Matching BiHom-Rota-Baxter Algebras and Related Structures

Abstract

In this paper, we introduce the notions of matching BiHom-Rota-Baxter algebras, matching BiHom-(tri)dendriform algebras, matching BiHom-Zinbiel algebras and matching BiHom-pre-Lie algebras. Moreover, we study the properties and relationships between categories of these matching BiHom-algebraic structures.

1. Introduction

The Hom-Lie algebra was introduced by Hartwig, Larsson and Silvestrov [1]. Further research on Hom-Lie algebras could be found in [2,3,4,5,6,7] and references cited therein. As a generalization of Hom- Lie algebra, Graziani etc. [8] also introduced BiHom-Lie algebras. More precisely, a BiHom-Lie algebra is a $\mathbf{k}$-module $\mathfrak{g}$ together with bilinear operation $[,]$ and two commuting linear transformations $p,q$, for $a,b,c\in \mathfrak{g}$, satisfying

$$\begin{array}{cc}& p\left(\right[a,b\left]\right)=\left[p\right(a),p(b\left)\right],q\left(\right[a,b\left]\right)=\left[q\right(a),q(b\left)\right],\end{array}$$

(1)

$$\begin{array}{cc}& \left[q\right(a),p(b\left)\right]=-\left[q\right(b),p(a\left)\right],\end{array}$$

(2)

$$\begin{array}{cc}& [q^2\left(a\right),[q\left(b\right),p\left(c\right)]]+[q^2\left(b\right),[q\left(c\right),p\left(a\right)]]+[q^2\left(c\right),[q\left(a\right),p\left(b\right)]]=0.\end{array}$$

(3)

Identities (1) is called multiplicative, identity (2) is called BiHom-skew symmetry and identity (3) is called BiHom-Jacobi identity. Recently, BiHom-type algebras have been further developed in mathematics and mathematical physics, including BiHom-pre-Lie algebra [9], BiHom-(tri)dendriform algebra [10], BiHom-Zinbiel algebra [10] and BiHom-quadri-algebra [10].

Rota-Baxter operator was introduced by Baxter [11] in the study of the fluctuation problems in probability, and further developed by Rota [12] in combinatorics. It is widely used in many fields, such as mathematics and mathematical physics. The partial study of Rota-Baxter algebra can be seen in [9,10,13,14,15]. More precisely, a Rota-Baxter algebra of weights $\lambda$ is an algebra A together with a linear operator P, and for all $a,b\in A$, it satisfies the following Rota-Baxter identity:

$$\begin{array}{c}\hfill P\left(a\right)P\left(b\right)=P\left(aP\right(b)+P(a)b+\lambda ab).\end{array}$$

For more detailed introduction of Rota-Baxter algebra, please refer to Guo’s book [16]. Recently, as a generalization of Rota-Baxter algebras, the concept of matching Rota-Baxter algebra [17] came from the study of matching pre-Lie algebra [18], and from the pioneering work of Bruned, Hairer and Zambotti [19] on the algebraic renormalization of regularization structure. The research shows that the matching Rota-Baxter algebra is formed by associative Yang-Baxter equation [20] and linear structure of Rota-Baxter operators [17].

The notions of matching Hom-Rota-Baxter algebras, matching Hom-(tri)dendriform algebras and matching Hom-pre-Lie algebras are introduced by [21]. The main purpose of this paper is to extend these matching algebraic structures to the BiHom-algebras setting and study the connections between these categories of BiHom-algebras. Thus the purpose of this paper is to obtain the following categories and functor commutative graph.

This paper is organized as follows. In Section 2, we introduce the matching BiHom-associative, matching BiHom-pre-Lie and matching BiHom-Lie algebras. In Section 3, we explore the relationships between the matching Rota-Baxter algebras and BiHom-associative algebras. In Section 4, we give the relationships between the matching BiHom-dendriform algebras, BiHom-Zinbiel algebras and matching BiHom-tridendriform algebras. In Section 5, we study the properties and relationships between matching Rota-Baxter operators and BiHom-Lie algebras.

Throughout this paper, we work on a unitary commutative ring $\mathbf{k}$ of characteristic different 2, which will be the base ring of all modules, algebras, tensor products, operations as well as linear maps, where tensor products will be denoted by $“\otimes ”$. We always suppose that $\mathsf{\Omega }$ is a non-empty set. We denote ${\odot }_{\mathsf{\Omega }}:=\{{\odot }_{\omega }:\omega \in \mathsf{\Omega }\},\odot \in \{P,·,\ast ,[,],\star ,\prec ,\succ ,•\}$, where $\mathsf{\Omega }$ is a set indexing the linear operators or bilinear operations.

2. Matching BiHom-Associative, Matching BiHom-Pre-Lie and Matching BiHom-Lie Algebras

In this section, we study the relationship among a kind of matching BiHom-type algebras, including matching BiHom-associative algebras, compatible BiHom-associative algebras, compatible BiHom-pre-Lie algebras and compatible BiHom-Lie algebras. The main results of this section are given as the following categories commutative graph.

Definition 1.

A matching BiHom-associative algebra is a 4-tuple $(A,{·}_{\mathsf{\Omega }},p,q)$ consisting of a $\mathbf{k}$-module A, a family of bilinear operations ${·}_{\omega }:A\otimes A\to A,\omega \in \mathsf{\Omega }$ and two linear transformations $p,q:A\to A$, satisfying

$$\begin{array}{c}p\circ q=q\circ p,\hfill \\ p\left(a{·}_{\omega }b\right)=p\left(a\right){·}_{\omega }p\left(b\right),q\left(a{·}_{\omega }b\right)=q\left(a\right){·}_{\omega }q\left(b\right),\hfill \\ \left(a{·}_{\alpha }b\right){·}_{\beta }q\left(c\right)=p\left(a\right){·}_{\alpha }\left(b{·}_{\beta }c\right),\hfill \end{array}$$

(4)

for all $a,b,c\in A$ and $\alpha ,\beta ,\omega \in \mathsf{\Omega }$. In particular, if p and q are algebra automorphisms, we call it a regular matching BiHom-associative algebra.

A matching BiHom-associative algebra is called totally compatible if it satisfies

$$\begin{array}{ccc}& & \left(a{·}_{\alpha }b\right){·}_{\beta }q\left(c\right)=p\left(a\right){·}_{\beta }\left(b{·}_{\alpha }c\right)\hfill \end{array}$$

(5)

for all $a,b,c\in A$ and $\alpha ,\beta \in \mathsf{\Omega }$.

Definition 2.

