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Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests

  • Jinwei Wang , Yuanyuan Zhang EMAIL logo , Zhicheng Zhu and Dan Chen
Published/Copyright: November 5, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

Abstract

In this paper, we decorate leaves and edges of planar rooted forests simultaneously and use a part of them to construct free nonunitary Rota-Baxter family algebras. As a corollary, we obtain the construction of free nonunitary Rota-Baxter algebras.

Keywords: Rota-Baxter algebras; Rota-Baxter family algebras; typed decorated rooted forests; leaf-spaced; parallelly typed leaf-spaced decorated
MSC 2010: 16W99; 16S10; 08B20; 16T30; 81R15

1 Introduction

1.1 Rota-Baxter (family) algebras

An associative algebra R together with a k-linear operator P : R → R is called a Rota-Baxter algebra of weight λ , if

(1) P ( x ) P ( y ) = P ( P ( x ) y ) + P ( x P ( y ) ) + λ P ( x y ) , for x , y ∈ R ,

where λ is a fixed element in the basis ring k. When λ = 0 , the operator P is indeed an algebraic abstraction of the usual integration operation in analysis [1]. The mathematician Glen E. Baxter first studied the Rota-Baxter algebras [2] in 1960 in his probability study. Some combinatoric properties of Rota-Baxter algebras were studied by Rota [3] and Cartier [4]. Free Rota-Baxter associative algebras were constructed on both commutative and noncommutative cases by using different methods, which appeared in [1,3,4,5,6,7,8]. A Rota-Baxter algebra naturally carries a dendriform or tridendriform algebra structure [9]. Nowadays, Rota-Baxter algebra has become a new branch with broad connections to other objects in mathematics, such as pre-Lie algebras, pre-Poisson algebras [10,11], quantum field theory [12,13,14], Hopf algebras [15,16], commutative algebras [17,18], Loday’s dendriform algebras [9,19], and Aguiar’s associative analogue of the classical Yang-Baxter equation [20,21,22].

In 2007, K. Ebrahimi-Fard, J. Gracia-Bondia and F. Patras [23, Proposition 9.1] (see also [25, Theorem 3.7.2]) introduced the first example of Rota-Baxter family about algebraic aspects of renormalization in quantum field theory, where a “Rota-Baxter family” appears: this terminology was suggested to the authors by Li Guo (see Footnote following Proposition 9.2 therein), who further discussed the underlying structure under the name Rota-Baxter family algebra in [24]. Namely, let Ω be a semigroup and λ ∈ k be given. A Rota-Baxter family of weight λ on an algebra R is a collection of k-linear operators ( P ω ) ω ∈ Ω on R such that

(2) P α ( a ) P β ( b ) = P α β ( P α ( a ) b + a P β ( b ) + λ a b ) , for a , b ∈ R and α , β ∈ Ω .

Then the pair ( R , ( P ω ) ω ∈ Ω ) is called a Rota-Baxter family algebra of weight λ . Rota-Baxter family algebra arises naturally in renormalization of quantum field theory. It is worthwhile to study the algebraic structure of Rota-Baxter family algebras. As the construction of free objects in a category is always interesting and important, the author in [26] constructed, respectively, free commutative unitary Rota-Baxter family algebras, and free noncommutative unitary Rota-Baxter family algebras by the method of Gröbner-Shirshov bases.

1.2 Algebraic structures on (typed decorated) rooted forests

Rooted trees/forests are a useful tool for studying many interesting algebraic structures. It appeared in the work of Arthur Cayley [27] in the 1850s considered rooted trees as a representation of combinatorial structures related to the free pre-Lie algebra. More than a century later, these structures formed the foundation of John Butcher’s theory of B-series [28,29], which has become an indispensable tool in the analysis of numerical integration. Many Hopf algebraic structures have been built up on top of rooted forests, such as Connes-Kreimer Hopf algebra [15], Loday-Ronco [19], Grossman-Larson [30] and Foissy-Holtkamp [31,32]. In particular, the famous Connes-Kreimer Hopf algebra was employed to deal with renormalization in quantum field theory [13,14]. Pre-Lie structures on non-planar rooted trees lead to Hopf algebras of combinatorial nature, which appeared in the works in [15,30,33]. Free pre-Lie algebras can also be described as the space of non-planar rooted trees with product given by grafting of trees [34].

The multi pre-Lie structures were first introduced in [35] (see Section 4 and Appendix A) in a more general setting before [36] which considered only the non-deformed structures. A recent preprint [37] used typed decorated rooted forests in numerical analysis for developing a general scheme for dispersive partial differential equations (PDEs) which strengths the universal aspect of these structures. Typed decorated rooted forests also appeared in a context of low-dimension topology [38] and in a context of the description of combinatorial species [39].

1.3 Motivation and layout of the paper

Our motivations come from two points. The first point is that it is almost natural to construct free nonunitary Rota-Baxter family algebras, parallel to the unitary case done in [26]. The second point is along the line of typed decorated rooted forests. Recall that Guo [24] constructed free nonunitary Rota-Baxter algebras in terms of leaf-spaced decorated planar rooted forests. In the present paper, combining typed decorated and leaf-spaced decorated planar rooted forests, we construct free nonunitary Rota-Baxter family algebras, as a generalization of the work in [24].

