of noncrossing tree partitions. The oriented flip graph inherits a cyclic action from its congruence-uniform structure. On noncrossing tree partitions, this cyclic action generalizes the classical Kreweras complementation on noncrossing set partitions. We show that the data of a noncrossing tree partition and its Kreweras complement is equivalent to a 2-term simple-minded collection of the associated tiling algebra.

Triangulations of marked surfaces provide an incredibly useful model for studying the combinatorics and representation theory related to cluster algebras [24]. The arcs on a surface are in bijection with the cluster variables, the triangulations are in bijection with clusters, and moving between two triangulations by flipping an arc corresponds to performing a single mutation on the corresponding clusters. In addition, compatibility of two cluster variables means their corresponding arcs are noncrossing. On the representation theory side, the additive categorification of cluster algebras [11,12] has been described using marked surfaces (see [12,10,44]).
More recent work (see [9] and the many references therein) has shown that exchange graphs of many cluster algebras can be modeled using many different representation theoretic objects related to certain Jacobian algebras Λ [21]. In particular, the poset of functorially finite torsion pairs in the module category of Λ and the poset of 2-term simple-minded collections in the bounded derived category of Λ are isomorphic to the oriented exchange graph [7] of the cluster algebra defined by the quiver of Λ. We remark that any simple-minded collection in the bounded derived category of Λ can be realized as the set of simple objects in the heart of a bounded t-structure on the bounded derived category of Λ obtained via Happel-Reiten-Smalø tilting [32], as is shown in [36].
The connection between these geometric and representation theoretic objects can be seen as follows. Let pS, Mq be a marked surface, and add to it a certain lamination L (see [25]). By choosing a triangulation ∆ of the laminated marked surface, one computes the shear coordinates to obtain the c-matrix C of the cluster corresponding to ∆. The row vectors of C are the signed dimension vectors of the objects in a 2-term simpleminded collection. In this way, c-matrices of the cluster algebra associated to pS, M, Lq are in bijection with the 2-term simple-minded collections of the Jacobian algebra Λ associated with pS, M, Lq. The torsion pair corresponding to C has torsion part (resp. torsion-free part) generated (resp. cogenerated) by indecomposable Λ-modules M with dimpM q (resp.´dimpM q) a row vector of C (see [9] for a proof that these maps are bijections when Λ is a finite dimensional Jacobian algebra).
It is natural to ask if the correspondences just described make sense when one considers polygonal subdivisions (equivalently, partial triangulations) of a marked surface. The goal of this paper is to address this question when the surface is a disk with marked points on its boundary. We obtain analogous isomorphisms between oriented flip graphs of such polygonal subdivisions and posets of torsion pairs and 2-term simple-minded collections, using lattice theory and the combinatorics of string modules. As there is not a known cluster structure on polygonal subdivisions, we do not know whether all of our results have cluster theoretic interpretations.
1.1. Overview. The purpose of this work is to understand the combinatorics and representation theory associated with lattices of polygonal subdivisions of a convex polygon. Given such a polygonal subdivision, one naturally associates to it a finite dimensional algebra Λ, which we will refer to as a tiling algebra [52]. Our aim in this paper is to: ‚ provide a combinatorial model for the torsion pairs in the module category of Λ and ‚ classify 2-term simple-minded collections in the bounded derived category of Λ. Tiling algebras are a class of representation-finite gentle algebras that were very recently introduced in [52]. These algebras also form a subclass of the algebras of partial triangulations introduced in [20]. The class of tiling algebras contains nice families of algebras including Jacobian algebras [21] of type A and m-cluster-tilted algebras [40] of type A, both of which naturally arise in the study of cluster algebras [26] and in the additive categorification of cluster algebras [11,12].
We refer to the lattices of polygonal subdivisions we study as oriented flip graphs (see Definition 3.11). Special cases of these posets include the Tamari order, type A Cambrian lattices [46], oriented exchange graphs of type A cluster algebras [7], and the Stokes poset of quadrangulations defined by Chapoton [15].
Rather than directly studying polygonal subdivisions, it turns out to be more convenient to formulate our theory in terms of trees that are dual to polygonal subdivisions of a polygon. That is, our work begins with the initial data of a tree T embedded in a disk so that its leaves lie on the boundary and its other vertices lie in the interior of the disk. This data gives rise to a simplicial complex of noncrossing sets of arcs on this tree that we call the reduced noncrossing complex (see Section 3 for the precise definitions of these notions). The combinatorics of the facets of this pure, thin simplcial complex (see Corollary 3.10) allow us to define our oriented flip graphs, which we denote by Ý Ý Ñ F GpT q. Our first main combinatorial result (Theorem 4.11), which sets the stage for the rest of the paper, is that these oriented flip graphs are congruence-uniform lattices. The Tamari order is a standard example of a congruence-uniform lattice [31]; see also [14], [46]. Nathan Reading gave a proof of congruence-uniformity of the Tamari order by proving that the weak order on permutations is congruence-uniform and applying the lattice quotient map from the weak order to the Tamari order defined by Björner and Wachs in [5]. To prove our congruence-uniformity result, we take a similar approach. We define a congruence-uniform lattice of biclosed sets of T , denoted BicpT q, and identify the oriented flip graph Ý Ý Ñ F GpT q with a lattice quotient of BicpT q. This method was applied to some other Tamari-like lattices in [30], [39]. The technique of studying a lattice by realizing it as a quotient lattice is not new, see for example [42], [43].
Congruence-uniform lattices admit an alternate poset structure called the shard intersection order [49]. For example, the shard intersection order of the Tamari lattice is the lattice of noncrossing set partitions [48]. We introduce a new family of objects called noncrossing tree partitions of T , and identify the shard intersection order of Ý Ý Ñ F GpT q with the lattice of noncrossing tree partitions of T , denoted NCPpT q (Theorem 5.14).

1.2.
Organization and main results. In Section 2.1, we recall the definition of oriented exchange graphs, which are defined by the initial data of a quiver. When the quiver is in the mutation-class of a type A Dynkin quiver, its oriented exchange graph is isomorphic to an oriented flip graph (see Theorem 7.8). In Sections 2.2 and 2.3, we review the lattice theory that we will use to obtain many of our results.
Our main combinatorial and lattice-theoretic results appear Sections 3, 4, 5, and 7. In Section 3, we introduce the noncrossing complex and reduced noncrossing complex of arcs on a tree. We then develop the combinatorics of these complexes, which is an important part of the definition of oriented flip graphs. In Section 4, we introduce the lattice of biclosed sets of T and we show how the oriented flip graph Ý Ý Ñ F GpT q is both a sublattice and quotient lattice of BicpT q.
In Section 5, we introduce noncrossing tree partitions of T , which generalize the classical noncrossing set partitions. We show that, as in the classical case, noncrossing tree partitions form a lattice NCPpT q under refinement. Furthermore, we show that NCPpT q is isomorphic to the shard intersection order of the oriented flip graph of T (Theorem 5.14). In Section 7, we show that the top element of Ý Ý Ñ F GpT q is obtained by rotating arcs in the bottom element of Ý Ý Ñ F GpT q (see Theorem 7.7). This result recovers one of Brüstle and Qiu (see [8]) in the case where the surface is a disk without punctures.
In Sections 6 and 8, we interpret the combinatorics of oriented flip graphs and noncrossing tree partitions in terms of the representation theory of the tiling algebra Λ T defined by T . In Section 6.4, we show that the lattice of torsion-free classes (resp. torsion classes) of Λ T ordered by inclusion (resp. reverse inclusion) is isomorphic to Ý Ý Ñ F GpT q (see Theorem 6.22). To obtain this result, we make use of the lattice quotient description of Ý Ý Ñ F GpT q from Section 4 and the classification of extensions between indecomposable Λ T -modules found in Section 6.2. In Section 6.5, we show that the poset of noncrossing tree partitions of T is isomorphic to the poset of wide subcategories of Λ T -mod. Wide subcategories have already been used in [33] to model the lattice of noncrossing partitions associated with a Dynkin quiver.
In Section 8, we show that the data of a noncrossing tree partition and its Kreweras complement is equivalent to a 2-term simple-minded collection of objects in the bounded derived category of Λ T (see Theorem 8.4). This theorem relies on the description of extensions between indecomposable Λ T -modules found in Section 6.2 and on a combinatorial description of the operation of left-and right-mutation on simple-minded collections found in Section 8.1 (see Lemma 8.6).
We conclude the paper with a classification of c-matrices of quivers defined by triangulations of polygons (see Theorem 9.1). This classification is similar to the classification obtained in [53] for acyclic quivers and to the classification found in [29] for type A Dynkin quivers.
Acknowledgements. Alexander Garver thanks Kiyoshi Igusa, Gregg Musiker, Ralf Schiffler, Sibylle Schroll, and Hugh Thomas for helpful conversations. The authors thank Emily Barnard for useful discussions. They also thank Hugh Thomas for noticing the connection between noncrossing tree partitions and wide subcategories.

2.1.
Oriented exchange graphs. The oriented flip graphs that we will introduce in Section 3 generalize a certain subclass of oriented exchange graphs of quivers, which are important objects in representation theory of finite dimensional algebras. We present the definition of oriented exchange graphs to motivate the introduction of the former.
A quiver Q is a directed graph. In other words, Q is a 4-tuple pQ 0 , Q 1 , s, tq, where Q 0 " rms :" t1, 2, . . . , mu is a set of vertices, Q 1 is a set of arrows, and two functions s, t : Q 1 Ñ Q 0 defined so that for every α P Q 1 , we have spαq α Ý Ñ tpαq. An ice quiver is a pair pQ, F q with Q a quiver and F Ă Q 0 a set of frozen vertices with the restriction that any i, j P F have no arrows of Q connecting them. By convention, we assume Q 0 zF " rns and F " rn`1, ms :" tn`1, n`2, . . . , mu. Any quiver Q is regarded as an ice quiver by setting Q " pQ, Hq.
If a given ice quiver pQ, F q has no loops or 2-cycles, we can define a local transformation of pQ, F q called mutation. The mutation of an ice quiver pQ, F q at a nonfrozen vertex k, denoted µ k , produces a new ice quiver pµ k Q, F q by the three step process: (1) For every 2-path i Ñ k Ñ j in Q, adjoin a new arrow i Ñ j.
(2) Delete any 2-cycles created during the first steps.
(3) Reverse the direction of all arrows incident to k in Q. We show an example of mutation below with the nonfrozen (resp. frozen) vertices in black (resp. blue).
The information of an ice quiver can be equivalently described by its (skew-symmetric) exchange matrix. Given pQ, F q, we define B " B pQ,F q " pb ij q P Z nˆm :" tnˆm integer matricesu by b ij :" #ti α Ñ j P Q 1 u´#tj α Ñ i P Q 1 u. Furthermore, ice quiver mutation can equivalently be defined as matrix mutation of the corresponding exchange matrix. Given an exchange matrix B P Z nˆm , the mutation of B at k P rns, also denoted µ k , produces a new exchange matrix µ k pBq " pb 1 ij q with entries b 1 ij :" "´b ij : if i " k or j " k b ij`| b ik |b kj`bik |b kj | 2 : otherwise.
For example, the mutation of the ice quiver above (here m " 4 and n " 3) translates into the following matrix mutation. Note that mutation of matrices and of ice quivers is an involution (i.e. µ k µ k pBq " B). Let Mut(pQ, F q) denote the collection of ice quivers obtainable from pQ, F q by finitely many mutations where such ice quivers are considered up to an isomorphism of quivers that fixes the frozen vertices. We refer to Mut(pQ, F q) as the mutation-class of Q. Such an isomorphism is equivalent to a simultaneous permutation of the rows and first n columns of the corresponding exchange matrices.
Given a quiver Q, we define its framed quiver to be the ice quiver p Q where p Q 0 :" Q 0 \ rn`1, 2ns, F " rn`1, 2ns, and p Q 1 :" Q 1 \ ti Ñ n`i : i P rnsu. We define the exchange graph of p Q, denoted EGp p Qq, to be the (a priori, infinite) graph whose vertices are elements of Mutp p Qq and two vertices are connected by an edge if the corresponding quivers differ by a single mutation.
The exchange graph of p Q has natural acyclic orientation using the notion of c-vectors. We refer to this directed graph as the oriented exchange graph of Q, denoted Ý Ý Ñ EGp p Qq. Given p Q, we say that C " C R is a c-matrix of Q if there exists R P EGp p Qq such that C is the nˆn submatrix of B R " pb ij q iPrns,jPr2ns containing its last n columns. That is, C " pb ij q iPrns,jPrn`1,2ns . We let c-mat(Q) :" tC R : R P EGp p Qqu. A row vector of a c-matrix, c i , is known as a c-vector. Since a c-matrix C is only defined up to a permutations of its rows, C can be regarded simply as a set of c-vectors.
The celebrated theorem of Derksen, Weyman, and Zelevinsky [22,Theorem 1.7], known as the sign-coherence of c-vectors, states that for any R P EGp p Qq and i P rns the c-vector c i is a nonzero element of Z n ě0 or Z n ď0 . If c i P Z n ě0 (resp. c i P Z n ď0 ) we say it is positive (resp. negative). It turns out that for any quiver Q one has c-vec(Q) :" tc-vectors of Qu " c-vec(Q)`\´c-vec(Q)`where c-vec(Q)`:" tpositive c-vectors of Qu.  The oriented exchange graph of a quiver Q, denoted Ý Ý Ñ EGp p Qq, is the directed graph whose underlying unoriented graph is EGp p Qq with directed edges pR 1 , F q ÝÑ pµ k R 1 , F q if and only if c k is positive in C R 1 . In Figure 1, we show Ý Ý Ñ EGp p Qq and we also show all of the c-matrices in c-mat(Q) where Q " 2 Ð 1.