A compatible BiHom-associative algebra is a 4-tuple $(A,{·}_{\mathsf{\Omega }},p,q)$ consisting of a $\mathbf{k}$-module A, a family of bilinear operations ${·}_{\omega }:A\otimes A\to A,\omega \in \mathsf{\Omega }$ and two linear transformations $p,q:A\to A$, satisfying the following conditions,

$$\begin{array}{c}p\circ q=q\circ p,\hfill \\ p\left(a{·}_{\omega }b\right)=p\left(a\right){·}_{\omega }p\left(b\right),q\left(a{·}_{\omega }b\right)=q\left(a\right){·}_{\omega }q\left(b\right),\hfill \\ \left(a{·}_{\alpha }b\right){·}_{\beta }q\left(c\right)+\left(a{·}_{\beta }b\right){·}_{\alpha }q\left(c\right)=p\left(a\right){·}_{\alpha }\left(b{·}_{\beta }c\right)+p\left(a\right){·}_{\beta }\left(b{·}_{\alpha }c\right)\hfill \end{array}$$

(6)

for all $a,b,c\in A$ and $\alpha ,\beta ,\omega \in \mathsf{\Omega }$. In particular, if p and q are algebra automorphisms, we call it a regular compatible BiHom-associative algebra.

Remark 1.

(i)

Obviously, compatible BiHom-associative algebras include matching BiHom-associative algebras or totally compatible BiHom-associative algebras.

(ii)

Matching (Hom)-associative algebras, totally compatible (Hom)-associative algebras and compatible (Hom)-associative algebras given in [17,21] are special cases of above definitions, when $p=q=Id(p=q)$.

(iii)

If Ω is a singleton, BiHom-associative algebras introduced in [8] are special cases of matching BiHom-associative algebras and compatible BiHom-associative algebras.

Definition 3.

A matching BiHom-Lie algebra is a 4-tuple $(\mathfrak{g},{[,]}_{\mathsf{\Omega }},p,q)$ consisting of a $\mathbf{k}$-module $\mathfrak{g}$, a family of bilinear operations ${[,]}_{\omega }:\mathfrak{g}\otimes \mathfrak{g}\to \mathfrak{g},\omega \in \mathsf{\Omega }$ and two linear transformations $p,q:\mathfrak{g}\to \mathfrak{g}$ satisfying the following conditions,

$$\begin{array}{c}p\circ q=q\circ p,\hfill \\ p\left({[a,b]}_{\omega }\right)={[p\left(a\right),p\left(b\right)]}_{\omega },q\left({[a,b]}_{\omega }\right)={[q\left(a\right),q\left(b\right)]}_{\omega },\hfill \\ {[q\left(a\right),p\left(b\right)]}_{\omega }=-{[q\left(b\right),p\left(a\right)]}_{\omega },\hfill \\ {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\beta }]}_{\alpha }+{[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\alpha }]}_{\beta }+{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\alpha }]}_{\beta }=0,\hfill \end{array}$$

(7)

for all $a,b,c\in \mathfrak{g}$ and $\alpha ,\beta ,\omega \in \mathsf{\Omega }$. In particular, if p and q are algebra automorphisms, we call it a regular matching BiHom-Lie algebra.

Proposition 1.

Let $(A,{·}_{\mathsf{\Omega }},p,q)$ be a regular matching BiHom-associative algebra. One can define the Lie bracket defined by

$$\begin{array}{ccc}& & {[a,b]}_{\omega }:=a{·}_{\omega }b-p^{-1}q\left(b\right){·}_{\omega }p{q}^{-1}\left(a\right)\hfill \end{array}$$

for $a,b\in A$ and $\omega \in \mathsf{\Omega }$. Then $(A,{[,]}_{\mathsf{\Omega }},p,q)$ is a matching BiHom-Lie algebra.

Proof. 

Similar to [8]. □

Lemma 1.

Let $(\mathfrak{g},{[,]}_{\mathsf{\Omega }},p,q)$ be a matching BiHom-Lie algebra. Consider linear combinations

$$\begin{array}{ccc}& & {[,]}_A:=\sum _{\alpha \in \mathsf{\Omega }}{x}_{\alpha }{[,]}_{\alpha },{[,]}_B:=\sum _{\beta \in \mathsf{\Omega }}{y}_{\beta }{[,]}_{\beta },\hfill \end{array}$$

(8)

where $x_{\alpha },y_{\beta }\in \mathbf{k}$ for $\alpha ,\beta \in \mathsf{\Omega }$ with finite supports. Then

$$\begin{array}{ccc}& & {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_B]}_A+{[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_A]}_B+{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_A]}_B=0,\hfill \end{array}$$

for $a,b,c\in \mathfrak{g}$.

Proof. 

For $a,b,c\in \mathfrak{g}$, by (8), we have

$$\begin{array}{ccc}\hfill {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_B]}_A& =& {[q^2\left(a\right),\sum _{\beta \in \mathsf{\Omega }}{y}_{\beta }{[q\left(b\right),p\left(c\right)]}_{\beta }]}_A\hfill \\ & =& \sum _{\alpha \in \mathsf{\Omega }}{x}_{\alpha }{[q^2\left(a\right),\sum _{\beta \in \mathsf{\Omega }}{y}_{\beta }{[q\left(b\right),p\left(c\right)]}_{\beta }]}_{\alpha }\hfill \\ & =& \sum _{\alpha \in \mathsf{\Omega }}\sum _{\beta \in \mathsf{\Omega }}{x}_{\alpha }{y}_{\beta }{[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\beta }]}_{\alpha }.\hfill \end{array}$$

Similarly, we also have

$$\begin{array}{ccc}& & {[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_A]}_B=\sum _{\alpha \in \mathsf{\Omega }}\sum _{\beta \in \mathsf{\Omega }}{y}_{\beta }{x}_{\alpha }{[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\alpha }]}_{\beta }\hfill \\ & & {[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_A]}_B=\sum _{\alpha \in \mathsf{\Omega }}\sum _{\beta \in \mathsf{\Omega }}{y}_{\beta }{x}_{\alpha }{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\alpha }]}_{\beta }.\hfill \end{array}$$

Since $(\mathfrak{g},{[,]}_{\mathsf{\Omega }},p,q)$ is a matching BiHom-Lie algebra, then

$$\begin{array}{ccc}& & {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\beta }]}_{\alpha }+{[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\alpha }]}_{\beta }+{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\alpha }]}_{\beta }=0.\hfill \end{array}$$

Thus

$$\begin{array}{ccc}& & {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_B]}_A+{[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_A]}_B+{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_A]}_B=0\hfill \end{array}$$

as desired. □

Proposition 2.

