Structural Interoperability Between π-Graded Hom-Type Algebras and Rota-Baxter Operators

1. Introduction

Hom-type algebras, proposed by Yau, are important noncommutative algebras that generalize classical algebras via twisting maps [12]. Their core idea is to replace the identity map with a compatible linear endomorphism. The construction of twisted tensor products for Hom-type algebras has broadened their applications in mathematical physics, representation theory and other fields [10]. Dendriform algebras and tridendriform algebras were introduced by Loday and Vallette [4,5]. By decomposing the associative multiplication, they reveal the internal rules of algebras and play significant roles in algebraic K-theory, operad theory and combinatorics [6].

Originating from problems in analysis and probability theory, Rota-Baxter operators were formally defined and systematically developed by Rota and Baxter [3,9]. They have become a vital bridge connecting multiple disciplines. The relations between Rota-Baxter operators and dendriform algebras have been thoroughly investigated [1], and relevant theories are widely applied in quantum field theory and the characterization of Hopf algebras. In recent years, the interdisciplinary research on Rota-Baxter operators and Hom-type algebras has gradually emerged, which gives rise to new structures such as Rota-Baxter Hom-algebras [2].

The $\pi$-graded structure is a classical tool in algebra research. It decomposes algebraic objects into direct sums of subspaces indexed by a group $\pi$ while preserving the homogeneity of operations, and is widely used in algebraic geometry, topology and homological algebra [11]. Combining $\pi$-gradation with Rota-Baxter algebras, dendriform algebras and tridendriform algebras yields new structures including $\pi$-graded Rota-Baxter algebras, $\pi$-graded dendriform algebras and $\pi$-graded tridendriform algebras [7], which inherit the essential properties of the two kinds of fundamental algebraic structures.

This paper focuses on the structural interoperability of the above three types of algebras. In Section 2, we present the definitions and constructions of $\pi$-graded Hom-tridendriform algebras, $\pi$-graded Hom-dendriform algebras and $\pi$-graded Hom-algebras, and establish the derivation relations among them. In Section 3, we give the definition, examples and several construction methods of Rota-Baxter operators on $\pi$-graded Hom-algebras, and prove that such operators can induce the three classes of $\pi$-graded Hom-type algebras proposed in Section 2. In Section 4, we define a generalized structure of $\pi$-graded Rota-Baxter Hom-algebras, namely the $\pi$-graded Rota-Baxter Hom-system, and prove the structural equivalence between this system and $\pi$-graded Hom-dendriform algebras.

Throughout this paper, $\pi$ always denotes a group used as the index set, and $\mathrm{id}$ stands for the identity map on a linear space.

2. $\pi$-Graded Hom-Tridendriform Algebras, $\pi$-Graded Hom-Dendriform Algebras, $\pi$-Graded Hom-Algebras and Structural Intercommunication

Definition 1.

A π-graded Hom-tridendriform algebra $({\left\{A_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}\right\}}_{p,q\in \pi },$${\left\{{•}_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi })$ consists of a family of linear spaces ${\left\{A_p\right\}}_{p\in \pi }$ and four families of linear maps ${\{{\succ }_{p,q},{\prec }_{p,q},{•}_{p,q}:A_p\otimes A_q⟶A_{pq}\}}_{p,q\in \pi },{\{{\alpha }_p:A_p⟶A_p\}}_{p\in \pi }$ such that for all $p,q,r\in \pi ,a\in A_p,b\in A_q,d\in A_r,$ the following identities hold:

$$\begin{array}{cc}\hfill \left(a{\prec }_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)& ={\alpha }_p\left(a\right){\prec }_{p,qr}\left(b{\prec }_{q,r}d+b{\succ }_{q,r}d+b{•}_{q,r}d\right),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \left(a{\succ }_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)& ={\alpha }_p\left(a\right){\succ }_{p,qr}\left(b{\prec }_{q,r}d\right),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\alpha }_p\left(a\right){\succ }_{p,qr}\left(b{\succ }_{q,r}d\right)& =\left(a{\prec }_{p,q}b+a{\succ }_{p,q}b+a{•}_{p,q}b\right){\succ }_{pq,r}{\alpha }_r\left(d\right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \left(a{\prec }_{p,q}b\right){•}_{pq,r}{\alpha }_r\left(d\right)& ={\alpha }_p\left(a\right){•}_{p,qr}\left(b{\succ }_{q,r}d\right),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \left(a{\succ }_{p,q}b\right){•}_{pq,r}{\alpha }_r\left(d\right)& ={\alpha }_p\left(a\right){\succ }_{p,qr}\left(b{•}_{q,r}d\right),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \left(a{•}_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)& ={\alpha }_p\left(a\right){•}_{p,qr}\left(b{\prec }_{q,r}d\right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \left(a{•}_{p,q}b\right){•}_{pq,r}{\alpha }_r\left(d\right)& ={\alpha }_p\left(a\right){•}_{p,qr}\left(b{•}_{q,r}d\right).\hfill \end{array}$$

(7)

A homomorphism from $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}^A\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}^A\right\}}_{p,q\in \pi },{\left\{{•}_{p,q}^A\right\}}_{p,q\in \pi },{\left\{{\alpha }_p^A\right\}}_{p\in \pi }\right)$ to $\left({\left\{B_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}^B\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}^B\right\}}_{p,q\in \pi },{\left\{{•}_{p,q}^B\right\}}_{p,q\in \pi },{\left\{{\alpha }_p^B\right\}}_{p\in \pi }\right)$ is a family of linear operators ${\{f_p:A_p⟶B_p\}}_{p\in \pi }$ such that for all $p,q\in \pi$, we have $f_{pq}{\prec }_{p,q}^A={\prec }_{p,q}^B\left(f_p\otimes f_q\right),f_{pq}{\succ }_{p,q}^A={\succ }_{p,q}^B\left(f_p\otimes f_q\right),f_{pq}{•}_{p,q}^A={•}_{p,q}^B\left(f_p\otimes f_q\right),{\alpha }_p^B{f}_p=f_p{\alpha }_p^A.$

π-graded triple dendriform algebras and their homomorphisms [7] are special cases of π-graded Hom-tridendriform algebras and their homomorphisms (with ${\alpha }_p={\mathrm{id}}_p$ for all $p\in \pi$); they also induce π-graded Hom-tridendriform algebras and their homomorphisms.

Theorem 1.

If ${\left\{{\alpha }_p\right\}}_{p\in \pi }$ is an endomorphism of the π-graded triple dendriform algebra $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}\right\}}_{p,q\in \pi },{\left\{{•}_{p,q}\right\}}_{p,q\in \pi }\right)$, then $\left({\left\{A_p\right\}}_{p\in \pi },{\{{\prec }_{p,q}^{\ast }={\alpha }_{pq}{\prec }_{p,q}\}}_{p,q\in \pi },{\{{\succ }_{p,q}^{\ast }={\alpha }_{pq}{\succ }_{p,q}\}}_{p,q\in \pi },{\{{•}_{p,q}^{\ast }={\alpha }_{pq}{•}_{p,q}\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a π-graded Hom-tridendriform algebra, called the Yau twist of the former.

If ${\left\{f_p\right\}}_{p\in \pi }$ is a π-graded triple dendriform algebra homomorphism from $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}^A\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}^A\right\}}_{p,q\in \pi },{\left\{{•}_{p,q}^A\right\}}_{p,q\in \pi }\right)$ to $\left({\left\{B_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}^B\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}^B\right\}}_{p,q\in \pi },{\left\{{•}_{p,q}^B\right\}}_{p,q\in \pi }\right)$ and satisfies ${\alpha }_p^B{f}_p=f_p{\alpha }_p^A$ for all $p\in \pi$, then ${\left\{f_p\right\}}_{p\in \pi }$ is a π-graded Hom-tridendriform algebra homomorphism between the two Yau twists $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}^{A\ast }\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}^{A\ast }\right\}}_{p,q\in \pi },{\left\{{•}_{p,q}^{A\ast }\right\}}_{p,q\in \pi },{\left\{{\alpha }_p^A\right\}}_{p\in \pi }\right)$ and $\left({\left\{B_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}^{B\ast }\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}^{B\ast }\right\}}_{p,q\in \pi },{\left\{{•}_{p,q}^{B\ast }\right\}}_{p,q\in \pi },{\left\{{\alpha }_p^B\right\}}_{p\in \pi }\right)$.