2.2.
Lattices. We will see that many properties of oriented flip graphs can be deduced from their lattice structure.
In this section, we review some background on lattice theory, following [49]. Unless stated otherwise, we assume that all lattices considered are finite. Given a poset pP, ďq, the dual poset pP˚, ď˚q has the same underlying set with x ď˚y if and only if y ď x. A chain in P is a totally ordered subposet of P . A chain x 0 ă¨¨¨ă x N is saturated if there does not exist y P P such that x i´1 ă y ă x i for some i. A saturated chain is maximal if x 0 is a minimal element of P and x N is a maximal element of P .
A lattice is a poset for which any two elements x, y have a least upper bound x _ y called the join and a greatest lower bound x^y called the meet. Any finite lattice has a lower and an upper bound, denoted0 and1, respectively. Unless stated otherwise, we will assume that our lattices are finite. An element j is join-irreducible if j ‰0 and whenever j " x _ y either j " x or j " y holds. Meet-irreducible elements are defined dually. Let JIpLq and MIpLq be the sets of join-irreducibles and meet-irreducibles of L, respectively. For Dually, one may define canonical meet-representations. In Figure 2, we define a lattice with 5 elements. The set of join-irreducibles is ta, b, cu. The top element1 has two irredundant expressions as a join of join-irreducibles, namely a _ c "1 and b _ c "1. Since ta, cu ď tb, cu, the expression a _ c "1 is the canonical join-representation for1.
A lattice L is meet-semidistributive if for any three elements x, y, z P L, x^z " y^z implies px _ yq^z " x^z. A lattice L is semidistributive if both L and L˚are meet-semidistributive. It is known that a lattice is semidistributive if and only if it has canonical join-representations and canonical meet-representations for each of its elements.
A lattice congruence Θ is an equivalence relation such that if x " y mod Θ then x^z " y^y mod Θ and x _ z " y _ z mod Θ for all x, y, z P L. If Θ is a lattice congruence of L, the set of equivalence classes L{Θ inherits a lattice structure from L. Namely, rxs _ rys " rx _ ys and rxs^rys " rx^ys for x, y P L. The lattice L{Θ is called a lattice quotient of L, and the natural map L Ñ L{Θ is a lattice quotient map. Although lattice quotients are easiest to describe in algebraic terms, it is often more useful to give the following order-theoretic definition.
Lemma 2.2. An equivalence relation Θ on a finite lattice L is a lattice congruence if (1) every equivalence class of Θ is a closed interval of L, and (2) the maps x Þ Ñ minrxs Θ and x Þ Ñ maxrxs Θ are order-preserving.
Lemma 2.2 has been proven several times in the literature. For our purposes, it is more convenient to use the following modification; see [30,Lemma 3.1] or [23,Lemma 4.2]. Lemma 2.3. Let L be a lattice with idempotent, order-preserving maps π Ó : L Ñ L, π Ò : π Ò pπ Ó pxqq " π Ò pxq, then the equivalence relation x " y mod Θ if π Ó pxq " π Ó pyq is a lattice congruence. Given x, y in a poset P , we say y covers x, denoted x Ì y, if x ă y and there does not exist z P P such that x ă z ă y. We let CovpP q denote the set of all covering relations of P . If P is finite, then the partial order on P is the transitive closure of its covering relations. In a finite lattice L, if j P JIpLq, then j covers a unique element j˚. Dually, if m P MIpLq, then m is covered by a unique element m˚. It should be clear from context whether m˚is an element of the dual lattice L˚or is the unique element covering a meet-irreducible m. We describe the behavior of covering relations under lattice quotients in Lemma 2.4. A proof of this lemma may be found in Section 1-5 of [49].
Lemma 2.4. Let L be a lattice with a lattice congruence Θ.
The set of lattice congruences ConpLq of a lattice L is partially ordered by refinement. The top element of ConpLq is the congruence that identifies all of the elements of L, whereas the bottom element does not identify any elements of L. It is known that ConpLq is a distributive lattice. By Birkhoff's representation theorem for distributive lattices, ConpLq is isomorphic to the poset of order-ideals of JIpConpLqq, where the set of joinirreducibles is viewed as a subposet of ConpLq.
Given x Ì y in L, let conpx, yq denote the most refined lattice congruence for which x " y. These congruences are join-irreducible, and if L is finite, then every join-irreducible lattice congruence is of the form conpj˚, jq for some j P JIpLq [28,Theorem 2.30]. Consequently, there is a natural surjective map of sets JIpLq Ñ JIpConpLqq given by j Þ Ñ conpj˚, jq. Dually, there is a natural surjection MIpLq Ñ MIpConpLqq given by m Þ Ñ conpm, m˚q. If both maps are bijections, then we say L is congruence-uniform (or bounded). Congruence-uniform lattices are the topic of the next section.
2.3. Congruence-uniform lattices. Given a subset I of a poset P , let P ďI " tx P P : pDy P Iq x ď yu. If I is a closed interval of a poset P , the doubling P rIs of P at I is the induced subposet of Pˆ2 consisting of the elements in P ďIˆt 0u \ ppP zP ďI q Y Iqˆt1u. Some doublings are shown in Figure 3. Day proved that a lattice is congruence-uniform if and only if it may be constructed from a 1-element lattice by a sequence of interval doublings [19].
Let L be a lattice and P a poset. A function λ : CovpLq Ñ P is a CN-labeling of L if L and its dual Ls atisfy the following condition (see [45]): For elements x, y, z P L with z Ì x, z Ì y, and maximal chains C 1 , C 2 in rz, x _ ys with x P C 1 and y P C 2 , (CN1) the elements x 1 P C 1 , y 1 P C 2 such that x 1 Ì x _ y and y 1 Ì x _ y satisfy λpx 1 , x _ yq " λpz, yq, λpy 1 , x _ yq " λpz, xq; (CN2) if pu, vq P CovpC 1 q with z ă u, v ă x _ y then λpz, xq ă λpu, vq and λpz, yq ă λpu, vq; (CN3) the labels on CovpC 1 q are all distinct.
A lattice is congruence-normal if it has a CN-labeling. Alternatively, a lattice is congruence-normal if it may be constructed from a 1-element lattice by a doubling sequence of order-convex sets; see [45].
(2) The restriction of a CN-labeling to an interval rx, ys is a CN-labeling of rx, ys.
For example, the colors on the edges of Figure 3 form a CU-labeling, where the color set is ordered s ď t if color s appears before t in the sequence of doublings.
In [45], Reading characterized congruence-normal lattices as those lattices that admit a CN-labeling. From his proof, it is straight-forward to show that a lattice is congruence-uniform if and only if it admits a CU-labeling. Proposition 2.6. A lattice is congruence-uniform if and only if it admits a CU-labeling. If x Ì y and w Ì z, then covers px, yq and pw, zq are associates if either y^w " x and y _ w " z or x^z " w and x _ z " y. Such a notion is useful for lattice congruences. Namely, if px, yq and pw, zq are associates and Θ is a lattice congruence, then x " y mod Θ if and only if w " z mod Θ.
Lemma 2.7. Let L be a congruence-uniform lattice with CU-labeling λ : CovpLq Ñ P . For any s P P , if j is a minimal element with the property s P λ Ó pjq, then j is a join-irreducible. Moreover, if px, yq P CovpLq such that λpx, yq " s, then pj˚, jq and px, yq are associates. Conversely, if pj˚, jq and px, yq are associates, then they have the same label. Dually, if m is a maximal element with the property s P λ Ò pmq, then m is meet-irreducible, and the cover pm, m˚q is associates with every other cover with the label s.
Proof. Let s P P be given, and let j be minimal such that s P λ Ó pjq, and let w P L with λpw, jq " s. If j is not join-irreducible, then there exists some z covered by j distinct from w. By (CN1), there exists an element w 1 ă j such that λpw^z, w 1 q " s, which is a contradiction to the minimality of j. Hence, j is join-irreducible. Let x, y P L such that x Ì y and λpx, yq " s. If y is join-irreducible, then y " j by (CU1). Otherwise, by the previous argument, px, yq is associates with some cover px 1 , y 1 q such that y ą y 1 . Applying this several times, we get a sequence y ą y 1 ą¨¨¨ą y N and covers px i , y i q such that px, yq is associates with px i , y i q for all i. This terminates if y N is minimal. But that forces y N " j, so px, yq is associates with pj˚, jq. Now let j P JIpLq and px, yq P CovpLq such that pj˚, jq and px, yq are associates. If y is a join-irreducible, then it is clear that y " j. Otherwise, we may construct a sequence px i , y i q P CovpLq such that any two covers are associates, λpx i , y i q " λpx, yq and y 1 ą y 2 ą¨¨¨ą y N with y N P JIpLq. Since associates pairs induce the same lattice congruence, we have conpj˚, jq " conpx N , y N q, so j " y N .
The dual statement may be proved in a similar manner. Lemma 2.7 shows that a CU-labeling is essentially unique if it exists. Using the proof, one can construct the following labeling.
It is known that congruence-uniformity is preserved under lattice quotients. Corollary 2.9. If L is a (finite) congruence-uniform lattice and Θ is a lattice congruence, then L{Θ is congruenceuniform.
Proposition 2.10. Let L be a congruence-uniform lattice with CU-labeling λ. For x P L, the canonical joinrepresentation of x is Ž D j, where D is the set of join-irreducibles such that λpj˚, jq P λ Ó pxq. Dually, for x P L, the canonical meet-representation of x is Ź U m, where U is the set of meet-irreducibles such that λpm, m˚q P λ Ò pxq. Proof. We prove that x " Ž D j is a canonical join-representation of x. The dual statement may be proved similarly.
We first show that the equality x " Ž D j holds. For j P D, the pair pj˚, jq is associates with some cover pc, xq, so j ă x. Hence, Ž D j ď x. If they are unequal, then there exists an element c covered by x for which Ž D j ď c. But pc, xq is associates with pj˚, jq for some j P D, which implies j ę c. Hence, x " Let c 1 be covered by x with λpc 1 , xq " λppj 1 q˚, j 1 q. By (CN1), there exists c 1 1 with c 0^c1 Ì c 1 1 ď c 0 such that λpc 0^c1 , c 1 1 q " λppj 1 q˚, j 1 q. But this means j 1 ď c 1 1 ď c 0 holds, which is a contradiction. Now let E Ď JIpLq such that x " Ž E j is irredundant, and suppose D ‰ E. Let j 0 P DzE, and let c 0 be the element covered by x such that λppj 0 q˚, j 0 q " λpc 0 , xq. Since c 0 ă Ž E j, there exists j 1 P E such that j 1 ę c 0 . Since j 1 ‰ j 0 , the cover pj 1 , j 1 q is not associates with pc 0 , xq. In particular, c 0^j 1 ă j 1 holds. Let a 0 be an element covering c 0^j 1 with a 0 ă j 1 . Then a 0 _ c 0 " x, so pc 0^j 1 , a 0 q and pc 0 , xq are associates. This means j 0 ď a 0 ă j 1 . Hence, D ĺ E, as desired.
Lemma 2.11. Let L be a congruence-uniform lattice with CU-labeling λ. For x P L, there exists a unique element y such that λ Ò pxq " λ Ó pyq.
Proof. We prove the lemma by induction on |L|. If |L| " 1, then the statement is immediate. If not, let L 1 be a congruence-uniform lattice with interval I such that L 1 rIs -L. Let Θ be the lattice congruence whose equivalence classes are the fibers of L Ñ L 1 . Let s be the label in each Θ-equivalence class.
For x P L, if x " maxrxs Θ , then the upper covers of x in L are in correspondence with the upper covers of rxs Θ in L{Θ. This correspondence preserves labels. Hence, there is a unique element rys Θ in L{Θ with λ Ó prys Θ q " λ Ò prxs Θ q. Taking y to be the minimum element in rys Θ , we have λ Ó pyq " λ Ó prys Θ q " λ Ò prxs Θ q " λ Ò pxq.
By the uniqueness of rys Θ , if y is not unique in L, then there exists an element y 1 such that y 1 ‰ minry 1 s Θ . But s P λ Ó py 1 q and s R λ Ò pxq. Hence, the element y is unique in L.
Now let x be an element of L such that x ‰ maxrxs Θ . Then the upper covers of x are in correspondence with upper covers of rxs Θ restricted to the interval I and one additional element, maxrxs Θ . Since s P λ Ò pxq, any element y with λ Ó pyq " λ Ò pxq satisfies rys Θ P I and y " maxrys Θ . Since I inherits a CU-labeling from L{Θ, there exists a unique element rys Θ in I whose lower covers in I have the same labels as the upper covers of rxs Θ (restricted to I). Taking y " maxrys Θ , we deduce that λ Ó pyq " λ Ò pxq. The uniqueness of y follows from the uniqueness of rys Θ .
We define the Kreweras map Kr : L Ñ L where Krpxq " y if x and y are defined as in Lemma 2.11. A dual statement to Lemma 2.11 shows that Kr is a bijection. A special case of this bijection was originally defined by Kreweras on the lattice of noncrossing set partitions [38]. Using a standard bijection between noncrossing partitions and bracketings of a word, the bijection defined by Kreweras is equivalent to the Kreweras map on the Tamari order.
Lemma 2.11 may be restated using Proposition 2.10 to define a bijection L Ñ L that switches canonical join-representations with canonical meet-representations. In these terms, this bijection can be shown to exist more generally for semidistributive lattices [3]. 8 Lemma 2.12. Let L be a congruence-uniform lattice with CU-labeling λ : CovpLq Ñ P . Let rx, ys be an interval of L for which y " Ž l i"1 a i for some elements a 1 , . . . , a l that cover x. Then there exist elements c 1 , . . . , c l covered by y such that x " Ź l i"1 c i and λpx, a i q " λpc i , yq for all i. Proof. Since the restriction of a CU-labeling to an interval rx, ys is a CU-labeling of rx, ys, we may assume x "0, y "1. Let U be the set of meet-irreducibles m such that λpm, m˚q P λ Ò p0q. Then0 " Ź U m is a canonical meet-representation. Then Krp0q " Ž U κpmq is a canonical join-representation. But tκpmq : m P U u is the set of atoms of L, so1 where A is the set of atoms of L. As this is the canonical join-representation of1, we must have A " ta 1 , . . . , a l u, and there exist c 1 , . . . , c l covered by y with λp0, a i q " λpc i ,1q for all i. As each c i is meet-irreducible, we have κpc i q " a i for all i. Hence, x " Ź l i"1 c i . Given a congruence-uniform lattice L, the shard intersection order can be defined from the labeling λ : CovpLq Ñ S as follows. For x P L, let y 1 , . . . , y k be the set of elements in L such that py i , xq P CovpLq. Define ψpxq " tλpw, zq : The shard intersection order ΨpLq is the collection of sets ψpxq for x P L, ordered by inclusion. The shard intersection order was defined at this level of generality by Nathan Reading following Theorem 1-7.24 in [49].
The poset ΨpLq derives its name from a related construction on hyperplane arrangements. If A is a real, central, simplicial hyperplane arrangement, then the poset of regions with respect to any choice of fundamental chamber is a semidistributive lattice. Each hyperplane is divided into several cones, called shards. The shard intersection order is the poset of intersections of shards, ordered by reverse inclusion. When the poset of regions is a congruence-uniform lattice, the resulting poset is isomorphic to ΨpLq. However, while any shard intersection order coming from a congruence-uniform poset of regions is a lattice, this does not hold for arbitrary congruence-uniform lattices.

The noncrossing complex
In this section, we introduce the noncrossing complex of arcs on a tree. This simplicial complex gives rise to a pure, thin simplicial complex that we refer to as the reduced noncrossing complex. We use the facets of the reduced noncrossing complex to define our main object of study, the oriented flip graph of a tree.
A tree is a finite, connected acyclic graph. Any tree may be embedded in a disk D 2 in such a way that a vertex is on the boundary if and only if it is a leaf. Unless specified otherwise, we will assume that any tree comes equipped with such an embedding. We will refer to non-leaf vertices of a tree as interior vertices. We assume that any interior vertex of a tree has degree at least 3. We say two trees T and T 1 to be equivalent if there is an isotopy between the spaces D 2 zT and D 2 zT 1 .
A tree T embedded in D 2 determines a collection of 2-dimensional regions in D 2 that we will refer to as faces. A corner of a tree is a pair pv, F q consisting of an interior vertex v and a 2-dimensional face F containing v. We let CorpT q denote the set of corners of T . The embedding that accompanies T also endows each interior vertex with a cyclic ordering. Given two corners pu, F q, pu, Gq P CorpT q, we say that pu, Gq is immediately clockwise (resp. immediately counterclockwise) from pu, F q if F X G ‰ H and G is clockwise (resp. counterclockwise) from F according to the cyclic ordering at u.
An acyclic path (or chordless path) supported by a tree T is a sequence pv 0 , . . . , v t q of vertices of T such that v i and v j are adjacent if and only if |i´j| " 1. We typically identify acyclic paths with their underlying vertex sets; that is, we do not distinguish between acyclic paths of the form pv 0 , . . . , v t q and pv t , . . . , v 0 q. We will refer to v 0 and v t as the endpoints of the acyclic path pv 0 , . . . , v t q. Note that an acyclic path is determined by its endpoints, and thus we can write rv 0 , v t s " pv 0 , . . . , v t q. As an acyclic path pv 0 , . . . , v t q defines a subgraph of T (namely, the induced subgraph on the vertices v 0 , . . . , v t ), it makes sense to refer to an edge of pv 0 , . . . , v t q. Additionally, if pv 0 , . . . , v t q and pv t , . . . , v t`s q are acyclic paths that agree only at v t and where rv 0 , v t`s s is an acyclic path, we define their composition as rv 0 , v t s˝rv t , v t`s s :" rv 0 , v t`s s.
An arc p " pv 0 , . . . , v t q is an acyclic path whose endpoints are distinct leaves and any two edges pv i´1 , v i q and pv i , v i`1 q are incident to a common face. We say p traverses a corner or contains a corner pv, F q if v " v i for for some i " 0, 1, . . . , t and F is the face that is incident to both pv i´1 , v i q and pv i , v i`1 q. Since an arc p divides D 2 into two components, it determines two disjoint subsets of the set of faces of T that we will call regions. We let Regpp, F q denote the region defined by p that contains the face F .
A segment is an acyclic path consisting of at least two vertices and with the same incidence condition that is required of arcs, but whose endpoints are not leaves. Observe that interior vertices of T are not considered to be segments. Since trees have unique geodesics between any two vertices, if the endpoints of a segment or arc are v, w, we may denote it by rv, ws.
Example 3.1. Let T denote the tree shown in Figure 4 and let p " p7, 10, 11, 12, 5q be the arc of T shown in blue. The arc p contains the corners p10, F 2 q, p11, F 5 q, and p12, F 5 q. The two regions defined by p are Regpp, F 1 q " tF 1 , F 2 , F 3 , F 6 , F 7 , F 8 u and Regpp, F 4 q " tF 4 , F 5 u.
Definition 3.2. We say that two arcs p " pv 0 , . . . , v t1 q, q " pw 0 , . . . , w t2 q are crossing along a segment s " pu 0 , . . . , u r q if iq each vertex of s appears in p and in q and iiq if R p and R q are regions defined by p and q, respectively, then R p Ć R q and R q Ć R p . We say they are noncrossing otherwise. The noncrossing complex ∆ N C pT q is defined to be the abstract simplicial complex whose simplices are pairwise noncrossing collections of arcs supported by the tree T .  Figure 4. Let p " p7, 10, 11, 12, 5q and q " p6, 10, 11, 9, 1q denote the arcs of T shown in blue and red, respectively. The arcs p and q cross along the segment s " p10, 11q shown in purple. 12  If every internal vertex of T has degree 3, then r ∆ N C pT q is isomorphic to the dual associahedron. By this identification, our notion of performing a flip on a facet of the reduced noncrossing complex of T translates into the well-known operation of performing a diagonal flip on the corresponding triangulation (see Figure 5).
If p is an arc whose vertices all lie on a common face, then p is noncrossing with every arc supported by T . We call such an arc a boundary arc. Observe that boundary arcs are exactly those arcs that define a region consisting of a single face. This implies that the faces of T are in bijection with boundary arcs of T . Using this fact, at times we will refer to the boundary arc corresponding to a given face. The reduced noncrossing complex r ∆ N C pT q is the abstract simplicial complex consisting of the faces of ∆ N C pT q containing no boundary arcs.
We now introduce a partial ordering on arcs that contain a particular corner of T . This partial ordering enables us to understand the the combinatorial structure of the noncrossing complex and the reduced noncrossing complex of T . Let F be a face of ∆ N C pT q and let pv, F q be a corner that is contained in at least one arc of F. The arcs of F that contain pv, F q are partially ordered in the following way: p ď pv,F q q if and only if Regpp, F q Ă Regpq, F q. Lemma 3.5. If F is a face of ∆ N C pT q and pv, F q is a corner contained in at least one arc of F, then the partially ordered set ptp P F : p contains pv, F qu, ď pv,F q q is a linearly ordered set. Proof. Since any pair p 1 , p 2 P tp P F : p contains pv, F qu are noncrossing and since each p i defines a region that contains F , one has that p 1 ď pv,F q p 2 or p 2 ď pv,F q p 1 . Thus ptp P F : p contains pv, F qu, ď pv,F q q is a linearly ordered set.
It follows from Lemma 3.5 that the partially order set ptp P F : p contains pv, F qu, ď pv,F q q has a unique maximal element, which we will denote by ppv, F q. We say that an arc p of F is marked at pv, F q if p " ppv, F q.
The following proposition enables us to show that the simplicial complex r ∆ N C pT q is a pure (i.e. any two facets have the same cardinality) and thin (i.e. every codimension 1 simplex is a face of exactly two facets) in Corollary 3.10.
Proposition 3.6. Let F be a face of ∆ N C pT q, let p P F, and let Reg 1 , Reg 2 denote the regions defined by p.
(1) The arc p is marked at some corner of T .
(2) In p is not a boundary arc, then p is marked at a corner in Reg 1 and at a corner in Reg 2 .
(3) Assume that p is marked at two distinct corners pv, F q, pw, Gq P CorpT q and that F and G belong to the same region defined by p. Then there exists an arc p 1 R F that contains pv, F q and pw, G 1 q where G 1 ‰ G and where F Y tp 1 u P ∆ N C pT q. (4) If F is a facet and p P F is not a boundary arc, then there exists a unique arc q R F such that pFztpuqYtqu is a facet. Moreover, if p is marked at two distinct corners pv, F q, pu, Gq P CorpT q, then rv, us is the unique longest segment along which p and q cross.
Proof. (1) Let pv, F q P CorpT q be a corner contained in p. If p " ppv, F q, then we are done. Otherwise, let q P F be the arc containing pv, F q such that p Ì pv,F q q. Let w be an interior vertex at which p and q separate, let pw, Gq be the corner traversed by p at w, and let p 1 " ppw, Gq P F. Since p 1 and q are noncrossing and p ď pw,Gq p 1 , p 1 must contain the corner pv, F q and G P Regpp, F q. Now this implies p ď pv,F q p 1 so p ď pv,F q p 1 ă pv,F q q. Thus p " p 1 .
(2) In the proof of (1), we showed that if p contains a corner pw i , G i q with G i P Reg i , then there exists a corner pv i , F i q with F i P Reg i such that p " ppv i , F i q. If p is not a boundary arc, then it contains such a corner pw i , G i q with G i P Reg i for i " 1, 2.
(3) Assume that p contains two distinct corners pv, F q, pw, Gq P CorpT q where p " ppv, F q and p " ppw, Gq and where F and G belong to the same region defined by p. Let G 1 be the face containing w such that G X G 1 is an edge of the segment rv, ws. We can assume that at least one arc of F contains pw, G 1 q P CorpT q, otherwise define p 1 to be the boundary arc corresponding to G 1 and we obtain that F Y tp 1 u P ∆ N C pT q.
Let q :" ppw, G 1 q P F. The arc p is expressible as the composition p " rv 0 , vs˝rv, ws˝rw, w 0 s. Similarly, q is the composition q " rv 1 , ws˝rw, w 1 s where rw, w 1 s and p do not agree along any edges. Let p 1 be the arc p 1 :" rv 0 , ws˝rw, w 1 s. Clearly, p 1 and p do not cross.
Next, we show that F Y tp 1 u P ∆ N C pT q. Let q 1 P F and suppose that q 1 and p 1 cross along a segment s. It is enough to assume that s is contained in either rv 0 , ws or rw, w 1 s. If s is contained in rv 0 , ws, then since p and p 1 agree along rv 0 , ws we have that q 1 and p cross along s, a contradiction. Similarly, q 1 and p 1 cannot cross along a segment s contained in rw, w 1 s. We conclude that F Y tp 1 u P ∆ N C pT q.
(4) By p2q, there exist distinct corners pv 1 , F 1 q, pv 2 , F 2 q P CorpT q contained in p where F i P Reg i and such that p " ppv i , F i q for i " 1, 2. Let p 1 and p 2 be arcs of Fztpu P ∆ N C pT q where p 1 and p 2 are marked at ppv 1 , F 1 q and ppv 2 , F 2 q, respectively, with respect to the other arcs of Fztpu. Since F is a facet, it contains each boundary arc. As p is not a boundary arc, there does exist the desired arcs p 1 and p 2 in Fztpu. , we indicate where the arc p appeared before it was removed.
Lemma 3.7. In the face Fztpu, p 1 " ppv 2 , G 2 q and p 2 " ppv 1 , G 1 q where G i is the unique face of the tree T such that pv i , G i q is immediately clockwise from pv i , F i q (see Figure 6).
Proof of Lemma 3.7. We show that p 1 " ppv 2 , G 2 q and the proof that p 2 " ppv 1 , G 1 q is similar. Write p 1 " s 1˝r v 1 , w 1 s and ppv 2 , G 2 q " s 2˝r v 2 , w 2 s where s 1 , rv 1 , w 1 s, s 2 , and rv 2 , w 2 s are acyclic paths of T , w 1 and w 2 are leaf vertices of T , and where we require that rv 1 , w 1 s and rv 2 , w 2 s each contain part of the segment rv 1 , v 2 s. Now consider the arc p 1 :" s 1˝r v 1 , v 2 s˝s 2 . Since p 1 (resp. ppv 2 , G 2 q) does not cross any arcs of F along s 1 (resp. s 2 ), the same is true for p 1 . Similarly, p does not cross any arcs of F along rv 1 , v 2 s so the same is true for p 1 . As F is a facet of ∆ N C pT q, we have that p 1 P F. Now it is clear that p 1 " ppv 1 , F 1 q and p 1 " ppv 2 , G 2 q, and the result follows.
Next, let p 1 " s 1˝r v 2 , w 1 s and let p 2 " rw 2 , v 1 s˝s 2 for some acyclic paths s 1 and s 2 and some leaf vertices w 1 and w 2 of T . Define q :" rw 2 , v 1 s˝rv 1 , v 2 s˝rv 2 , w 1 s ‰ p. By Lemma 3.7 and the proof of Proposition 3.6 p3q, we have that pFztpuq Y tqu P ∆ N C pT q. Furthermore, it is clear that q " ppv 1 , G 1 q " ppv 2 , G 2 q in pFztpuq Y tqu and that rv 1 , v 2 s is the unique longest segment along which p and q cross.
Next, we show that F and pFztpuq Y tqu are the unique faces of ∆ N C pT q that contain Fztpu. Note that from this it also follows that pFztpuq Y tqu is a facet of ∆ N C pT q. Suppose there exists an arc p 1 R Fztpu such that pFztpuq Y tp 1 u is a facet. Then p 1 " ppv 2 , F 2 q " ppv 1 , F 1 q or p 1 " ppv 2 , G 2 q " ppv 1 , G 1 q, otherwise by combining Proposition 3.6 (3) and Lemma 3.7 we have that pFztpuq Y tp 1 u is not a facet. In particular, we obtain that p 1 contains the segment rv 1 , v 2 s. The following lemma shows that if p 1 " ppv 2 , F 2 q " ppv 1 , F 1 q (resp. p 1 " ppv 2 , G 2 q " ppv 1 , G 1 q), then p 1 " p (resp. p 1 " q). This establishes the uniqueness of p and q.
i) If p 1 contains the corner pv 2 , F 2 q, then p 1 and p agree along ru 1 , v 2 s˝rv 2 , v 1 s. ii) If p 1 contains the corner pv 1 , F 1 q, then p 1 and p agree along ru 2 , v 1 s˝rv 1 , v 2 s. iii) If p 1 contains the corner pv 2 , G 2 q, then p 1 and q agree along rw 1 , v 2 s˝rv 2 , v 1 s. iv) If p 1 contains the corner pv 1 , G 1 q, then p 1 and q agree along rw 2 , v 1 s˝rv 1 , v 2 s.
Proof of Lemma 3.8. We prove part iq, and the proofs of the other parts are analogous. Suppose there exists an interior vertex x P ru 1 , v 2 s where p and p 1 separate. Let px, Hq P CorpT q be the corner contained in p. Since p " ppv 1 , F 1 q " ppv 2 , F 2 q in F and since F is a facet, there exists an arc a P Fztpu where a " ppx, Hq in F. There are two cases: H P Regpp, F 2 q or H P Regpp, F 1 q (see Figure 7). Without loss of generality, we assume H P Regpp, F 2 q. If a contains pv 2 , F 2 q, then Regpp, F 2 q " Regpp, Hq Ĺ Regpa, Hq " Regpa, F 2 q, contradicting that p " ppv 2 , F 2 q in F. Thus a does not contain pv 2 , F 2 q. This implies that there exists y P rx, v 2 s such that p and a separate at y. Since a and p are noncrossing and since p ă px,Hq a, any edge of a that is not an edge of p is only incident to faces in Regpp, F 1 q. We conclude that p 1 and a cross along rx, ys, a contradiction. Note that in this situation we do not know if p 1 contains pv 1 , F 2 q or pv 1 , F 1 q, which is why it appears to terminate at v 1 in paq and pbq. The arc a " ppx, Hq has the property that H P Regpp, F 2 q or H P Regpp, F 1 q. We indicate that a is marked at corner px, Hq by marking it with a black dot in paq and pbq.
In the proof of Proposition 3.6 (4), we explained how for a given facet F P ∆ N C pT q and a given arc p P F that is not a boundary arc there is a unique way to produce another facet of ∆ N C pT q. To summarize our construction, suppose that p " ru 1 , us˝ru, vs˝rv, v 1 s in a facet F is a nonboundary arc of T where p " ppu, F q and p " ppv, Gq are the unique corners of T where p is maximal. Then there is a unique nonboundary arc q such that pFztpuqYtqu is a facet of ∆ N C pT q. The arc q " ru 2 , us˝ru, vs˝rv, v 2 s for some leaf vertices u 2 and v 2 so that q " ppu, F 1 q and q " ppv, G 1 q where the vertices of F X F 1 and G X G 1 are contained in both p and q. Example 3.9. Figure 8 shows an example of the construction in Proposition 3.6 (4) for the tree T (the tree depicted in black). A black dot appears in an arc if it is the largest arc containing the corresponding corner in that facet. The boundary arcs of T are p1, 5, 2q, p1, 5, 4q, p2, 6, 3q, and p3, 4, 6q. These appear in gold. Flipping the green arc produces the red arc. Proof. Any facet F of ∆ N C pT q has #F " #tnonboundary arcs of Fu`#tboundary arcs of Fu. Note that #tboundary arcs of Fu " #tfaces of T u. Thus to show r ∆ N C pT q is pure, it is enough to prove that ∆ N C pT q is pure. Assume F P ∆ N C pT q is a facet. Each corner of T is contained in a boundary arc of F, and thus each corner of T has a unique maximal arc containing it. Since F is a facet, by Proposition 3.6 (1), each boundary arc is maximal at exactly one corner of T . Similarly, since F is a facet, by Proposition 3.6 (2), each nonboundary arc Figure 8. The two facets of ∆ N C pT q.
of T is maximal at exactly two corners of T . This implies that #CorpT q " #tboundary arcs in Fu`2#tnonboundary arcs in Fu " #tfaces of T u`2#tnonboundary arcs in Fu.
Thus #tnonboundary arcs in Fu " 1 2 p#CorpT q´#tfaces of T uq. As the latter number is independent of F, we have that ∆ N C pT q is pure and thus so is r ∆ N C pT q. The simplicial complex r ∆ N C pT q is thin because the move between facets of ∆ N C pT q described in Proposition 3.6 (4) only involves nonboundary arcs.
We refer to the operation F Þ ÝÑ pFztpuq Y tqu sending facet F of r ∆ N C pT q to a new facet of r ∆ N C pT q as a flip of F at p (see Figure 8) and denote it by µ p . We define the flip graph of T , denoted F GpT q, to be the graph whose vertices are facets of r ∆ N C pT q and such that two vertices are connected by an edge if and only if the corresponding facets can be obtained from each other by a single flip.
We now define the following object, which is fundamental to our work in this paper.
Definition 3.11. Let F 1 , F 2 P F GpT q and assume that F 1 and F 2 are connected by an edge in F GpT q. Let If p " ppu, F q " ppv, Gq and q " ppu, F 1 q " ppv, G 1 q, we orient the edge connecting F 1 and F 2 so that F 1 ÝÑ F 2 if the corner pu, F 1 q (resp. pv, G 1 q) is immediately clockwise from the corner pu, F q (resp. pv, Gq) about vertex u (resp. v). Otherwise, we orient the edge so that F 2 ÝÑ F 1 . We refer to the resulting directed graph as the oriented flip graph of T and denote it by Ý Ý Ñ F GpT q. Additionally, any edge of Ý Ý Ñ F GpT q connecting F and µ p F is naturally labeled by the segment determined by the marked corners of p in F (or in µ p F).
Example 3.12. In Figure 9, we show the oriented flip graph (without edge labels) of the tree T from