The following is the outline of the paper. In Section 2.1, we recall some basic concepts of planar rooted forests used in this paper. Combining leaf decorated and typed decorated planar rooted forests, we propose the concept of (parallelly) typed leaf-spaced decorated planar rooted forests in Section 2.2. Based on this concept, we construct the free nonunitary Rota-Baxter family algebra on a set in Section 2.3. As a corollary, the construction of free nonunitary Rota-Baxter algebra on a set is obtained.

  1. Notation: Throughout this paper, let k be a nonunitary commutative ring which will be the base ring of all modules, algebras, as well as linear maps. Algebras are nonunitary associative algebras but not necessary commutative. For a set Y, we denote by k Y and S ( Y ) the free k-module with a basis Y and free semigroup on Y, respectively.

2 Free nonunitary Rota-Baxter family algebras

In this section, we first propose the concept of parallelly typed leaf-spaced decorated planar rooted forests and then use them to construct free nonunitary Rota-Baxter family algebras.

2.1 Planar rooted forests

In this subsection, let us recall some basic concepts of planar rooted forests. A rooted tree is a connected and simply connected set of vertices and oriented edges such that there is precisely one distinguished vertex, called the root, with no incoming edge. The only vertex of the tree • is taken to be a leaf. If two vertices of a rooted tree are connected by an edge, then the vertex on the side of the root is called the parent and the vertex on the opposite side of the root is called a child.

A rooted tree is called planar rooted tree if it is endowed with an embedding in the plane. Here are some examples.where the root in a planar rooted tree is at the top. A subforest of a planar rooted tree T is the forest consisting of a set of vertices of T together with their descents and edges connecting all these vertices.

Let T be the set of planar rooted trees and ℱ ≔ S ( T ) the free semigroup generated by T . Thus, an element in S ( T ) , called a planar rooted forest, is a noncommutative product of planar rooted trees in T . Here are some examples of planar rooted forests.

The following concepts are standard.

Definition 2.1

  1. For a planar rooted tree T, the depth dep ( T ) of T is the maximal length of paths from the root to leaves of the tree. The depth dep ( F ) of a forest F is the maximal depth of trees in F.

  2. For a planar rooted forest F = T 1 … T b with T 1 , … , T b ∈ T , define bre ( F ) ≔ b to be the breadth of F.

For example,

In the noncommutative version of the well-known Connes-Kreimer Hopf algebra [31,32], there is a linear grafting operation

(3) B + : k ℱ → k ℱ , T 1 … T n ↦ B + ( T 1 … T n ) ,

where B + ( T 1 … T n ) is obtained by adding a new root together with an edge from the new root to the root of each of the planar rooted trees T 1 , … , T n . For example,

2.2 Parallelly typed leaf-spaced decorated planar rooted forests

In this subsection, we first recall the concept of leaf-spaced decorated planar rooted forests [24] and then generalize it to parallelly typed version with an eye toward constructing free nonunitary Rota-Baxter family algebras.

Guo utilized leaf-spaced decorated rooted forests to construct free nonunitary Rota-Baxter algebras [24].

Definition 2.2

Let X be a set and T a planar rooted tree.

  1. The tree T is called leaf-decorated if its each leaf is decorated by an element of X.

  2. A subtree starting from a vertex v of T is the planar rooted tree consisting of v, as the root, together with all descents of v and edges connecting all these vertices. If we write the subtree starting from v in the form B + ( T 1 … T n ) with T 1 , … , T n being planar rooted trees, we call T i and T i + 1 adjacent branches of v, where 1 ≤ i ≤ n − 1 .

  3. A planar rooted tree is called leaf-spaced if it does not have a vertex with adjacent non-leaf branches.

  4. A leaf-spaced decorated planar rooted tree is a leaf-decorated planar rooted tree which is also leaf-spaced.

Let us expose some examples for better understanding.

Example 2.3

The following are leaf-spaced planar rooted trees

while

are not leaf-spaced planar rooted trees, since two right most branches are not separated by a leaf branch.

Example 2.4

The leaf-decorated planar rooted tree

is not leaf-spaced since two right most branches, with leaves decorated by s and v, are not separated by a leaf branch. While the leaf-decorated planar rooted treeis leaf-spaced.

Remark 2.5

For a leaf-spaced planar rooted tree, it doesn’t have a vertex with adjacent non-leaf branches. In the view point of rewriting system, it means that we rewrite the Rota-Baxter equation (1) from left to right. So it is natural to use leaf-spaced planar rooted trees to construct free nonunitary Rota-Baxter algebra in [25].

Now we generalize Definition 2.2 to planar rooted forests.

Definition 2.6

Let X be a set and F = T 1 … T n a decorated planar rooted forest, where T 1 , … , T n are decorated planar rooted trees. We call F leaf-spaced if T 1 , … , T n are leaf-spaced and for each i, at least one of dep ( T i ) and dep ( T i + 1 ) is 0.

Example 2.7

The following are some examples of leaf-spaced decorated planar rooted forests:

while the following are some counterexamples:Here decorations are from X.

Typed decorated planar rooted trees are planar rooted trees with vertices decorated by elements of a set X and edges decorated by elements of a set Ω , which are applied to give a systematic description of a canonical renormalization procedure of stochastic PDEs [40]. Several algebraic structures have been built up on these planar rooted trees [36].

For a rooted tree T, denote by V ( T ) (resp. E ( T ) ) the set of its vertices (resp. edges).