Proof. 

For any $p,q,r\in \pi ,a\in A_p,b\in A_q,d\in A_r$, we have

$$\begin{array}{cc}\hfill \left(a{\prec }_{p,q}^{\ast }b\right){\prec }_{pq,r}^{\ast }{\alpha }_r\left(d\right)& ={\alpha }_{pq}\left(a{\prec }_{p,q}b\right){\prec }_{pq,r}^{\ast }{\alpha }_r\left(d\right)\hfill \\ \hfill & ={\alpha }_{pqr}\left({\alpha }_{pq}\left(a{\prec }_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)\right)={\alpha }_{pqr}^2\left(\left(a{\prec }_{p,q}b\right){\prec }_{pq,r}d\right)\hfill \\ \hfill & ={\alpha }_{pqr}^2\left(a{\prec }_{p,qr}\left(b{\prec }_{q,r}d+b{\succ }_{q,r}d+b{•}_{q,r}d\right)\right)\hfill \\ \hfill & ={\alpha }_{pqr}^2\left(a{\prec }_{p,qr}\left(b{\prec }_{q,r}d\right)\right)+{\alpha }_{pqr}^2\left(a{\prec }_{p,qr}\left(b{\succ }_{q,r}d\right)\right)\hfill \\ \hfill & +{\alpha }_{pqr}^2\left(a{\prec }_{p,qr}\left(b{•}_{q,r}d\right)\right)\hfill \\ \hfill & ={\alpha }_{pqr}\left({\alpha }_p\left(a\right){\prec }_{p,qr}{\alpha }_{qr}\left(b{\prec }_{q,r}d\right)\right)+{\alpha }_{pqr}\left({\alpha }_p\left(a\right){\prec }_{p,qr}{\alpha }_{qr}\left(b{\succ }_{q,r}d\right)\right)\hfill \\ \hfill & +{\alpha }_{pqr}\left({\alpha }_p\left(a\right){\prec }_{p,qr}{\alpha }_{qr}\left(b{•}_{q,r}d\right)\right)\hfill \\ \hfill & ={\alpha }_p\left(a\right){\prec }_{p,qr}^{\ast }\left(b{\prec }_{q,r}^{\ast }d\right)+{\alpha }_p\left(a\right){\prec }_{p,qr}^{\ast }\left(b{\succ }_{q,r}^{\ast }d\right)\hfill \\ \hfill & +{\alpha }_p\left(a\right){\prec }_{p,qr}^{\ast }\left(b{•}_{q,r}^{\ast }d\right)\hfill \\ \hfill & ={\alpha }_p\left(a\right){\prec }_{p,qr}^{\ast }\left(b{\prec }_{q,r}^{\ast }d+b{\succ }_{q,r}^{\ast }d+b{•}_{q,r}^{\ast }d\right).\hfill \end{array}$$

The other defining identities of the $\pi$-graded Hom-tridendriform algebra can be verified similarly. Moreover,

$$f_{pq}{\prec }_{p,q}^{A^{\ast }}=f_{pq}{\alpha }_{pq}{\prec }_{p,q}^A={\alpha }_{pq}^B{f}_{pq}{\prec }_{p,q}^A={\alpha }_{pq}^B{\prec }_{p,q}^B\left(f_p\otimes f_q\right)={\prec }_{p,q}^{B^{\ast }}\left(f_p\otimes f_q\right).$$

The other defining identities of the $\pi$-graded Hom-tridendriform algebra homomorphi

sm can be verified similarly. □

Definition 2.

A π-graded Hom-dendriform algebra $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ consists of a family of linear spaces ${\left\{A_p\right\}}_{p\in \pi }$ and three families of linear maps ${\{{\prec }_{p,q},{\succ }_{p,q}:A_p\otimes A_q⟶A_{pq}\}}_{p,q\in \pi },{\{{\alpha }_p:A_p⟶A_p\}}_{p,q\in \pi }$ such that for all $p,q,r\in \pi ,a\in A_p,b\in A_q,d\in A_r,$ the following identities hold:

$$\left\{\begin{array}{c}\left(a{\prec }_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)={\alpha }_p\left(a\right){\prec }_{p,qr}\left(b{\prec }_{q,r}d\right)+{\alpha }_p\left(a\right){\prec }_{p,qr}\left(b{\succ }_{q,r}d\right),\hfill \\ \left(a{\succ }_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)={\alpha }_p\left(a\right){\succ }_{p,qr}\left(b{\prec }_{q,r}d\right),\hfill \\ {\alpha }_p\left(a\right){\succ }_{p,qr}\left(b{\succ }_{q,r}d\right)=\left(a{\succ }_{p,q}b\right){\succ }_{pq,r}{\alpha }_r\left(d\right)+\left(a{\prec }_{p,q}b\right){\succ }_{pq,r}{\alpha }_r\left(d\right).\hfill \end{array}\right.$$

A π-graded Hom-dendriform algebra is a special case of a π-graded Hom-tridendriform algebra (with $a{•}_{p,q}b=0$ for all $p,q\in \pi ,a\in A_p,b\in A_q$). Therefore the conclusions in Theorem 1 also hold for π-graded Hom-dendriform algebras, i.e., they can be derived from π-graded dendriform algebras and their endomorphisms [7]. Conversely, a π-graded Hom-tridendriform algebra induces a π-graded Hom-dendriform algebra.

Theorem 2.

Let $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}\right\}}_{p,q\in \pi },{\left\{{•}_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ be a π-graded Hom-tridendriform algebra. For any $p,q\in \pi ,a\in A_p,b\in A_q$, define $a{\prec }_{p,q}^{\prime }b=a{\prec }_{p,q}b+a{•}_{p,q}b,a{\succ }_{p,q}^{\prime }b=a{\succ }_{p,q}b.$ Then $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}^{\prime }\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}^{\prime }\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a π-graded Hom-dendriform algebra.

Proof. 

For any $p,q,r\in \pi ,a\in A_p,b\in A_q,d\in A_r$, we have

$$\begin{array}{cc}\hfill \left(a{\prec }_{p,q}^{\prime }b\right){\prec }_{pq,r}^{\prime }{\alpha }_r\left(d\right)& =\left(a{\prec }_{p,q}b+a{•}_{p,q}b\right){\prec }_{pq,r}^{\prime }{\alpha }_r\left(d\right),\hfill \\ \hfill & =\left(a{\prec }_{p,q}b\right){\prec }_{pq,r}^{\prime }{\alpha }_r\left(d\right)+\left(a{•}_{p,q}b\right){\prec }_{pq,r}^{\prime }{\alpha }_r\left(d\right),\hfill \\ \hfill & =\left(a{\prec }_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)+\left(a{\prec }_{p,q}b\right){•}_{pq,r}{\alpha }_r\left(d\right)\hfill \\ \hfill & +\left(a{•}_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)+\left(a{•}_{p,q}b\right){•}_{pq,r}{\alpha }_r\left(d\right),\hfill \\ \hfill & ={\alpha }_p\left(a\right){\prec }_{p,qr}\left(b{\prec }_{q,r}d\right)+{\alpha }_p\left(a\right){\prec }_{p,qr}\left(b{\succ }_{q,r}d\right)\hfill \\ \hfill & +{\alpha }_p\left(a\right){\prec }_{p,qr}\left(b{•}_{q,r}d\right)+{\alpha }_p\left(a\right){•}_{p,qr}\left(b{\succ }_{q,r}d\right)\hfill \\ \hfill & +{\alpha }_p\left(a\right){•}_{p,qr}\left(b{\prec }_{q,r}d\right)+{\alpha }_p\left(a\right){•}_{p,qr}\left(b{•}_{q,r}d\right),\hfill \\ \hfill & ={\alpha }_p\left(a\right){\prec }_{p,qr}^{\prime }\left(b{\prec }_{q,r}d\right)+{\alpha }_p\left(a\right){\prec }_{p,qr}^{\prime }\left(b{•}_{q,r}d\right)\hfill \\ \hfill & +{\alpha }_p\left(a\right){\prec }_{p,qr}^{\prime }\left(b{\succ }_{q,r}d\right),\hfill \\ \hfill & ={\alpha }_p\left(a\right){\prec }_{p,qr}^{\prime }\left(b{\prec }_{q,r}^{\prime }d\right)+{\alpha }_p\left(a\right){\prec }_{p,qr}^{\prime }\left(b{\succ }_{q,r}^{\prime }d\right).\hfill \end{array}$$

The other defining identities of the $\pi$-graded Hom-dendriform algebra can be verified similarly. □

Definition 3.