Sublattice and quotient lattice description of the oriented flip graph
In this section, we identify the oriented flip graph Ý Ý Ñ F GpT q as both a sublattice and quotient lattice of another lattice. In Section 4.1 we define a closure operator on segments, and introduce a poset of biclosed sets of segments, denoted BicpT q. It was shown in [30] that BicpT q is a congruence-uniform lattice. We define a distinguished lattice congruence Θ on BicpT q.
In Section 4.3, we define maps η : BicpT q Ñ Ý Ý Ñ F GpT q and φ : The map η is a surjective lattice map such that ηpXq " ηpY q exactly when X " Y mod Θ. The map φ is a lattice map such that η˝φ is the identity on Ý Ý Ñ F GpT q. Since congruence-uniformity and polygonality are preserved by lattice quotient maps, we deduce that Ý Ý Ñ F GpT q is a congruence-uniform and polygonal lattice.

4.1.
Biclosed collections of segments. Let SegpT q be the set of segments supported by a tree T . For X Ď SegpT q, we say X is closed if for segments s, t P SegpT q, if s, t P X and s˝t P SegpT q then s˝t P X. If X is any subset of SegpT q, its closure X is the smallest closed set containing X. Say X is biclosed if X and SegpT qzX are both closed. For example, the collection of red segments in the left part of Figure 11 is biclosed. We let BicpT q denote the poset of biclosed subsets of SegpT q, ordered by inclusion. Let Q be the graph whose vertices are the edges between interior vertices of T , where e and e 1 are adjacent in Q if they meet at a corner pv, F q. Later, we will give Q an orientation and view it as a quiver. An acyclic path (or chordless path) of Q is a sequence of vertices pv 0 , . . . , v t q such that v i and v j are adjacent if and only if |i´j| " 1. We view acyclic paths as undirected, so they are determined by the set of vertices they visit.
A segment of T is naturally regarded as an acyclic path of Q. The set of segments of T thus forms some of the acyclic paths of Q. In Theorem 5.4 of [30], we proved that the set of biclosed subsets of acyclic paths of Q under inclusion forms a congruence-uniform, semidistributive, and polygonal lattice. By a minor modification of the proof, this can be shown to hold for biclosed subsets of any order ideal of acyclic paths, where paths are ordered by inclusion. As SegpT q is naturally regarded as an order ideal of acyclic paths of Q, we deduce the following result.
Theorem 4.1. The poset BicpT q is a semidistributive, congruence-uniform, and polygonal lattice. Furthermore: A lattice of biclosed sets of segments is given in Figure 10 (see also the upper lattice in Figure 7 in [30]). To simplify the figure, we only show the edges of the tree connecting two interior vertices in Figure 10. The Hasse diagram of this lattice is the skeleton of a zonotope with 26 vertices. Although one can find examples where BicpT q is isomorphic to the weak order on permutations, Figure 10 shows this is not true for all trees T .
Any subset S 1 of a closure space S inherits a closure operator X Þ Ñ pX X S 1 q. In general, biclosed subsets of S 1 may not be biclosed as subsets of S. For spaces of segments, some intervals of BicpT q are isomorphic to BicpS 1 q for some subset S 1 of segments. We state this precisely as the following proposition.  Let W Ă SegpT q be a biclosed set of segments, and let s 1 , . . . , s k P SegpT qzW such that W Yts i u is biclosed for all i. Let pB 1 , . . . , B l q be the finest partition on ts 1 , . . . , s k u such that if s i˝sj is a segment then s i and s j lie in the same block. Then the interval rW, W Y ts 1 , . . . , s k us is isomorphic to BicpB 1 qˆ¨¨¨ˆBicpB l q.
Proof. We first prove that the sets W, B 1 , . . . , B l are all disjoint. Suppose W X B i is nonempty for some i, and let t P W X B i be of minimum length. Since s j R W for all j, t must be a concatenation t 1˝t2 of elements of B i . By minimality, t 1 and t 2 are not in W . But W is co-closed, a contradiction. Now suppose there are two blocks, say B 1 , B 2 , such that B 1 X B 2 contains an element t. Then t is the concatenation of some elements of B 1 and of some elements of B 2 . Relabeling if necessary, let s i P B i , t i P B i for i " 1, 2 such that s 1˝t1 " t " s 2˝t2 . Then either s 1 is a subsegment of s 2 or vice versa. Without loss of generality, we assume s 1 Ĺ s 2 . Let s 1 be the segment such that s 1˝s 1 " s 2 . Since W Y ts 1 u is closed, s 1 must not be in W . But s 1 is in W since W Y ts 2 u is co-closed. Hence, we have shown that the closures of the blocks are disjoint.
Since the biclosed property is preserved under restriction, the map X Þ Ñ pX X B 1 , . . . , X X B l q from rW, W Y ts 1 , . . . , s k u to BicpB 1 qˆ¨¨¨ˆBicpB l q is well-defined. It remains to show that the inverse is also well-defined. Namely, given pX 1 , . . . , X l q P BicpB 1 qˆ¨¨¨ˆBicpB l q, we prove that W Y Ť l i"1 X i is biclosed in SegpT q. Suppose this does not always hold, and choose pX 1 , . . . , there is some nonempty X j . As BicpB j q is ordered by single-step inclusion, there is some s P X j such that X j ztsu is biclosed. By the minimality assumption, pW Y Xqztsu is biclosed.
Assume W Y X is not co-closed. Then there exist segments t, t 1 not in W Y X such that s " t˝t 1 . As X j is co-closed in B j , the segment t is not in ts 1 , . . . , s k u. Since W Y ts i u is co-closed for any i, the segment s can be factored as s i˝s 1 for some s i P X j and s 1 P ts 1 , . . . , s k u. There are two cases to consider: either t is contained in If s i Ĺ t, then there exists a segment t 2 with s i˝t After relabeling, we may assume s " s 1˝¨¨¨˝sm for some m ď k. Since W Y ts m u is closed, the segment s m˝s 1 is in W . Similarly, s i˝¨¨¨˝sm˝s 1 is in W for any i. This contradicts the assumption that t R W .
We may refer to intervals of BicpT q as in Proposition 4.2 as facial intervals.

4.2.
A lattice congruence on biclosed sets. In this section, we define a lattice congruence Θ on BicpT q. The quotient lattice BicpT q{Θ will be shown to be isomorphic to Ý Ý Ñ F GpT q in Section 4.3. Let s " pv 0 , . . . , v l q be a segment, and orient the segment from v 0 to v l . Let C s be the set of segments pv i , . . . , v j q such that ‚ if i ą 0 then s turns right at v i , and ‚ if j ă l then s turns left at v j . We note that s is always in C s since the above conditions are vacuously true. Furthermore if t P C s , then The following simple statement is used frequently in later proofs, so we state it explicitly.
The dual statement about K s follows from the same reasoning.
Given a tree T embedded in a disk, we let T _ be a reflection of T (i.e. T _ is the image of T under a Euclidean reflection performed on D 2 ). The choice of reflection is immaterial since the noncrossing complex and oriented flip graph are invariant under rotations of T . The tree T _ has the same set of segments and defines the same noncrossing complex as T . Since reflection switches left and right, Ý Ý Ñ F GpT _ q has the opposite orientation of Ý Ý Ñ F GpT q, and for any segment s, C s _ " K _ s . Let π Ó , π Ò be functions on BicpT q such that for X P BicpT q, π Ó pXq " ts P X : C s Ď Xu π Ò pXq " ts P S : K s X X ‰ Hu These maps are closely related to the maps labeled π Ó and π Ò in [30]. For completeness, we prove their main properties here.
Proof. Let s P π Ó pXq. Then C s Ď X. Since C t Ď C s for t P C s , it follows that C s Ď π Ó pXq. If s " t˝u, then either t P C s or u P C s , so either t P π Ó pXq or u P π Ó pXq. Hence, π Ó pXq is co-closed.
Let s, t P π Ó pXq such that s˝t is a segment. For u P C s˝t if u is a subsegment of s or t, then u P C s or u P C t , respectively. Otherwise, u " u 1˝u2 where u 1 is a subsegment of s and u 2 is a subsegment of t. In this case u 1 P C s and u 2 P C t . In either case, u P X holds. Consequently s˝t P π Ó pXq. Therefore, π Ó pXq is biclosed.
The fact that π Ò pXq is biclosed may be proved by a similar argument. Alternatively, it follows from the fact that π Ó pXq is biclosed and Lemma 4.5(1).
Proof. Both (4) and (7) are clear from the definitions. (3) and (6) follow from (2) and (5) by taking the complement of the reflection of X and applying (1). It remains to prove (1), (2), and (5). For (1), we have the following set of equalities: For (2), the reverse inclusion is clear. Suppose π Ó pπ Ò pXqq ‰ π Ó pXq and let s P π Ó pπ Ò pXqqzπ Ó pXq be of minimum length. Since C t Ď C s for t P C s , this implies s P π Ò pXqzX. Let u P K s X X. Then either s " t˝u, s " u˝t 1 , or s " t˝u˝t 1 holds for some segments t, t 1 P C s . But this implies s P X, a contradiction.
For the remainder of the paper, we let Θ be the equivalence relation on BicpT q such that X " Y mod Θ if π Ó pXq " π Ó pY q. Using Lemmas 2.3 and 4.5, we deduce the following proposition.
Proposition 4.6. The equivalence relation Θ is a lattice congruence on BicpT q.