Definition 2.8

[40] Let X and Ω be two sets. An X- decorated Ω -typed (abbreviated typed decorated) rooted tree is a triple T = ( T , dec, type) , where

  1. T is a rooted tree,

  2. dec : V ( T ) → X is a map,

  3. type : E ( T ) → Ω is a map.

Here are some examples of typed decorated rooted trees.where x , y , z , u , v ∈ X and α , β , γ ∈ Ω .

Combining Definitions 2.2 and 2.8, we propose the following concept. For a planar rooted tree T, denote by L ( T ) the set of its leaves.

Definition 2.9

Let X and Ω be two sets. A typed leaf-spaced decorated planar rooted tree (resp. forest) is a triple T = ( T , dec , type ) , where

  1. T is a leaf-spaced planar rooted tree (resp. forest),

  2. dec : L ( T ) → X is a map,

  3. type : E ( T ) → Ω is a map.

Example 2.10

Let X and Ω be two sets. For x , y ∈ X and α , β , γ ∈ Ω ,

are typed leaf-spaced decorated planar rooted trees.

Now we come to the key concept used in this paper.

Definition 2.11

Let X and Ω be two sets. A typed leaf-spaced decorated planar rooted tree (resp. forest) is called parallelly typed if type ( e 1 ) = type ( e 2 ) whenever edges e 1 and e 2 share the same parent vertex. Denote by T ℓ ( X , Ω ) (resp. ℱ ℓ ( X , Ω ) ) the set of parallelly typed leaf-spaced decorated planar rooted trees (resp. forests).

Example 2.12

The following are some elements in ℱ ℓ ( X , Ω ) :

whilewith α ≠ β are two counterexamples not in ℱ ℓ ( X , Ω ) . The first one is because it is not a leaf-spaced decorated planar rooted forest. The second one is because the right most two edges don’t have the same edge decoration. Here x , y , z , s , t , u , v ∈ X and α , β ∈ Ω .

Remark 2.13

Typed decorated planar rooted forests are allowed different decorations for edges sharing the same parent and are used to construct free Rota-Baxter family algebras [41] and free (tri)dendriform family algebras [42].

The classical grafting operation in Eq. (3) can be adapted to a linear operator B ω + for each ω ∈ Ω :

(4) B ω + : k ℱ ℓ ( X , Ω ) → k ℱ ℓ ( X , Ω ) , T 1 … T n ↦ B ω + ( T 1 … T n ) ,

where B ω + ( T 1 … T n ) is the parallelly typed leaf-spaced decorated planar rooted tree obtained from B + ( T 1 … T n ) by decorating all the edges connecting the new root by ω . For example,Then k ℱ ℓ ( X , Ω ) is closed under the operators B ω + with ω ∈ Ω .

2.3 Free nonunitary Rota-Baxter family algebras

This subsection is devoted to construct free nonunitary Rota-Baxter family algebras in terms of parallelly typed leaf-spaced decorated planar rooted forests.

Now we are going to equip k ℱ ℓ ( X , Ω ) with a free nonunitary Rota-Baxter family algebra structure. Let us first define a multiplication

⋄ ℓ : k ℱ ℓ ( X , Ω ) ⊗ k ℱ ℓ ( X , Ω ) → k ℱ ℓ ( X , Ω ) .

Definition 2.14

Let X be a set and Ω a semigroup. Let F , F ′ ∈ ℱ ℓ ( X , Ω ) .

  1. If bre ( F ) = 1 = bre ( F ′ ) , then define

    (5) F ⋄ ℓ F ′ ≔ F F ′ ( concatenation of planar rooted trees ) , if F = • x or F ′ = • y ; B α β + ( F ¯ ⋄ ℓ F ′ ) + B α β + ( F ⋄ ℓ F ¯ ′ ) + λ B α β + ( F ¯ ⋄ ℓ F ¯ ′ ) , if F = B α + ( F ¯ ) , F ′ = B β + ( F ¯ ′ ) .

  2. In general, if bre ( F ) = b and bre ( F ′ ) = b ′ , write

F = T 1 … T b and F ′ = T 1 ′ … T b ′ ′

and define

(6) F ⋄ ℓ F ′ ≔ T 1 … T b − 1 ( T b ⋄ ℓ T 1 ′ ) T 2 ′ … T b ′ ′ .

Let us give an example.

Example 2.15

The concept of the free nonunitary Rota-Baxter family algebra is given as usual.

Definition 2.16

Let X be a set and let Ω be a semigroup. Let λ ∈ k be given. A free nonunitary Rota-Baxter family algebra of weight λ on X is a nonunitary Rota-Baxter family algebra F RBF ( X ) of weight λ together with a set map i X : X → F RBF ( X ) that satisfies the following universal property: for any nonunitary Rota-Baxter family algebra ( R , ⋄ R , ( P ω ) ω ∈ Ω ) of weight λ and any set map f : X → R , there is a unique Rota-Baxter family algebra morphism f ¯ : F RBF ( X ) → R such that f = f ¯ ∘ i X .

We are ready for our main result. Let us define the set map i X by:

i X : X → ℱ ℓ ( X , Ω ) , x ↦ • x .

Theorem 2.17

Let X be a set and Ω a semigroup. The triple ( k ℱ ℓ ( X , Ω ) , ⋄ ℓ , ( B ω + ) ω ∈ Ω ) , together with i X , is the free nonunitary Rota-Baxter family algebra on X.

Proof

We divide the proof into two steps.