A π-graded Hom-algebra $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ consists of a family of linear spaces ${\left\{A_p\right\}}_{p\in \pi }$ and two families of linear maps ${\{m_{p,q}:A_p\otimes A_q⟶A_{pq}\}}_{p,q\in \pi },{\{{\alpha }_p:A_p⟶A_p\}}_{p,q\in \pi }$ such that for all $p,q,r\in \pi ,a\in A_p,b\in A_q,$$d\in A_r,$ the identity ${\alpha }_p\left(a\right)\left(bd\right)=\left(ab\right){\alpha }_r\left(d\right)$ holds.

A π-graded Hom-algebra homomorphism from $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}^A\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ to $({\left\{B_p\right\}}_{p\in \pi },$${\left\{m_{p,q}^B\right\}}_{p,q\in \pi },{\left\{{\beta }_p\right\}}_{p\in \pi })$ is a family of linear operators ${\{f_p:A_p⟶B_p\}}_{p\in \pi }$ such that for all $p,q\in \pi$, we have $f_{pq}{m}_{p,q}^A=m_{p,q}^B(f_p\otimes f_q),f_p{\alpha }_p={\beta }_p{f}_p.$

Definition 4.

A π-graded Hom-subalgebra of a π-graded Hom-algebra $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ $\left({\left\{B_p\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ satisfies:

1. 

For all $p\in \pi$, $B_p$ is a subspace of $A_p$;

2. 

For all $p,q\in \pi$, $a\in B_p,b\in B_q$, we have $m_{p,q}(B_p\otimes B_q)\subseteq B_{pq}$;

3. 

For all $p\in \pi$, we have ${\alpha }_p\left(B_p\right)\subseteq B_p$.

Definition 5.

Let $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ be a π-graded Hom-algebra. If for all $p\in \pi$, $a\in A_p$, the condition “for all $q\in \pi$, $b\in A_q$, we have $ab=0$ (or $ba=0$)” implies $a=0$, then $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is called a non-degenerate π-graded Hom-algebra.

Theorem 3.

If ${\left\{{\alpha }_p\right\}}_{p\in \pi }$ is a π-algebra endomorphism on $({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi })$, then $\left({\left\{A_p\right\}}_{p\in \pi },{\{m_{p,q}^{\ast }={\alpha }_{pq}{m}_{p,q}\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a π-graded Hom-algebra.

If ${\{f_p:A_p\to B_p\}}_{p\in \pi }$ is a π-algebra homomorphism from $({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}^A\right\}}_{p,q\in \pi })$ to $({\left\{B_p\right\}}_{p\in \pi },$${\left\{m_{p,q}^B\right\}}_{p,q\in \pi })$ and satisfies ${\alpha }_p^B{f}_p=f_p{\alpha }_p^A$ for all $p\in \pi$, then ${\left\{f_p\right\}}_{p\in \pi }$ is a π-graded Hom-algebra homomorphism between the two Yau twists $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}^{A^{\ast }}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p^A\right\}}_{p\in \pi }\right)$ and $\left({\left\{B_p\right\}}_{p\in \pi },{\left\{m_{p,q}^{B^{\ast }}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p^B\right\}}_{p\in \pi }\right).$

Proof. 

For any $p,q,r\in \pi$, $a\in A_p$, $b\in A_q$, $d\in A_r$, we have (denoting $m_{p,q}^{\ast }(a\otimes b)=a{\ast }_{p,q}b$)

$$\begin{array}{cc}\hfill {\alpha }_p\left(a\right){\ast }_{p,qr}\left(b{\ast }_{q,r}d\right)& ={\alpha }_p\left(a\right){\ast }_{p,qr}{\alpha }_{qr}\left(bd\right)={\alpha }_{pqr}\left({\alpha }_p\left(a\right){\alpha }_{qr}\left(bd\right)\right)\hfill \\ \hfill & ={\alpha }_{pqr}^2\left(a\left(bd\right)\right)={\alpha }_{pqr}^2\left(\left(ab\right)d\right)={\alpha }_{pqr}\left({\alpha }_{pq}\left(ab\right){\alpha }_r\left(d\right)\right)\hfill \\ \hfill & ={\alpha }_{pq}\left(ab\right){\ast }_{pq,r}{\alpha }_r\left(d\right)=\left(a{\ast }_{p,q}b\right){\ast }_{pq,r}{\alpha }_r\left(d\right).\hfill \end{array}$$

For any $p,q\in \pi$, $a\in A_p$, $b\in A_q$, we have

$$\begin{array}{cc}\hfill f_{pq}{m}_{p,q}^{A^{\ast }}(a\otimes b)& =f_{pq}{\alpha }_{pq}^A\left(ab\right)={\alpha }_{pq}^B{f}_{pq}\left(ab\right)={\alpha }_{pq}^B\left(f_p\left(a\right)f_q\left(b\right)\right)=m_{p,q}^{B^{\ast }}\left(f_p\otimes f_q\right)(a\otimes b).\hfill \end{array}$$

□

Theorem 4.

Let $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ be a π-graded Hom-dendriform algebra. Define a family of linear operators ${\{{\ast }_{p,q}:A_p\otimes A_q⟶A_{pq},a\otimes b⟼a{\ast }_{p,q}b=a{\prec }_{p,q}b+a{\succ }_{p,q}b\}}_{p,q\in \pi }.$ Then $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\ast }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a π-graded Hom-algebra.

Proof. 

For any $p,q,r\in \pi ,a\in A_p,b\in A_q,d\in A_r$, we have

$$\begin{array}{cc}\hfill {\alpha }_p\left(a\right){\ast }_{p,qr}\left(b{\ast }_{q,r}d\right)& ={\alpha }_p\left(a\right){\ast }_{p,qr}\left(b{\prec }_{q,r}d+b{\succ }_{q,r}d\right),\hfill \\ \hfill & ={\alpha }_p\left(a\right){\prec }_{p,qr}\left(b{\prec }_{q,r}d\right)+{\alpha }_p\left(a\right){\prec }_{p,qr}\left(b{\succ }_{q,r}d\right)\hfill \\ \hfill & +{\alpha }_p\left(a\right){\succ }_{p,qr}\left(b{\prec }_{q,r}d\right)+{\alpha }_p\left(a\right){\succ }_{p,qr}\left(b{\succ }_{q,r}d\right),\hfill \\ \hfill & ={\alpha }_p\left(a\right){\prec }_{p,qr}\left(b{\prec }_{q,r}d\right)+{\alpha }_p\left(a\right){\prec }_{p,qr}\left(b{\succ }_{q,r}d\right)\hfill \\ \hfill & +\left(a{\succ }_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)+\left(a{\succ }_{p,q}b\right){\succ }_{pq,r}{\alpha }_r\left(d\right)\hfill \\ \hfill & +\left(a{\prec }_{p,q}b\right){\succ }_{pq,r}{\alpha }_r\left(d\right),\hfill \\ \hfill & =\left(a{\prec }_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)+\left(a{\succ }_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)\hfill \\ \hfill & +\left(a{\succ }_{p,q}b\right){\succ }_{pq,r}{\alpha }_r\left(d\right)+\left(a{\prec }_{p,q}b\right){\succ }_{pq,r}{\alpha }_r\left(d\right),\hfill \\ \hfill & =\left(a{\prec }_{p,q}b+a{\succ }_{p,q}b\right){\ast }_{pq,r}{\alpha }_r\left(d\right)=\left(a{\ast }_{p,q}b\right){\ast }_{pq,r}{\alpha }_r\left(d\right).\hfill \end{array}$$

□

Combining Theorem 2, we obtain:

Corollary 1.