4.3.
Map from biclosed sets to the oriented flip graph. In this section, we define a surjective map η : BicpT q Ñ Ý Ý Ñ F GpT q and prove that it is a lattice quotient map. Let X P BicpT q. Given a corner pv, F q, let p pv,F q be the (unique) arc supported by T such that for any interior vertex u of p pv,F q distinct from v, the following condition holds: ‚ Orienting p pv,F q from v to u, the arc p pv,F q turns left at u if and only if rv, us is in X. In Lemmas 4.7 and 4.8, we prove that the this collection of arcs is a facet of the noncrossing complex. Before proving this, we set up some notation.
For an arc p " pv 0 , . . . , v l q oriented from v 0 to v l , let C p be the set of segments pv i , . . . , v j q, 0 ă i ă j ă l such that ‚ p turns right at v i , and ‚ p turns left at v j . Define K p in the same way, switching the roles of left and right.
Lemma 4.7. Let X and tp pv,F q u pv,F q be defined as above. For p P tp pv,F q u pv,F q , C p Ď X and K p X X " H.
Proof. Let p " p pv,F q for some corner pv, F q of T . Let s P C p , and set s " ru, ws. We show that s P X by considering several cases on the location of v relative to s.
If v is an endpoint of s, then s P X by the defining rule of p pv,F q .
If v is in the interior of s, then s " ru, vs˝rv, ws. Since p extends left through both endpoints of s, both ru, vs and rv, ws are in X. Since X is closed, this implies s P X.
If v is not in s, then there exists a segment rv, us such that rv, us˝ru, ws is a segment of p. Since p extends left through both endpoints of s, rv, ws P X but rv, us R X. Since X is co-closed, this implies s P X.
The fact that K p X X " H follows from a dual argument.
Lemma 4.8. The set tp pv,F q u pv,F q is a facet of ∆ N C pT q. Moreover, p pv,F q is the arc marked at the corner pv, F q.
Proof. Let pv, F q, pv 1 , F 1 q be two corners of T and let p 1 " p pv,F q and p 2 " p pv 1 ,F 1 q . Suppose p 1 and p 2 cross along a segment s. We may assume that p 1 leaves each of the endpoints of s to the right while p 2 leaves s to the left. Then s P K p1 and s P C p2 . By Lemma 4.7, K p1 X X " H and C p2 Ď X, a contradiction. Let F " tp pv,F q u pv,F q , and let pv, F q be a corner of T . Let q P F be the arc marked at pv, F q. If q ‰ p pv,F q , then they agree on some segment rv, ws and diverge at w. Orient both arcs from v to w. Then p pv,F q turns in the same direction at both v and w whereas q turns in different directions.
If q turns left at v and right at w, then rv, ws R X since K q X X " H. As p pv,F q turns left at w, this contradicts the rule defining p pv,F q .
If q turns right at v and left at w, then rv, ws P X since C q Ď X. As p pv,F q turns right at w, this again contradicts the rule defining p pv,F q .
In either case, we obtain a contradiction. Hence, p pv,F q is the arc marked at pv, F q. It remains to show that F is maximal. If not, then there exist two corners pv, F q, pv 1 , F 1 q such that p pv,F q " If p turns right at both v and v 1 , then rv, v 1 s P X by the definition of p pv 1 ,F 1 q but rv, v 1 s R X by definition of p pv,F q . If p turns left at both v and v 1 , then rv, v 1 s P X by definition of p pv,F q but rv, v 1 s R X by definition of p pv 1 ,F 1 q . In either case, we obtain a contradiction. Hence, F is a facet of ∆ N C pT q.
We let η : BicpT q Ñ Ý Ý Ñ F GpT q be the map ηpXq " tp pv,F q u pv,F q where X P BicpT q and arcs p pv,F q are defined as above. An example of this map is given in Figure 11.
Proof. (1): Let X P BicpT q be given. Set F " ηpXq. If s P φpFq, then there exist s 1 , . . . , s l such that s " s 1˝¨¨¨˝sl and s i P C p for some p P F. For each i, C si Ď C p Ď X, so s i P π Ó pXq. As π Ó pXq is closed, this implies s P π Ó pXq. Hence φpηpXqq Ď π Ó pXq.
We prove the reverse inclusion π Ó pXq Ď φpηpXqq by induction on the length. Let s P π Ó pXq and assume that t P C s , t ‰ s implies t P φpηpXqq. Let v be an endpoint of s. Orienting s away from v, let F be the face to the right of s. Let p " p pv,F q and orient p in the same direction as s. Let v 1 be the last vertex along s at which s and p meet. Let t " rv, v 1 s. If v 1 is an endpoint of s, then p must turn left at v 1 by definition, and s P C p . If v 1 is not an endpoint of s, we consider two cases: (i) If s turns left at v 1 , then t P C s . By the inductive hypothesis, t P φpηpXqq holds, which contradicts the definition of p pv,F q .
(ii) If s turns right at v 1 , then s " t˝t 1 and t P C p . Since t 1 P C s , t 1 P φpηpXqq. Hence s P φpηpXqq holds.
(2): Let F P Ý Ý Ñ F GpT q and set X " φpFq. Let pv, F q be a corner of T . Let p be the arc in ηpφpFqq marked at pv, F q and let q be the arc in F marked at pv, F q. We prove that p " q and conclude that ηpφpFqq " F.
Suppose p and q diverge at some vertex v 1 . Orient both paths from v to v 1 . Let s " rv, v 1 s. Assume p turns left at v 1 and q turns right at v 1 . Then s P φpFq, so there exist s 1 , . . . , s l such that s " s 1˝¨¨¨˝sl and s i P C qi for some arcs q i P F. Orient each q i in the same direction as q. We may assume v P s 1 and v 1 P s l . Let v i be the first vertex of s i for each i. Since q 1 and q do not cross and q is marked at pv, F q, we conclude that both q 1 and q turn left at v 2 . By similar reasoning, q 2 and q both turn left at v 3 . By induction, q turns left at v 1 , a contradiction. Now assume p turns right at v 1 and q turns left. Then s R φpFq. Since s R C q , q must turn left at v. Let F 1 be the face to the right of q containing v and the first edge of s. Let q 1 be the arc of F marked at pv, F 1 q. Then q 1 and q agree after v. Hence s P C q 1 , a contradiction.
By Lemma 4.9, the equivalence relation on BicpT q induced by η is equal to Θ. That is, X " Y mod Θ holds if and only if ηpXq " ηpY q. By Proposition 4.6, we may identify the facets of the noncrossing complex with the elements of the quotient lattice BicpT q{Θ. It remains to show that this ordering is isomorphic to Ý Ý Ñ F GpT q. To this end, it is enough to check that the Hasse diagram of BicpT q{Θ is Ý Ý Ñ F GpT q, as in the following lemma. Recall the edge-labeling of the oriented flip graph from Definition 3.11. Lemma 4.10. The Hasse diagram of Ý Ý Ñ F GpT q is isomorphic to that of BicpT q{Θ. More precisely, we have the following.
(1) Let X P BicpT q such that X " π Ó pXq. If s is a segment in X such that Xztsu is biclosed, then ηpXztsuq Ñ F for some arcs p,p and segment s, then φpFqztsu is biclosed and ηpφpFqztsuq " Fztpu Y tpu.
Proof. (1): Let X P BicpT q such that X " π Ó pXq. Let s be a segment in X such that Xztsu is biclosed. Then C s Ď X and K s X X " H. Let v, v 1 be the endpoints of s. Orient s from v to v 1 . Let F be the face to the right of s incident to v and the first edge of s. Let p be the arc of ηpXq marked at pv, F q. Since C s Ď X and K s X X " H, p contains s and turns left at v 1 . Let F 1 be the face left of s incident to v 1 and the last edge of s. Let p 1 be the arc of ηpXq marked at pv 1 , F 1 q. Reversing the orientation on s, the previous argument implies that p 1 contains s.
We claim that p " p 1 . If not, then p and p 1 must diverge at a vertex v 2 . Let t " rv 1 , v 2 s and u " rv, v 2 s. Without loss of generality, we may assume that s˝t " u. Since p and p 1 do not cross, p turns left at v 2 and p 1 turns right at v 2 . Hence u P X and t R X. But then Xztsu is not co-closed, a contradiction.
Letp be the arc obtained by flipping p in ηpXq. Then p andp meet along s. We show that ηpXztsuq " ηpXqztpu Y tpu.
Let G be the face left of s containing v and the first edge of s. Similarly, let G 1 be the face right of s containing v 1 and the last edge of s. Let q be the arc marked at pv, Gq in ηpXq, and let q 1 be the arc marked at pv 1 , G 1 q in ηpXq.
By the definition of η, the only arcs that can be different between ηpXq and ηpXztsuq are those arcs marked at pv, F q, pv, Gq, pv 1 , F 1 q, or pv 1 , G 1 q. Just as we proved that p is the arc in ηpXq marked at pv, F q and pv 1 , F 1 q, a similar argument shows thatp is the arc in ηpXztsuq marked at pv, Gq and pv 1 , G 1 q.
We show that q is in ηpXztsuq and is marked at pv 1 , F 1 q. Similarly we claim that q 1 is in ηpXztsuq and is marked at pv, F q. As these two proofs are nearly identical, we only write the first.
Letq be the arc in ηpXztsuq marked at pv 1 , F 1 q, and assume q ‰q. Let v 2 be a vertex at which q andq diverge. Orientq from v 1 to v 2 . Thenq turns right at If v 2 is strictly between v and v 1 , then q andq turn in the same direction at v 2 . This is impossible since C s Ď X and K s X X " H, and either rv, v 2 s P C s and rv 2 , v 1 s P K s or rv, v 2 s P K s and rv 2 , v 1 s P C s .
If v 1 is strictly between v and v 2 , then either rv 1 , v 2 s P X and rv, v 2 s R X or rv 1 , v 2 s P X but rv, v 2 s R X. The first case implies Xztsu is not co-closed, and the second case implies X is not closed.
If v is strictly between v 1 and v 2 , then we again deduce a contradiction in a similar way as the previous case. This completes the proof.
(2): Let F P Ý Ý Ñ F GpT q. Assume Fztpu Y tpu s Ñ F for some arcs p,p and segment s. Then s P C p , so s P φpF q. Suppose φpFqztsu is not closed. Then there exist segments t, u P φpFq such that t˝u " s. We may assume t P K s and u P C s . Let v, v 1 be the endpoints of t. Assume t and u meet at v 1 , and orient the arcs containing t from v to v 1 . Let t 1 , . . . , t l be segments such that t " t 1˝¨¨¨˝tl and t i P C pi for some arcs p i P F. We assume v is in t 1 and v 1 is in t l . For each i, let v i be the first vertex in t i with the orientation induced by t. Since p 1 andp do not cross along t 1 ,p must turn left at v 2 . Similarly,p turns left at v 3 , . . . , v l . But since t P K s ,p turns right at v 1 , so it crosses p l , a contradiction. We deduce that φpFqztsu is closed.
Suppose φpFqztsu is not co-closed. Then there exist segments t, u such that t R φpFq, u P φpFq, and s˝t " u. Since π Ó pφpFqq " φpFq, we deduce that s P C u and t P K u . Let u 1 , . . . , u l be segments with arcs p 1 , . . . , p l in F such that u i P C pi and u " u 1˝¨¨¨˝ul . Orient u from u 1 to u l . By similar reasoning as before, sincep and p i do not cross along u i for each i, if u i is a subsegment of s, thenp turns left at the end of u i . As t R φpFq, there exists some segment u j that is neither a subsegment of s or t. Let v 1 be the common endpoint of s and t, and let v be the endpoint of u j contained in s. Since s P C u , u j turns left at v. Hence, p j turns right at v and left at v 1 , whereasp turns left at v and right at v 1 . But this meansp and p j cross along rv, v 1 s, an impossibility. Therefore, φpFqztsu is biclosed. From (1), the equality ηpφpFqztsuq " Fztpu Y tpu holds.
Theorem 4.11. The maps η and φ identify Ý Ý Ñ F GpT q as a quotient lattice and a sublattice of BicpT q as follows. (1) The map η is a surjective lattice map such that ηpXq " ηpY q if and only X " Y mod Θ.
(2) The map φ is an injective lattice map whose image is π Ó pBicpT qq.
Proof. We have already established that η is lattice quotient map. It remains to show that φ preserves the lattice operations.
Since φ is order-preserving, φpF^F 1 q ď φpFq^φpF 1 q holds. Let X " φpFq^φpF 1 q. Since it suffices to show that π Ó pXq " X. Let s P X and t P C s . Since φpFq " π Ó pφpFqq, C s Ď φpFq X φpF 1 q. If t R X then there exist u 1 , . . . , u l R φpFq X φpF 1 q such that t " u 1˝¨¨¨˝ul . But u i P C t for some i. Since C t Ď C s , we deduce u i P φpFq X φpF 1 q, a contradiction.
By Lemma 2.5 and Theorem 4.1(3), it follows that the labeling F 1 s Ñ F of the covering relations of Ý Ý Ñ F GpT q by segments is a CN-labeling. To see that this is a CU-labeling, we observe that if there is a flip F 1 s Ñ F, then ηpC s q ď F. The following corollary is a consequence of Proposition 2.10.

Noncrossing tree partitions
In this section, we introduce noncrossing tree partitions, which are partitions of the interior vertices of a tree embedded in a disk whose blocks are noncrossing as defined in Section 5.1. In Section 5.2, we define a bijection on the set of noncrossing tree partitions, which we call Kreweras complementation. The equivalence of this definition of Kreweras complementation with the lattice-theoretic definition in Section 2.2 is given in Section 5.3. Our main result in this section is that the lattice of noncrossing tree partitions is isomorphic to the shard intersection order of Ý Ý Ñ F GpT q, which we prove in Section 5.5.
5.1. Admissible curves. Fix a tree T " pV, Eq embedded in a disk D 2 with the Euclidean metric. Let V o denote the set of interior vertices of T . We fix a small ą 0 such that the -ball centered at any interior vertex of T is contained in D 2 , and no two such -balls intersect. For each corner pv, F q, we fix a point zpv, F q in the interior of F of distance from v. Let In words, T is the embedded tree T plus the open -ball around each interior vertex. If s is a segment of T , let s denote the set of points on an edge of s of distance at least from any interior vertex of T . It will be convenient to represent segments as certain curves in the disk as follows. A flag is a triple pv, e, F q of a vertex v incident to an edge e, which is incident to a face F . Orienting e away from v, we say a flag is green if F is left of e. Otherwise, the flag is red. Let pu, e, F q, pv, e 1 , Gq be two green flags such that ru, vs is a segment containing the edges e, e 1 as in Figure 12. A green admissible curve γ : r0, 1s Ñ D 2 for ru, vs is a simple curve for which γp0q " zpu, F q, γp1q " zpv, Gq and γpr0, 1sq Ď D 2 zpT zru, vs q. Similarly, if pu, e, F 1 q and pv, e 1 , G 1 q are red flags, then a red admissible curve is defined the same way, with γp0q " zpu, F 1 q, γp1q " zpv, G 1 q. We say a segment is green if it is represented by a green admissible curve. Similarly, a segment is red if it is represented by a red admissible curve. We may also refer to an admissible curve for a segment without specifying a color. Such a curve may be either green or red.
If a colored segment s is represented by a curve with endpoints zpu, F q and zpv, Gq, we say that pu, F q and pv, Gq are the endpoints of s, and we write Endptpsq " tpu, F q, pv, Gqu. If S is a collection of colored segments, we let EndptpSq " Ť sPS Endptpsq. We refer to corners or vertices as the endpoints of a segment at different parts of this paper. The distinction should be clear from context. Given an interior vertex v P V o incident to faces F and F 1 , let α F,F 1 v : r0, 1s Ñ D 2 be a simple path contained in F Y F 1 with α F,F 1 v p0q " zpv, F q, α F,F 1 v p1q " zpv, F 1 q and |α F,F 1 v ptq´v| " for t P r0, 1s. We use the paths α F,F 1 v to concatenate admissible curves.
Two colored segments are noncrossing if they admit admissible curves that do not intersect. Otherwise, they are crossing. We remark that if two curves share an endpoint zpu, F q then they are considered to be crossing. To determine whether two colored segments s, t cross, one must check whether the endpoints of t lie in different connected components of pD 2 zpT zt qqzγ for some admissible curve γ for s. We will find it convenient to distinguish several cases of crossing as in the following lemma. The three cases correspond to the three columns of Figure 13.
Lemma 5.1. Let γ and γ 1 be two (left or right) admissible curves corresponding to segments s and s 1 that meet along a common segment t. Let t " ra, bs and orient γ and γ 1 from a to b. Assume that γ and γ 1 do not share a corner. Then γ and γ 1 are noncrossing if and only if one of the following holds: (1) γ (or γ 1 ) does not share an endpoint with t, and γ turns left (or right) at both endpoints of t; (2) γ starts at a and turns left (resp. right) at b, and γ 1 ends at b and turns right (resp. left) at a; (3) γ and γ 1 both start at a (resp. both end at b) where γ leaves a (resp. b) to the left, and γ turns left at b (resp. a) or γ 1 turns right at b (resp. a). If γ and γ 1 are both left admissible or both right admissible, then the third case does not occur.
22 Figure 14. A lattice of noncrossing tree partitions Lemma 5.2. If red segments s and s 1 are noncrosssing, then K s X C s 1 is empty.
Proof. Suppose K s X C s 1 contains an element ru, vs. Orient s and s 1 from u to v. Then s either starts at u or turns left at u, and it either ends at v or turns right at v. On the other hand, s 1 either starts at u or turns right at u, and it either ends at v or turns left at v. In each case, the segments s and s 1 are crossing.
By a similar analysis, the same result holds for segments of different color. We note that Lemma 5.3 is asymmetric in red and green. Lemma 5.3. If a green segment s and a red segment s 1 are noncrossing, then K s X C s 1 is empty.
For B Ď V o , let Seg g pBq be the set of inclusion-minimal segments whose endpoints lie in B. We say B is segment-connected if for any two elements u, v of B, there exists a sequence u " u 0 , . . . , u N " v of elements of B such that ru i´1 , u i s P SegpBq for all i. If B " pB 1 , . . . , B l q is a partition of V o , we let SegpBq " Ť l i"1 SegpB i q. We let Seg g pBq (resp. Seg r pBq) denote the same set of segments, all colored green (resp. red).
A noncrossing tree partition B is a set partition of V o such that any two segments of Seg r pBq are noncrossing and each block of B is segment-connected. Note that we intentionally define noncrossing tree partitions using only red segments. Let NCPpT q be the poset of noncrossing tree partitions of T , ordered by refinement. We give an example of NCPpT q in Figure 14 where T is the tree whose biclosed sets appear in Figure 10. We remark that the lattice of noncrossing tree partitions is not isomorphic to the lattice of noncrossing set partitions in this example.
We record some basic properties of noncrossing tree partitions in the rest of this section.
Lemma 5.4. Let B be a noncrossing tree partition containing a block B. If u, v P B are distinct vertices such that ru, vs is not a segment, then there exists a vertex w P B distinct from u and v such that w P ru, vs.
Proof. Let w P ru, vs such that ru, ws is a segment of maximum length. Since B is segment-connected, there exists a sequence u " u 0 , u 1 , . . . , u l " v of elements of B such that ru i´1 , u i s is a segment for all i. We further assume that each segment ru i´1 , u i s is in SegpBq and that l is minimal with this property.
Since w is in ru, vs, there exists some segment ru i´1 , u i s containing w such that u i´1 P ru, ws. Then w P ru i , vs, so the noncrossing property forces w P B.
Lemma 5.5. Let B be a noncrossing tree partition. If B is a block of B, then for any distinct vertices u, v P B, there exists a sequence u " u 0 , . . . , u l " v such that ru i´1 , u i s is in SegpBq for all i and ru, vs " ru 0 , u 1 s˝¨¨¨r u l´1 , u l s.
Proof. Let B be a block of B with at least two elements, and fix distinct vertices u, v P B. We proceed by induction on the length of ru, vs. Among the vertices of ru, vs, let u 1 be the element of Bztuu minimizing the length of ru, u 1 s. By Lemma 5.4, ru, u 1 s is a segment. By assumption, it is inclusion-minimal, so it is in SegpBq. If u 1 ‰ v, then by the inductive hypothesis, there exists a sequence u 1 , u 2 , . . . , u l of elements of B such that u l " v and ru i´1 , u i s P SegpBq for all i. Lemma 5.6. Let b be a vertex in a green segment ra, cs, and let rd, es be some green segment that crosses ra, cs. Then either ra, bs or rb, cs crosses rd, es, where ra, bs and rb, cs are both green.
Proof. Suppose neither ra, bs nor rb, cs crosses rd, es. Let γ 1 , γ 2 and γ 1 be green admissible curves for ra, bs, rb, cs, and rd, es, respectively. Let e, e 1 be the edges of ra, bs and rb, cs incident to b. Orienting e and e 1 away from b, let F be the face left of e and F 1 the face left of e 1 . Then γ 1 (resp. γ 2 ) has an endpoint at zpb, F q (resp. zpb, F 1 q). Since ra, cs is a segment, the faces F and F 1 are adjacent. Let γ " γ 1˝α F,F 1 b˝γ 2 . Then γ is a green admissible curve for ra, cs and γ does not intersect γ 1 , a contradiction.