Let $(\mathfrak{g},{[,]}_{\mathsf{\Omega }},p,q)$ be a matching BiHom-Lie algebra. Consider linear combinations

$$\begin{array}{ccc}& & {[,]}_A:=\sum _{\alpha \in \mathsf{\Omega }}{x}_{\alpha }{[,]}_{\alpha },x_{\alpha }\in \mathbf{k},\hfill \end{array}$$

(9)

with a finite support. Then $(\mathfrak{g},{[,]}_A,p,q)$ is a BiHom-Lie algebra.

Proof. 

It follows from Lemma 1 by taking ${\left(x_{\alpha }\right)}_{\alpha \in \mathsf{\Omega }}={\left(y_{\beta }\right)}_{\beta \in \mathsf{\Omega }}$. □

More generally.

Definition 4.

A compatible BiHom-Lie algebra is a 4-tuple $(\mathfrak{g},{[,]}_{\mathsf{\Omega }},p,q)$ consisting of a $\mathbf{k}$-module $\mathfrak{g}$, a family of bilinear operations ${[,]}_{\omega }:\mathfrak{g}\otimes \mathfrak{g}\to \mathfrak{g},\omega \in \mathsf{\Omega }$ and two linear transformations $p,q:\mathfrak{g}\to \mathfrak{g}$ satisfying the following conditions:

$$\begin{array}{c}p\circ q=q\circ p,\hfill \\ p\left({[a,b]}_{\omega }\right)={[p\left(a\right),p\left(b\right)]}_{\omega },q\left({[a,b]}_{\omega }\right)={[q\left(a\right),q\left(b\right)]}_{\omega },\hfill \\ {[q\left(a\right),p\left(b\right)]}_{\omega }=-{[q\left(b\right),p\left(a\right)]}_{\omega },\hfill \\ {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\alpha }]}_{\beta }+{[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\alpha }]}_{\beta }+{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\alpha }]}_{\beta }\hfill \\ +{[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\beta }]}_{\alpha }+{[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\beta }]}_{\alpha }+{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\beta }]}_{\alpha }=0,\hfill \end{array}$$

(10)

for all $a,b,c\in \mathfrak{g}$ and $\alpha ,\beta ,\omega \in \mathsf{\Omega }$. In particular, if p and q are algebra automorphisms, we call it a regular compatible BiHom-Lie algebra.

Remark 2.

(i)

Matching BiHom-Lie algebras are special cases of compatible BiHom-Lie algebras.

(ii)

Given two BiHom-Lie algebras $(\mathfrak{g},{[,]}_{\alpha },p,q)$ and $(\mathfrak{g},{[,]}_{\beta },p,q)$. Define a new bracket $[,]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$ as follows:

$$\begin{array}{c}\hfill [a,b]:=x_{\alpha }{[a,b]}_{\alpha }+y_{\beta }{[a,b]}_{\beta }\end{array}$$

for some $x_{\alpha },y_{\beta }\in \mathbf{k}$. Easily to check that this new bracket is BiHom-skew symmetry. By a direct calculation, we can check that $(\mathfrak{g},[,],p,q)$ is a BiHom-Lie algebra. In fact, the BiHom-Jacobi identity

$$\begin{array}{c}\hfill [q^2\left(a\right),[q\left(b\right),p\left(c\right)]]+[q^2\left(b\right),[q\left(c\right),p\left(a\right)]]+[q^2\left(c\right),[q\left(a\right),p\left(b\right)]]=0\end{array}$$

is equivalent to identity (10).

Proposition 3.

Let $(\mathfrak{g},{·}_{\mathsf{\Omega }},p,q)$ be a matching BiHom-Lie algebra. Then for $a,b,c\in \mathfrak{g}$ and $\alpha ,\beta \in \mathsf{\Omega },$ we have

$$\begin{array}{ccc}& & {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\alpha }]}_{\beta }={[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\beta }]}_{\alpha },\hfill \\ & & {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\alpha }]}_{\beta }+{[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\alpha }]}_{\beta }+{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\alpha }]}_{\beta }=0.\hfill \end{array}$$

Proof. 

Since (7) holds for any $a,b,c\in \mathfrak{g}$ and $\alpha ,\beta \in \mathsf{\Omega }$, we get

$$\begin{array}{ccc}& & {[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\alpha }]}_{\beta }+{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\beta }]}_{\alpha }+{[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\beta }]}_{\alpha }=0.\hfill \end{array}$$

(11)

By (7) and (11), we have

$$\begin{array}{ccc}& & {[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\alpha }]}_{\beta }-{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\beta }]}_{\alpha }=0,\hfill \end{array}$$

By the arbitrariness of $a,b,c$, we have

$$\begin{array}{ccc}& & {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\alpha }]}_{\beta }={[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\beta }]}_{\alpha }.\hfill \end{array}$$

Thus

$$\begin{array}{ccc}& & {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\alpha }]}_{\beta }+{[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\alpha }]}_{\beta }+{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\alpha }]}_{\beta }=0.\hfill \end{array}$$

□

Proposition 4.

Let $(A,{·}_{\mathsf{\Omega }},p,q)$ be a regular compatible BiHom-associative algebra. Then $(A,{[,]}_{\mathsf{\Omega }},p,q)$ is a compatible BiHom-Lie algebra, where

$$\begin{array}{ccc}& & {[,]}_{\omega }:A\times A\to A,{[a,b]}_{\omega }:=a{·}_{\omega }b-p^{-1}q\left(b\right){·}_{\omega }p{q}^{-1}\left(a\right)\hfill \end{array}$$

(12)

for $a,b\in A$ and $\omega \in \mathsf{\Omega }.$

Proof. 

Obviously, p and q are multiplicative with respect to ${[,]}_{\omega }$. Clearly, ${[,]}_{\omega }$ is BiHom-skew symmetry.