If ${\left\{{\alpha }_p\right\}}_{p\in \pi }$ is a π-graded triple dendriform algebra endomorphism on $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}\right\}}_{p,q\in \pi },{\left\{{•}_{p,q}\right\}}_{p,q\in \pi }\right)$, then we obtain:

1. 

a π-graded Hom-dendriform algebra $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}^{\prime }\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}^{\prime }\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$, where for any $p,q\in \pi$, $a\in A_p$, $b\in A_q$, $a{\prec }_{p,q}^{\prime }b={\alpha }_{pq}\left(a{\prec }_{p,q}b\right)+{\alpha }_{pq}\left(a{•}_{p,q}b\right),a{\succ }_{p,q}^{\prime }b={\alpha }_{pq}\left(a{\succ }_{p,q}b\right);$

2. 

a π-graded Hom-algebra $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\ast }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$, where for any $p,q\in \pi$, $a\in A_p$, $b\in A_q$, $a{\ast }_{p,q}b={\alpha }_{pq}\left(a{\prec }_{p,q}b+a{\succ }_{p,q}b+a{•}_{p,q}b\right).$

3. Construction of Rota-Baxter Operators on $\pi$-Graded Hom-Algebras

Definition 6.

A Rota-Baxter operator of weight λ on a π-graded Hom-algebra $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a family of linear operators ${\{R_p:A_p⟶A_p\}}_{p\in \pi }$ such that for all $p,q\in \pi$, $a\in A_p$, $b\in A_q$, we have

$$R_p\left(a\right)R_q\left(b\right)=R_{pq}\left(R_p\left(a\right)b+a{R}_q\left(b\right)+\lambda ab\right).$$

In this case, $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is called a π-graded Rota-Baxter Hom-algebra of weight λ. (Unless otherwise specified, the weight of all π-graded Rota-Baxter Hom-algebras below is taken to be λ.)

A π-graded Rota-Baxter Hom-algebra homomorphism from $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}^A\right\}}_{p,q\in \pi },{\left\{R_p^A\right\}}_{p\in \pi },{\left\{{\alpha }_p^A\right\}}_{p\in \pi }\right)$ to $\left({\left\{B_p\right\}}_{p\in \pi },{\left\{m_{p,q}^B\right\}}_{p,q\in \pi },{\left\{R_p^B\right\}}_{p\in \pi },{\left\{{\alpha }_p^B\right\}}_{p\in \pi }\right)$ is a π-graded Hom-algebra homomorphism ${\{f_p:A_p⟶B_p\}}_{p\in \pi }$ such that for all $p\in \pi$ we have $f_p{R}_p^A=R_p^B{f}_p.$

Example 1.

If ${\left\{R_p\right\}}_{p\in \pi }$ is a Rota-Baxter operator of weight ${\lambda }_1$ on a π-graded Hom-algebra, then

1. 

${\{-{\lambda }_1{\mathrm{id}}_p-R_p\}}_{p\in \pi }$ is a Rota-Baxter operator of weight ${\lambda }_1$;

2. 

${\left\{{\lambda }_2{R}_p\right\}}_{p\in \pi }$ is a Rota-Baxter operator of weight ${\lambda }_1{\lambda }_2$.

Below we demonstrate the interconnections between Rota-Baxter operators on $\pi$-graded Hom-algebras and the several types of structures in Section 2:

Theorem 5.

If $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a π-graded Rota-Baxter Hom-algebra and ${\alpha }_p{R}_p=R_p{\alpha }_p$ for all $p\in \pi$, then we obtain

1. 

a π-graded Hom-subalgebra $\left({\left\{R_p\left(A_p\right)\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$;

2. 

a π-graded Hom-tridendriform algebra $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}\right\}}_{p,q\in \pi },{\left\{{•}_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$: where for all $p,q\in \pi$, $a\in A_p$, $b\in A_q$, $a{\prec }_{p,q}b=a{R}_q\left(b\right)$, $a{\succ }_{p,q}b=R_p\left(a\right)b$, $a{•}_{p,q}b=\lambda ab;$

3. 

a π-graded Hom-dendriform algebra $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$: where for all $p,q\in \pi$, $a\in A_p$, $b\in A_q$, $a{\prec }_{p,q}b=a{R}_q\left(b\right)+\lambda ab,a{\succ }_{p,q}b=R_p\left(a\right)b;$

4. 

a π-graded Hom-algebra $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\ast }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$: where for all $p,q\in \pi$, $a\in A_p$, $b\in A_q$, $a\ast b=R_p\left(a\right)b+a{R}_q\left(b\right)+\lambda ab.$

Proof. (1) For all $p,q\in \pi$, $R_p\left(a\right)\in R_p\left(A_p\right)$, $R_q\left(b\right)\in R_q\left(A_q\right)$, we have

$$R_p\left(a\right)R_q\left(b\right)=R_{pq}\left(R_p\left(a\right)b+a{R}_q\left(b\right)+\lambda ab\right)\in R_{pq}\left(A_{pq}\right).$$

(2) For any $p,q,r\in \pi$, $a\in A_p$, $b\in A_q$, $d\in A_r$, we have

$$\begin{array}{cc}\hfill \left(a{\prec }_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)& =\left(a{R}_q\left(b\right)\right){\prec }_{pq,r}{\alpha }_r\left(d\right)=\left[a{R}_q\left(b\right)\right]R_r{\alpha }_r\left(d\right)\hfill \\ \hfill & =\left[a{R}_q\left(b\right)\right]{\alpha }_r{R}_r\left(d\right)={\alpha }_p\left(a\right)\left[R_q\left(b\right)R_r\left(d\right)\right]\hfill \\ \hfill & ={\alpha }_p\left(a\right)R_{qr}\left(R_q\left(b\right)d+b{R}_r\left(d\right)+\lambda bd\right)\hfill \\ \hfill & ={\alpha }_p\left(a\right)R_{qr}\left(b{R}_r\left(d\right)\right)+{\alpha }_p\left(a\right)R_{qr}\left(R_q\left(b\right)d\right)+\lambda {\alpha }_p\left(a\right)R_{qr}\left(bd\right)\hfill \\ \hfill & ={\alpha }_p\left(a\right){\prec }_{p,qr}\left(b{R}_r\left(d\right)+R_q\left(b\right)d+\lambda bd\right)\hfill \\ \hfill & ={\alpha }_p\left(a\right){\prec }_{p,qr}\left(b{\prec }_{q,r}d+b{\succ }_{q,r}d+b{•}_{q,r}d\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left(a{\succ }_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)& =\left[R_p\left(a\right)b\right]{\prec }_{pq,r}{\alpha }_r\left(d\right)=\left[R_p\left(a\right)b\right]R_r{\alpha }_r\left(d\right)=\left[R_p\left(a\right)b\right]{\alpha }_r{R}_r\left(d\right)\hfill \\ \hfill & ={\alpha }_p{R}_p\left(a\right)\left[b{R}_r\left(d\right)\right]=R_p{\alpha }_p\left(a\right)\left[b{R}_r\left(d\right)\right]\hfill \\ \hfill & ={\alpha }_p\left(a\right){\succ }_{p,qr}\left[b{R}_r\left(d\right)\right]={\alpha }_p\left(a\right){\succ }_{p,qr}\left(b{\prec }_{q,r}d\right).\hfill \end{array}$$

Thus equations (1) and (2) are proved. Equation (3) can be proved similarly. Moreover,

$$\begin{array}{cc}\hfill \left(a{\prec }_{p,q}b\right){•}_{pq,r}{\alpha }_r\left(d\right)& =\left[a{R}_q\left(b\right)\right]{•}_{pq,r}{\alpha }_r\left(d\right)=\lambda \left[a{R}_q\left(b\right)\right]{\alpha }_r\left(d\right)=\lambda {\alpha }_p\left(a\right)\left[R_q\left(b\right)d\right]\hfill \\ \hfill & ={\alpha }_p\left(a\right){•}_{p,qr}\left[R_q\left(b\right)d\right]={\alpha }_p\left(a\right){•}_{p,qr}\left(b{\succ }_{q,r}d\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left(a{\succ }_{p,q}b\right){•}_{pq,r}{\alpha }_r\left(d\right)& =\left[R_p\left(a\right)b\right]{•}_{pq,r}{\alpha }_r\left(d\right)=\lambda \left[R_p\left(a\right)b\right]{\alpha }_r\left(d\right)=\lambda {\alpha }_p{R}_p\left(a\right)\left(bd\right)\hfill \\ \hfill & =\lambda R_p{\alpha }_p\left(a\right)\left(bd\right)={\alpha }_p\left(a\right){\succ }_{p,qr}\left(\lambda bd\right)={\alpha }_p\left(a\right){\succ }_{p,qr}\left(b{•}_{q,r}d\right).\hfill \end{array}$$

