# Prismatic Sets Associated with Planar Rooted Trees

Authors: S. Kaan Gürbüzer; Bedia Akyar
Journal: Journal of Mathematics
 (2019)

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## Abstract

We construct the almost strong prismatic structure on the set of planar rooted trees and the bicomplex of planar rooted trees. Furthermore, we study the prismatic properties of Loday’s algebraic operations on the set of planar rooted trees.


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## Body

## 1. Introduction

Simplicial sets play an important role in simplicial category generated by two particularly important families of morphisms, whose images under given simplicial set functors are called face maps and degeneracy maps of those simplicial sets. Later, the concept of simplicial set was generalized by Dupont and Ljungman [1] by the name of prismatic sets which was derived from a simplicial complex, in order to construct an explicit fibre integration map in smooth Deligne cohomology [2]. Also, Akyar and Dupont [3] worked on the role of prismatic set in Lattice Gauge Field theory.In his paper [4] about arithmetree, Loday introduced planar binary trees in homology theory related to dialgebras and constructed addition and multiplication operations on the set of planar binary rooted trees giving rise to a new kind of arithmetic theory. This theory has also been extended to all planar rooted trees. On the other hand, Frabetti [5] worked on the simplicial properties of the set of planar binary rooted trees and constructed a decomposition of the towers on the bicomplex related to planar binary rooted trees.In this paper, we generalize the simplicial structure defined by Frabetti [5] of planar binary trees to the planar rooted trees. This structure is not simplicial but an almost simplicial structure called after Frabetti [5]. We construct an almost prismatic structure on the set of planar rooted trees and define a bicomplex on it. Furthermore, we investigate the combination of prismatic face and degeneracy maps with algebraic operations on trees defined by Loday [4].