Now we prove that (10) holds. For $a,b,c\in A$ and $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}& & {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\alpha }]}_{\beta }\hfill \\ & =& {[q^2\left(a\right),q\left(b\right){·}_{\alpha }p\left(c\right)-p^{-1}q\left(p\left(c\right)\right){·}_{\alpha }p{q}^{-1}\left(q\left(b\right)\right)]}_{\beta }\hfill \\ & =& {[q^2\left(a\right),q\left(b\right){·}_{\alpha }p\left(c\right)-q\left(c\right){·}_{\alpha }p\left(b\right)]}_{\beta }\hfill \\ & =& q^2\left(a\right){·}_{\beta }(q\left(b\right){·}_{\alpha }p\left(c\right)-q\left(c\right){·}_{\alpha }p\left(b\right))-p^{-1}q(q\left(b\right){·}_{\alpha }p\left(c\right)-q\left(c\right){·}_{\alpha }p\left(b\right)){·}_{\beta }p{q}^{-1}\left(q^2\left(a\right)\right)\hfill \\ & =& q^2\left(a\right){·}_{\beta }\left(q\left(b\right){·}_{\alpha }p\left(c\right)\right)-q^2\left(a\right){·}_{\beta }\left(q\left(c\right){·}_{\alpha }p\left(b\right)\right)\hfill \\ & & -p^{-1}q\left(q\left(b\right){·}_{\alpha }p\left(c\right)\right){·}_{\beta }p{q}^{-1}\left(q^2\left(a\right)\right)+p^{-1}q\left(q\left(c\right){·}_{\alpha }p\left(b\right)\right){·}_{\beta }p{q}^{-1}\left(q^2\left(a\right)\right)\hfill \\ & =& p\left(p^{-1}{q}^2\left(a\right)\right){·}_{\beta }\left(q\left(b\right){·}_{\alpha }p\left(c\right)\right)-p\left(p^{-1}{q}^2\left(a\right)\right){·}_{\beta }\left(q\left(c\right){·}_{\alpha }p\left(b\right)\right)\hfill \\ & & -\left(p^{-1}{q}^2\left(b\right){·}_{\alpha }q\left(c\right)\right){·}_{\beta }qp\left(a\right)+\left(p^{-1}{q}^2\left(c\right){·}_{\alpha }q\left(b\right)\right){·}_{\beta }qp\left(a\right).\hfill \end{array}$$

Similarly, we have

$$\begin{array}{ccc}\hfill {[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\alpha }]}_{\beta }& =& p\left(p^{-1}{q}^2\left(b\right)\right){·}_{\beta }\left(q\left(c\right){·}_{\alpha }p\left(a\right)\right)-p\left(p^{-1}{q}^2\left(b\right)\right){·}_{\beta }\left(q\left(a\right){·}_{\alpha }p\left(c\right)\right)\hfill \\ & & -\left(p^{-1}{q}^2\left(c\right){·}_{\alpha }q\left(a\right)\right){·}_{\beta }qp\left(b\right)+\left(p^{-1}{q}^2\left(a\right){·}_{\alpha }q\left(c\right)\right){·}_{\beta }qp\left(b\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\alpha }]}_{\beta }& =& p\left(p^{-1}{q}^2\left(c\right)\right){·}_{\beta }\left(q\left(a\right){·}_{\alpha }p\left(b\right)\right)-p\left(p^{-1}{q}^2\left(c\right)\right){·}_{\beta }\left(q\left(b\right){·}_{\alpha }p\left(a\right)\right)\hfill \\ & & -\left(p^{-1}{q}^2\left(a\right){·}_{\alpha }q\left(b\right)\right){·}_{\beta }qp\left(c\right)+\left(p^{-1}{q}^2\left(b\right){·}_{\alpha }q\left(a\right)\right){·}_{\beta }qp\left(c\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\beta }]}_{\alpha }& =& p\left(p^{-1}{q}^2\left(a\right)\right){·}_{\alpha }\left(q\left(b\right){·}_{\beta }p\left(c\right)\right)-p\left(p^{-1}{q}^2\left(a\right)\right){·}_{\alpha }\left(q\left(c\right){·}_{\beta }p\left(b\right)\right)\hfill \\ & & -\left(p^{-1}{q}^2\left(b\right){·}_{\beta }q\left(c\right)\right){·}_{\alpha }qp\left(a\right)+\left(p^{-1}{q}^2\left(c\right){·}_{\beta }q\left(b\right)\right){·}_{\alpha }qp\left(a\right).\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\beta }]}_{\alpha }& =& p\left(p^{-1}{q}^2\left(b\right)\right){·}_{\alpha }\left(q\left(c\right){·}_{\beta }p\left(a\right)\right)-p\left(p^{-1}{q}^2\left(b\right)\right){·}_{\alpha }\left(q\left(a\right){·}_{\beta }p\left(c\right)\right)\hfill \\ & & -\left(p^{-1}{q}^2\left(c\right){·}_{\beta }q\left(a\right)\right){·}_{\alpha }qp\left(b\right)+\left(p^{-1}{q}^2\left(a\right){·}_{\beta }q\left(c\right)\right){·}_{\alpha }qp\left(b\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\beta }]}_{\alpha }& =& p\left(p^{-1}{q}^2\left(c\right)\right){·}_{\alpha }\left(q\left(a\right){·}_{\beta }p\left(b\right)\right)-p\left(p^{-1}{q}^2\left(c\right)\right){·}_{\alpha }\left(q\left(b\right){·}_{\beta }p\left(a\right)\right)\hfill \\ & & -\left(p^{-1}{q}^2\left(a\right){·}_{\beta }q\left(b\right)\right){·}_{\alpha }qp\left(c\right)+\left(p^{-1}{q}^2\left(b\right){·}_{\beta }q\left(a\right)\right){·}_{\alpha }qp\left(c\right).\hfill \end{array}$$

By (6), we get

$$\begin{array}{ccc}& & {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\alpha }]}_{\beta }+{[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\alpha }]}_{\beta }+{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\alpha }]}_{\beta }\hfill \\ & & +{[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\beta }]}_{\alpha }+{[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\beta }]}_{\alpha }+{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\beta }]}_{\alpha }=0.\hfill \end{array}$$

Hence $(A,{[,]}_{\mathsf{\Omega }},p,q)$ is a compatible BiHom-Lie algebra. □

Definition 5.

A matching BiHom-pre-Lie algebra is a 4-tuple $(A,{\ast }_{\mathsf{\Omega }},p,q)$ consisting of a $\mathbf{k}$-module A, a family of bilinear operations ${\ast }_{\omega }:A\otimes A\to A,\omega \in \mathsf{\Omega }$ and two linear transformations $p,q:A\to A$ satisfying the following condition,

$$\begin{array}{c}p\circ q=q\circ p,\hfill \\ p\left(a{\ast }_{\omega }b\right)=p\left(a\right){\ast }_{\omega }p\left(b\right),q\left(a{\ast }_{\omega }b\right)=q\left(a\right){\ast }_{\omega }q\left(b\right),\hfill \\ pq\left(a\right){\ast }_{\alpha }\left(p\left(b\right){\ast }_{\beta }c\right)-\left(q\left(a\right){\ast }_{\alpha }p\left(b\right)\right){\ast }_{\beta }q\left(c\right)\hfill \\ =pq\left(b\right){\ast }_{\beta }\left(p\left(a\right){\ast }_{\alpha }c\right)-\left(q\left(b\right){\ast }_{\beta }p\left(a\right)\right){\ast }_{\alpha }q\left(c\right),\end{array}$$

(13)

for all $a,b,c\in A$ and $\alpha ,\beta ,\omega \in \mathsf{\Omega }$. In particular, if p and q are algebra automorphisms, we call it regular matching BiHom-pre-Lie algebra.