Thus equations (4) and (5) are proved. Equation (6) can be proved similarly. Now we prove Equation (7):

$$\begin{array}{cc}\hfill \left(a{•}_{p,q}b\right){•}_{pq,r}{\alpha }_r\left(d\right)& =\left[\lambda ab\right]{•}_{pq,r}{\alpha }_r\left(d\right)={\lambda }^2\left(ab\right){\alpha }_r\left(d\right)={\lambda }^2{\alpha }_p\left(a\right)\left(bd\right)\hfill \\ \hfill & =\lambda {\alpha }_p\left(a\right){•}_{p,qr}\left(bd\right)={\alpha }_p\left(a\right){•}_{p,qr}\left(b{•}_{q,r}d\right).\hfill \end{array}$$

Combining the conclusion of (2) with Theorem 2 and Theorem 4, we obtain (3) and (4). □

Similar to Theorem 1, $\pi$-graded Rota-Baxter algebras and $\pi$-graded Rota-Baxter Hom-algebras can be derived from each other via endomorphisms:

Theorem 6.

1. 

Let ${\{{\alpha }_p:A_p⟶A_p\}}_{p\in \pi }$ be an endomorphism of the π-graded Rota-Baxter algebra $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi }\right)$ such that ${\alpha }_p{R}_p=R_p{\alpha }_p$ for all $p\in \pi$. Then $({\left\{A_p\right\}}_{p\in \pi },{\{m_{p,q}^{\ast }={\alpha }_{pq}{m}_{p,q}\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi }$ $,{\left\{{\alpha }_p\right\}}_{p\in \pi })$ is a π-graded Rota-Baxter Hom-algebra.

2. 

Let $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ be a π-graded Rota-Baxter Hom-algebra such that for all $p,q\in \pi$, each ${\alpha }_p:A_p⟶A_p$ is bijective, and we have ${\alpha }_p{R}_p=R_p{\alpha }_p$ and ${\alpha }_{pq}{m}_{p,q}=m_{p,q}({\alpha }_p\otimes {\alpha }_q)$. Then $\left({\left\{A_p\right\}}_{p\in \pi },{\{m_{p,q}^{\#}={\alpha }_{pq}^{-1}{m}_{p,q}\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi }\right)$ is a π-graded Rota-Baxter algebra.

Proof. (1) It is easy to verify that $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}^{\ast }\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a $\pi$-graded Hom-algebra. Moreover, for any $p,q\in \pi$, $a\in A_p$, $b\in A_q$, we have

$$\begin{array}{cc}\hfill R_p\left(a\right){\ast }_{p,q}{R}_q\left(b\right)& ={\alpha }_{pq}\left(R_p\left(a\right)R_q\left(b\right)\right)={\alpha }_{pq}{R}_{pq}\left(R_p\left(a\right)b+a{R}_q\left(b\right)+\lambda ab\right)\hfill \\ \hfill & =R_{pq}{\alpha }_{pq}\left(R_p\left(a\right)b+a{R}_q\left(b\right)+\lambda ab\right)\hfill \\ \hfill & =R_{pq}\left(R_p\left(a\right){\ast }_{p,q}b+a{\ast }_{p,q}{R}_q\left(b\right)+\lambda a{\ast }_{p,q}b\right).\hfill \end{array}$$

(2) For any $p,q,r\in \pi$, $a\in A_p$, $b\in A_q$, $d\in A_r$, we have

$$\begin{array}{cc}\hfill \left(a{\#}_{p,q}b\right){\#}_{pq,r}d& ={\alpha }_{pq}^{-1}\left(ab\right){\#}_{pq,r}d={\alpha }_{pq}^{-1}\left({\alpha }_{pq}^{-1}\left(ab\right)d\right)={\left({\alpha }_{pq}^{-1}\right)}^2\left(\left(ab\right){\alpha }_r\left(d\right)\right)\hfill \\ \hfill & ={\left({\alpha }_{pqr}^{-1}\right)}^2\left({\alpha }_p\left(a\right)\left(bd\right)\right)={\left({\alpha }_{pqr}^{-1}\right)}^2{\alpha }_{pqr}\left(a{\alpha }_{qr}^{-1}\left(bd\right)\right)\hfill \\ \hfill & ={\alpha }_{pqr}^{-1}\left(a{\alpha }_{qr}^{-1}\left(bd\right)\right)=a{\#}_{p,qr}{\alpha }_{qr}^{-1}\left(bd\right)=a{\#}_{p,qr}\left(b{\#}_{q,r}d\right),\hfill \end{array}$$

hence $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}^{\#}\right\}}_{p,q\in \pi }\right)$ is a $\pi$-graded algebra. Now for any $p\in \pi$, we have ${\alpha }_p^{-1}{R}_p={\alpha }_p^{-1}{R}_p{\alpha }_p{\alpha }_p^{-1}={\alpha }_p^{-1}{\alpha }_p{R}_p{\alpha }_p^{-1}=R_p{\alpha }_p^{-1}.$ Therefore, for any $p,q\in \pi$, $a\in A_p$, $b\in A_q$, we have

$$\begin{array}{cc}\hfill R_p\left(a\right){\#}_{p,q}{R}_q\left(b\right)& ={\alpha }_{pq}^{-1}\left(R_p\left(a\right)R_q\left(b\right)\right)={\alpha }_{pq}^{-1}{R}_{pq}\left(R_p\left(a\right)b+a{R}_q\left(b\right)+\lambda ab\right)\hfill \\ \hfill & =R_{pq}{\alpha }_{pq}^{-1}\left(R_p\left(a\right)b+a{R}_q\left(b\right)+\lambda ab\right)\hfill \\ \hfill & =R_{pq}\left(R_p\left(a\right){\#}_{p,q}b+a{\#}_{p,q}{R}_q\left(b\right)+\lambda a{\#}_{p,q}b\right),\hfill \end{array}$$

hence $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}^{\#}\right\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi }\right)$ is a $\pi$-graded Rota-Baxter algebra. Note that in this case we have

$$\begin{array}{cc}\hfill {\alpha }_{pq}\left(a{\#}_{p,q}b\right)& ={\alpha }_{pq}{\alpha }_{pq}^{-1}\left(ab\right)=ab={\alpha }_{pq}^{-1}{\alpha }_{pq}\left(ab\right)={\alpha }_{pq}^{-1}\left({\alpha }_p\left(a\right){\alpha }_q\left(b\right)\right)={\alpha }_p\left(a\right){\#}_{p,q}{\alpha }_q\left(b\right),\hfill \end{array}$$

i.e., ${\left\{{\alpha }_p\right\}}_{p\in \pi }$ is a $\pi$-graded algebra endomorphism on $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}^{\#}\right\}}_{p,q\in \pi }\right)$, so its Yau twist is exactly $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right).$ □

$\pi$-graded Rota-Baxter Hom-algebras can also be constructed iteratively via endomorp

hisms:

Theorem 7.

Let $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ be a π-graded Rota-Baxter Hom-algebra such that for all $p,q\in \pi$ we have ${\alpha }_p{R}_p=R_p{\alpha }_p$ and ${\alpha }_{pq}{m}_{p,q}=m_{p,q}({\alpha }_p\otimes {\alpha }_q)$. Then for any positive integer n, $\left({\left\{A_p\right\}}_{p\in \pi },{\{m_{p,q}^{\left(n\right)}={\alpha }_{pq}^n{m}_{p,q}\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi },\left\{{\alpha }_p^{n+1}\right\}\right)$ is a π-graded Rota-Baxter Hom-algebra.

Proof. 