Kreweras complementation.
In this section, we define a bijection on NCPpT q, which we call Kreweras complementation. A representation-theoretic interpretation of this bijection is given in Section 8.
We define a function ρ : Ý Ý Ñ F GpT q Ñ NCPpT q as follows. Let F P Ý Ý Ñ F GpT q, and let S be the set of segments for which there exists F 1 with F 1 s Ñ F. Since the arcs in F are pairwise noncrossing, there exists a realization by simple curves tγ p : p P Fu such that the following conditions hold.
‚ If s is the largest segment contained in an arc p P F, then the image of γ p is contained in D 2 zpT zs q. ‚ For distinct p, q P F, γ p and γ q are disjoint except possibly at the endpoints. ‚ For p P F, if γ p is marked at pv, F q, then γ p contains the point zpv, F q. For s P S, let p be the arc marked at the endpoints of s. If p is marked at the corners pv, F q, pv 1 , F 1 q, we let γ s be the subpath of γ p with endpoints zpv, F q and zpv 1 , F 1 q. Since s is a lower label of F, the curve γ s is a red admissible curve for s. Since γ p and γ q are disjoint for distinct arcs p, q, the collection tγ s : s P Su is a noncrossing set of red admissible curves. Hence, S defines a noncrossing tree partition B.
Given F and B as above, we set ρpFq " B. We prove that ρ is a bijection.
Proposition 5.7. The map ρ is a bijection.
To show that ρ is surjective, it suffices to prove thatφ is injective and its image is π Ó pBicpT qq. The latter statement is clear since C s P π Ó pBicpT qq for any segment s, and π Ó pBicpT qq is a sublattice of BicpT q by Theorem 4.11.
Let B P NCPpT q and set X "φpBq. Let S " ts P SegpT q : Xztsu P BicpT qu. To prove thatφ is injective, we show that SegpBq " S.
Suppose Sz SegpBq is nonempty, and let t P Sz SegpBq. Since Xzttu is biclosed, the segment t is not the concatenation of any two segments in X. Consequently, t P C s for some s P SegpBq and s ‰ t. Then s " t p1q˝t˝tp2q where t p1q or t p2q is a nonempty segment (or both). Moreover, since Xzttu is co-closed and s P X, either t p1q P X or t p2q P X holds. We may assume without loss of generality that t p1q is a segment in X. Since t P C s , we have t p1q P K s . By definition, t p1q " t 1˝¨¨¨˝tl where each t i is in C s 1 for some s 1 P SegpBq. By repeated application of Lemma 4.3, some t i is in K s . Hence, K s X C s 1 is nonempty for some s 1 P SegpBq. By Lemma 5.2, the red segments s and s 1 are crossing, which is a contradiction. Now assume SegpBqzS is nonempty, and let s P SegpBqzS. Then Xztsu is not biclosed. Suppose Xztsu is not closed. Then there exists t, t 1 P X such that s " t˝t 1 . Without loss of generality, we may assume t P C s and t 1 P K s . Since t 1 P X, there exist a decomposition t 1 " t 1˝¨¨¨˝tl such that for all i, t i P C s 1 for some s 1 P SegpBq. By Lemma 4.3, some t i is in K s . Hence, K s X C s 1 is nonempty for some s 1 P SegpBq, and we again deduce that s and s 1 are crossing.
Suppose Xztsu is not co-closed. Then there exists t P SegpT qzX such that s˝t P X. We choose the segment t to be minimal with those properties.
Suppose s˝t P C s 1 for some s 1 P SegpBq. Since t R X, t is not in C s 1 . Hence, t P K s˝t and s P C s˝t . But this implies s P C s 1 , so s and s 1 are crossing. Now assume s˝t " s 1˝¨¨¨˝sl , l ą 1 where for all i, s i P C s 1 for some s 1 P SegpBq. Since t R X and X " π Ó pXq, the segment t is not in C s1˝¨¨¨˝s l . Consequently, s P C s1˝¨¨¨˝s l . We consider two cases: either s l is a proper subsegment of t or t is a proper subsegment of s l . We note that t is not equal to s l since t R X and s l P X.
If s l is a subsegment of t, then there exists a segment t 1 such that s˝t 1 " s 1˝¨¨¨˝sl´1 . Then t 1 P X by minimality of t. But t " t 1˝s l , contrary to X being closed. If t is a subsegment of s l , then there exists t 1 such that t 1˝t " s l . Then t P K s l since t P K s1˝¨¨¨˝s l , so t 1 P C s l . Since s l P C s 1 for some s 1 P SegpBq, we have t 1 P C s 1 . Since t 1 is a proper subsegment of s that shares an endpoint with s, either t 1 P K s or t 1 P C s . If t 1 P K s , then s and s 1 cross by Lemma 5.2. If t 1 P C s , then s 1˝¨¨¨˝sl´1 P K s . By repeated application of Lemma 4.3, some s i is in K s . But s i P C s 1 for some s 1 P SegpBq, which is again a contradiction.
We have established that S and SegpBq are identical. Hence, the mapφ is injective, and the result follows.
Proposition 5.8. For s, t P SegpT q, if s and t are noncrossing as green segments, then they are noncrossing as red segments.
Proof. Let γ s and γ t be green admissible curves for s and t that do not intersect. Suppose s has corners pu, F q, pv, Gq as a green segment and pu, F 1 q, pv, G 1 q as a red segment. Define γ 1 s to be the curve We apply a slight homotopy to γ 1 s so that γ 1 s is a simple curve and zpu, F 1 q and zpv, G 1 q are the unique points of distance at most from some interior vertex of T . Then γ 1 s is a red admissible curve for s. If γ 1 t is defined in a similar manner, then it is a red admissible curve for t that does not intersect γ 1 s . Hence, s and t are noncrossing as red segments.
Theorem 5.9. Let B be a noncrossing tree partition, and let F " ρ´1pBq. Setting S " ts P SegpT q : DF s Ñ F 1 u, we have S " SegpB 1 q for some noncrossing tree partition B 1 .
The noncrossing tree partition B 1 of Theorem 5.9 is called the Kreweras complement of B. Kreweras complementation is a bijection Kr : NCPpT q Ñ NCPpT q.

Red-green trees.
A red-green tree T is a collection of pairwise noncrossing colored segments such that every pair of vertices in V o is connected by a sequence of curves in T . The segments in T are allowed to be red or green. Let T r (resp. T g ) be the subset of red (resp. green) segments of T .
That red-green trees are actual trees (i.e. acyclic) will be a consequence of Theorem 5.10. Given F P Ý Ý Ñ F GpT q, let S r " ts : DF 1 s Ñ Fu and S g " ts : DF s Ñ F 1 u.
Theorem 5.10. The sets S r and S g form the red and green segments of a red-green tree. Conversely, every red-green tree is of this form.
Proof. In the same way that a nonintersecting collection of red admissible curves for segments of S r was constructed in the definition of ρ in Section 5.2, one may construct a family of nonintersecting red and green admissible curves for S r Y S g . It remains to show that the graph on the interior vertices of T with edge set S r Y S g is connected. This follows from the fact that the graph of facets is connected, and flips preserve connectivity of S r Y S g ; see Proposition 8.8 and Figure 21. Now let T be a red-green tree. Then T r is the set of minimal segments of a noncrossing tree partition B. Let X "φpBq, whereφ is the map to biclosed sets from the proof of Proposition 5.7. By definition, X " Ť sPTr C s . We prove that SegpT qzπ Ò pXq " ď sPTg K s .
Since Ž sPTg ηpK s q is the canonical join-representation of an element in Ý Ý Ñ F GpT _ q, this equality uniquely identifies T g .
By definition, the set SegpT qzπ Ò pXq consists of segments t for which K t X X is empty. We first show that Ť sPTg K s is a subset of SegpT qzπ Ò pXq. To this end, it suffices to show that K s X X " H holds whenever s P T g . If not, then let s P T g such that K s X X is nonempty, and let t P K s X X. Since t P X, there exist segments t 1 , . . . , t l , s 1 , . . . , s l such that t " t 1˝¨¨¨˝tl and t i P C si for all i. Then t i P K t for some i. Since K t Ď K s , t i is in K s . But since s i and s do not cross, K s X C si is empty by Lemma 5.3, a contradiction. Now we prove that SegpT qzπ Ò pXq is a subset of Ť sPTg K s . Let t " ru, vs be a segment for which K t X X " H. Since T is a red-green tree, there is a path in T with edges s 1 , . . . , s l such that s 1 starts at u and s l ends at v. We consider two cases: either t is the concatenation of s 1 , . . . , s l (i.e. t " s 1˝¨¨¨˝sl ), or it is not.
Assume that t is not equal to the concatenation of s 1 , . . . , s l . Then there exists a vertex w incident to an edge e such that two adjacent segments s i , s i`1 both contain e and share an endpoint at w. Then s i and s i`1 must have different colors. Up to reversing the order of the segments, we may assume s i is red and s i`1 is green. Let rw 1 , ws be the largest common subsegment of s i and s i`1 . Since s i and s i`1 are noncrossing, the segment rw 1 , ws is in K si X C si`1 . Let s 1 i , s 1 i`1 be segments such that s 1 i˝r w 1 , ws " s i and s 1 i`1˝r w 1 , ws " s i`1 . It is possible that rw 1 , ws is equal to s i or s i`1 (but not both), in which case s 1 i or s 1 i`1 is a lazy path and thus not a segment. If s 1 i is a segment and i ą 1, we claim that it does not cross s i´1 . Indeed, if s 1 i and s i´1 cross, then s i´1 must be green and C s 1 i X K si´1 is nonempty. But this implies C si X K si´1 is nonempty, a contradiction. If s 1 i is not a segment and i ą 1, we claim that s 1 i`1 does not cross s i´1 . If s 1 i`1 and s i´1 do cross, then s i´1 must be red and K s 1 i`1 X C si´1 is nonempty. But this implies K si`1 X C si´1 is nonempty, a contradiction. Hence, s 1 , . . . , s 1 i , s 1 i`1 , . . . , s l is a sequence of red and green segments connecting the endpoints of t such that the red segments are in Ť sPTr C s and the green segments are in Ť sPTg K s . Moreover, adjacent segments are noncrossing. Proceeding inductively, we may assume that t is the concatenation of noncrossing colored segments t 1 , . . . , t l where each t i is either a red segment in Ť sPTr C s or a green segment in Ť sPTg K s . If t 1 , . . . , t l are all green segments, then t P Ť sPTg K s , as desired. Assume at least one segment is red, and let t i , . . . , t j be a maximal subsequence of red segments. We prove that t i˝¨¨¨˝tj is in K t . If i ą 1, then t i´1 is a green segment not crossing t i such that the concatenation t i´1˝ti is a segment. This implies t i P K ti´1˝ti . Similarly, if j ă l, then t j`1 is a green segment not crossing t j , and t j is in K tj˝tj`1 . Hence, t i˝¨¨¨˝tj is in K t . But this implies t m is in K t for some i ď m ď j. As t m P X, this contradicts the assumption that K t X X is empty.
Since ρ is a bijection that only depends on the red segments of a facet, Theorem 5.10 gives a bijection between noncrossing tree partitions and red-green trees. This correspondence encodes Kreweras complementation in a nice way.
Corollary 5.11. Let B be a noncrossing tree partition. There exists a unique red-green tree T whose set of red segments is SegpBq. Moreover, the set of green segments of T is SegpKrpBqq.

Lattice property.
Let ΠpV o q be the lattice of all set partitions of V o , ordered by refinement. Recall that the meet of any two set partitions is their common refinement. We prove that NCPpT q is a meet-subsemilattice of ΠpV o q in Theorem 5.12. Since NCPpT q has a top and bottom element, this implies that it is a lattice.
Theorem 5.12. The poset NCPpT q is a lattice.
Proof. Let B, B 1 be two noncrossing tree partitions, and let B 2 be the common refinement of B and B 1 . We claim that B 2 is a noncrossing tree partition and deduce that NCPpT q is a meet-subsemilattice of ΠpV o q. We first prove that every block of B 2 is segment-connected.
Let B 2 be a block of B 2 , and let u, v P B 2 . There exist blocks B P B, B 1 P B 1 , each containing u and v. We prove by induction that there exists a sequence u " u 0 , . . . , u l " v of elements of B 2 such that ru i´1 , u i s is a segment for all i. Among vertices of ru, vs, choose u 1 such that ru, u 1 s is a segment of maximum length. If u 1 " v, we are done. Since B is segment-connected, there exists a sequence u " w 0 , . . . , w m " v of elements of B such that rw i´1 , w i s is a segment for all i. Moreover, these segments may be chosen so that ru, vs is the concatenation of the segments rw i´1 , w i s. Then u 1 is a vertex in rw i´1 , w i s for some i. As rw i´1 , w i s is a segment, this forces u 1 " w i´1 or u 1 " w i . Hence, u 1 P B. By a similar argument u 1 P B 1 so u 1 is an element of B 2 . By induction, we conclude that B 2 is segment-connected.
Let S " Ť B 2 PB 2 SegpB 2 q and suppose ra, bs, rc, ds P S such that ra, bs and rc, ds are crossing. Assume that these segments share a common endpoint, say b " c, then they intersect in a common segment rb, es. As B and B 1 are noncrossing tree partitions, there exist blocks B P B, B 1 P B 1 such that a, b, d, e P B and a, b, d, e P B 1 . Hence, e P B 2 . But rb, es is a subsegment of ra, bs and rb, ds, contradicting the minimality of segments in SegpB 2 q. Now assume that the endpoints are all distinct. Let B 2 1 , B 2 2 be blocks in B 2 such that a, b P B 2 1 and c, d P B 2 2 . Since B 2 is the common refinement of B and B 1 , we may assume without loss of generality that B contains distinct blocks B 1 and B 2 such that a, b P B 1 and c, d P B 2 . Since B is noncrossing, either ra, bs R SegpB 1 q or rc, ds R SegpB 2 q. Suppose ra, bs R SegpB 1 q. Then there exists a 1 P ra, bs such that ra, a 1 s P SegpB 1 q. Then either ra, a 1 s or ra 1 , bs cross rc, ds by Lemma 5.6. By induction, there exists segments ra 1 , b 1 s P SegpB 1 q, rc 1 , d 1 s P SegpB 2 q such that ra 1 , b 1 s and rc 1 , d 1 s cross, a contradiction.

Shard intersection order.
In this section, we prove that the shard intersection order of Ý Ý Ñ F GpT q is naturally isomorphic to NCPpT q.
Let B be a segment-connected subset of T o , and let S " Seg r pBq. We define the contracted tree T B such that ‚ B is the set of interior vertices of T B , ‚ S is the set of interior edges of T B , and ‚ for edges e with one endpoint u in B and the other endpoint not between two vertices of B, there is an edge from u to the boundary in the direction of e. As in Proposition 4.2, we may compute the facial intervals of Ý Ý Ñ F GpT q as follows.
Proposition 5.13. Let F P Ý Ý Ñ F GpT q, and let s 1 , . . . , s k be a set of segments for which there exists flips F si Ñ F 1 for each i. Let B " pB 1 , . . . , B l q be the noncrossing tree partition with segments SegpBq " ts 1 , . . . , s k u. Let T i denote the contracted tree T Bi . Then rF, where the join is taken over F 1 for which F si Ñ F 1 for some s i (see Figure 15).
Proof. Let X be the biclosed set π Ò pφpFqq. Then ts 1 , . . . , s k u is the set of segments for which X Y ts i u is biclosed, and ηpX Y ts i uq " F 1 where F 1 is the facet obtained by flipping F at s i . Set Y " X Y ts 1 , . . . , s k u. Let B " tB 1 , . . . , B l u be the noncrossing tree partition with SegpBq " ts 1 , . . . , s k u, and let T i be the contracted tree T Bi . By Proposition 4.2, the interval rX, Y s is isomorphic to BicpT 1 qˆ¨¨¨ˆBicpT l q.
As usual, we let Θ denote the lattice congruence that identifies Ý Ý Ñ F GpT q with BicpT q{Θ. We let Θ i denote the corresponding lattice congruence on BicpT i q. By Lemma 2.4, the quotient interval rX, Y s{Θ is isomorphic to rF, Ž F 1 s. Hence, we prove rX, Y s{Θ -BicpT 1 q{Θ 1ˆ¨¨¨ˆB icpT l q{Θ l . Given a segment s supported by T i , we let C i s (resp. K i s ) denote the intersection C s X SegpT i q (resp. K s X SegpT i q), and we define maps π i Ó and π Ò i by the congruence Θ i . Explicitly, we have π i Ó pZq " ts P SegpT i q : C i s Ď Zu and π Ò i pZq " ts P SegpT i q : K i s X Z ‰ Hu.
Let Z, Z 1 P rX, Y s. Then Z " X Y Ť l i"1 Z i and Z 1 " X Y Ť l i"1 Z 1 i for some (unique) Z i , Z 1 i P BicpT i q. We prove that Z " Z 1 mod Θ if and only if Z i " Z 1 i mod Θ i for all i.
Suppose Z " Z 1 mod Θ, and fix i P t1, . . . , lu. To prove that Z i " Z 1 i mod Θ i , it suffices to show that π Ò i pZ i q " π Ò pZq X SegpT i q. If s P π Ò i pZ i q, then K i s X Z i is nonempty. But this implies K s X Z is nonempty, so s P π Ò pZq X SegpT i q. Conversely, if s P π Ò pZq X SegpT i q, then K s X Z is nonempty. Since π Ò pXq " X, we deduce that K s X Ť l j"1 Z j is nonempty. But K s X Z j " H whenever j ‰ i since blocks B i and B j are noncrossing. Hence, s P π Ò i pZ i q, as desired. Now assume Z i " Z 1 i mod Θ i for all i. Since π Ò i pZ i q Ď π Ò pZq and π Ò is idempotent, we have Therefore, Z " Z 1 mod Θ.
Proof. Let F be an element of Ý Ý Ñ F GpT q, and let Let S " SegpρpFqq. By Lemma 2.12, F is equal to Let ρpFq " pB 1 , . . . , B l q, and let T i be the contracted tree T Bi . By Proposition 5.13, the interval rF 1 , Fs is isomorphic to Ý Ý Ñ F GpT 1 qˆ¨¨¨ˆÝ Ý Ñ F GpT l q. The set ψpFq is defined to be the set of labels s such that there exists a covering relation From this description, it is clear that ψ is a bijection. Hence, the inverse ψ´1 exists, and the composite map ρ˝ψ´1 is a bijection. Since the Kreweras complement is defined for both Ψp Ý Ý Ñ F GpT qq and NCPpT q via the bijections ρ and ψ, the Kreweras-equivariance is immediate. If F 1 , F 2 P Ý Ý Ñ F GpT q satisfy ψpF 1 q Ď ψpF 2 q, then the corresponding noncrossing tree partitions are ordered by refinement. Conversely, it is clear that if ρpF 1 q ď ρpF 2 q, then any segment in ψpF 1 q is contained in ψpF 2 q. Hence, the bijection ρ˝ψ´1 is an isomorphism of posets.