## 2. Preliminaries

Definition 1.
Asimplicial setS={Sn} is a collection of sets Sn with the face maps di:Sn→Sn-1,0≤i≤n and degeneracy maps si:Sn→Sn+1,0≤i≤n (with n>0 in the case of di) satisfying thesimplicial identities(1)ddidj=dj-1di,i<j;djdi+1,i≥jssisj=sj+1si,i≤j;sjsi-1,i>jdsdisj=sj-1di,i<j;id,i=j,j+1;sjdi-1,i>j+1.Remark 2.
In order to define the unique face mapd0 on degree zero, one can also append the empty set ∅ as S-1 to a simplicial set. Instead of this, in this paper, we consider the face map d0 on degree zero as the identity map on S0.One can weaken the conditions of simplicial sets. A collection of sets with only face maps satisfying the relation (d) is called apre-simplicial set. Apseudo-simplicial set is a collection of sets with face and degeneracies satisfying the relations (d) and (ds). A subcollection of a pseudo-simplicial set whose degeneracies satisfy relation (s) except for i=j, that is, sisi≠si+1si, is called analmost simplicial set.(2)simplicialsets⊂almostsimplicialsets⊂pseudo-simplicialsets⊂pre-simplicialsetsAs given in Akyar [3], a strong prismatic set is a sequence P≔{Pp}={Sp,n0n1…np} of (p+1)-multisimplicial sets together with face di:Sp,n0…np→Sp-1,n0…n^i…np and degeneracy si:Sp,n0…np→Sp+1,n0…nini…np operators (with p>0 in the case of di) satisfying the following conditions:(3)didjk=djkdi,k<idi,k=idjk-1di,k>i.disjk=sjkdi,k<idi,k=isjk-1di,k>i.(4)sidjk=djksi,k<idjk+1djksi,k=idjk+1si,k>i.sisjk=sjksi,k<isjk+1sjksi,k=isjk+1si,k>i,where the maps djk and sjk areinternal  j-th face and degeneracy maps of the simplicial sets for each index k.We recall that agraphG is an ordered pairing G=VG,EG comprising the set V(G) of vertices or nodes together with the set E(G) of edges and atree is a nonempty connected finite simple graph. Two vertices in a graph are called adjacent if and only if there exists an edge connecting them. If a vertex is adjacent to only one vertex in a graph then it is also called anexternal vertex. The nonexternal vertices are calledinternal vertices. Specially, a tree with a distinguished external vertex (root vertex) is calleda rooted tree. For a rooted tree, theleaves are the external vertices different from the root. The degree or valency of an internal vertex is the number of edges incident to it. In this paper, we work on the planar rooted trees having at least one vertex and whose internal vertices have degree at least 3. For any natural number n, we denote the set of all planar rooted trees with n+1 leaves by Tn. In particular, it can be set that T-1 contains the unique planar rooted tree having only the root vertex. We define thedegree of a planar rooted tree in Tn by the number n. For a generic planar rooted tree in Tn, we assume that the leaves are labelled with the numbers, in the set {0,…,n} ordered from left to right and we denote it by tn to specify the degree of it.In order to study on the simplicial properties of the collection of planar rooted trees, we consider the sequenceT={Tn}n≥0 of the sets of planar rooted trees with the face operators di:Tn→Tn-1 deleting the i-th leaf for n>0 and the degeneracy operators si:Tn→Tn+1 making a bifurcation on the i-th leaf for all i∈{0,…n}, where n≥0. We give an example in Figure 1 of second face and degeneracy maps on T4. The face and degeneracy maps satisfy the simplicial identities (d) and (s) except for the case i=j, that is, sjsj≠sj+1sj. Let tn∈Tn and i<j, the i-th leaf of t is again the i-th leaf of the tree sj(tn), and if we apply first the face map di, the j-th leaf of tn becomes the j-1-st leaf of the tree di(tn), which means that the equality disj=sj-1di holds. When i=j,j+1, the bifurcation and deletion operations behave like inverses of each other. For the last case, if i>j+1, the face map di deletes the i-th leaf of sj(tn) which is the i-1-st leaf of tn. So the equality disj=sjdi-1 also holds and hence the sequence T={Tn}n≥0 is an almost simplicial set.Figure 1