Denote $m_{p,q}^{\left(n\right)}(a\otimes b)$ by $a{\ast }_{p,q}^{\left(n\right)}b$. For any $p,q,r\in \pi$, $a\in A_p$, $b\in A_q$, $d\in A_r$, we have

$$\begin{array}{cc}\hfill {\alpha }_p^2\left(a\right){\ast }_{p,qr}^{\left(1\right)}\left(b{\ast }_{q,r}^{\left(1\right)}d\right)& ={\alpha }_p^2\left(a\right){\ast }_{p,qr}{\alpha }_{qr}\left(bd\right)={\alpha }_{pqr}\left({\alpha }_p^2\left(a\right){\alpha }_{qr}\left(bd\right)\right)\hfill \\ \hfill & ={\alpha }_{pqr}\left({\alpha }_p^2\left(a\right)\left({\alpha }_q\left(b\right){\alpha }_r\left(d\right)\right)\right)={\alpha }_{pqr}\left(\left({\alpha }_p\left(a\right){\alpha }_q\left(b\right)\right){\alpha }_r^2\left(d\right)\right)\hfill \\ \hfill & ={\alpha }_{pqr}\left({\alpha }_{pq}\left(ab\right){\alpha }_r^2\left(d\right)\right)={\alpha }_{pq}\left(ab\right){\ast }_{pq,r}^{\left(1\right)}{\alpha }_r^2\left(d\right)=\left(a{\ast }_{p,q}^{\left(1\right)}b\right){\ast }_{pq,r}^{\left(1\right)}{\alpha }_r^2\left(d\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill R_p\left(a\right){\ast }_{p,q}^{\left(1\right)}{R}_q\left(b\right)& ={\alpha }_{pq}\left(R_p\left(a\right)R_q\left(b\right)\right)={\alpha }_{pq}{R}_{pq}\left(R_p\left(a\right)b+a{R}_q\left(b\right)+\lambda ab\right)\hfill \\ \hfill & =R_{pq}{\alpha }_{pq}\left(R_p\left(a\right)b+a{R}_q\left(b\right)+\lambda ab\right)\hfill \\ \hfill & =R_{pq}\left(R_p\left(a\right){\ast }_{p,q}^{\left(1\right)}b+a{\ast }_{p,q}^{\left(1\right)}{R}_q\left(b\right)+\lambda a{\ast }_{p,q}^{\left(1\right)}b\right).\hfill \end{array}$$

Thus for $n=1$, $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}^{\left(1\right)}\right\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi },\left\{{\alpha }_p^2\right\}\right)$ is a $\pi$-graded Rota-Baxter Hom-algebra.

Assume that for $n=k$, $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}^{\left(k\right)}\right\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi },\left\{{\alpha }_p^{k+1}\right\}\right)$ is a $\pi$-graded Rota-Baxter Hom-algebra, i.e., assume that we have ${\alpha }_p^{k+1}\left(a\right){\ast }_{p,qr}^{\left(k\right)}\left(b{\ast }_{q,r}^{\left(k\right)}d\right)=\left(a{\ast }_{p,q}^{\left(k\right)}b\right){\ast }_{pq,r}^{\left(k\right)}{\alpha }_r^{k+1}\left(d\right)$ and $R_p\left(a\right){\ast }_{p,q}^{\left(k\right)}{R}_q\left(b\right)=R_{pq}\left(R_p\left(a\right){\ast }_{p,q}^{\left(k\right)}b+a{\ast }_{p,q}^{\left(k\right)}{R}_q\left(b\right)+\lambda a{\ast }_{p,q}^{\left(k\right)}b\right).$ Then for $n=k+1$, we have

$$\begin{array}{cc}\hfill {\alpha }_p^{(k+2)}\left(a\right){\ast }_{p,qr}^{(k+1)}\left(b{\ast }_{q,r}^{(k+1)}d\right)& ={\alpha }_{pqr}\left({\alpha }_p^{(k+2)}\left(a\right){\ast }_{p,qr}^{\left(k\right)}\left(b{\ast }_{q,r}^{(k+1)}d\right)\right)\hfill \\ \hfill & ={\alpha }_{pqr}\left({\alpha }_p^{(k+2)}\left(a\right){\ast }_{p,qr}^{\left(k\right)}{\alpha }_{qr}\left(b{\ast }_{q,r}^{\left(k\right)}d\right)\right)\hfill \\ \hfill & ={\alpha }_{pqr}^2\left({\alpha }_p^{(k+1)}\left(a\right){\ast }_{p,qr}^{\left(k\right)}\left(b{\ast }_{q,r}^{\left(k\right)}d\right)\right)\hfill \\ \hfill & ={\alpha }_{pqr}^2\left(\left(a{\ast }_{p,q}^{\left(k\right)}b\right){\ast }_{pq,r}^{\left(k\right)}{\alpha }_r^{(k+1)}\left(d\right)\right)\hfill \\ \hfill & ={\alpha }_{pqr}\left({\alpha }_{pq}\left(a{\ast }_{p,q}^{\left(k\right)}b\right){\ast }_{pq,r}^{\left(k\right)}{\alpha }_r^{(k+2)}\left(d\right)\right)\hfill \\ \hfill & ={\alpha }_{pqr}\left(\left(a{\ast }_{p,q}^{(k+1)}b\right){\ast }_{pq,r}^{\left(k\right)}{\alpha }_r^{(k+2)}\left(d\right)\right)\hfill \\ \hfill & =\left(a{\ast }_{p,q}^{(k+1)}b\right){\ast }_{pq,r}^{(k+1)}{\alpha }_r^{(k+2)}\left(d\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill R_p\left(a\right){\ast }_{p,q}^{(k+1)}{R}_q\left(b\right)& ={\alpha }_{pq}\left(R_p\left(a\right){\ast }_{p,q}^{\left(k\right)}{R}_q\left(b\right)\right)\hfill \\ \hfill & ={\alpha }_{pq}{R}_{pq}\left(R_p\left(a\right){\ast }_{p,q}^{\left(k\right)}b+a{\ast }_{p,q}^{\left(k\right)}{R}_q\left(b\right)+\lambda a{\ast }_{p,q}^{\left(k\right)}b\right)\hfill \\ \hfill & =R_{pq}{\alpha }_{pq}\left(R_p\left(a\right){\ast }_{p,q}^{\left(k\right)}b+a{\ast }_{p,q}^{\left(k\right)}{R}_q\left(b\right)+\lambda a{\ast }_{p,q}^{\left(k\right)}b\right)\hfill \\ \hfill & =R_{pq}\left(R_p\left(a\right){\ast }_{p,q}^{(k+1)}b+a{\ast }_{p,q}^{(k+1)}{R}_q\left(b\right)+\lambda a{\ast }_{p,q}^{(k+1)}b\right).\hfill \end{array}$$

Thus $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}^{(k+1)}\right\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi },\left\{{\alpha }_p^{(k+2)}\right\}\right)$ is a $\pi$-graded Rota-Baxter Hom-algebra. By the induction principle, the proof is complete. □

$\pi$-graded Rota-Baxter algebras can also be used to construct $\pi$-graded Rota-Baxter Hom-algebras via their centroid elements [8].

Theorem 8.

Let ${\{{\alpha }_p:A_p⟶A_p\}}_{p\in \pi }$ be a family of linear operators on the π-graded Rota-Baxter algebra $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}\right\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi }\right)$ such that for all $p,q\in \pi$, $a\in A_p$, $b\in A_q$, we have ${\alpha }_{pq}\left(ab\right)={\alpha }_p\left(a\right)b=a{\alpha }_q\left(b\right)$ and ${\alpha }_p{R}_p=R_p{\alpha }_p.$ Then $\left({\left\{A_p\right\}}_{p\in \pi },{\{m_{p,q}^{\ast }=m_{p,q}({\alpha }_p\otimes {\mathrm{id}}_q)\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ and $\left({\left\{A_p\right\}}_{p\in \pi },{\{m_{p,q}^{\#}=m_{p,q}({\alpha }_p\otimes {\alpha }_q)\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ are both π-graded Rota-Baxter Hom-algebras.

Proof. 