Step 1: We prove that ( k ℱ ℓ ( X , Ω ) , ⋄ ℓ , ( B ω + ) ω ∈ Ω ) is a Rota-Baxter family algebra. From Eq. (5), we obtain immediately ( B ω + ) ω ∈ Ω is a Rota-Baxter family of weight λ . It remains to prove that ⋄ ℓ satisfies the associativity:

( F 1 ⋄ ℓ F 2 ) ⋄ ℓ F 3 = F 1 ⋄ ℓ ( F 2 ⋄ ℓ F 3 ) , for F 1 , F 2 , F 3 ∈ ℱ ℓ ( X , Ω ) .

We use induction on dep ( F 1 ) + dep ( F 2 ) + dep ( F 3 ) ≥ 0 . For the initial step of dep ( F 1 ) + dep ( F 2 ) + dep ( F 3 ) = 0 , we have dep ( F 1 ) = dep ( F 2 ) = dep ( F 3 ) = 0 . Let

F 1 = • x 1 , 1 • x 1 , 2 … • x 1 , n 1 , F 2 = • x 2 , 1 • x 2 , 2 … • x 2 , n 2 and F 3 = • x 3 , 1 • x 3 , 2 … • x 3 , n 3 with x i , n i ∈ X for 1 ≤ i ≤ 3 .

Then by Eq. (6), we have

( F 1 ⋄ ℓ F 2 ) ⋄ ℓ F 3 = ( • x 1 , 1 • x 1 , 2 … • x 1 , n 1 ⋄ ℓ • x 2 , 1 • x 2 , 2 … • x 2 , n 2 ) ⋄ ℓ • x 3 , 1 • x 3 , 2 … • x 3 , n 3 = • x 1 , 1 • x 1 , 2 … • x 1 , n 1 • x 2 , 1 • x 2 , 2 … • x 2 , n 2 • x 3 , 1 • x 3 , 2 … • x 3 , n 3 = • x 1 , 1 • x 1 , 2 … • x 1 , n 1 ⋄ ℓ ( • x 2 , 1 • x 2 , 2 … • x 2 , n 2 ⋄ ℓ • x 3 , 1 • x 3 , 2 … • x 3 , n 3 ) = F 1 ⋄ ℓ ( F 2 ⋄ ℓ F 3 ) .

For the inductive step of dep ( F 1 ) + dep ( F 2 ) + dep ( F 3 ) ≥ 1 , we have dep ( F i ) ≥ 1 for some i = 1 , 2 , 3 and we use induction on bre ( F 1 ) + bre ( F 2 ) + bre ( F 3 ) ≥ 3 . For the initial step of bre ( F 1 ) + bre ( F 2 ) + bre ( F 3 ) = 3 , we have bre ( F 1 ) = bre ( F 2 ) = bre ( F 3 )=1 . There are three cases to consider.

Case 1: F 1 , F 2 and F 3 are of depth greater than zero. Write F 1 = B α + ( F ¯ 1 ) , F 2 = B β + ( F ¯ 2 ) and F 3 = B γ + ( F ¯ 3 ) . Then

( F 1 ⋄ ℓ F 2 ) ⋄ ℓ F 3 = ( B α + ( F ¯ 1 ) ⋄ ℓ B β + ( F ¯ 2 ) ) ⋄ ℓ B γ + ( F ¯ 3 ) = ( B α β + ( B α + ( F ¯ 1 ) ⋄ ℓ F ¯ 2 ) + B α β + ( F ¯ 1 ⋄ ℓ B β + ( F ¯ 2 ) ) + λ B α β + ( F ¯ 1 ⋄ ℓ F ¯ 2 ) ) ⋄ ℓ B γ + ( F ¯ 3 ) = B α β + ( B α + ( F ¯ 1 ) ⋄ ℓ F ¯ 2 ) ⋄ ℓ B γ + ( F ¯ 3 ) + B α β + ( F ¯ 1 ⋄ ℓ B β + ( F ¯ 2 ) ) ⋄ ℓ B γ + ( F ¯ 3 ) + λ B α β + ( F ¯ 1 ⋄ ℓ F ¯ 2 ) ⋄ ℓ B γ + ( F ¯ 3 ) = B α β γ + ( B α β + ( B α + ( F ¯ 1 ) ⋄ ℓ F ¯ 2 ) ⋄ ℓ F ¯ 3 ) + B α β γ + ( ( B α + ( F ¯ 1 ) ⋄ ℓ F ¯ 2 ) ⋄ ℓ B γ + ( F ¯ 3 ) ) + λ B α β γ + ( ( B α + ( F ¯ 1 ) ⋄ ℓ F ¯ 2 ) ⋄ ℓ F ¯ 3 ) + B α β γ + ( B α β + ( F ¯ 1 ⋄ ℓ B β + ( F ¯ 2 ) ) ⋄ ℓ F ¯ 3 ) + B α β γ + ( ( F ¯ 1 ⋄ ℓ B β + ( F ¯ 2 ) ) ⋄ ℓ B γ + ( F ¯ 3 ) ) + λ B α β γ + ( ( F ¯ 1 ⋄ ℓ B β + ( F ¯ 2 ) ) ⋄ ℓ F ¯ 3 ) + λ B α β γ + ( B α β + ( F ¯ 1 ⋄ ℓ F ¯ 2 ) ⋄ ℓ F ¯ 3 ) + λ B α β γ + ( ( F ¯ 1 ⋄ ℓ F ¯ 2 ) ⋄ ℓ B γ + ( F ¯ 3 ) ) + λ 2 B α β γ + ( ( F ¯ 1 ⋄ ℓ F ¯ 2 ) ⋄ ℓ F ¯ 3 ) = B α β γ + ( B α β + ( B α + ( F ¯ 1 ) ⋄ ℓ F ¯ 2 ) ⋄ ℓ F ¯ 3 ) + B α β γ + ( ( B α + ( F ¯ 1 ) ⋄ ℓ F ¯ 2 ) ⋄ ℓ B γ + ( F ¯ 3 ) ) + λ B α β γ + ( ( B α + ( F ¯ 1 ) ⋄ ℓ F ¯ 2 ) ⋄ ℓ F ¯ 3 ) + B α β γ + ( B α β + ( F ¯ 1 ⋄ ℓ B β + ( F ¯ 2 ) ) ⋄ ℓ F ¯ 3 ) + B α β γ + ( F ¯ 1 ⋄ ℓ B β γ + ( B β + ( F ¯ 2 ) ⋄ ℓ ( F ¯ 3 ) ) + B α β γ + ( F ¯ 1 ⋄ ℓ B β γ + ( F ¯ 2 ⋄ ℓ B γ + ( F ¯ 3 ) ) ) + λ B α β γ + ( F ¯ 1 ⋄ ℓ B β γ + ( F ¯ 2 ⋄ ℓ F ¯ 3 ) ) + λ B α β γ + ( ( F ¯ 1 ⋄ ℓ B β + ( F ¯ 2 ) ) ⋄ ℓ F ¯ 3 ) + λ B α β γ + ( B α β + ( F ¯ 1 ⋄ ℓ F ¯ 2 ) ⋄ ℓ F ¯ 3 ) + λ B α β γ + ( ( F ¯ 1 ⋄ ℓ F ¯ 2 ) ⋄ ℓ B γ + ( F ¯ 3 ) ) + λ 2 B α β γ + ( ( F ¯ 1 ⋄ ℓ F ¯ 2 ) ⋄ ℓ F ¯ 3 ) ( by the induction on dep ( F 1 ) + dep ( F 2 ) + dep ( F 3 ) and Eq . ( 5 ) ) .