Examples of face and degeneracy mapsd2,s2.Definition 3.
Combining the roots of two planar rooted trees on a vertex and creating a new root is called grafting operation and denoted by “∨”.The grafting operation is not associative, that is, for different treestn1,tn2, and tn3, the relation (tn1∨tn2)∨tn3=tn1∨(tn2∨tn3) is not satisfied and it is not commutative either. Without any confusion, we denote the grafting of more than two planar rooted trees by tq0∨⋯∨tqp, where tqi∈Tqi, and it is the planar rooted tree obtained by joining the roots of them to a new vertex and creating a new root; to imagine, see Figure 2. Note that the degree of the planar rooted tree tq0∨⋯∨tqp is ∑i=0p(qi+1)-1. In this perspective, for any tree tn∈Tn with positive degree, there is a unique representation such that tn=tq0∨⋯∨tqp where tqi∈Tqi. For n>0, this allows us to decompose each set Tn into the form ⋃q+p=nTp,q0…qp, where 1≤p≤n,qi≥0,∀i,q=q0+⋯+qp and Tp,q0…qp is the set of all possible planar rooted trees in the form tq0∨⋯∨tqp with ∑i=0p(qi+1) leaves. For n=0, we take attention that p must be equal to 0 and hence T0,0=T0.Figure 2
A general planar rooted tree of the formtq0∨⋯∨tqp.Let us recall the operations called over and under similar to the grafting from Loday [4]. The operationover is defined by grafting a tree to the most left leaf of another tree and similarly the operationunder is defined by grafting a tree to the most right leaf of an another tree. For tn1∈Tn1 and tn2∈Tn2, tn1 over tn2 and tn1 under tn2 are denoted by tn1/tn2 and tn1∖tn2, respectively. Figure 3 helps us to imagine these operations.Figure 3
Example of grafting, over and under operations.In [4], Loday constructed the dendriform trialgebra of planar rooted trees with some operations such as the sum of two planar rooted trees recursively. Note that Loday defined the sum of two planar rooted trees as not a planar rooted tree but a grove which is a disjoint union of planar rooted trees. By definition, the unique planar rooted tree in T0, denoted by 0, can be considered as a unit element of the sum and for k,l≥1, the sum of two planar rooted trees tn1∈Tk,q0…qk and tn2∈Tl,r0…rl is explicitly given by a recursive formula, (5)tn1+tn2≔tq0∨⋯∨tqk+tn2∪tq0∨⋯∨tqk+tr0∨⋯∨trl∪tn1+tr0∨⋯∨trl,with the initial steps 0+0=0 and (6)t1+t1=0∨0+t1∪0∨0+0∨0∪t1+0∨0=0∨0∨0∪0∨0∨0∪0∨0∨0,where t1=0∨0 is the unique planar rooted tree with two leaves in T1.In this formula, the disjoint union operation is used to obtain the grove and the grafting operation is considered as distribution on the disjoint union. Loday separated this sum into three parts and considered each part as a result of another sum. These three sums are called left sum⊣, middle sum ⊥ and right sum ⊢ and defined by (7)⊣:Tk,q0…qk×Tl,r0…rl→Tk,q0…qk-1qk+r0+…rl+l,tn1⊣tn2=tq0∨⋯∨tqk+t2,⊥:Tk,q0…qk×Tl,r0…rl→Tk+l,q0…qk-1qk+r0r1…rl,tn1⊥tn2=tq0∨⋯∨tqk+tr0∨⋯∨trl,⊢:Tk,q0…qk×Tl,r0…rl→Tl,q0+⋯+qk+k+r0r1…rl,tn1⊢tn2=t1+tr0∨⋯∨trl,respectively, where tn1≠0 and tn2≠0 and Tp,q0…qp denotes the set of groves, that is, the set of all disjoint union of the planar rooted trees of the form tq0∨⋯∨tqp with degree (∑i=0pqp)+p. The extension of these sums for 0 is given by tn1⊣0=tn1,0⊣tn2=0,tn1⊢0=0,0⊢tn2=tn2, and tn1⊥0=0,0⊥tn2=0.