For any $p,q,r\in \pi$, $a\in A_p$, $b\in A_q$, $d\in A_r$, denote $m_{p,q}^{\ast }(a\otimes b)$ and $m_{p,q}^{\#}(a\otimes b)$ by $a{\ast }_{p,q}b$ and $a{\#}_{p,q}b$ respectively. We have

$$\begin{array}{cc}\hfill R_p\left(a\right){\ast }_{p,q}{R}_q\left(b\right)& ={\alpha }_p{R}_p\left(a\right)R_q\left(b\right)=R_p{\alpha }_p\left(a\right)R_q\left(b\right)\hfill \\ \hfill & =R_{pq}\left(R_p{\alpha }_p\left(a\right)b+{\alpha }_p\left(a\right)R_q\left(b\right)+\lambda {\alpha }_p\left(a\right)b\right)\hfill \\ \hfill & =R_{pq}\left({\alpha }_p{R}_p\left(a\right)b+{\alpha }_p\left(a\right)R_q\left(b\right)+\lambda {\alpha }_p\left(a\right)b\right)\hfill \\ \hfill & =R_{pq}\left(R_p\left(a\right){\ast }_{p,q}b+a{\ast }_{p,q}{R}_q\left(b\right)+\lambda a{\ast }_{p,q}b\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\alpha }_p\left(a\right){\ast }_{p,qr}\left(b{\ast }_{q,r}d\right)& ={\alpha }_p\left(a\right){\ast }_{p,qr}\left({\alpha }_q\left(b\right)d\right)={\alpha }_p^2\left(a\right)\left({\alpha }_q\left(b\right)d\right)=\left({\alpha }_p\left(a\right){\alpha }_q\left(b\right)\right){\alpha }_r\left(d\right)\hfill \\ \hfill & ={\alpha }_{pq}\left({\alpha }_p\left(a\right)b\right){\alpha }_r\left(d\right)=\left({\alpha }_p\left(a\right)b\right){\ast }_{pq,r}{\alpha }_r\left(d\right)=\left(a{\ast }_{p,q}b\right){\ast }_{pq,r}{\alpha }_r\left(d\right),\hfill \end{array}$$

Therefore $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}^{\ast }\right\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a $\pi$-graded Rota-Baxter Hom-algebra. Moreover,

$$\begin{array}{cc}\hfill {\alpha }_p\left(a\right){\#}_{p,qr}\left(b{\#}_{q,r}d\right)& ={\alpha }_p\left(a\right){\#}_{p,qr}\left({\alpha }_q\left(b\right){\alpha }_r\left(d\right)\right)={\alpha }_p^2\left(a\right){\alpha }_{qr}\left({\alpha }_q\left(b\right){\alpha }_r\left(d\right)\right)\hfill \\ \hfill & ={\alpha }_p^2\left(a\right)\left({\alpha }_q^2\left(b\right){\alpha }_r\left(d\right)\right)=\left({\alpha }_p\left(a\right){\alpha }_q^2\left(b\right)\right){\alpha }_r^2\left(d\right)={\alpha }_{pq}\left({\alpha }_p\left(a\right){\alpha }_q\left(b\right)\right){\alpha }_r^2\left(d\right)\hfill \\ \hfill & =\left({\alpha }_p\left(a\right){\alpha }_q\left(b\right)\right){\#}_{pq,r}{\alpha }_r\left(d\right)=\left(a{\#}_{p,q}b\right){\#}_{pq,r}{\alpha }_r\left(d\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill R_p\left(a\right){\#}_{p,q}{R}_q\left(b\right)& ={\alpha }_p{R}_p\left(a\right){\alpha }_q{R}_q\left(b\right)=R_p{\alpha }_p\left(a\right)R_q{\alpha }_q\left(b\right)\hfill \\ \hfill & =R_{pq}\left(R_p{\alpha }_p\left(a\right){\alpha }_q\left(b\right)+{\alpha }_p\left(a\right)R_q{\alpha }_q\left(b\right)+\lambda {\alpha }_p\left(a\right){\alpha }_q\left(b\right)\right)\hfill \\ \hfill & =R_{pq}\left({\alpha }_p{R}_p\left(a\right){\alpha }_q\left(b\right)+{\alpha }_p\left(a\right){\alpha }_q{R}_q\left(b\right)+\lambda {\alpha }_p\left(a\right){\alpha }_q\left(b\right)\right)\hfill \\ \hfill & =R_{pq}\left(R_p\left(a\right){\#}_{p,q}b+a{\#}_{p,q}{R}_q\left(b\right)+\lambda a{\#}_{p,q}b\right),\hfill \end{array}$$

Therefore $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}^{\#}\right\}}_{p,q\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a $\pi$-graded Rota-Baxter Hom-algebra. □

4. Structural Intercommunication between $\pi$-Graded Rota-Baxter Hom-Systems and $\pi$-Graded Hom-Dendriform Algebras

Definition 7.

A π-graded Rota-Baxter Hom-system $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{S_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ consists of a π-graded Hom-algebra $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ and two families of linear operators ${\{R_p,S_p:A_p⟶A_p\}}_{p\in \pi }$ such that for all $p,q\in \pi$, $a\in A_p$, $b\in A_q$, we have

$$\begin{array}{cc}\hfill R_p\left(a\right)R_q\left(b\right)& =R_{pq}\left(R_p\left(a\right)b+a{S}_q\left(b\right)\right),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill S_p\left(a\right)S_q\left(b\right)& =S_{pq}\left(R_p\left(a\right)b+a{S}_q\left(b\right)\right).\hfill \end{array}$$

(9)

The $\pi$-graded Rota-Baxter Hom-system can be regarded as a generalization of the $\pi$-graded Rota-Baxter Hom-algebra:

Proposition 1.

If $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a π-graded Rota-Baxter Hom-algebra, th

en $({\left\{A_p\right\}}_{p\in \pi },$ ${\left\{R_p\right\}}_{p\in \pi },{\{R_p+\lambda {\mathrm{id}}_p\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi })$ and $\left({\left\{A_p\right\}}_{p\in \pi },{\{R_p+\lambda {\mathrm{id}}_p\}}_{p\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ are both π-graded Rota-Baxter systems.

Proof. 

For $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\{R_p+\lambda {\mathrm{id}}_p\}}_{p\in \pi }\right),$ Equation (8) is precisely the definition of a $\pi$-graded Rota-Baxter algebra. We now verify Equation (9):

$$\begin{array}{cc}\hfill \left(R_p+\lambda {\mathrm{id}}_p\right)\left(a\right)\left(R_q+\lambda {\mathrm{id}}_q\right)\left(b\right)& =\left(R_p\left(a\right)+\lambda a\right)\left(R_q\left(b\right)+\lambda b\right)\hfill \\ \hfill & =R_p\left(a\right)R_q\left(b\right)+\lambda a{R}_q\left(b\right)+\lambda R_p\left(a\right)b+{\lambda }^{2}ab\hfill \\ \hfill & =R_{pq}\left(R_p\left(a\right)b+a{R}_q\left(b\right)+\lambda ab\right)+\lambda R_p\left(a\right)b\hfill \\ \hfill & +\lambda a{R}_q\left(b\right)+{\lambda }^{2}ab\hfill \\ \hfill & =\left(R_{pq}+\lambda {\mathrm{id}}_{pq}\right)\left(R_p\left(a\right)b+a{R}_q\left(b\right)+\lambda ab\right)\hfill \\ \hfill & =\left(R_{pq}+\lambda {\mathrm{id}}_{pq}\right)\left(R_p\left(a\right)b+a\left(R_q+\lambda {\mathrm{id}}_q\right)\left(b\right)\right).\hfill \end{array}$$

Thus $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\{R_p+\lambda {\mathrm{id}}_p\}}_{p\in \pi }\right)$ forms a $\pi$-graded Rota-Baxter system. Similarly, one can prove that $\left({\left\{A_p\right\}}_{p\in \pi },{\{R_p+\lambda {\mathrm{id}}_p\}}_{p\in \pi },{\left\{R_p\right\}}_{p\in \pi }\right)$ is also a $\pi$-graded Rota-Baxter system. □

Theorem 9.

Let ${\left\{A_p\right\}}_{p\in \pi }$ be a π-graded algebra, and let ${\{R_p,S_p,{\alpha }_p:A_p⟶A_p\}}_{p\in \pi }$ be three families of linear operators such that for all $p,q\in \pi$, $a\in A_p$, $b\in A_q$, we have $R_{pq}\left(ab\right)=a{R}_q\left(b\right),S_{pq}\left(ab\right)=S_p\left(a\right)b,R_p{\alpha }_p={\alpha }_p{R}_p,S_p{\alpha }_p={\alpha }_p{S}_p.$ Then

1. 