Remark 3.

Obviously, every compatible BiHom-associative algebra is a matching BiHom-pre-Lie algebra.

In the following, we check that matching BiHom-pre-Lie algebras can induce compatible BiHom-Lie algebras.

Proposition 5.

Let $(A,{\ast }_{\mathsf{\Omega }},p,q)$ be a regular matching BiHom-pre-Lie algebra. Then $(A,{[,]}_{\mathsf{\Omega }},p,q)$ is a compatible BiHom-Lie algebra, where

$$\begin{array}{ccc}& & {[,]}_{\omega }:A\times A\to A,{[a,b]}_{\omega }:=a{\ast }_{\omega }b-p^{-1}q\left(b\right){\ast }_{\omega }p{q}^{-1}\left(a\right),\hfill \end{array}$$

(14)

for $a,b\in A$ and $\omega \in \mathsf{\Omega }.$

Proof. 

Obviously, p and q are multiplicative with respect to ${[,]}_{\omega }$. Clearly, ${[,]}_{\omega }$ is BiHom-skew symmetry.

Now we prove that (10) holds. For $a,b,c\in A$ and $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}& & {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\alpha }]}_{\beta }\hfill \\ & =& {[q^2\left(a\right),q\left(b\right){\ast }_{\alpha }p\left(c\right)-p^{-1}q\left(p\left(c\right)\right){\ast }_{\alpha }p{q}^{-1}\left(q\left(b\right)\right)]}_{\beta }\hfill \\ & =& {[q^2\left(a\right),q\left(b\right){\ast }_{\alpha }p\left(c\right)-q\left(c\right){\ast }_{\alpha }p\left(b\right)]}_{\beta }\hfill \\ & =& q^2\left(a\right){\ast }_{\beta }(q\left(b\right){\ast }_{\alpha }p\left(c\right)-q\left(c\right){\ast }_{\alpha }p\left(b\right))-p^{-1}q(q\left(b\right){\ast }_{\alpha }p\left(c\right)-q\left(c\right){\ast }_{\alpha }p\left(b\right)){\ast }_{\beta }p{q}^{-1}\left(q^2\left(a\right)\right)\hfill \\ & =& q^2\left(a\right){\ast }_{\beta }\left(q\left(b\right){\ast }_{\alpha }p\left(c\right)\right)-q^2\left(a\right){\ast }_{\beta }\left(q\left(c\right){\ast }_{\alpha }p\left(b\right)\right)\hfill \\ & & -p^{-1}q\left(q\left(b\right){\ast }_{\alpha }p\left(c\right)\right){\ast }_{\beta }p{q}^{-1}\left(q^2\left(a\right)\right)+p^{-1}q\left(q\left(c\right){\ast }_{\alpha }p\left(b\right)\right){\ast }_{\beta }p{q}^{-1}\left(q^2\left(a\right)\right)\hfill \\ & =& pq\left(p^{-1}q\left(a\right)\right){\ast }_{\beta }\left(p\left(p^{-1}q\left(b\right)\right){\ast }_{\alpha }p\left(c\right)\right)-pq\left(p^{-1}q\left(a\right)\right){\ast }_{\beta }\left(p\left(p^{-1}q\left(c\right)\right){\ast }_{\alpha }p\left(b\right)\right)\hfill \\ & & -\left(q\left(p^{-1}q\left(b\right)\right){\ast }_{\alpha }p\left(p^{-1}q\left(c\right)\right)\right){\ast }_{\beta }qp\left(a\right)+\left(q\left(p^{-1}q\left(c\right)\right){\ast }_{\alpha }p\left(p^{-1}q\left(b\right)\right)\right){\ast }_{\beta }qp\left(a\right).\hfill \end{array}$$

Similarly, we have

$$\begin{array}{ccc}& & {[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\alpha }]}_{\beta }\hfill \\ & =& pq\left(p^{-1}q\left(b\right)\right){\ast }_{\beta }\left(p\left(p^{-1}q\left(c\right)\right){\ast }_{\alpha }p\left(a\right)\right)-pq\left(p^{-1}q\left(b\right)\right){\ast }_{\beta }\left(p\left(p^{-1}q\left(a\right)\right){\ast }_{\alpha }p\left(c\right)\right)\hfill \\ & & -\left(q\left(p^{-1}q\left(c\right)\right){\ast }_{\alpha }p\left(p^{-1}q\left(a\right)\right)\right){\ast }_{\beta }qp\left(b\right)+\left(q\left(p^{-1}q\left(a\right)\right){\ast }_{\alpha }p\left(p^{-1}q\left(c\right)\right)\right){\ast }_{\beta }qp\left(b\right),\hfill \end{array}$$

$$\begin{array}{ccc}& & {[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\alpha }]}_{\beta }\hfill \\ & =& pq\left(p^{-1}q\left(c\right)\right){\ast }_{\beta }\left(p\left(p^{-1}q\left(a\right)\right){\ast }_{\alpha }p\left(b\right)\right)-pq\left(p^{-1}q\left(c\right)\right){\ast }_{\beta }\left(p\left(p^{-1}q\left(b\right)\right){\ast }_{\alpha }p\left(a\right)\right)\hfill \\ & & -\left(q\left(p^{-1}q\left(a\right)\right){\ast }_{\alpha }p\left(p^{-1}q\left(b\right)\right)\right){\ast }_{\beta }qp\left(c\right)+\left(q\left(p^{-1}q\left(b\right)\right){\ast }_{\alpha }p\left(p^{-1}q\left(a\right)\right)\right){\ast }_{\beta }qp\left(c\right),\hfill \end{array}$$

$$\begin{array}{ccc}& & {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\beta }]}_{\alpha }\hfill \\ & =& pq\left(p^{-1}q\left(a\right)\right){\ast }_{\alpha }\left(p\left(p^{-1}q\left(b\right)\right){\ast }_{\beta }p\left(c\right)\right)-pq\left(p^{-1}q\left(a\right)\right){\ast }_{\alpha }\left(p\left(p^{-1}q\left(c\right)\right){\ast }_{\beta }p\left(b\right)\right)\hfill \\ & & -\left(q\left(p^{-1}q\left(b\right)\right){\ast }_{\beta }p\left(p^{-1}q\left(c\right)\right)\right){\ast }_{\alpha }qp\left(a\right)+\left(q\left(p^{-1}q\left(c\right)\right){\ast }_{\beta }p\left(p^{-1}q\left(b\right)\right)\right){\ast }_{\alpha }qp\left(a\right),\hfill \end{array}$$