## 3. Main Results

Theorem 4.
For any fieldF, let F[Tn] be the vector space over F with basis Tn, the sequence 0←F[T0]←F[T1]←⋯←F[Tn]←⋯ is a chain complex F[T∗] with the boundary operator ∂=∑i=0n(-1)idi.Theorem 5.
For any fieldF, the chain complex F[T∗] is acyclic, that is, HnFT∗,∂=F,n=0;0,n≠0.In order to construct a bicomplex on the set of all planar rooted trees, we observe an almost strong prismatic structure of the collectionT. The decomposition of Tn given in previous section gives us the following theorem.Theorem 6.
The collection of all planar rooted trees becomes an almost strong prismatic set with the decomposition(8)T=⋃n⋃q+p=nTp,q0…qp,where the face maps dk:Tp,q0…qp→Tp-1,q0…qk^…qp deleting the k-th tree for p>0 and the degeneracies sk:Tp,q0…qp→Tp+1,q0…qkqk…qp repeating the k-th tree; that is, (9)dktq0∨⋯∨tqp=tq0∨⋯∨tqk-1∨tqk+1∨⋯∨tqpsktq0∨⋯∨tqp=tq0∨⋯∨tqk-1∨tqk∨tqk∨tqk+1∨⋯∨tqp.The internal face and degeneracy maps are the face and degeneracy maps given in the almost simplicial structure.Theorem 7.
Fortn1∈Tk,q0…qk and tn2∈Tl,r0…rl, prismatic face and degeneracy maps have effect to over and under operations as follows: (10)ditn1/tn2=ditn2,i=0;tn1/ditn2,i≠0.ditn1∖tn2=ditn1,i=k;ditn1∖tn2,i≠k.sitn1/tn2=tn1/tr0∨tn1/tr0∨tr1∨⋯∨trl,i=0;tn1/sitn2,i≠0.sitn1∖tn2=tq0∨⋯∨tqk∖tn2∨tqk∖tn2,i=k;sitn1∖tn2,i≠k.Theorem 8.
Fortn1∈Tk,q0…qk and tn2∈Tl,r0…rl different from 0, the effects of prismatic external face and degeneracy maps on these tree sum operations are given as follows: (11)ditn1⊣tn2=ditn1⊣tn2,i≠k;dktn1,i=k.ditn1⊥tn2=ditn1⊥tn2,i<ktq0∨⋯∨tqk-1∨tr1∨⋯∨trl,i=ktn1⊥di-ktn2,i>k.ditn1⊢tn2=d0tn2,i=0tn1⊢ditn2,i≠0.sitn1⊣tn2=sitn1⊣tn2,i≠ktq0∨⋯∨tqk-1∨tqk+tn2∨tqk+tn2,i=k.sitn1⊥tn2=sitn1⊥tn2,i<ktq0∨⋯∨tqk-1∨tqk+tr0∨tqk+tr0∨tr1∨⋯∨trl,i=ktn1⊥si-ktn2,i>k.sitn1⊢tn2=tn1⊢sitn2,i≠0;tn1+tr0∨tn1+tr0∨tr1∨⋯∨trl,i=0.Taking a symmetry on the axes passing through the root edge gives an involution on the set of trees. We denote this involution byσ and it is clear that there is a distribution property of the involution on grafting and hence the sums of trees. For tn∈Tk,q0…qk, it can be checked that the relations diσ(tn)=σ(dk-itn) and siσ(tn)=σ(sk-itn) hold.The prismatic structure of the collectionT allows us to divide the chain complex F[T∗], given in Theorem 4, into two parts called horizontal and vertical.The boundary operators∂vk:F[Tp,q0…qk…qp]→F[Tp,q0…qk-1…qp] of the vertical complex are defined by ∂vk=0 whenever qk=0; otherwise ∂vk=∑j=0qk(-1)jdjk. The total boundary operator of the vertical complexes is given by (12)∂v≔∂v0+-1q0+1∂v1+⋯+-1q0+⋯+qp-1+p∂vp.For each horizontal complex, the boundary operator∂h:F[Tp,q0…qp]→F[Tp-1,q0′…qp-1′] is defined by (13)∂h=∂0+-1q0+1∂1+⋯+-1q0+⋯+qp-1+p∂pwhere ∂i=0,ifqi>0;di,ifqi=0.Theorem 9.
The vertical and horizontal boundary operators satisfy the identities∂v∂v=0 and ∂h∂h=0. For p>0 and q=q0+q1+⋯+qp, the complexes (F[T∗,q],∂h) and (F[Tp,∗],∂v) are chain complexes. In addition, the triple (F[T∗,∗],∂v,∂h) forms a chain bicomplex as shown in Figure 4 and the total complex F[T∗] is given by the boundary operator ∂~=∂v+∂h.Figure 4
The bicomplex of planar rooted trees.Corollary 10.
For anytn1∈Tn1 and tn2∈Tn2, the vertical and horizontal boundary operators satisfy the followings (14)∂vtn1∨tn2=∂tn1∨tn2+-1n1+1tn1∨∂tn2,∂htn1∨tn2=0,n1=n2=0tn2,n1=0andn2≠0-1n1+1tn1,n1≠0andn2=00,n1≠0andn2≠0.Furthermore, the relation ∂~(tn1∨tn2)=∂tn1∨tn2+(-1)n1+1tn1∨∂tn2 holds.