By a similar calculation, we have

F 1 ⋄ ℓ ( F 2 ⋄ ℓ F 3 ) = B α + ( F ¯ 1 ) ⋄ ℓ ( B β + ( F ¯ 2 ) ⋄ ℓ B γ + ( F ¯ 3 ) ) = B α β γ + ( B α β + ( B α + ( F ¯ 1 ) ⋄ ℓ F ¯ 2 ) ⋄ ℓ F ¯ 3 ) + B α β γ + ( B α β + ( F ¯ 1 ⋄ ℓ B β + ( F ¯ 2 ) ) ⋄ ℓ F ¯ 3 ) + λ B α β γ + ( B α β + ( F ¯ 1 ⋄ ℓ F ¯ 2 ) ⋄ ℓ F ¯ 3 ) + B α β γ + ( F ¯ 1 ⋄ ℓ B β γ + ( B β + ( F ¯ 2 ) ⋄ ℓ F ¯ 3 ) ) + λ B α β γ + ( F ¯ 1 ⋄ ℓ ( B β + ( F ¯ 2 ) ⋄ ℓ F ¯ 3 ) ) + B α β γ + ( B α + ( F ¯ 1 ) ⋄ ℓ ( F ¯ 2 ⋄ ℓ B γ + ( F ¯ 3 ) ) ) + B α β γ + ( F ¯ 1 ⋄ ℓ B β γ + ( F ¯ 2 ⋄ ℓ B γ + ( F ¯ 3 ) ) ) + λ B α β γ + ( F ¯ 1 ⋄ ℓ ( F ¯ 2 ⋄ ℓ B γ + ( F ¯ 3 ) ) ) + λ B α β γ + ( B α + ( F ¯ 1 ) ⋄ ℓ ( F ¯ 2 ⋄ ℓ F ¯ 3 ) ) + λ B α β γ + ( F ¯ 1 ⋄ ℓ B β γ + ( F ¯ 2 ⋄ ℓ F ¯ 3 ) ) + λ 2 B α β γ + ( F ¯ 1 ⋄ ℓ ( F ¯ 2 ⋄ ℓ F ¯ 3 ) ) .

The ith term in the expansion of ( F 1 ⋄ ℓ F 2 ) ⋄ ℓ F 3 matches with the σ ( i ) th term in the expansion of F 1 ⋄ ℓ ( F 2 ⋄ ℓ F 3 ) . Here σ is a permutation of order 11:

i σ ( i ) = 1 2 3 4 5 6 7 8 9 10 11 1 6 9 2 4 7 10 5 3 8 11 .

Case 2: Exactly two of F 1 , F 2 and F 3 are of depth greater than zero. There are three subcases to consider.

Subcase 2.1: F 1 = B α + ( F ¯ 1 ) , F 2 = B β + ( F ¯ 2 ) and F 3 = • x . Then

( F 1 ⋄ ℓ F 2 ) ⋄ ℓ F 3 = ( B α + ( F ¯ 1 ) ⋄ ℓ B β + ( F ¯ 2 ) ) ⋄ ℓ • x = ( B α + ( F ¯ 1 ) ⋄ ℓ B β + ( F ¯ 2 ) ) • x ( by Eq . (5) ) = B α + ( F ¯ 1 ) ⋄ ℓ ( B β + ( F ¯ 2 ) • x ) ( by Eq . (6) ) = B α + ( F ¯ 1 ) ⋄ ℓ ( B β + ( F ¯ 2 ) ⋄ ℓ • x ) ( by Eq . (5) ) = F 1 ⋄ ℓ ( F 2 ⋄ ℓ F 3 ) .