$$\begin{array}{ccc}& & {[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\beta }]}_{\alpha }\hfill \\ & =& pq\left(p^{-1}q\left(b\right)\right){\ast }_{\alpha }\left(p\left(p^{-1}q\left(c\right)\right){\ast }_{\beta }p\left(a\right)\right)-pq\left(p^{-1}q\left(b\right)\right){\ast }_{\alpha }\left(p\left(p^{-1}q\left(a\right)\right){\ast }_{\beta }p\left(c\right)\right)\hfill \\ & & -\left(q\left(p^{-1}q\left(c\right)\right){\ast }_{\beta }p\left(p^{-1}q\left(a\right)\right)\right){\ast }_{\alpha }qp\left(b\right)+\left(q\left(p^{-1}q\left(a\right)\right){\ast }_{\beta }p\left(p^{-1}q\left(c\right)\right)\right){\ast }_{\alpha }qp\left(b\right),\hfill \end{array}$$

$$\begin{array}{ccc}& & {[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\beta }]}_{\alpha }\hfill \\ & =& pq\left(p^{-1}q\left(c\right)\right){\ast }_{\alpha }\left(p\left(p^{-1}q\left(a\right)\right){\ast }_{\beta }p\left(b\right)\right)-pq\left(p^{-1}q\left(c\right)\right){\ast }_{\alpha }\left(p\left(p^{-1}q\left(b\right)\right){\ast }_{\beta }p\left(a\right)\right)\hfill \\ & & -\left(q\left(p^{-1}q\left(a\right)\right){\ast }_{\beta }p\left(p^{-1}q\left(b\right)\right)\right){\ast }_{\alpha }qp\left(c\right)+\left(q\left(p^{-1}q\left(b\right)\right){\ast }_{\beta }p\left(p^{-1}q\left(a\right)\right)\right){\ast }_{\alpha }qp\left(c\right).\hfill \end{array}$$

By (13), we have

$$\begin{array}{ccc}& & {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\alpha }]}_{\beta }+{[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\alpha }]}_{\beta }+{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\alpha }]}_{\beta }\hfill \\ & & +{[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{\beta }]}_{\alpha }+{[q^2\left(b\right),{[q\left(c\right),p\left(a\right)]}_{\beta }]}_{\alpha }+{[q^2\left(c\right),{[q\left(a\right),p\left(b\right)]}_{\beta }]}_{\alpha }=0.\hfill \end{array}$$

Hence $(A,{[,]}_{\mathsf{\Omega }},p,q)$ is a compatible BiHom-Lie algebra. □

3. Matching BiHom-Associative Rota-Baxter Algebras

In this section, we introduce the notion of matching BiHom-associative Rota-Baxter algebras and discuss its properties.

Definition 6

([17]). Let ${\lambda }_{\mathsf{\Omega }}:={\left({\lambda }_{\omega }\right)}_{\omega \in \mathsf{\Omega }}\subseteq \mathbf{k}$ be a set of scalars indexed by Ω. A matching associative Rota-Baxter algebra of weight ${\lambda }_{\mathsf{\Omega }}$ is a 3-tuple $(A,·,P_{\mathsf{\Omega }})$ consisting of an associative algebra $(A,·)$, a family of linear operators $P_{\omega }:A\to A,\omega \in \mathsf{\Omega }$ satisfying the matching Rota-Baxter identity,

$$\begin{array}{ccc}& & P_{\alpha }\left(a\right)·P_{\beta }\left(b\right)=P_{\alpha }(a·P_{\beta }\left(b\right))+P_{\beta }(P_{\alpha }\left(a\right)·b)+{\lambda }_{\beta }{P}_{\alpha }(a·b),\hfill \end{array}$$

(15)

for all $a,b\in A$ and $\alpha ,\beta \in \mathsf{\Omega }$.

Definition 7.

A matching BiHom-associative Rota-Baxter algebra is a 5-tuple $(A,·,P_{\mathsf{\Omega }},p,q)$ consisting of a matching Rota-Baxter algebra $(A,·,P_{\mathsf{\Omega }})$, a BiHom-associative algebra $(A,·,p,q)$ satisfying $p\circ P_{\omega }=P_{\omega }\circ p$, $q\circ P_{\omega }=P_{\omega }\circ q$. In particular, if p and q are algebra automorphisms, we call it regular matching BiHom-associative Rota-Baxter algebra.

Theorem 1.

Let $(A,·,P_{\mathsf{\Omega }})$ be a matching associative Rota-Baxter algebra and $p,q:A\to A$ be two commuting algebra endomorphisms such that $p\circ P_{\omega }=P_{\omega }\circ p$ and $q\circ P_{\omega }=P_{\omega }\circ q$ for all $\omega \in \mathsf{\Omega }$. Then $(A,{·}_{pq}=·\circ (p\otimes q),P_{\mathsf{\Omega }},p,q)$ is a matching BiHom-associative Rota-Baxter algebra.

Proof. 

For $a,b,c\in A$, we have

$$\begin{array}{ccc}& & p\left(a{·}_{pq}b\right)=p(p\left(a\right)·q\left(b\right))=p\left(p\left(a\right)\right)·q\left(p\left(b\right)\right)=p\left(a\right){·}_{pq}q\left(b\right).\hfill \end{array}$$

Similarly, $q\left(a{·}_{pq}b\right)=q\left(a\right){·}_{pq}q\left(b\right)$.

$$\begin{array}{ccc}& & \left(a{·}_{pq}b\right){·}_{pq}q\left(c\right)-p\left(a\right){·}_{pq}\left(b{·}_{pq}c\right)\hfill \\ & =& p(p\left(a\right)·q\left(b\right))·q^2\left(c\right)-p^2\left(a\right)·q(p\left(b\right)·q\left(c\right))\hfill \\ & =& (p^2\left(a\right)·pq\left(b\right))·q^2\left(c\right)-p^2\left(a\right)·(pq\left(b\right)·q^2\left(c\right))\hfill \\ & =& 0.\hfill \end{array}$$