Trees and their tiling algebras
Given a tree T embedded in D 2 , we explain how one associates to it a finite dimensional algebra Λ T . The construction we present is useful in that the indecomposable modules of the resulting algebra Λ T , as we will show (see Corollary 6.5), are parameterized by the segments of T. We also classify the extensions between indecomposable Λ T -modules (see Propositions 6.6 and 6.7 and Theorems 6.8 and 6.9), which will be useful in our applications. Before presenting the definition of Λ T , we review some background on path algebras, quiver representations, and string modules. At the end of this section, we show that the oriented flip graph of T is isomorphic to the lattice of torsion-free classes in Λ T -mod, and we show that the lattice of noncrossing tree partitions is isomorphic to the lattice of wide subcategories of Λ T -mod. 6.1. Path algebras and quiver representations. Following [2], let Q be a given quiver. We define a path of length ě 1 to be an expression α 1 α 2¨¨¨α where α i P Q 1 for all i P r s and spα i q " tpα i`1 q for all i P r ´1s. We may visualize such a path in the following waÿ¨¨¨¨¨¨¨¨α Furthermore, the source (resp. target) of the path α 1 α 2¨¨¨α is spα q (resp. tpα 1 q). Let Q denote the set of all paths in Q of length . We also associate to each vertex i P Q 0 a path of length " 0, denoted ε i , that we will refer to as the lazy path at i. 28 Definition 6.1. Let Q be a quiver. The path algebra of Q is the k-algebra generated by all paths of length ě 0. Throughout this paper, we assume that k is algebraically closed. The multiplication of two paths α 1¨¨¨α P Q and β 1¨¨¨βk P Q k is given by the following rule α 1¨¨¨α ¨β 1¨¨¨βk " " α 1¨¨¨α β 1¨¨¨βk P Q `k : spα q " tpβ 1 q 0 : spα q ‰ tpβ 1 q.
We will denote the path algebra of Q by kQ. Note also that as k-vector spaces we have kQ " where kQ is the k-vector space of all paths of length .
In this paper, we study certain quivers Q which have oriented cycles. We say a path of length ě 0 α 1¨¨¨α P Q is an oriented cycle if tpα 1 q " spα q. We denote by kQ ,cyc Ă kQ the subspace of all oriented cycles of length ě 0. If a quiver Q possesses any oriented cycles of length ě 1, we see that kQ is infinite dimensional. If Q has no oriented cycles, we say that Q is acyclic.
In order to avoid studying infinite dimensional algebras, we will add relations to path algebras whose quivers contain oriented cycles in such a way that we obtain finite dimensional quotients of path algebras. The relations we add are those coming from an admissible ideal I of kQ meaning that If I is an admissible ideal of kQ, we say that pQ, Iq is a bound quiver and that kQ{I is a bound quiver algebra.
In this paper, we study modules over a bound quiver algebra kQ{I by studying certain representations of Q that are "compatible" with the relations coming from I. A representation V " ppV i q iPQ0 , pϕ α q αPQ1 q of a quiver Q is an assignment of a k-vector space V i to each vertex i and a k-linear map ϕ α : V spαq Ñ V tpαq to each arrow α P Q 1 . If ρ P kQ, it can be expressed as where c i P k and α piq 1¨¨¨α piq ki P Q i k so when considering a representation V of Q, we define If we have a bound quiver pQ, Iq, we define a representation of Q bound by I to be a representation of Q where ϕ ρ " 0 if ρ P I. We say a representation of Q bound by I is finite dimensional if dim k V i ă 8 for all i P Q 0 . It turns out that kQ{I-mod is equivalent to the category of finite dimensional representations of Q bound by I. In the sequel, we use this fact without mentioning it further. Additionally, the dimension vector of V P kQ{Imod is the vector dimpV q :" pdim k V i q iPQ0 and the dimension of V is defined as dim k pV q " ř iPQ0 dim k V i . The support of V P kQ{I-mod is the set supppV q :" ti P Q 0 : V i ‰ 0u.
In this paper, we will focus on a special type of bound quiver algebras known as gentle algebras. Gentle algebras have a simple combinatorial parameterization of their indecomposable modules in terms of string modules. The string modules and the homological properties of string modules of the tiling algebra Λ T (see Section 6.2) defined by T will be closely related to the combinatorics T that we have developed in the preceding sections. A gentle algebra Λ " kQ{I is a bound quiver algebra that satisfies the following conditions: i) For each vertex of Q is the starting point of at most two arrows and the ending point of at most two arrows. ii) For each arrow β P Q 1 there is at most one arrow α P Q 1 such that βα R I, and there is at most one arrow γ P Q 1 such that γβ R I. iii) For each arrow β P Q 1 , there is at most one arrow δ P Q 1 such that βδ P I, and there is at most one arrow µ P Q 1 such that µβ P I. iv) I is generated by paths of length 2.
A string in Λ is a sequence where each x i P Q 0 and each α i P Q 1 or α i P Q´1 1 :" tformal inverses of arrows of Qu. We require that each α i connects x i and x i`1 (i.e. either spα i q " x i and tpα i q " x i`1 or spα i q " x i`1 and tpα i q " x i where if α i P Q´1 1 we define spα i q :" tpα´1 i q and tpα i q :" spα´1 i q) and that w contains no substrings of w of the following forms: γs ÐÝ x is`1 where β s¨¨¨β1 , γ 1¨¨¨γs P I. In other words, w is an irredundant walk in Q that avoids the relations imposed by I. By convention, we consider w to be a different word in the vertices of Q than w´1 :" x m`1 αm ÐÑ x m αm´1 ÐÑ¨¨¨α 1 ÐÑ x 1 . We say the string w is cyclic if x 1 " x m`1 and we say a cyclic string is a band if ÐÑ¨¨¨α m ÐÑ x 1 k copies of w is a string but w is not a proper power of another string u (i.e. there does not exist an integer s ě 2 such that w " u s ). Let w be a string in Λ. The string module defined by w is the bound quiver representation M pwq :" k sj : i " x j for some j P rm`1s 0 : otherwise where s j :" #tk P rm`1s : x k " x j u and the action of ϕ α is induced by the relevant identity morphisms if α lies on w and is zero otherwise. One observes that M pwq -M pw´1q. If, in addition, w is a band, it defines a band module M pw, n, φq :" ppV i q iPQ0 , pϕ α q αPQ1 q where V i :" " k n : i " x j for some j P rm`1s 0 : otherwise for each choice of n P N and φ P Autpk n q. The action of ϕ α is induced by relevant identity morphisms (resp. by φ) if α " α j for some j P rm´1s (resp. α " α m ).
If kQ{I is a representation-finite gentle algebra, it follows from [54] that set of indecomposable kQ{I-modules, denoted indpkQ{I-modq, consists of exactly the string modules M pwq where w is a string in kQ{I.
The algebra kQ{I has the following string modules.
The tiling algebra of a tree. Let T be a tree embedded in D 2 . Then T defines a bound quiver, denoted pQ T , I T q, as follows. Let Q T be quiver whose vertices are in bijection with the edges of T that contain no leaves and whose arrows are exactly those of the form e 1 α ÝÑ e 2 satisfying: iq e 1 and e 2 define a corner of T , iiq e 2 is counterclockwise from e 1 . The admissible ideal I T is, by definition, generated by the relations αβ where α : e 2 ÝÑ e 3 defines the corner pv, F q and β : e 1 ÝÑ e 2 defines the corner pv, Gq. We define Λ T :" kQ T {I T and refer to this as the tiling algebra of T . Example 6.3. In Figure 16, we show three trees. In the left tree in Figure 16, we illustrate how T 1 determines the quiver Q T1 " 1 β ÝÑ 2 α ÝÑ 3. The algebra defined by T 1 is Λ T1 " kQ T1 {I T1 where I T1 " xαβy. Also note 30 that Q T2 -Q T3 -Q and Λ T2 -Λ T3 -Λ where Q is the quiver from Example 6.2 and Λ is the algebra from Example 6.2. 12 11 10 Figure 16.
We remark that the term tiling algebra first appeared in [52] where a tiling algebra is defined by a partial triangulation of a polygon. The definition of a tiling algebra from [52] agrees with our definition of Λ P in Section 7 which is canonically isomorphic to Λ T . Proposition 6.4. The algebra Λ T is a gentle algebra. Furthermore, the algebra Λ T is representation-finite and its indecomposables are exactly the string modules.
Proof. The first assertion follows from [52,Proposition 3.2]. To prove the second assertion, it is enough to observe that any string w in Λ T can be regarded as a full, connected subquiver of Q T that avoids the relations imposed by I T . In particular, w has at most one arrow from any cycle in Q T so w is not a cyclic string. This description of the strings in Λ T implies that there are only finitely many strings in Λ T and there are no bands in Λ T . Thus Λ T is representation-finite.
Corollary 6.5. The following hold for the tiling algebra Λ T .
1. Assume M pwq :" ppV i q iPQ0 , pϕ α q αPQ1 q is a string module of Λ T . Then dim k pV i q " 1 if i P supppM pwqq and dim k pV i q " 0 otherwise. 2. The map indpΛ T q ÝÑ SegpT q defined by M pwq Þ ÝÑ s w :" pv 0 , . . . , v t q where each v i is a vertex of T belonging to some e j P supppM pwqq and where each pair v i and v i`1 belongs to a common e j P supppM pwqq is a bijection.
Proof. Assertion 1. follows from the proof of Proposition 6.4.
To prove assertion 2., we note that, as in the proof of Proposition 6.4, any string module M pwq P indpΛ T q can be regarded as a full, connected subquiver of Q T that avoids the relations imposed by I T . With this identification, we observe that M pwq is equivalent to a sequence of interior vertices pv 0 , . . . , v t q of Q T with the property that any two edges pv i´1 , v i q and pv i , v i`1 q are contained in a common face of T . Thus the given map is a bijection.
We now present a description of the spaces of extensions between indecomposable Λ T -modules. These results, especially Theorems 6.9, appear to be new. These results generalize, in the finite representation type case, the description of extensions between indecomposables found in [13]. The proofs of the following results depend on several lemmas presented in Section 6.3. Proof. Since s u and s v have no common vertices, there is no arrow α P pQ T q 1 such that u α Ð v is a string in Λ T . By exactness of the given sequence and by Lemma 6.15, it is clear that X " M puq ' M pvq. Thus the given sequence is split. Proposition 6.7. Let M puq, M pvq P indpΛ T -modq where s u and s v either share an endpoint and agree along a segment or they have a common vertex that is an endpoint of at most one of s u and s v . Then Ext 1 Λ T pM pvq, M puqq " 0.
Proof. Let 0 Ñ M puq f Ñ X g Ñ M pvq Ñ 0 be an extension. By Lemma 6.18 iq, X has at least two summands M pyq and M pzq for some nonempty strings y and z in Λ T . By Lemma 6.18 iiq, without loss of generality, we have that M pyq " M puq and M pzq " M pvq so the given sequence is split. Proof. Assume that there exists an arrow α P pQ T q 1 such that u α Ð v is a string in Λ T . Thus M pu α Ð vq is a string module and so ξ is a nonsplit extension.
Assume that there does not exist an arrow α P pQ T q 1 such that u α Ð v is a string in Λ T . Let 0 Ñ M puq f Ñ X g Ñ M pvq Ñ 0 be an extension. Lemma 6.15 implies that X " M puq ' M pvq so all such extensions are split. The last assertion follows from the fact that dim k Ext 1 Λ T pM pvq, M puqq " 1 by Lemma 6.14.
Theorem 6.9. Suppose that supppM puqq X supppM pvqq ‰ H, and let w denote the unique maximal string supported on supppM puqq X supppM pvqq. Furthermore, assume that the segments s u and s v do not have any common endpoints. Write u " u p1q Ø w Ø u p2q and v " v p1q Ø w Ø v p2q for some strings u p1q , u p2q , v p1q , and v p2q in Λ T some of which may be empty. Then Ext 1 Λ T pM pvq, M puqq ‰ 0 if and only if u " u p1q Ð w Ñ u p2q and v " v p1q Ñ w Ð v p2q . Additionally, in this case, Proof. Assume that u " u p1q Ð w Ñ u p2q and v " v p1q Ñ w Ð v p2q for some strings u p1q , u p2q , v p1q , and v p2q in Λ T . Note that the segments s u and s v have no common endpoints. This means that M pu p1q Ð w Ð v p2q q is not isomorphic to M puq or M pvq and the same is true for M pv p1q Ñ w Ñ u p2q q. Thus is a nonsplit extension. This implies that Ext 1 Λ T pM pvq, M puqq ‰ 0. Conversely, assume that Ext 1 Λ T pM pvq, M puqq ‰ 0. Let 0 Ñ M puq f Ñ X g Ñ M pvq Ñ 0 be a nonsplit extension and let X " ' k i"1 X i be a direct sum decomposition of X into indecomposables. By Corollary 6.17, we have that X " M pu p1q Ø w Ø v p2q q ' M pv p1q Ø w Ø u p2q q. Since the given sequence is exact, we must have that pu p1q Ø w Ø v p2q q " pu p1q Ð w Ð v p2q q and pv p1q Ø w Ø u p2q q " pv p1q Ñ w Ñ u p2q q. Thus u " u p1q Ð w Ñ u p2q The last assertion follows from the fact that dim k Ext 1 Λ T pM pvq, M puqq " 1 by Lemma 6.14.
6.3. Homomorphisms and extensions between string modules. In this section, we present the technical facts required to prove Propositions 6.6 and 6.7 and Theorems 6.8 and 6.9. We prove Lemma 6.10, which is used in the statement of Theorem 6.9, Lemma 6.16, and Corollary 6.17. We omit the proofs of Lemma 6.11, 6.12, and 6.14 as they are nearly identical to that of [ Lemma 6.10. Let M puq, M pvq P indpΛ T -modq with supppM puqq X supppM pvqq ‰ H. Then w " x 1 Ø x 2¨¨¨xk´1 Ø x k where supppM puqq X supppM pvqq " tx i u iPrks is a string in Λ. Furthermore, w is the unique maximal string along which u and v agree.
Proof. Any string in Λ T includes at most two vertices from any oriented cycle in Q T . Thus a string u " y 1 Ø y 2¨¨¨ys´1 Ø y s is the shortest path connecting y 1 and y s in the underlying graph of Q T . This implies that for any y i and y j appearing in u, the string y i Ø y i`1¨¨¨yj´1 Ø y j is the shortest path connecting y i and y j in the underlying graph of Q T . Therefore if supppM puqq X supppM pvqq ‰ H, then w " x 1 Ø x 2¨¨¨xk´1 Ø x k where supppM puqq X supppM pvqq " tx i u iPrks is a string in Λ T . Clearly, w is the unique maximal string along which u and v agree. Proof. Assume s u and s v agree along a segment s w . By Lemma 6.10, assume that s w is the unique largest segment along which s u and s v agree. We have that either u " u p1q Ð w and v " v p1q Ñ w or u " u p1q Ñ w and v " v p1q Ð w. In the former case, Hom Λ T pM puq, M pvqq ‰ 0. In the latter case, Hom Λ T pM pvq, M puqq ‰ 0. The converse statement is obvious.
Lemma 6.14. Let M puq, M pvq P indpΛ T -modq. Then dim k Ext 1 Λ T pM puq, M pvqq ď 1. Next, we present four results, each of which is crucial to classifying extensions between indecomposable Λ Tmodules. Lemma 6.15 is used in the proof of Proposition 6.6 and Theorem 6.8. Corollary 6.17, which is used in the proof Theorem 6.9, follows from Lemma 6.16. Lemma 6.16 establishes several restrictions on which indecomposable Λ T -modules can appear as middle terms of a nonsplit extension between two indecomposables whose corresponding segments agree along a segment, but have no shared endpoints. Lastly, Lemma 6.18 is used in the proof of Proposition 6.7. Assume that there does not exist an arrow α P pQ T q 1 such that u α Ð v is a string in Λ T and let X " ' k i"1 X i be a direct sum decomposition of X in to indecomposables (i.e. X i P indpΛ T -modq for each i P rks). Then none of the modules X i have any of the following properties iq supppX i q X supppM puqq ‰ H and supppX i q X supppM pvqq ‰ H iiq supppX i q Ĺ supppM puqq iiiq supppX i q Ĺ supppM pvqq.
Proof. Suppose some X i satisfies iq. Then we can write X i " M pwq, u " u 1 Øw 1 , and v " w 2 Ø v 2 where w " w 1 Øw 2 is a string in Λ T . By assumption, w " w 1 Øw 2 " w 1 β Ñ w 2 . Observe that the direction of β implies that Hom Λ T pM puq, M pwqq " 0 and Hom Λ T pM pwq, M pvqq " 0. Since supppM puqq X supppM pvqq " H, tsupppX i qu k i"1 is a set partition of the set supppXq. Thus we have that M pwq X impf q " 0, but M pwq Ă kerpgq. This contradicts that the given sequence is exact.
As none of the X i satisfy iq, we can separate these modules into those supported on M puq and those supported on M pvq. We denote the former modules by tM pu pjq qu s j"1 and the latter by tM pv p q qu t "1 . Suppose M pu pjq q satisfies iiq. Then there exist M pu pj 1 q q for some j 1 ‰ j such that u pjq β ÐÑ u pj 1 q is a string in Λ T supported on u. Thus if u pjq β ÐÝ u pj 1 q (resp. u pjq β ÝÑ u pj 1 q ) is a string in Λ T , we have that Hom Λ T pM puq, M pu pjq qq " 0 (resp. Hom Λ T pM puq, M pu pj 1 q qq " 0). This implies that there exists a summand M pu pj 2 q q of X such that M pu pj 2 q qXimpf q " 0. However, M pu pj 2 q q Ă kerpgq since supppM pu pj 2 q qqXsupppM pvqq " H. This contradicts that the given sequence is exact. The proof that there are no summands M pv p q q of X that satisfy iiiq is similar so we omit it. Ñ M pvq Ñ 0 be a nonsplit extension where supppM puqq X supppM pvqq ‰ H, and let w denote the unique maximal string supported on supppM puqq X supppM pvqq. Let X " ' k i"1 X i be a direct sum decomposition of X into indecomposables and write u " u p1q Ø w Ø u p2q and v " v p1q Ø w Ø v p2q for some strings u p1q , u p2q , v p1q , and v p2q in Λ T some of which may be empty. Then the following hold.
iq X is not indecomposable. iiq There is no X i such that supppX i q X supppM pxqq ‰ H for any x P tw, u p1q , u p2q u, assuming that both u p1q and u p2q are nonempty strings. iiiq There is no X i such that supppX i q X supppM pxqq ‰ H for any x P tw, v p1q , v p2q u, assuming that both v p1q and v p2q are nonempty strings. ivq There is no X i such that supppX i q Ĺ supppM pxqq where x P tu p1q , u p2q , v p1q , v p2q u. Thus each X i satisfies supppX i q X supppM pwqq ‰ H. vq If X i and x P tw, u p1q , u p2q , v p1q , v p2q u satisfy supppX i q X supppM pxqq ‰ H, then supppM pxqq Ă supppX i q.
Proof. We first show that each X i satisfies X i fl M puq and X i fl M pvq since the given extension is nonsplit. Without loss of generality, suppose a summand X i of X satisfies X i -M puq. Since s u and s v have no common endpoints, impf q " X i . By dimension considerations and the fact that g is surjective, M pvq is also a summand of X. Thus the given sequence is split, a contradiction.
iq We observe that by exactness, dim k pXq " dim k pM puqq`dim k pM pvqq. Since supppM puqqXsupppM pvqq ‰ H, Lemma 6.5 1. implies that X is not a string module and therefore not indecomposable.
iiq Suppose that such an X i exists. Then supppM pwqq Ă supppX i q. Now note that since X i fl M puq and X i fl M pvq, we can assume that dim k pM pu p1q qq ě 1 or dim k pM pu p2q qq ě 1. Without loss of generality, we assume the former. This implies that supppX i q X supppM pu p1q qq Ĺ supppM pu p1q qq and supppX i q X supppM pu p1q qq ‰ H. The fact that dim k pM pu p1q qq ě 1 also implies that we can write u p1q " x p1q Ø x p2q for some nonempty strings x p1q and x p2q in Λ T where supppM px p2q qq " supppX i q X supppM pu p1q qq and u " In this case, Hom Λ T pM puq, X j q " 0 if X j is any summand of X where supppX j q Ă supppM px p1q qq and x p1q P supppX j q. Thus any such X j satisfies X j X impf q " 0. One also observes that supppM px p1q qq X supppM pvqq " H so Hom Λ T pX j , M pvqq " 0. Therefore, any such X j Ă kerpgq. This means that if such a summand X j exists, then the given sequence is not exact.
We show that there must be a summand X j of X satisfying supppX j q Ă supppM px p1q qq and whose string contains x p1q . First note that by the exactness of the given sequence, there must exist a summand X j of X whose support contains x p1q and thus intersects supppM px p1q qq. To complete the proof, it is enough to show that, without loss of generality, there is no string y in Λ T such that supppM pyqq X supppM px p1q qq ‰ H and supppM pyqq X supppM pv p1q qq ‰ H. To show this, it is enough to observe that the segments s x p1q and s v p1q have no common vertices, since x p2q is a nonempty string. We obtain a contradiction.
We now have that u p1q " x p1q Ñ x p2q . This implies that Hom Λ T pM puq, X i q " 0. Let us express X i as X i " ppV i q iPQ0 , pϕ α q αPQ1 q. By exactness and dimension considerations, the module X i is the only summand of X satisfying supppX i q X supppM px p2q qq ‰ H. Thus if λ P V i is nonzero and i P supppX i q X supppM px p2q qq, then λ R impf q. This contradicts that f is injective.
iiiq The proof of this assertion is similar to the proof of assertion iiq so we omit it.
ivq It suffices to show that there does not exist a summand X i of X such that supppX i q Ĺ supppM pv p1q qq. Suppose there exists such a summand X i . Then there exist summands M pxq and M pyq of X where x Ø y is a string in Λ T where supppM pxqq Ĺ supppM pv p1q qq and supppM pyqq X supppM pv p1q qq ‰ H. If the string px Ø yq " px Ð yq, then Hom Λ T pM pyq, M pvqq " 0. Let us express M pyq as M pyq " ppV i q iPQ0 , pϕ α q αPQ1 q. Then any nonzero λ P V i where i P supppM pyqq X supppM pv p1q qq satisfies λ P kerpgq. Since λ does not belong to any summand besides M pyq, we have that g is not surjective, a contradiction. If the string px Ø yq " px Ñ yq, then Hom Λ T pM pxq, M pvqq " 0. Similarly, this implies that M pxq Ă kerpgq, which contradicts that g is surjective.
vq We first prove the assertion for any x P tu p1q , u p2q , v p1q , v p2q u. As in the proof of ivq, it suffices to prove this for x " v p1q . Suppose that there exists X i such that supppX i q X supppM pv p1q qq ‰ H and supppM pv p1q qq Ć supppX i q. By ivq, we have that supppX i q X supppM pwqq ‰ H. Now by exactness of the given sequence, there exists another summand X j of X such that supppX j q Ă supppM pv p1q qqzsupppX i q Ă supppM pv p1q qq. This contradicts ivq.
By assertion ivq, each summand X i satisfies supppX i qXsupppM pwqq ‰ H. Thus it is enough to show that there are no summands X i such that supppX i q Ĺ supppM pwqq. Suppose there exists such a summand X i " M py p2q q. We can assume, without loss of generality, that there is another summand X j " M py p1q q of X such that ‚ y p1q Ø y p2q is a string in Λ T , ‚ supppM py p1q qq X supppM pv p1q qq ‰ H, ‚ supppM py p1q qq X supppM pwqq ‰ H. Suppose that py p1q Ø y p2q q " py p1q Ñ y p2q q. Then Hom Λ T pM py p1q q, M pvqq " 0. Let us express M py p1q q as M py p1q q " ppV i q iPQ0 , pϕ α q αPQ1 q. The for any nonzero λ P V i where i P supppM py p1q qq X supppM pv p1q qq satisfies λ P kerpgq. Since M py p1q q, is the only summand containing λ, this contradicts that g is surjective.
Now suppose py p1q Ø y p2q q " py p1q Ð y p2q q and write y p2q " y p2q 1 Ø¨¨¨Ø y p2q . Then Hom Λ T pM py p2q q, M pvqq " 0. This means that any other summand M py p3q q of X where py p1q Ð y p3q q is a string in Λ T and y p2q 1 P supppM py p3q qq has the property that Hom Λ T pM py p3q q, M pvqq " 0. Since M py p1q q is the only summand of X whose support intersects supppM py p1q qq X supppM pv p1q qq and since supppM py p1q qq Ă supppM pvqq, we have that there is an inclusion M py p1q q ãÑ M pvq. Since the given sequence is exact, there must exist a summand M pzq " ppV i q iPQ0 , pϕ α q αPQ1 q of X where z satisfies ‚ supppM pzqq X supppM py p1q qq ‰ H where any nonzero λ P V i for i P supppM pzqq X supppM py p1q qq satisfies λ R kerpgq " impf q, and ‚ supppM pzqq X supppM py p2q qq ‰ H where any nonzero λ P V i for i P supppM pzqq X supppM py p2q qq satisfies λ P impf q.
However, since py p1q Ø y p2q q " py p1q Ð y p2q q there are no homomorphisms from M puq to M pzq satisfying these properties. Thus there are no summands X i of X such that supppX i q Ĺ supppM pwqq.
Corollary 6.17. Let M puq, M pvq P indpΛ T -modq where s u and s v have no common endpoints. Let 0 Ñ M puq f Ñ X g Ñ M pvq Ñ 0 be a nonsplit extension where supppM puqq X supppM pvqq ‰ H, and let w denote the unique maximal string supported on supppM puqq X supppM pvqq. Let X " ' k i"1 X i be a direct sum decomposition of X into indecomposables and write u " u p1q Ø w Ø u p2q and v " v p1q Ø w Ø v p2q for some strings u p1q , u p2q , v p1q , and v p2q in Λ T some of which may be empty. Then Proof. By Lemma 6.16 iq, X has at least two indecomposable summands. By Lemma 6.16 ivq and vq, X has exactly two summands, M pyq and M pzq, where supppM pwqq Ă supppM pyqq and supppM pwqq Ă supppM pzqq. By exactness of the given sequence and by Lemma 6.16 vq, for any x P tu p1q , u p2q , v p1q , v p2q u we have that supppM pxqq is contained in supppM pyqq or supppM pzqq. By combining Lemma 6.16 iiq and iiiq, we have that M pyq " M pu p1q Ø w Ø v p2q q and M pzq " M pv p1q Ø w Ø u p2q q. Lemma 6.18. Let M puq, M pvq P indpΛ T -modq where s u and s v either share an endpoint and agree along a segment or they have a common vertex that is an endpoint of at most one of s u and s v . If 0 Ñ M puq f Ñ X g Ñ M pvq Ñ 0 is an extension and X " ' k i"1 X i is a direct sum decomposition into indecomposables, then the following hold.
iq X is not indecomposable.
iiq There is no X i such that supppX i q Ĺ supppM pxqq where x P tu, vu.
Proof. Note that only Lemma 6.16 iiq and iiiq relied on the assumption that the given extension was nonsplit. Thus one proves these assertions by adapting the proofs of Lemmas 6.16 iq, ivq, and vq, since these did not depend on Lemma 6.16 iiq and iiiq.

