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.
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. A simplicial set is a collection of sets with the face maps and degeneracy maps (with in the case of ) satisfying the simplicial identities
Remark 2. In order to define the unique face map on degree zero, one can also append the empty set as to a simplicial set. Instead of this, in this paper, we consider the face map on degree zero as the identity map on .
One can weaken the conditions of simplicial sets. A collection of sets with only face maps satisfying the relation (d) is called a pre-simplicial set. A pseudo-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 , that is, , is called an almost simplicial set.As given in Akyar [3], a strong prismatic set is a sequence of -multisimplicial sets together with face and degeneracy operators (with in the case of ) satisfying the following conditions:where the maps and are internal -th face and degeneracy maps of the simplicial sets for each index
We recall that a graph is an ordered pairing comprising the set of vertices or nodes together with the set of edges and a tree 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 an external vertex. The nonexternal vertices are called internal vertices. Specially, a tree with a distinguished external vertex (root vertex) is called a rooted tree. For a rooted tree, the leaves 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 , we denote the set of all planar rooted trees with leaves by In particular, it can be set that contains the unique planar rooted tree having only the root vertex. We define the degree of a planar rooted tree in by the number For a generic planar rooted tree in , we assume that the leaves are labelled with the numbers, in the set ordered from left to right and we denote it by to specify the degree of it.
In order to study on the simplicial properties of the collection of planar rooted trees, we consider the sequence of the sets of planar rooted trees with the face operators deleting the -th leaf for and the degeneracy operators making a bifurcation on the -th leaf for all , where We give an example in Figure 1 of second face and degeneracy maps on The face and degeneracy maps satisfy the simplicial identities (d) and (s) except for the case , that is, Let and , the -th leaf of is again the -th leaf of the tree , and if we apply first the face map , the -th leaf of becomes the -st leaf of the tree , which means that the equality holds. When , the bifurcation and deletion operations behave like inverses of each other. For the last case, if , the face map deletes the -th leaf of which is the -st leaf of So the equality also holds and hence the sequence is an almost simplicial set.
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 trees , and , the relation is not satisfied and it is not commutative either. Without any confusion, we denote the grafting of more than two planar rooted trees by , where , 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 is In this perspective, for any tree with positive degree, there is a unique representation such that where For , this allows us to decompose each set into the form , where and is the set of all possible planar rooted trees in the form with leaves. For , we take attention that must be equal to 0 and hence
Let us recall the operations called over and under similar to the grafting from Loday [4]. The operation over is defined by grafting a tree to the most left leaf of another tree and similarly the operation under is defined by grafting a tree to the most right leaf of an another tree. For and , over and under are denoted by and , respectively. Figure 3 helps us to imagine these 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 , denoted by 0, can be considered as a unit element of the sum and for , the sum of two planar rooted trees and is explicitly given by a recursive formula, with the initial steps and where is the unique planar rooted tree with two leaves in
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 respectively, where and and denotes the set of groves, that is, the set of all disjoint union of the planar rooted trees of the form with degree The extension of these sums for 0 is given by , and
3. Main Results
Theorem 4. For any field , let be the vector space over with basis , the sequence is a chain complex with the boundary operator
Theorem 5. For any field , the chain complex is acyclic, that is,
In order to construct a bicomplex on the set of all planar rooted trees, we observe an almost strong prismatic structure of the collection The decomposition of 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 where the face maps deleting the th tree for and the degeneracies repeating the th tree; that is, The internal face and degeneracy maps are the face and degeneracy maps given in the almost simplicial structure.
Theorem 7. For and , prismatic face and degeneracy maps have effect to over and under operations as follows:
Theorem 8. For and different from 0, the effects of prismatic external face and degeneracy maps on these tree sum operations are given as follows:
Taking a symmetry on the axes passing through the root edge gives an involution on the set of trees. We denote this involution by and it is clear that there is a distribution property of the involution on grafting and hence the sums of trees. For , it can be checked that the relations and hold.
The prismatic structure of the collection allows us to divide the chain complex , given in Theorem 4, into two parts called horizontal and vertical.
The boundary operators of the vertical complex are defined by whenever ; otherwise The total boundary operator of the vertical complexes is given by
For each horizontal complex, the boundary operator is defined by where
Theorem 9. The vertical and horizontal boundary operators satisfy the identities and For and , the complexes and are chain complexes. In addition, the triple forms a chain bicomplex as shown in Figure 4 and the total complex is given by the boundary operator
Corollary 10. For any and , the vertical and horizontal boundary operators satisfy the followings Furthermore, the relation holds.
3.1. Proofs
Proof of Theorem 4. Since the effects of the maps and are the same for , the boundary operator satisfies the condition and hence the sequence becomes a chain complex.
Proof of Theorem 5. Let us define a map which grafts a leaf to the root edge; that is, for , For any , it can be seen that the relation is satisfied and the composition becomes identity. Then, we get So, and the induced map becomes a homotopy between identity and zero map.
Proof of Theorem 6. First we check that the simplicial identities are satisfied for external face and degeneracy maps. For , Now, we check the identities (3) and (4). The prismatic identities between external maps and internal degeneracies can be verified in a similar way.
Proof of Theorem 9. To simplify the notation, let denote the sum for and For and , in the composition , we have the sum which is equal to 0, because after applying , the index of the -th part of the planar rooted tree becomes and so and For the case , the proof induces the boundary operator given in Theorem 4 on the -th part of the planar rooted tree and hence
In order to prove that , we only need to consider the planar rooted trees which have 0 at least for two indices. Now, we assume that and In this case, for the composition includes the sums of the form Since , we obtain that and Hence this sum is equal to 0 and
Last, similarly the composition can be rewritten by shifting one the indices of the composition and hence the equality 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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.