Hence, $(A,{·}_{pq},p,q)$ is a BiHom-associative algebra. For $\alpha ,\beta \in \mathsf{\Omega }$,

$$\begin{array}{ccc}& & P_{\alpha }\left(a\right){·}_{pq}{P}_{\beta }\left(b\right)\hfill \\ & =& p\left(P_{\alpha }\left(a\right)\right)·q\left(P_{\beta }\left(b\right)\right)\hfill \\ & =& P_{\alpha }\left(p\left(a\right)\right)·P_{\beta }\left(q\left(b\right)\right)\hfill \\ & =& P_{\alpha }(p\left(a\right)·P_{\beta }\left(q\left(b\right)\right))+P_{\beta }(P_{\alpha }\left(p\left(a\right)\right)·q\left(b\right))+{\lambda }_{\beta }{P}_{\alpha }(p\left(a\right)·q\left(b\right))\hfill \\ & =& P_{\alpha }(p\left(a\right)·q\left(P_{\beta }\left(b\right)\right))+P_{\beta }(p\left(P_{\alpha }\left(a\right)\right)·q\left(b\right))+{\lambda }_{\beta }{P}_{\alpha }(p\left(a\right)·q\left(b\right))\hfill \\ & =& P_{\alpha }\left(a{·}_{pq}{P}_{\beta }\left(b\right)\right)+P_{\beta }\left(P_{\alpha }\left(a\right){·}_{pq}b\right)+{\lambda }_{\beta }{P}_{\alpha }\left(a{·}_{pq}b\right).\hfill \end{array}$$

Hence $(A,{·}_{pq},P_{\mathsf{\Omega }},p,q)$ is a matching BiHom-associative Rota-Baxter algebra. □

Conversely, we have.

Proposition 6.

Let $(A,·,P_{\mathsf{\Omega }},p,q)$ be a regular matching BiHom-assoicative Rota-Baxter algebra. Then $(A,{·}^{\prime }=·\circ (p^{-1}\otimes q^{-1}),P_{\mathsf{\Omega }})$ is a matching associative Rota-Baxter algebra.

Proof. 

For $a,b,c\in A$, we have

$$\begin{array}{ccc}& & \left(a{·}^{\prime }b\right){·}^{\prime }c-a{·}^{\prime }\left(b{·}^{\prime }c\right)\hfill \\ & =& p^{-1}((p^{-1}\left(a\right)·q^{-1}\left(b\right))·q^{-1}\left(c\right)-p^{-1}\left(a\right)·q^{-1}(p^{-1}\left(b\right)·q^{-1}\left(c\right))\hfill \\ & =& (p^{-2}\left(a\right)·p^{-1}{q}^{-1}\left(b\right))·q\left(q^{-2}\left(c\right)\right)-p\left(p^{-2}\left(a\right)\right)·(p^{-1}{q}^{-1}\left(b\right)·q^{-2}\left(c\right)))\hfill \\ & =& 0.\hfill \end{array}$$

Thus the associativity condition holds. For $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}& & P_{\alpha }\left(a\right){·}^{\prime }{P}_{\beta }\left(b\right)=p^{-1}\left(P_{\alpha }\left(a\right)\right)·q^{-1}\left(P_{\beta }\left(b\right)\right)\hfill \\ & =& P_{\alpha }\left(p^{-1}\left(a\right)\right)·P_{\beta }\left(q^{-1}\left(b\right)\right)\hfill \\ & =& P_{\alpha }(p^{-1}\left(a\right)·P_{\beta }\left(q^{-1}\left(b\right)\right))+P_{\beta }(P_{\alpha }\left(p^{-1}\left(a\right)\right)·q^{-1}\left(b\right))+{\lambda }_{\beta }{P}_{\alpha }(p^{-1}\left(a\right)·q^{-1}\left(b\right))\hfill \\ & =& P_{\alpha }(p^{-1}\left(a\right)·q^{-1}\left(P_{\beta }\left(b\right)\right))+P_{\beta }(p^{-1}\left(P_{\alpha }\left(a\right)\right)·q^{-1}\left(b\right))+{\lambda }_{\beta }{P}_{\alpha }(p^{-1}\left(a\right)·q^{-1}\left(b\right))\hfill \\ & =& P_{\alpha }\left(a{·}^{\prime }{P}_{\beta }\left(b\right)\right)+P_{\beta }\left(P_{\alpha }\left(a\right){·}^{\prime }b\right)+{\lambda }_{\beta }{P}_{\alpha }\left(a{·}^{\prime }b\right).\hfill \end{array}$$

Hence $(A,{·}^{\prime },P_{\mathsf{\Omega }})$ is a matching associative Rota-Baxter algebra. □

Remark 4.

With the method of Proposition 6, we can give other types of regular BiHom-algebras, which can construct this type of non-Hom-algebras, for example, matching BiHom-pre-Lie Rota-Baxter algebras, matching BiHom-Leibniz Rota-Baxter algebras and so on.

Definition 8.

Let $(A,·,p,q)$ be a BiHom-associative algebra and $n\ge 0$. The n-th derived BiHom-associative algebra of A is defined by $(A,{·}^n=·\circ (p^n\otimes q^n),p^{n+1},q^{n+1}).$

Theorem 2.

Let $(A,·,P_{\mathsf{\Omega }},p,q)$ be a matching BiHom-associative Rota-Baxter algebra. Then

$$(A,{·}^n,P_{\mathsf{\Omega }},p^{n+1},q^{n+1})$$

is also a matching BiHom-associative Rota-Baxter algebra.

Proof. 

Obviously, $(A,{·}^n,p^{n+1},q^{n+1})$ is a BiHom-associative algebra. Now we show the matching Rota-Baxter identity holds. For $a,b,c\in A$ and $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}& & P_{\alpha }\left(a\right){·}^n{P}_{\beta }\left(b\right)\hfill \\ & =& p^n\left(P_{\alpha }\left(a\right)\right)·q^n\left(P_{\beta }\left(b\right)\right)\hfill \\ & =& P_{\alpha }\left(p^n\left(a\right)\right)·P_{\beta }\left(q^n\left(b\right)\right)\hfill \\ & =& P_{\alpha }(p^n\left(a\right)·P_{\beta }\left(q^n\left(b\right)\right))+P_{\beta }(P_{\alpha }\left(p^n\left(a\right)\right)·q^n\left(b\right))+{\lambda }_{\beta }{P}_{\alpha }(p^n\left(a\right)·q^n\left(b\right))\hfill \\ & =& P_{\alpha }(p^n\left(a\right)·q^n\left(P_{\beta }\left(b\right)\right))+P_{\beta }(p^n\left(P_{\alpha }\left(a\right)\right)·q^n\left(b\right))+{\lambda }_{\beta }{P}_{\alpha }(p^n\left(a\right)·q^n\left(b\right))\hfill \\ & =& P_{\alpha }\left(a{·}^n{P}_{\beta }\left(b\right)\right)+P_{\beta }\left(P_{\alpha }\left(a\right){·}^{n}b\right)+{\lambda }_{\beta }{P}_{\alpha }\left(a{·}^{n}b\right).\hfill \end{array}$$

Hence $(A,{·}^n,P_{\mathsf{\Omega }},p^{n+1},q^{n+1})$ is a matching BiHom-associative Rota-Baxter algebra. □

Remark 5.