### 3.1. Proofs

Proof of Theorem4.
Since the effects of the mapsdidj and dj-1di are the same for i<j, the boundary operator satisfies the condition ∂2=0 and hence the sequence becomes a chain complex.Proof of Theorem5.
Let us define a maph:Tn→Tn+1 which grafts a leaf to the root edge; that is, for tn∈Tn, h(tn)=0∨tn. For any j>0, it can be seen that the relation djh=hdj-1 is satisfied and the composition d0h becomes identity. Then, we get (15)∂h=∑i=0n+1-1idi∘h=d0h+∑i=1n+1-1idih=d0h+∑i=0n-1i+1di+1h=d0h-∑i=0n-1idi+1h=d0h-∑i=0n-1ihdi=d0h-h∂.So, ∂h+h∂=identity and the induced map h:F[Tn]→F[Tn+1] becomes a homotopy between identity and zero map.Proof of Theorem6.
First we check that the simplicial identities are satisfied for external face and degeneracy maps. Fortn=tq0∨⋯∨tqp, (16)didjtn=ditq0∨⋯∨tqj-1∨tqj+1∨⋯∨tqp=tq0∨⋯∨tqi-1∨tqi+1∨⋯∨tqj-1∨tqj+1∨⋯∨tqp=dj-1tq0∨⋯∨tqi-1∨tqi+1∨⋯∨tqp=dj-1ditn,fori<j;didjtn=ditq0∨⋯∨tqj-1∨tqj+1∨⋯∨tqp=tq0∨⋯∨tqj-1∨tqj+1∨⋯∨tqi∨tqi+2∨⋯∨tqp=djtq0∨⋯∨tqi∨tqi+2∨⋯∨tqp=djdi+1tn,fori≥j;sisjtn=sitq0∨⋯∨tqj∨tqj∨⋯∨tqp=tq0∨⋯∨tqi∨tqi∨⋯∨tqj∨tqj∨⋯∨tqp=sj+1tq0∨⋯∨tqi∨tqi∨⋯∨tqp=sj+1sitn,fori<j;sisjtn=sitq0∨⋯∨tqj∨tqj∨⋯∨tqp=tq0∨⋯∨tqj∨tqj∨⋯∨tqi-1∨tqi-1∨⋯∨tqp=si-1tq0∨⋯∨tqj∨tqj∨⋯∨tqp=sjsi-1tn,fori>j;sisitn=sitq0∨⋯∨tqi∨tqi∨⋯∨tqp=tq0∨⋯∨tqi∨tqi∨tqi∨⋯∨tqp=si+1tq0∨⋯∨tqi∨tqi∨⋯∨tqp=si+1sitn.Now, we check the identities (3) and (4). (17)didjktn=ditq0∨⋯∨djtqk∨⋯∨tqp=tq0∨⋯∨djtqk∨⋯∨tqi-1∨tqi+1∨⋯∨tqp=djkditn;fork<i;didjktn=ditq0∨⋯∨djtqk∨⋯∨tqp=tq0∨⋯∨tqi-1∨tqi+1∨⋯∨tqp=ditn;fork=i;didjktn=ditq0∨⋯∨djtqk∨⋯∨tqp=tq0∨⋯∨tqi-1∨tqi+1∨⋯∨djtqk∨⋯∨tqp=djk-1ditn;fork>i;sidjktn=sitq0∨⋯∨djtqk∨⋯∨tqp=tq0∨⋯∨djtqk∨⋯∨tqi∨tqi∨⋯∨tqp=djksitn;fork<i;sidjktn=sitq0∨⋯∨djtqk∨⋯∨tqp=tq0∨⋯∨djtqk∨djtqk∨⋯∨tqp=djk+1djksitn;fork=i;sidjktn=sitq0∨⋯∨djtqk∨⋯∨tqp=tq0∨⋯∨tqi∨tqi∨⋯∨djtqk∨⋯∨tqp=djk+1sitn;fork>i.The prismatic identities between external maps and internal degeneracies can be verified in a similar way.Proof of Theorem9.
To simplify the notation, letQk denote the sum q0+⋯+qk-1+k for k>0 and Q0=0. For l<k and qk≠0≠ql, in the composition ∂v∂v, we have the sum (18)-1Qk′-1Ql∂vk∂vl+-1Ql′-1Qk∂vl∂vk,which is equal to 0, because after applying ∂vl, the index of the l-th part of the planar rooted tree becomes ql-1 and so Qk′=Qk-1 and Ql=Ql′. For the case l=k, the proof induces the boundary operator ∂ given in Theorem 4 on the k-th part of the planar rooted tree and hence ∂v∂v=0.
In order to prove that∂h∂h=0, we only need to consider the planar rooted trees which have 0 at least for two indices. Now, we assume that l<k and ql=qk=0. In this case, for the composition ∂h∂h includes the sums of the form (19)-1Ql′-1Qk∂l∂k+-1Qk-1′-1Ql∂k-1∂l.Since ql=0, we obtain that Qk-1′=Qk-1 and Ql′=Ql. Hence this sum is equal to 0 and ∂h∂h=0.
Last, similarly the composition∂h∂v can be rewritten by shifting one the indices of the composition ∂v∂h and hence the equality ∂~∂~=0 holds.