6.4.
Oriented flip graphs and torsion-free classes. In this section, we recall the definition of torsion-free classes and their lattice structure. After that, we show that oriented flip graphs are isomorphic as posets to the lattice of torsion-free classes of Λ T ordered by inclusion and torsion classes of Λ T ordered by reverse inclusion. Let Λ be a finite dimensional k-algebra. A full, additive subcategory C Ă Λ-mod is extension closed if for any objects X, Y P C satisfying 0 Ñ X Ñ Z Ñ Y Ñ 0 one has Z P C. We say C is quotient closed (resp. submodule closed) if for any X P C satisfying X α ÝÑ Z where α is a surjection (resp. Z β ÝÑ X where β is an injection), then Z P C. A full, additive subcategory T Ă Λ-mod is called a torsion class if T is quotient closed and extension closed. Dually, a full, additive subcategory F Ă Λ-mod is called a torsion-free class if F is extension closed and submodule closed.
Let torspΛq (resp. torsfpΛq) denote the lattice of torsion classes (resp. of torsion-free classes) of Λ ordered by inclusion. We have the following proposition, which shows that a torsion class of Λ uniquely determines a torsion-free class of Λ and vice versa. Given T a torsion class and F its corresponding torsion-free class, we say that the data pT , Fq is a torsion pair.  aq Let tT i u iPI Ă torspΛq be a collection of torsion classes. Then we have bq Let tF i u iPI Ă torsfpΛq be a collection of torsion-free classes. Then we have Proof. By Lemma 6.21, it is enough to prove that Ý Ý Ñ F GpT q -torsfpΛ T q. Furthermore, by Theorem 4.11 (2), we have that Ý Ý Ñ F GpT q -π Ó pBicpT qq so it is enough to show that the latter is isomorphic to torsfpΛ T q. We claim that the map π Ó pBicpT qq ζ ÝÑ torsfpΛ T q π Ó pXq Þ ÝÑ F :" addp' su M puq : s u P π Ó pXqq is an isomorphism of posets where addp' k i"1 X i q for any finite set of Λ T -modules X i denotes the smallest full, additive subcategory of Λ T -mod closed under taking summands of ' k i"1 X i . Furthermore, we claim that the inverse of this map is given by torsfpΛ T q δ ÝÑ π Ó pBicpT qq F " addp' iPrks M pw piq qq Þ ÝÑ π Ó pts w p1q , . . . , s w pkq uq.
We can see that these maps are order-preserving, since π Ó is order-preserving by Lemma 4.5 (7). Assuming that ζpπ Ó pXqq is a torsion-free class and that δpFq P π Ó pBicpT qq, we have that δ " ζ´1 as π Ó is an idempotent map (see Lemma 4.5 (5)).
We first show that δpFq P π Ó pBicpT qq where F " addp' iPrks M pw piq qq. Let s u , s v P ts w p1q , . . . , s w pkq u and assume s u˝sv P SegpT q. Then, up to reversing the roles of u and v, u Ð v is a string in Λ T so there is an extension 0 Ñ M puq Ñ M pu Ð vq Ñ M pvq Ñ 0. Since F is extension closed, M pu Ð vq P F so s u˝sv " s puÐvq P ts w p1q , . . . , s w pkq u. Thus ts w p1q , . . . , s w pkq u is closed. Since F is submodule closed, there are no extensions of the form 0 Ñ M puq Ñ M pu Ð vq Ñ M pvq Ñ 0 where s u , s v R ts w p1q , . . . , s w pkq u, but s puÐvq P ts w p1q , . . . , s w pkq u. Thus ts w p1q , . . . , s w pkq u is co-closed.
Next, we show that F :" addp' su M puq : s u P π Ó pXqq is a torsion-free class. We begin by showing that it is submodule closed. Assume that there is an inclusion M pvq ãÑ M puq where M puq P F. Write s u " px 0 , . . . , x q and orient this segment from x 0 to x . Let s v " px i , . . . , x j q where we can assume that 0 ă i and j ă . The inclusion M pvq ãÑ M puq implies that u " u p1q Ñ v Ð u p2q for some nonempty strings u p1q and u p2q in Λ T . Now we have that s v turns right (resp. left) at x i (resp. at x j ). Thus s v P C su Ă X. This implies that C sv Ă C su Ă X so s v P π Ó pXq. We obtain that M pvq P F. Now suppose f : M pvq ãÑ X " ' iPrks M pw piq q ai for some a i ě 0 and M pvq does not include into any summand of X. Furthermore, suppose any indecomposable M puq with dim k pM puqq ă dim k pM pvqq that includes into an object of F belongs to F. Let M pw piq q be a summand of X where the component map g : M pvq Ñ M pw piq q of f is nonzero. By Lemma 6.12, we can assume that there exists a nonempty string w in Λ T not equal to u or w piq such that M pvq M pwq ãÑ M pw piq q. By the previous paragraph, M pwq P F. Now express v as v " v p1q Ð w Ñ v p2q where, without loss of generality, both v p1q and v p2q are nonempty. This implies that M pv piq q ãÑ X so M pv piq q P F for i " 1, 2 since dim k pM pv piq qq ă dim k pM pvqq. Observe that we have an extension 0 Ñ M pv p2q q Ñ M pw Ñ v p2q q Ñ M pwq Ñ 0, which shows that M pw Ñ v p2q q P F since s pwÑv p2q q " s w˝sv p2q P π Ó pXq. This implies that we have an extension 0 Ñ M pv p1q q Ñ M pvq Ñ M pw Ñ v p2q q Ñ 0, which shows that M pvq P F since s v " s v p1q˝s w˝sv p2q P π Ó pXq. We conclude that F is submodule closed.
Lastly, we show that F is extension closed. Since π Ó pXq is closed, it is easy to see that F is extension closed with respect to extensions whose nonzero terms are indecomposable. By our description of nonsplit extensions in Λ T -mod (see Section 6.2), it suffices to show that if M puq, M pvq P F where u " u p1q Ð w Ñ u p2q and v " v p1q Ñ w Ð v p2q and is the nonsplit extension defined by these modules, then M pu p1q Ð w Ð v p2q q, M pv p1q Ñ w Ñ u p2q q P F. We show M pu p1q Ð w Ð v p2q q P F and the proof that M pv p1q Ñ w Ñ u p2q q P F is very similar. Notice that M pu p1q q ãÑ M puq and M pw Ð v p2q q ãÑ M pvq so M pu p1q q, M pw Ð v p2q q P F. Thus we obtain a nonsplit extension 6.5. Noncrossing tree partitions and wide subcategories. In this section, we show that noncrossing tree partitions of a tree T provide a combinatorial model for the wide subcategories of Λ T -mod.
If Λ is a finite dimensional k-algebra, we say that a full, additive subcategory W Ă Λ-mod is a wide subcategory if it is abelian and extension closed. We let widepΛq denote the poset of wide subcategories of Λ-mod partially ordered by inclusion. It is easy to see that the intersection of two wide subcategories is also a wide subcategory, and the zero subcategory (resp. Λ-mod) is the bottom (resp. top) element widepΛq. Thus if Λ is representation-finite, the poset widepΛq is a lattice.
Theorem 6.23. For any tree T , we have the following isomorphisms of posets: Proof. That the map on the left is an isomorphism follows from Theorem 5.14. It is clear that the map on the right is order-preserving so it is enough to show that this same map defines a wide subcategories and has an order-preserving inverse. We first show that W :" add´' su M puq : s u P SegpBq¯P widepΛ T q. By Lemma 6.24, we know that W is closed under taking kernels and cokernels of maps between modules M puq, M pvq P W where s u , s v P SegpBq. Now suppose that f P Hom Λ T pM puq, M pvqq is nonzero where s u P SegpBq, s v P SegpB 1 q, and where B and B 1 are blocks of B. We can further assume that f is neither injective nor surjective. We write s u " s u p1q˝¨¨¨˝s u pkq and s v " s v p1q˝¨¨¨˝s v p q where s u p1q , . . . , s u pkq P SegpBq and s v p1q , . . . , s v p q P SegpB 1 q.
Assume B ‰ B 1 . Since f is nonzero, then s u R SegpBq or s v R SegpB 1 q. Without loss of generality, we assume that s u R SegpBq. Thus we have an inclusion M pu ptq q ãÑ M puq for some t " 1, . . . , k. This implies that Hom Λ T pM pu ptq q, M pvqq ‰ 0, which contradicts Lemma 6.24. Now assume B " B 1 , and suppose that any g P Hom Λ T pX, Y q has kerpgq, cokerpgq P W for any X, Y P W with dim k pXq`dim k pY q ă dim k pM puqq`dim k pM pvqq. Define s w :" s w p1q˝¨¨¨˝s w ptq where ts w p1q , . . . , s w ptq u " ts u p1q , . . . , s u pkq u X ts v p1q , . . . , s v p q u. Now we have that f factors as M puq α M pwq β ãÑ M pvq. Since f is neither injective nor surjective, we know that dim k pM puqq`dim k pM pwqq and dim k pM pwqq`dim k pM pvqq are both less than dim k pM puqq`dim k pM pvqq. We thus obtain that kerpf q " kerpαq P W and cokerpf q " cokerpβq P W. We conclude that W is abelian.
To see that W is extension closed, it is enough to is the nonsplit extension defined by these modules, then M pu p´q Ð w Ð v p`q q, M pv p´q Ñ w Ñ u p`q q P W. Notice that the existence of such an extension implies that the segments s u and s v are crossing. Now using Lemma 5.6 and 5.8, we deduce the existence of a crossing between two segments in SegpBq, a contradiction. We thus have that s u , s v P SegpBq for some block B of B.
Write s u " s u p1q˝¨¨¨˝s u pkq and s v " s v p1q˝¨¨¨˝s v p q where s u p1q , . . . , s u pkq , s v p1q , . . . , s v p q P SegpBq. Note that up to reversing the expression, these are the unique expressions for s u and s v in terms of elements of SegpBq. This implies that that s w " s w p1q˝¨¨¨˝s w ptq where ts w p1q , . . . , s w ptq u " ts u p1q , . . . , s u pkq u X ts v p1q , . . . , s v p q u so that M pwq P W. We now observe that we can write s u " s u p1q˝¨¨¨˝s u piq s u p´q˝s w p1q˝¨¨¨˝s w ptq sw˝s u pi`tq˝¨¨¨˝s u pkq One can thus construct extensions showing that M pu p´q Ð w Ð v p`q q, M pv p´q Ñ w Ñ u p`q q P W. We conclude that W P widepΛ T q.
We now claim that the map ω : widepΛ T q ÝÑ Ψp Ý Ý Ñ F GpT qq defined by W Þ Ñ S :" ts u : M puq is a simple object of Wu is an order-preserving inverse to the map Ψp Ý Ý Ñ F GpT qq ÝÑ widepΛ T q. Assuming that ωpWq P Ψp Ý Ý Ñ F GpT qq, it is clear that ω is an inverse as a map of sets.
Our earlier argument shows that the elements M pu piq q are the simple objects of W where s u piq P SegpBq. Thus to prove that S " SegpBq for some B P NCPpT q, it is enough to show that any two distinct segments s u and s v are noncrossing where M puq, M pvq P W are simple objects in W. Note that Hom Λ T pM puq, M pvqq " Hom Λ T pM pvq, M puqq " 0, since M puq and M pvq are simple objects and W is a wide subcategory.
If s u and s v share an endpoint, then Lemma 6.13 implies that s u and s v do not agree along a segment. Thus they are noncrossing in this case.
If s u and s v are crossing, then by Theorem 6.9, and up to reversing the roles of u and v they define a unique nonsplit extension Using this description of the strings u and v, we notice that there is map f P Hom Λ T pM puq, M pvqq where impf q " M pwq, a contradiction. Thus s u and s v are noncrossing.
Next, we show that ω is order-preserving. Since any two simple objects of W P widepΛ T q correspond to noncrossing segments, the segment defined by any indecomposable object of W can be expressed as a concatenation of segments corresponding to simple objects of W. That is, the segments of ωpWq are in bijection with the indecomposable objects of W. Thus if W 1 Ă W 2 , one has ωpW 1 q Ă ωpW 2 q. Lemma 6.24. Let B P NCPpT q and let M puq, M pvq be two distinct indecomposable Λ T -modules whose corresponding segments appear in SegpBq and SegpB 1 q, respectively, for some blocks B and B 1 of B. Then one has Hom Λ T pM puq, M pvqq " 0 and Hom Λ T pM pvq, M puqq " 0.
Proof. First assume B " B 1 . Since M puq and M pvq are distinct, the corresponding segments s u and s v share at most one vertex of T . This means u and v are supported on disjoint sets of vertices of Q T so the statement holds. Thus we can assume that s u P SegpBq and s v P SegpB 1 q where B and B 1 are distinct blocks of B. Since B P NCPpT q, this implies that s u and s v have no common endpoints.
Let γ u and γ v be left admissible curves for s u and s v , respectively, witnessing that s u and s v are noncrossing. Write s w " ra, bs for the unique maximal segment along which s u and s v agree, if it exists, and orient γ u and γ v from a to b. Without loss of generality, we have two cases: i) supppM puqq Ĺ supppM pvqq ii) supppM pvqqzsupppM puqq ‰ H and supppM puqqzsupppM pvqq ‰ H Suppose supppM puqq Ĺ supppM pvqq. Here s w " s u . By Lemma 5.1 (1), with s u playing the role of t, we have that γ v either turns left at both a and b or it turns right at both a and b. This means that either v " v p1q Ð u Ð v p2q or v " v p1q Ñ u Ñ v p2q for some nonempty strings v p1q and v p2q in Λ T . Thus Hom Λ T pM puq, M pvqq " 0 and Hom Λ T pM pvq, M puqq " 0. Now suppose that supppM pvqqzsupppM puqq ‰ H and supppM puqqzsupppM pvqq ‰ H. We can assume that a (resp. b) is an endpoint of s v (resp. s u ). Thus we can write s v " ra, bs˝s v 1 and s u " s u 1˝ra, bs for some nonempty segments s v 1 , s u 1 P SegpT q. By Lemma 5.1 (2), with ra, bs playing the role of t, we have that either γ v turns right at b and γ u turns left at a or γ v turns left at b and γ u turns right at a. Thus either v " w Ñ v 1 and u " u 1 Ð w or v " w Ð v 1 and u " u 1 Ñ w. We conclude that Hom Λ T pM puq, M pvqq " 0 and Hom Λ T pM pvq, M puqq " 0.

Polygonal subdivisions
In this section, we show how oriented flip graphs can be equivalently described using certain decompositions of a convex polygon P Ă R 2 into smaller convex polygons called polygonal subdivisions. The notion of a flip between two facets of the reduced noncrossing complex will translate into a type of flip between polygonal subdivisions of P . After that, we show that the polygonal subdivision corresponding to the top element of an oriented flip graph is obtained by rotating the arcs in the polygonal subdivision corresponding to the bottom element. We show that oriented exchange graphs of quivers that are mutation-equivalent to type A Dynkin quivers are examples of oriented flip graphs. Lastly, we show that the Stokes poset of quadrangulations are also examples of oriented flip graphs.
A polygonal subdivision P " tP i u iPr s of a polygon P is a family of polygons P 1 , . . . , P such that ‚ l ď i"1 P i " P ‚ P i X P j is a face of P i and P j for all i, j, and ‚ every vertex of P i is a vertex of P for all i. Equivalently, we can define a polygonal subdivision of P to be a collection of pairwise noncrossing diagonals of P (i.e. curves in R 2 connecting two vertices of P ) up to endpoint fixing isotopy. 12 Figure 17. Two examples of polygonal subdivisions where the latter is drawn with its corresponding tree.
Remark 7.1. Trees and polygonal subdivisions are dual. Given any tree T embedded in D 2 , it defines a polygonal subdivision P as follows. Let P be a polygon with vertex set tv F : F is a face of T u and where v F1 is connected to v F2 by an edge of P if and only if there is an edge of T that is incident to both F 1 and F 2 . Using the data of the embedding of T , the resulting collection of polygons P is a polygonal subdivision of P . It is straightforward to verify that this construction can be reversed. We show an example of this duality in Figure 17.
Given a polygonal subdivision P " tP i u iPr s of a polygon P , there is a natural bound quiver pQ P , I P q that we associate to P. Define Q P to be the quiver whose vertices are in bijection with edges in P belonging to two distinct polygons P i , P j P P and whose arrows are exactly those of the form 1 α ÝÑ 2 satisfying: iq 1 and 2 share a vertex of P , iiq 2 is clockwise from 1 . The admissible ideal I P is, by definition, generated by the relations αβ where α : 2 ÝÑ 3 , β : 1 ÝÑ 2 , and 1 , 2 , and 3 all belong to a common polygon P i P P. We also define Λ P :" kQ P {I P . The following lemma is easy to verify using Remark 7.1.
Lemma 7.2. Let T be a tree embedded in D 2 and let P be the corresponding polygonal subdivision. Then there are natural isomorphisms Q T -Q P and Λ T -Λ P . Remark 7.3. In [52], it is shown that tiling algebras can be defined without reference to a combinatorial model such as a polygonal subdivision, and any tiling algebra so defined naturally gives rise to a polygonal subdivision. More generally, it is shown in [51] that any gentle algebra gives rise to a certain finite graph. If the algebra is a tiling algebra, then its polygonal subdivision from [52] is exactly the finite graph associated to it in [51]. Remark 7.4. When P " tP i u iPr s is a triangulation of a polygon P (i.e. each polygon P i is a triangle), the definition of the algebra Λ P agrees with the definition of the Jacobian algebra [21] associated to the triangulation. Moreover, the triangulations of P are exactly those polygonal subdivisions whose corresponding tree has only degree 3 interior vertices. When P " tP i u iPr s is an pm`2q-angulation of P where m ě 1 (i.e. each polygon P i is an pm`2q-gon), the algebra Λ P is an m-cluster-tilted algebra of type A as was shown in [40].
Additionally, the class of tiling algebras also contains the surface algebras when the surface is the disk. These algebras were introduced in [17] and studied further in [18] and [1]. Now let T be a tree embedded in D 2 . Using Remark 7.1, let P T be the polygonal subdivision of the polygon P T defined by T and let tv F : F is a face of T u be the set of vertices of the polygon P T . There is an obvious bijection between elements of tv F : F is a face of T u and the set of boundary vertices of T given by sending v F to the counterclockwise most leaf of T in face F . Using this bijection and the fact that any arc of T is completely determined by the leaves of T it connects, we obtain the following.
Proposition 7.5. Let T be a tree embedded in D 2 . The map sending each arc in a facet F P r ∆ N C pT q to its corresponding diagonal of P T defines a polygonal subdivision PpFq of P T . This map defines an injection from the facets of r ∆ N C pT q to the set of polygonal subdivisions of P T .

