Journal of Mathematics

Volume 2019, Article ID 1543201, 6 pages

https://doi.org/10.1155/2019/1543201

## Prismatic Sets Associated with Planar Rooted Trees

^{1}Dokuz Eylül University, Department of Mathematics, Tınaztepe, İzmir, Turkey^{2}Aalborg University, Department of Mathematical Sciences, Skjernvej 4, 9220 Aalborg Ø, Denmark

Correspondence should be addressed to S. Kaan Gürbüzer; rt.ude.ued@rezubrug.naak

Received 11 April 2019; Accepted 10 June 2019; Published 1 July 2019

Academic Editor: Li Guo

Copyright © 2019 S. Kaan Gürbüzer and Bedia Akyar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.