## 3.1. Proofs

Proof of Theorem4.
Since the effects of the mapsdidj and dj-1di are the same for i<j, the boundary operator satisfies the condition ∂2=0 and hence the sequence becomes a chain complex.Proof of Theorem5.
Let us define a maph:Tn→Tn+1 which grafts a leaf to the root edge; that is, for tn∈Tn, h(tn)=0∨tn. For any j>0, it can be seen that the relation djh=hdj-1 is satisfied and the composition d0h becomes identity. Then, we get (15)∂h=∑i=0n+1-1idi∘h=d0h+∑i=1n+1-1idih=d0h+∑i=0n-1i+1di+1h=d0h-∑i=0n-1idi+1h=d0h-∑i=0n-1ihdi=d0h-h∂.So, ∂h+h∂=identity and the induced map h:F[Tn]→F[Tn+1] becomes a homotopy between identity and zero map.Proof of Theorem6.
First we check that the simplicial identities are satisfied for external face and degeneracy maps. Fortn=tq0∨⋯∨tqp, (16)didjtn=ditq0∨⋯∨tqj-1∨tqj+1∨⋯∨tqp=tq0∨⋯∨tqi-1∨tqi+1∨⋯∨tqj-1∨tqj+1∨⋯∨tqp=dj-1tq0∨⋯∨tqi-1∨tqi+1∨⋯∨tqp=dj-1ditn,fori<j;didjtn=ditq0∨⋯∨tqj-1∨tqj+1∨⋯∨tqp=tq0∨⋯∨tqj-1∨tqj+1∨⋯∨tqi∨tqi+2∨⋯∨tqp=djtq0∨⋯∨tqi∨tqi+2∨⋯∨tqp=djdi+1tn,fori≥j;sisjtn=sitq0∨⋯∨tqj∨tqj∨⋯∨tqp=tq0∨⋯∨tqi∨tqi∨⋯∨tqj∨tqj∨⋯∨tqp=sj+1tq0∨⋯∨tqi∨tqi∨⋯∨tqp=sj+1sitn,fori<j;sisjtn=sitq0∨⋯∨tqj∨tqj∨⋯∨tqp=tq0∨⋯∨tqj∨tqj∨⋯∨tqi-1∨tqi-1∨⋯∨tqp=si-1tq0∨⋯∨tqj∨tqj∨⋯∨tqp=sjsi-1tn,fori>j;sisitn=sitq0∨⋯∨tqi∨tqi∨⋯∨tqp=tq0∨⋯∨tqi∨tqi∨tqi∨⋯∨tqp=si+1tq0∨⋯∨tqi∨tqi∨⋯∨tqp=si+1sitn.Now, we check the identities (3) and (4). (17)didjktn=ditq0∨⋯∨djtqk∨⋯∨tqp=tq0∨⋯∨djtqk∨⋯∨tqi-1∨tqi+1∨⋯∨tqp=djkditn;fork<i;didjktn=ditq0∨⋯∨djtqk∨⋯∨tqp=tq0∨⋯∨tqi-1∨tqi+1∨⋯∨tqp=ditn;fork=i;didjktn=ditq0∨⋯∨djtqk∨⋯∨tqp=tq0∨⋯∨tqi-1∨tqi+1∨⋯∨djtqk∨⋯∨tqp=djk-1ditn;fork>i;sidjktn=sitq0∨⋯∨djtqk∨⋯∨tqp=tq0∨⋯∨djtqk∨⋯∨tqi∨tqi∨⋯∨tqp=djksitn;fork<i;sidjktn=sitq0∨⋯∨djtqk∨⋯∨tqp=tq0∨⋯∨djtqk∨djtqk∨⋯∨tqp=djk+1djksitn;fork=i;sidjktn=sitq0∨⋯∨djtqk∨⋯∨tqp=tq0∨⋯∨tqi∨tqi∨⋯∨djtqk∨⋯∨tqp=djk+1sitn;fork>i.The prismatic identities between external maps and internal degeneracies can be verified in a similar way.Proof of Theorem9.
To simplify the notation, letQk denote the sum q0+⋯+qk-1+k for k>0 and Q0=0. For l<k and qk≠0≠ql, in the composition ∂v∂v, we have the sum (18)-1Qk′-1Ql∂vk∂vl+-1Ql′-1Qk∂vl∂vk,which is equal to 0, because after applying ∂vl, the index of the l-th part of the planar rooted tree becomes ql-1 and so Qk′=Qk-1 and Ql=Ql′. For the case l=k, the proof induces the boundary operator ∂ given in Theorem 4 on the k-th part of the planar rooted tree and hence ∂v∂v=0.
In order to prove that∂h∂h=0, we only need to consider the planar rooted trees which have 0 at least for two indices. Now, we assume that l<k and ql=qk=0. In this case, for the composition ∂h∂h includes the sums of the form (19)-1Ql′-1Qk∂l∂k+-1Qk-1′-1Ql∂k-1∂l.Since ql=0, we obtain that Qk-1′=Qk-1 and Ql′=Ql. Hence this sum is equal to 0 and ∂h∂h=0.
Last, similarly the composition∂h∂v can be rewritten by shifting one the indices of the composition ∂v∂h and hence the equality ∂~∂~=0 holds.


## 4. Concluding Remarks

Loday [6] and Frabetti [7] used the bicomplex on the collection of the sets of planar binary rooted trees to construct the homology and cohomology theory of dialgebras. In order to study on the homology or cohomology theory on dendriform trialgebras, the prismatic structure and the bicomplex given in this paper may be a useful tool.



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