Figure 18.
Example 7.6. By Proposition 7.5, we can identify the vertices of Ý Ý Ñ F GpT q with a certain subset of the polygonal subdivisions of P T . In Figure 18, we show the oriented flip graph from Figure 9 with its vertices represented by the corresponding polygonal subdivisions of P T . α (α) Figure 19. The effect of on a diagonal α.
Next, we show how the polygonal subdivisions corresponding to the top and bottom elements of an oriented flip graph compare to each other. Note that there is a natural cyclic action on the diagonals of the polygon P T . If α is a diagonal of P T , we define the rotation of α, denoted pαq, to be the diagonal of P whose endpoints are the vertices of P immediately clockwise from the endpoints of α (see Figure 19). If PpFq is a polygonal subdivision of P T , we let pPpFqq denote the polygonal subdivision of P T obtained by applying to each diagonal in PpFq.
Theorem 7.7. Let T be a tree embedded in D 2 . Then the bottom element (resp. top element) of Ý Ý Ñ F GpT q corresponds to the polygonal subdivision P T (resp. pP T q).
Proof. Let F 1 and F 2 be the facets of r ∆ N C pT q corresponding to the bottom and top elements Ý Ý Ñ F GpT q, respectively. Using Proposition 7.5, we let PpF 1 q and PpF 2 q be the corresponding polygonal subdivisions of P T . It is clear that F 2 " ηpSegpT qq and F 1 " ηpHq.
Let p pv,F q be any arc of T that appears in ηpSegpT qq (resp. ηpHq). Let u be any interior vertex of T that appears in p pv,F q , and orient the arc p pv,F q from v to u. By the definition of η, the arc p pv,F q must turn left (resp. right) at u.
Next, let e " pv 1 , v 2 q be an edge of T whose endpoints are internal vertices of T , and let F and G be the two faces of T that are incident to e and satisfy pv 1 , F q (resp. pv 2 , Gq) is immediately clockwise from pv 1 , Gq (resp. pv 2 , F q). Define p :" p pv1,Gq P ηpHq and q :" p pv1,F q P ηpSegpT qq and let α p P PpF 1 q and α q P PpF 2 q be the diagonals corresponding to p and q, respectively. If we write p " pu 1 , . . . , u k , v 1 , v 2 , u k`1 , . . . , u r q and q " pw 1 , . . . , w , v 1 , v 2 , w `1 , . . . , w s q, then the argument in the previous paragraph implies that the corners contained in q are pw 2 , F q, . . . , pw , F q, pv 1 , F q, pv 2 , Gq, pw `1 , Gq, . . . , pw s´1 , Gq and the corners contained in p are pu 2 , Gq, . . . , pu k , Gq, pv 1 , Gq, pv 2 , F q, pu k`1 , F q, . . . , pu r´1 , F q. Thus we have that α q " pα p q and PpF 1 q " P T . The desired result follows.
As we mentioned in Example 3.4, the flip graph of a tree T with only degree 3 internal vertices is isomorphic to the dual associahedron. By this identification and by Proposition 7.5, we obtain an orientation of the 1-skeleton of the associahedron. This orientation adds the data of a "sign" to the operation of performing a single flip between two triangulations PpF 1 q, PpF 2 q of P T .
It turns out that this oriented version of flipping between triangulations has been described by Fomin and Thurston (we refer the reader to [25] for more details). Given any triangulation PpF 1 q of P T , one adds some additional curves pL 1 , . . . , L n q to PpF 1 q (here n " #pQ T q 0 ), called an elementary lamination (see [25,Definition 17.2]), and records the shear coordinates [25, Definition 12.2] (i.e. integer vectors indicating the number of certain crossings of arcs in PpFq and the curves pL 1 , . . . , L n q). The elementary lamination is a collection of curves that are slightly deformed versions of the arcs in P T and the shear coordinates are the c-vectors appearing in the c-matrix of the ice quiver corresponding to PpF 1 q. Then there is a directed edge PpF 1 q Ñ PpF 2 q in Ý Ý Ñ F GpT q if and only if PpF 2 q is obtained from PpF 1 q by performing a single diagonal flip on an arc α in PpF 1 q and the shear coordinate of α is positive in PpF 1 q. We thus obtain following proposition. Proposition 7.8. If T is a tree whose internal vertices have degree 3, then Ý Ý Ñ F GpT q -Ý Ý Ñ EGp p Q T q and this isomorphism commutes with flips and mutations. Remark 7.9. A version of Theorem 7.7 has been established by Brüstle and Qiu (see [8]) for oriented exchange graphs defined by quivers arising from triangulations of marked surfaces (see [24] for more details). By identifying a convex polygon with an unpunctured disk, Theorem 7.7 recovers their result in the case where one considers oriented flip graphs of a tree arising from a polygonal subdivision of an unpunctured disk. In their language, is the universal tagged rotation of the marked surface.
The Stokes poset defined by Chapoton in [15] is a partial order on a family of quadrangulations which are "compatible" with a given quadrangulation Q. The compatibility condition was defined by Baryshnikov as follows [4]. Let P be a p2nq-gon whose vertices lie on a circle. The vertices of P are colored black and white, alternating in color around the circle. Let P 1 be the same polygon, rotated slightly clockwise. A quadrangulation is a polygonal subdivision into quadrilaterals. Fix a quadrangulation Q of P . A quadrangulation Q 1 of P 1 is compatible with Q if for each diagonal q P Q and q 1 P Q 1 such that q and q 1 intersect, the white endpoint of q 1 appears clockwise from the white endpoint of q before the black endpoint of q.
Let T be the tree dual to P . We may assume that the leaves of T are the vertices of Q 1 . If p is a geodesic between two leaves of T that does not take a sharp turn at an interior vertex then it crosses a pair of opposite sides of some quadrilateral in Q. As a result, p cannot be part of a quadrangulation compatible with Q. Let ∆ Q be the simplicial complex on the diagonals of Q 1 whose facets are quadrangulations compatible with Q. Then ∆ Q is a pure subcomplex of ∆ N C pT q of the same dimension. The complex ∆ Q is thin by Proposition 1.1 of [15]. Since the dual graph of ∆ N C pT q is connected, it follows that ∆ Q and ∆ N C pT q are isomorphic. Moreover, the orientation on the flips of quadrangulations defined in Section 1.3 of [15] coincides with Ý Ý Ñ F GpT q. Consequently, we deduce the following proposition.
Proposition 7.10. If every interior vertex of T has degree 4, then the poset Ý Ý Ñ F GpT q is isomorphic to the Stokes poset of quadrangulations compatible with the quadrangulation P T .

Simple-minded collections
In this section, we interpret noncrossing tree partitions in terms of the representation theory of Λ T using simple-minded collections in the bounded derived category of Λ T , denoted D b pΛ T q. We show that the data of a noncrossing tree partition and its Kreweras complement is equivalent to that of a certain type of simple-minded collection.
Simple-minded collections were originally used by Rickard [50] in the construction of derived equivalences of symmetric algebras from stable equivalences. A standard example of a simple-minded collection in representation theory is a complete set of non-isomorphic simple Λ-modules regarded as elements of D b pΛq. Note that any Λ-module X becomes an element of D b pΛq by mapping it to the stalk complex concentrated in degree 0 whose degree 0 term is X. Additionally, in [36], simple-minded collections were useful in computing spaces of Bridgeland stability conditions [6].
Here we recall some of the definitions we will need in order to study simple-minded collections. For a more complete presentation of the notions of derived categories and triangulated categories, we refer the reader to Chapter 1 of [35].
Let Λ be a finite dimensional k-algebra (or, more generally, a ring). By a complex, we mean a diagram of finitely generated Λ-modules ÝÑ¨¨ẗ hat satisfies d i`1 X˝d i X " 0 for each i P Z. We say that the Λ-module X i in the complex X is in degree i. We refer to the Λ-module homomorphisms d i X : X i Ñ X i´1 as differentials. If the only nonzero module of a complex X is in degree i, we say that X is a stalk complex concentrated in degree i. Given a complex X, it is natural to define the shift of X, denoted Xr1s, where ÝÑ¨¨ä nd where in Xr1s the module in degree i is X i`1 . Now let f : X Ñ Y be a morphism of complexes. We define the mapping cone or cone of f , denoted Conepf q, to be the componentwise direct sum of complexes Conepf q ÝÑ¨¨ẅ ith differential given by Dually, one defines the cocone of f , denoted Coconepf q.
The bounded derived category of Λ has objects given by complexes X of Λ-modules with X i " 0 when |i| is sufficiently large. Two objects X and Y in D b pΛq are isomorphic if and only if X and Y are quasi-isomorphic (i.e. there exists a morphism of complexes ϕ : X Ñ Y that induces an isomorphism H k pXq Ñ H k pY q for all k). The category D b pΛq, which is a triangulated category, also has the property that any triangle is isomorphic to a triangle of the form X f ÝÑ Y ÝÑ Conepf q ÝÑ Xr1s. 42 One can also show that any triangle in D b pΛq is isomorphic to one of the form Xr´1sÝÑCoconepf q ÝÑ X f ÝÑ Y. In this paper, we will be interested in understanding collections of objects from D b pΛq where the spaces of morphisms between any two objects in such a collection satisfy certain strong constraints. Morphism spaces between objects in derived categories can be very complicated. However, the objects in the collections we will study turn out to be stalk complexes. In this situation, the problem of understanding morphisms between such objects in D b pΛq is more tractable, as the following well-known proposition shows.
Proposition 8.1. Let X, Y P D b pΛq be stalk complexes concentrated in degree 0. Then Hom D b pΛq pXris, Y rjsq " Ext j´i Λ pX, Y q. We now give the main definition of this section.
Definition 8.2. Let C be a triangulated category. A collection tX 1 , . . . , X n u of objects of C is said to be simple-minded if the following hold for any i, j P rns: i) Hom C pX i , X j rksq " 0 for any k ă 0, ii) Hom C pX i , X j q " " k : if i " j 0 : otherwise, iii) C " thickxX 1 , . . . , X n y (i.e. the smallest triangulated category containing X 1 , . . . , X n and closed under taking summands of objects is C). One says that the objects tX 1 , . . . , X n u form a thick subcategory of C. Now let Λ be a finite dimensional k-algebra and consider a simple-minded collection tX 1 , . . . , X n u in D b pΛq. If for each i P rns one has H k pX i q " 0 for any k ‰ 0,´1, we say the collection is 2-term. We let 2-smc(Λ) denote the set of isomorphism classes of 2-term simple-minded collections of D b pΛq.
It turns out that, as the following lemma shows, it is easy to say what objects can appear in a 2-term simple minded collection in D b pΛ T q. Lemma 8.3. Let X " tX 1 , . . . , X n u P 2-smcpΛ T q. Each X i P X is isomorphic to a stalk complex of an indecomposable Λ T -module concentrated in degree 0 or´1.
Proof. By [9,Remark 4.11], each X P X is isomorphic to a stalk complex of a Λ T -module concentrated in degree 0 or´1. Suppose X P X is of the form X -M r1s where M P Λ T -mod. Now we have that End Λ T pM q " Hom Λ T pM, M q " Hom D b pΛ T q pM, M q " Hom D b pΛ T q pM r1s, M r1sq " k where the last equality follows from the fact that X P 2-smcpΛ T q. Since End Λ T pM q is a local ring, M is indecomposable. The proof is similar when X -M for some M P Λ T -mod. From Lemma 8.3, we have that any 2-term simple-minded collection X " tX 1 , . . . , X n u in D b pΛ T q can be regarded as a collection of segments of T . We define SegpX q " ts 1 , . . . , s n u to be this collection where s i P SegpX q corresponds to X i P X . Moreover, we can write SegpX q " Seg 0 pX q \ Seg´1pX q where Seg i pX q :" ts j P SegpX q : X j is concentrated in degree iu.
The simple-minded collection X also naturally defines a graph lying on D 2 as follows. Let SEGpX q be the graph whose vertices are the internal vertices of T and whose edges are admissible curves γ i defined by the segments s i P SegpX q up to endpoint fixing isotopy where if s i P Seg 0 pX q (resp. s i P Seg´1pX q) then γ i is a green-(resp. red-) admissible curve. By abuse of notation, we will write SEGpX q " tγ 1 , . . . , γ n u. It will also be useful to define SEG 0 pX q (resp. SEG´1pX q) to be the subgraph of SEGpX q consisting of green-(resp. red-) admissible curves from SEGpX q.
Proof. The image of θ lies in 2-smcpΛ T q by Lemma 6.24, Lemma 8.10, and Lemma 8.11. Next, decompose Seg 0 pX q and Seg´1pX q into segment-connected subsets of maximal size as follows: Seg 0 pX q " ğ i"1 Seg 0 i pX q and Seg´1pX q " Seg´1 i pX q.
In Section 8.2, we construct a map from : 2-smcpΛ T q ÝÑ tpB, KrpBqqu πPNCPpT q by X Þ ÝÑ pB X , KrpB X qq where B X :" pB 1 , . . . , B k q and where B i :" tvertices of T that are endpoints of segments in Seg´1 i pX qu. It follows from Proposition 8.9 that B X P NCPpT q and that any block B 1 i in KrpB X q " pB 1 1 , . . . , B 1 q satisfies B 1 i " tvertices of T that are endpoints of segments in Seg 0 i pX qu. It is easy to see that " θ´1.

Mutation of simple-minded collections.
Here we recall the notion of mutation of simple-minded collections and interpret this as a combinatorial operation on configurations of admissible curves. Our interpretation of mutation will be a key ingredient in showing that a 2-term simple-minded collection gives rise to a noncrossing tree partition paired with its Kreweras complement.
Mutation was first introduced in [37, Section 8.1] for spherical collections and generalized in [36] to Hom-finite, Krull-Schmidt triangulated categories. This notion is defined using the language of approximations, which we now briefly review.
Let C be an arbitrary category (not necessarily triangulated), and let A be any subcategory of C. We say that a morphism f : C Ñ A where C P C and A P A is a left A-approximation of C if for any morphism g : C Ñ A 1 where A 1 P A one has g " g 1 f for some morphism g 1 : A Ñ A 1 . Dually, one defines the notion of a right A-approximation of C. Additionally, we say that f : C Ñ A where C P C and A P A is left minimal morphism if for every morphism g : A Ñ A that satisfies gf " f one has that g is an isomorphism. Dually, one defines right minimal morphisms. A morphism f : C Ñ A (resp. f : A Ñ C) is a left minimal A-approximation (resp. right minimal A-approximation) if f is left minimal and is a left A-approximation (resp. right minimal and is a right A-approximation).
Let X " tX 1 , . . . , X n u be a simple-minded collection in D b pΛq where Λ is an arbitrary finite dimensional kalgebra. Let extpX k q denote the extension closure of X k in D b pΛq (i.e. the smallest subcategory of D b pΛq that contains X k and is closed under extensions). We define the left mutation of X to be µk pX q :" tX1 , . . . , Xǹ u where Xì :" " X k r1s : if i " k Conepgì : X i r´1s Ñ X k,i q : if i ‰ k where gì is a left minimal extpX k q-approximation. It is known that such approximations exist and that µk pX q is a simple-minded collection in D b pΛq (see [36,Section 7.2]). Dually, one defines the right mutation of X ,  Figure 22. The c-matrices of Q " 2 Ð 1 and the corresponding noncrossing tree partitions with their Kreweras complements.
respectively, define a segment ra 1 , b 1 s P SegpT q. This implies that s w p1q˝s w p2q P SegpT q. Thus, up to reversing the roles of w p1q and w p2q , there is a nonsplit extension 0 Ñ M pw p1q q Ñ M pw p1q Ð w p2q q Ñ M pw p2q q Ñ 0. This extension defines a triangle in D b pΛ T q given by M pw p1q q Ñ M pw p1q Ð w p2q q Ñ M pw p2q q Ñ M pw p1q qr1s. Thus M pw p1q Ð w p2q q P T . We obtain an admissible sequence ps v p1q , . . . , s v pi´1q , s w p1q Ðw p2q , s v pi`2q , . . . , s v pk`1q q for s v of length k. By induction, we obtain that M pvq P T .

Classification of c-matrices
We now apply our work to obtain a combinatorial classification of the c-matrices of quivers Q T (see Section 2.1) where the internal vertices of T are all of degree 3. By [41], the vertices of the oriented exchange graph of Q T index the clusters in the cluster algebra [26] defined by Q T . The c-matrices [27] of a quiver Q are related to noncrossing partitions of finite Coxeter groups [47] and many important objects in representation theory [9]. In [9], the c-matrices of quivers were interpreted representation theoretically as certain simple-minded collections in the bounded derived category of a finite dimensional algebra Λ. Our result is that c-matrices of Q T are classified by noncrossing tree partitions of T paired with their Kreweras complement.
Theorem 9.1. Assume that T is a tree whose internal vertices are of degree 3.
(1) The map ϕ : SegpT q Ñ c-vec(Q)`defined by s Þ Ñ pa 1 , . . . , a n q P Z n ě0 , where a i :" 1 if the edge corresponding to vertex i of Q T appears in s and a i :" 0 otherwise, is a bijection.
(2) The map tpB, KrpBqqu BPNCPpT q Ñ c-mat(Q) defined by sending pB, KrpBqq to the c-matrix C whose negative c-vectors are t´ϕpsq : s P SegpBq where B P Bu and whose positive c-vectors are tϕpsq : s P SegpB 1 q where B 1 P KrpBqu is a bijection (see Figure 22).
Proof. (1) By Corollary 6.5, there is a bijection between segments of T and the indecomposable modules of Λ T . This bijection sends a segment s to a string module M pwq of Λ T where w " w 1 Ø¨¨¨Ø w k has the property that each w i corresponds to an edge of T whose vertices both appear in s. Now consider the map dim : Λ T -mod Ñ Z n ě0 . By [16,Theorem 6], the restriction dim : indpΛ T q Ñ c-vecpQq`is a bijection. As the composition s Þ Ñ dimpM pwqq agrees with the map in the assertion, this completes the proof.
(2) By Theorem 8.4, there is a bijective map pB, KrpBqq θ Þ ÝÑ tM puqr1s : s u P SegpBq where B P Bu \ tM pvq : s v P SegpB 1 q where B 1 P KrpBqu where the latter belongs to 2-smcpΛ T q. Define a map Φ : 2-smcpΛ T q Ñ c-matpQq by tX 1 , . . . , X n u Þ Ñ tdimpX 1 q, . . . , dimpX n qu where dim : D b pΛ T q Ñ Z n is defined as dimpX i q :" ř jPZ p´1q j dimpX j i q. The latter map was shown to be a bijection in [9]. Using the proof of (1), we see that pB, KrpBqq Φ˝θ Þ ÝÑ t´ϕps u q : s u P SegpBq where B P Bu \ tϕps v q : s v P SegpB 1 q where B 1 P KrpBqu and the result follows.