$\left({\left\{A_p\right\}}_{p\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{S_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a π-graded Rota-Baxter Hom-system if and only if for all $p,q\in \pi$, $a\in A_p$, $b\in A_q$, we have $a{R}_q{S}_q\left(b\right)=0=S_p{R}_p\left(a\right)b.$

2. 

If ${\left\{A_p\right\}}_{p\in \pi }$ is a non-degenerate π-graded Hom-algebra, then $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{S_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a π-graded Rota-Baxter Hom-system if and only if for all $p\in \pi$, we have $R_p{S}_p=0=S_p{R}_p.$

Proof. (1) (⇒) From $R_p\left(a\right)R_q\left(b\right)=R_{pq}\left(R_p\left(a\right)b+a{S}_q\left(b\right)\right)=R_{pq}\left(R_p\left(a\right)b\right)+R_{pq}\left(a{S}_q\left(b\right)\right)=R_p\left(a\right)R_q\left(b\right)+a{R}_q{S}_q\left(b\right),$ we obtain $a{R}_q{S}_q\left(b\right)=0$. Similarly, $S_p{R}_p\left(a\right)b=0$ can be proved.

(⇐) $R_{pq}\left(R_p\left(a\right)b+a{S}_q\left(b\right)\right)=R_{pq}\left(R_p\left(a\right)b\right)+R_{pq}\left(a{S}_q\left(b\right)\right)=R_p\left(a\right)R_q\left(b\right)+a{R}_q{S}_q\left(b\right)=R_p\left(a\right)R_q\left(b\right),$ and Equation (9) can be proved similarly.

(2) The proof of (2) follows directly from (1) combined with the definition of a non-degenerate $\pi$-graded Hom-algebra. □

The following is the main result of this section: the structural intercommunication between $\pi$-graded Rota-Baxter Hom-systems and $\pi$-graded Hom-dendriform algebras.

Theorem 10.

Let ${\left\{A_p\right\}}_{p\in \pi }$ be a π-graded Hom-algebra, and let ${\{R_p,S_p,{\alpha }_p:A_p⟶A_p\}}_{p\in \pi }$ be three families of linear operators such that for all $p,q\in \pi$, we have $R_p{\alpha }_p={\alpha }_p{R}_p,S_p{\alpha }_p={\alpha }_p{S}_p.$ Define two families of linear operators ${\{{\prec }_{p,q}:A_p\otimes A_q⟶A_{pq},a\otimes b⟼a{\prec }_{p,q}b=a{S}_q\left(b\right)\}}_{p,q\in \pi }$ and ${\{{\succ }_{p,q}:A_p\otimes A_q⟶A_{pq},a\otimes b⟼a{\succ }_{p,q}b=R_p\left(a\right)b\}}_{p,q\in \pi }.$

1. 

If $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{S_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a π-graded Rota-Baxter Hom-system, then $({\left\{A_p\right\}}_{p\in \pi },$ ${\left\{{\prec }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi })$ is a π-graded Hom-dendriform algebra.

2. 

If $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{{\prec }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\succ }_{p,q}\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a π-graded Hom-dendriform algebra, and ${\left\{A_p\right\}}_{p\in \pi }$ is a non-degenerate π-graded Hom-algebra, then $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{S_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a π-graded Rota-Baxter Hom-system.

Proof. (1) For any $p,q,r\in \pi$, $a\in A_p$, $b\in A_q$, $d\in A_r$, we have

$$\begin{array}{cc}\hfill {\alpha }_p\left(a\right){\succ }_{p,qr}\left(b{\succ }_{q,r}d\right)& ={\alpha }_p\left(a\right){\succ }_{p,qr}\left(R_q\left(b\right)d\right)=\left[R_p{\alpha }_p\left(a\right)\right]\left[R_q\left(b\right)d\right]\hfill \\ \hfill & =\left({\alpha }_p{R}_p\left(a\right)\right)\left[R_q\left(b\right)d\right]=\left[R_p\left(a\right)R_q\left(b\right)\right]{\alpha }_r\left(d\right)\hfill \\ \hfill & =R_{pq}\left(R_p\left(a\right)b+a{S}_q\left(b\right)\right){\alpha }_r\left(d\right)\hfill \\ \hfill & =R_{pq}\left(R_p\left(a\right)b\right){\alpha }_r\left(d\right)+R_{pq}\left(a{S}_q\left(b\right)\right){\alpha }_r\left(d\right)\hfill \\ \hfill & =\left(R_p\left(a\right)b\right){\succ }_{pq,r}{\alpha }_r\left(d\right)+\left(a{S}_q\left(b\right)\right){\succ }_{pq,r}{\alpha }_r\left(d\right)\hfill \\ \hfill & =\left(a{\succ }_{p,q}b\right){\succ }_{pq,r}{\alpha }_r\left(d\right)+\left(a{\prec }_{p,q}b\right){\succ }_{pq,r}{\alpha }_r\left(d\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\alpha }_p\left(a\right){\succ }_{p,qr}\left(b{\prec }_{q,r}d\right)& ={\alpha }_p\left(a\right){\succ }_{p,qr}\left(b{S}_r\left(d\right)\right)=R_p{\alpha }_p\left(a\right)\left[b{S}_r\left(d\right)\right]\hfill \\ \hfill & ={\alpha }_p{R}_p\left(a\right)\left[b{S}_r\left(d\right)\right]=\left[R_p\left(a\right)b\right]\left[{\alpha }_r{S}_r\left(d\right)\right]\hfill \\ \hfill & =\left[R_p\left(a\right)b\right]\left[S_r{\alpha }_r\left(d\right)\right]=\left[R_p\left(a\right)b\right]{\prec }_{pq,r}{\alpha }_r\left(d\right)\hfill \\ \hfill & =\left(a{\succ }_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right).\hfill \end{array}$$

Similarly, one can prove $\left(a{\prec }_{p,q}b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)={\alpha }_p\left(a\right){\prec }_{p,qr}\left(b{\prec }_{q,r}d\right)+{\alpha }_p\left(a\right){\prec }_{p,qr}\left(b{\succ }_{q,r}d\right).$

(2) For any $p,q,r\in \pi$, $a\in A_p$, $b\in A_q$, $d\in A_r$, we have

$$\begin{array}{cc}\hfill \left[R_p\left(a\right)R_q\left(b\right)\right]{\alpha }_r\left(d\right)& ={\alpha }_p{R}_p\left(a\right)\left[R_q\left(b\right)d\right]=R_p{\alpha }_p\left(a\right)\left[R_q\left(b\right)d\right]\hfill \\ \hfill & ={\alpha }_p\left(a\right){\succ }_{p,qr}\left(R_q\left(b\right)d\right)={\alpha }_p\left(a\right){\succ }_{p,qr}\left(b{\succ }_{q,r}d\right)\hfill \\ \hfill & =\left(a{\succ }_{p,q}b\right){\succ }_{pq,r}{\alpha }_r\left(d\right)+\left(a{\prec }_{p,q}b\right){\succ }_{pq,r}{\alpha }_r\left(d\right)\hfill \\ \hfill & =\left(a{S}_q\left(b\right)\right){\succ }_{pq,r}{\alpha }_r\left(d\right)+\left(R_p\left(a\right)b\right){\prec }_{pq,r}{\alpha }_r\left(d\right)\hfill \\ \hfill & =R_{pq}\left(a{S}_q\left(b\right)\right){\alpha }_r\left(d\right)+R_{pq}\left(R_p\left(a\right)b\right){\alpha }_r\left(d\right).\hfill \end{array}$$

Combined with non-degeneracy, Equation (8) is proved. Equation (9) can be proved similarly.

Furthermore, by Theorem 3, we have □

Corollary 2.

If $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{R_p\right\}}_{p\in \pi },{\left\{S_p\right\}}_{p\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right)$ is a π-graded Rota-Baxter Hom-system, then we obtain a new π-graded Hom-algebra $\left({\left\{A_p\right\}}_{p\in \pi },{\left\{m_{p,q}^{\ast }\right\}}_{p,q\in \pi },{\left\{{\alpha }_p\right\}}_{p\in \pi }\right):$ for all $p,q\in \pi$, $a\in A_p$, $b\in A_q$, $m_{p,q}^{\ast }(a\otimes b)=R_p\left(a\right)b+a{S}_q\left(b\right).$