Subcase 2.2: F 1 = B α + ( F ¯ 1 ) , F 2 = • x and F 3 = B β + ( F ¯ 3 ) . Then

( F 1 ⋄ ℓ F 2 ) ⋄ ℓ F 3 = ( B α + ( F ¯ 1 ) ⋄ ℓ • x ) ⋄ ℓ B β + ( F ¯ 3 ) = ( B α + ( F ¯ 1 ) • x ) ⋄ ℓ B β + ( F ¯ 3 ) ( by Eq . (5) ) = B α + ( F ¯ 1 ) • x B β + ( F ¯ 3 ) ( by Eqs . (5) and (6)) = B α + ( F ¯ 1 ) ⋄ ℓ ( • x ⋄ ℓ B β + ( F ¯ 3 ) ) (by Eqs . (5) and (6)) = F 1 ⋄ ℓ ( F 2 ⋄ ℓ F 3 ) .

Subcase 2.3: F 1 = • x , F 2 = B α + ( F ¯ 2 ) and F 3 = B β + ( F ¯ 3 ) . This case is similar to Subcase 2.1.

Case 3: Exactly one of F 1 , F 2 and F 3 is of depth greater than zero. We have three subcases to consider.

Subcase 3.1: F 1 = B ω + ( F ¯ 1 ) , F 2 = • x and F 3 = • y . Then

( F 1 ⋄ ℓ F 2 ) ⋄ ℓ F 3 = ( B ω + ( F ¯ 1 ) ⋄ ℓ • x ) ⋄ ℓ • y = ( B ω + ( F ¯ 1 ) • x ) ⋄ ℓ • y (by Eq . (5)) = B ω + ( F ¯ 1 ) • x • y (by Eqs . (5) and (6)) = B ω + ( F ¯ 1 ) ⋄ ℓ ( • x ⋄ ℓ • y ) (by Eqs . (5) and (6)) = F 1 ⋄ ℓ ( F 2 ⋄ ℓ F 3 ) .

Subcase 3.2: F 1 = • x , F 2 = B ω + ( F ¯ 2 ) and F 3 = • y . Then

( F 1 ⋄ ℓ F 2 ) ⋄ ℓ F 3 = ( • x ⋄ ℓ B ω + ( F ¯ 2 ) ) ⋄ ℓ • y = • x B ω + ( F ¯ ) • y ( by Eqs . ( 5 ) and ( 6 ) ) = • x ⋄ ℓ ( B ω + ( F ¯ 2 ) ⋄ ℓ • y ) ( by Eqs . ( 5 ) and ( 6 ) ) = F 1 ⋄ ℓ ( F 2 ⋄ ℓ F 3 ) .

Subcase 3.3: F 1 = • x , F 2 = • y and F 3 = B ω + ( F ¯ 3 ) . This case is similar to Subcase 3.1.

For the inductive step of bre ( F 1 ) + bre ( F 2 ) + bre ( F 3 ) > 3 , we have bre ( F i ) ≥ 2 for some i = 1 , 2 , 3 . There are three cases to consider.

Case 4: bre ( F 1 ) ≥ 2 . Let F 1 = T 1 , 1 T 1 , 2 … T 1 , s 1 with T 1 , 1 , … , T 1 , s 1 ∈ T ℓ ( X , Ω ) and s 1 ≥ 2 . Then

( F 1 ⋄ ℓ F 2 ) ⋄ ℓ F 3 = ( ( T 1 , 1 T 1 , 2 … T 1 , s 1 ) ⋄ ℓ F 2 ) ⋄ ℓ F 3 = ( T 1 , 1 T 1 , 2 … T 1 , s 1 − 1 ( T 1 , s 1 ⋄ ℓ F 2 ) ) ⋄ ℓ F 3 ( by Eq . ( 6 ) ) = T 1 , 1 T 1 , 2 … T 1 , s 1 − 1 ( ( T 1 , s 1 ⋄ ℓ F 2 ) ⋄ ℓ F 3 ) ( by Eq . ( 6 ) ) = T 1 , 1 T 1 , 2 … T 1 , s 1 − 1 ( T 1 , s 1 ⋄ ℓ ( F 2 ⋄ ℓ F 3 ) ) ( by the induction on bre ( F 1 ) + bre ( F 2 ) + bre ( F 3 ) ) = ( T 1 , 1 T 1 , 2 … T 1 , s 1 ) ⋄ ℓ ( F 2 ⋄ ℓ F 3 ) ( by Eq . ( 6 ) ) = F 1 ⋄ ℓ ( F 2 ⋄ ℓ F 3 ) .

Case 5: bre ( F 2 ) ≥ 2 . Let F 2 = T 2 , 1 T 2 , 2 … T 2 , s 2 with T 2 , 1 , … , T 2 , s 2 ∈ T ℓ ( X , Ω ) and s 2 ≥ 2 . Then

( F 1 ⋄ ℓ F 2 ) ⋄ ℓ F 3 = ( F 1 ⋄ ℓ T 2 , 1 ) T 2 , 2 … T 2 , s 2 − 1 ( T 2 , s 2 ⋄ ℓ F 3 ) ( by Eq . ( 6 ) ) = F 1 ⋄ ℓ ( F 2 ⋄ ℓ F 3 ) .

Case 6: bre ( F 3 ) ≥ 2 . This case is similar to Case 4.

Step 2: We show that ( k ℱ ℓ ( X , Ω ) , ⋄ ℓ , ( B ω + ) ω ∈ Ω ) satisfies the universal property. For this, let ( R , ⋄ R , ( P ω ) ω ∈ Ω ) be a nonunitary Rota-Baxter family algebra and let f : X → R be a set map.

(Existence) We define a linear map

f ¯ : k ℱ ℓ ( X , Ω ) → R , F ↦ f ¯ ( F ) ,

by induction on dep ( F ) ≥ 0 . Consider the initial step of dep ( F ) = 0 . If bre ( F ) = 1 , then F = • x for some x ∈ X and define

(7) f ¯ ( • x ) ≔ f ¯ ∘ i X ( x ) ≔ f ( x ) .

If bre ( F ) ≥ 2 , then F = • x 1 • x 2 … • x b for some x 1 , … , x b ∈ X , and we define

(8) f ¯ ( F ) ≔ f ( x 1 ) ⋄ R ⋯ ⋄ R f ( x b ) .

Assume that f ¯ ( F ) has been defined for F ∈ ℱ ℓ ( X , Ω ) with dep ( F ) ≤ k for a k ≥ 0 and consider F ∈ ℱ ℓ ( X , Ω ) with dep ( F ) = k + 1 ≥ 1 . If bre ( F ) = 1 , we have F = B ω + ( F ¯ ) for some ω ∈ Ω and F ¯ ∈ ℱ ℓ ( X , Ω ) with dep ( F ¯ ) = k . We then define

(9) f ¯ ( F ) ≔ f ¯ ( B ω + ( F ¯ ) ) ≔ P ω ( f ¯ ( F ¯ ) ) .

If bre ( F ) > 1 , let F = T 1 … T b with T 1 , … , T b ∈ T ℓ ( X , Ω ) and define

(10) f ¯ ( F ) ≔ f ¯ ( T 1 ) ⋄ R ⋯ ⋄ R f ¯ ( T b ) ,

where each f ¯ ( T i ) , 1 ≤ i ≤ b is defined by Eq. (7) or Eq. (9).

Now we prove that f ¯ is an algebra homomorphism:

(11) f ¯ ( F 1 ⋄ ℓ F 2 ) = f ¯ ( F 1 ) ⋄ R f ¯ ( F 2 ) for F 1 , F 2 ∈ ℱ ℓ ( X , Ω ) ,

by induction on the sum of depth dep ( F 1 ) + dep ( F 2 ) ≥ 0 . Write

F 1 = T 1 , 1 … T 1 , s and F 2 = T 2 , 1 … T 2 , t .

If dep ( F 1 ) + dep ( F 2 ) = 0 , then

F 1 = • x 1 , 1 … • x 1 , s and F 2 = • x 2 , 1 … • x 2 , t ,

and so

F 1 ⋄ ℓ F 2 = • x 1 , 1 … • x 1 , s • x 2 , 1 … • x 2 , t .

It follows from Eq. (8) that

f ¯ ( F 1 ⋄ ℓ F 2 ) = f ¯ ( • x 1 , 1 … • x 1 , s • x 2 , 1 … • x 2 , t ) = f ( x 1 , 1 ) ⋄ R ⋯ ⋄ R f ( x 1 , s ) ⋄ R f ( x 2 , 1 ) ⋄ R ⋯ ⋄ R f ( x 2 , t ) = ( f ( x 1 , 1 ) ⋄ R ⋯ ⋄ R f ( x 1 , s ) ) ⋄ R ( f ( x 2 , 1 ) ⋄ R ⋯ ⋄ R f ( x 2 , t ) ) = f ¯ ( F 1 ) ⋄ R f ¯ ( F 2 ) .

For the inductive step, assume that Eq. (11) holds when dep ( F 1 ) + dep ( F 2 ) ≤ k for a given k ≥ 0 and consider the case of dep ( F 1 ) + dep ( F 2 ) = k + 1 . We reduce to the induction on bre ( F 1 ) + bre ( F 2 ) ≥ 2 . For the initial step of bre ( F 1 ) + bre ( F 2 ) = 2 , we have bre ( F 1 ) = 1 = bre ( F 2 ) and F 1 = B α + ( F ¯ 1 ) , F 2 = B β + ( F ¯ 2 ) .

f ¯ ( F 1 ⋄ ℓ F 2 ) = f ¯ ( B α + ( F ¯ 1 ) ⋄ ℓ B β + ( F ¯ 2 ) ) = f ¯ ( B α β + ( B α + ( F ¯ 1 ) ⋄ ℓ F ¯ 2 ) + B α β + ( F ¯ 1 ⋄ ℓ B β + ( F ¯ 2 ) ) + λ B α β + ( F ¯ 1 ⋄ ℓ F ¯ 2 ) ) ( by Eq . ( 5 ) ) = f ¯ ( B α β + ( B α + ( F ¯ 1 ) ⋄ ℓ F ¯ 2 ) ) + f ¯ ( B α β + ( F ¯ 1 ⋄ ℓ B β + ( F ¯ 2 ) ) ) + f ¯ ( λ B α β + ( F ¯ 1 ⋄ ℓ F ¯ 2 ) ) = P α β ( f ¯ ( B α + ( F ¯ 1 ) ⋄ ℓ F ¯ 2 ) ) + P α β ( f ¯ ( F ¯ 1 ⋄ ℓ B β + ( F ¯ 2 ) ) ) + λ P α β ( f ¯ ( F ¯ 1 ⋄ ℓ F ¯ 2 ) ) ( by Eq . ( 9 ) ) = P α β ( f ¯ ( B α + ( F ¯ 1 ) ) ⋄ R f ¯ ( F ¯ 2 ) ) + P α β ( f ¯ ( F ¯ 1 ) ⋄ R f ¯ ( B β + ( F ¯ 2 ) ) ) + λ P α β ( f ¯ ( F ¯ 1 ) ⋄ R f ¯ ( F ¯ 2 ) ) ( by the induction on dep ( F 1 ) + dep ( F 2 ) ) = P α β ( P α ( f ¯ ( F ¯ 1 ) ) ⋄ R f ¯ ( F ¯ 2 ) ) + P α β ( f ¯ ( F ¯ 1 ) ⋄ R P β ( f ¯ ( F ¯ 2 ) ) ) + λ P α β ( f ¯ ( F ¯ 1 ) ⋄ R f ¯ ( F ¯ 2 ) ) ( by Eq . ( 9 ) ) = P α ( f ¯ ( F ¯ 1 ) ) ⋄ R P β ( f ¯ ( F ¯ 2 ) ) ( by Eq . ( 2 ) ) = f ¯ ( B α + ( F ¯ 1 ) ) ⋄ R f ¯ ( B β + ( F ¯ 2 ) ) ( by Eq . ( 9 ) ) = f ¯ ( F 1 ) ⋄ R f ¯ ( F 2 ) .

For the induction step of bre ( F 1 ) + bre ( F 2 ) ≥ 3 , we write

F 1 = T 1 , 1 … T 1 , s and F 2 = T 2 , 1 … T 2 , t .

Then

f ¯ ( F 1 ⋄ ℓ F 2 ) = f ¯ ( ( T 1 , 1 … T 1 , s ) ⋄ ℓ ( T 2 , 1 … T 2 , t ) ) = f ¯ ( T 1 , 1 … T 1 , s − 1 ( T 1 , s ⋄ ℓ T 2 , 1 ) T 2 , 2 … T 2 , t ) ( by Eq . ( 6 ) ) = f ¯ ( T 1 , 1 ) ⋄ R ⋯ ⋄ R f ¯ ( T 1 , s − 1 ) ⋄ R f ¯ ( T 1 , s ⋄ ℓ T 2 , 1 ) ⋄ R f ¯ ( T 2 , 2 ) ⋄ R ⋯ ⋄ R f ¯ ( T 2 , t ) ( by Eq . ( 10 ) ) = f ¯ ( T 1 , 1 ) ⋄ R ⋯ ⋄ R f ¯ ( T 1 , s − 1 ) ⋄ R f ¯ ( T 1 , s ) ⋄ R f ¯ ( T 2 , 1 ) ⋄ R f ¯ ( T 2 , 2 ) ⋄ R ⋯ ⋄ R f ¯ ( T 2 , t ) ( by the induction hypothesis on bre ( F 1 ) + bre ( F 2 ) ) = f ¯ ( T 1 , 1 … T 1 , s ) ⋄ R f ¯ ( T 2 , 1 … f ¯ ( T 2 , t ) ) ( by Eq . ( 10 ) ) = f ¯ ( F 1 ) ⋄ R f ¯ ( F 2 ) .

Thus, f ¯ is an algebra homomorphism. Further by Eq. (9),

f ¯ ∘ B ω + = P ω ∘ f ¯ for ω ∈ Ω ,

whence f ¯ is a Rota-Baxter family algebra morphism. Finally, it follows from Eq. (7) that f ¯ ∘ i X = f . This completes the proof of existence.

(Uniqueness) Suppose that such f ¯ exists. Then, since f ¯ is a Rota-Baxter family algebra morphism such that f ¯ ∘ i X = f , f ¯ ( F ) must be of the forms in Eqs. (7)–(10) for F ∈ ℱ ℓ ( X , Ω ) .□

If Ω is a trivial semigroup, that is, Ω has only one element, then all edges of an element F in ℱ ℓ ( X , Ω ) are of the same decoration. So we can view F has no edge decoration. In this case, denote ℱ ℓ ( X ) ≔ ℱ ℓ ( X , Ω ) and notice that Rota-Baxter family Eq. (2) reduces to Rota-Baxter Eq. (1).

Corollary 2.18

Let X be a set. Then the triple ( k ℱ ℓ ( X ) , ⋄ ℓ , B + ) , together with the i X , is the free nonunitary Rota-Baxter algebra of weight λ on X.

Proof

It follows from Theorem 2.17 by taking Ω to be a trivial semigroup.□

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11771191 and 11861051).

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Received: 2020-01-21
Revised: 2020-06-23
Accepted: 2020-07-01
Published Online: 2020-11-05

© 2020 Jinwei Wang et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
https://doi.org/10.1515/math-2020-0059
Keywords for this article
Rota-Baxter algebras; Rota-Baxter family algebras; typed decorated rooted forests; leaf-spaced; parallelly typed leaf-spaced decorated
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  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
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  150. Special Issue on Evolution Equations, Theory and Applications
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  152. Three classes of decomposable distributions
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