The conclusion of Proposition 6 and Theorem 2 is that existence is not unique. For example. Let $(A,·,P_{\mathsf{\Omega }},p,q)$ be a matching BiHom-associative Rota-Baxter algebra.

(i)

Set ${·}^1=·\circ (p^{}\otimes q^{})$. Then $(A,{·}^1\circ (p^{-2}\otimes q^{-2}),P_{\mathsf{\Omega }})$ is a matching associative Rota-Baxter algebra.

(ii)

Then $(A,·\circ (p^{n+1}\otimes q^{n+1}),P_{\mathsf{\Omega }},p^{n+2},q^{n+2})$ is also a matching BiHom-associative Rota-Baxter algebra.

4. Matching BiHom-Zinbiel Algebras and Matching BiHom-(Tri)Dendriform Algebras

In this section, we study the relationship among a kinds of matching BiHom-type algebras, which includes matching BiHom-dendriform algebras, matching BiHom-Zinbiel and matching BiHom-tridendriform algebras. The main results of this section are given as the following categories commutative graph.

Definition 9.

A matching BiHom-dendriform algebra is a 5-tuple $(D,{\prec }_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }},p,q)$ consisting of a k-module D, a family of bilinear operations ${\odot }_{\omega }:D\otimes D\to D$, $\odot \in \{\prec ,\succ \}$, $\omega \in \mathsf{\Omega }$ and two linear transformations $p,q:D\to D$ satisfying the following conditions,

$$\begin{array}{ccc}\hfill p\circ q& =& q\circ p,\hfill \\ \hfill p\left(a{\prec }_{\omega }b\right)& =& p\left(a\right){\prec }_{\omega }p\left(b\right),p\left(a{\succ }_{\omega }b\right)=p\left(a\right){\succ }_{\omega }p\left(b\right),\hfill \\ \hfill q\left(a{\prec }_{\omega }b\right)& =& q\left(a\right){\prec }_{\omega }q\left(b\right),q\left(a{\succ }_{\omega }b\right)=q\left(a\right){\succ }_{\omega }q\left(b\right),\hfill \end{array}$$

$$\left(a{\prec }_{\alpha }b\right){\prec }_{\beta }q\left(c\right)=p\left(a\right){\prec }_{\alpha }\left(b{\prec }_{\beta }c\right)+p\left(a\right){\prec }_{\beta }\left(b{\succ }_{\alpha }c\right),$$

(16)

$$\begin{array}{ccc}\hfill \left(a{\succ }_{\alpha }b\right){\prec }_{\beta }q\left(c\right)& =& p\left(a\right){\succ }_{\alpha }\left(b{\prec }_{\beta }c\right),\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill p\left(a\right){\succ }_{\alpha }\left(b{\succ }_{\beta }c\right)& =& \left(a{\prec }_{\beta }b\right){\succ }_{\alpha }q\left(c\right)+\left(a{\succ }_{\alpha }b\right){\succ }_{\beta }q\left(c\right),\hfill \end{array}$$

(18)

for all $a,b,c\in D$ and $\alpha ,\beta ,\omega \in \mathsf{\Omega }.$ In particular, if $q\left(a\right){\succ }_{\omega }p\left(b\right)=q\left(b\right){\prec }_{\omega }p\left(a\right)$, we call it a commutative matching BiHom-dendriform algebra. If p and q are algebra automorphisms, we call it a regular matching BiHom-dendriform algebra.

Definition 10

([14]). A (left) matching Zinbiel algebra is a pair $(Z,{\star }_{\mathsf{\Omega }})$ consisting of a k-module Z and a family of bilinear operations ${\left({\star }_{\omega }\right)}_{\omega \in \mathsf{\Omega }}$ satisfying

$$\begin{array}{ccc}\hfill \left(a{\star }_{\alpha }b\right){\star }_{\beta }c& =& a{\star }_{\alpha }\left(b{\star }_{\beta }c\right)+a{\star }_{\beta }\left(c{\star }_{\alpha }b\right),\hfill \end{array}$$

for all $a,b,c\in Z$ and $\alpha ,\beta \in \mathsf{\Omega }$.

Definition 11.

A matching BiHom-Zinbiel algebra is a 4-tuple $(Z,{\diamond }_{\mathsf{\Omega }},p,q)$ consisting of a k-module Z, a family of bilinear operations ${\left({\diamond }_{\omega }\right)}_{\omega \in \mathsf{\Omega }}$ and two linear transformations $p,q:Z\to Z$ satisfying the following conditions,

$$\begin{array}{ccc}\hfill p\circ q& =& q\circ p,\hfill \\ \hfill p\left(a{\diamond }_{\omega }b\right)& =& p\left(a\right){\diamond }_{\omega }p\left(b\right),q\left(a{\diamond }_{\omega }b\right)=q\left(a\right){\diamond }_{\omega }q\left(b\right),\hfill \\ \hfill \left(a{\diamond }_{\alpha }q\left(b\right)\right){\diamond }_{\beta }pq\left(c\right)& =& p\left(a\right){\diamond }_{\alpha }\left(q\left(b\right){\diamond }_{\beta }p\left(c\right)\right)+p\left(a\right){\diamond }_{\beta }\left(q\left(c\right){\diamond }_{\alpha }p\left(b\right)\right),\hfill \\ \hfill \left(q\left(b\right){\diamond }_{\alpha }p\left(a\right)\right){\diamond }_{\beta }pq\left(c\right)& =& \left(q\left(b\right){\diamond }_{\beta }q\left(c\right)\right){\diamond }_{\alpha }{p}^2\left(a\right),\hfill \end{array}$$

for $a,b,c\in Z$ and $\alpha ,\beta \in \mathsf{\Omega }$. In particular, if p and q are bijective, we call it a regular matching BiHom-Zinbiel algebra.

Proposition 7.

Let $(Z,{\star }_{\mathsf{\Omega }})$ be a matching Zinbiel algebra and $p,q:Z\to Z$ be two commuting algebra endomorphisms. Define a new multiplication on Z by $a{\diamond }_{\omega }b=p\left(a\right){\star }_{\omega }q\left(b\right)$, for all $a,b\in Z$ and $\omega \in \mathsf{\Omega }$. Then $(Z,{\diamond }_{\mathsf{\Omega }},p,q)$ is a matching BiHom-Zinbiel algebra.

Proof. 

For $a,b,c\in Z$ we compute:

$$\begin{array}{ccc}\hfill \left(a{\diamond }_{\alpha }q\left(b\right)\right){\diamond }_{\beta }pq\left(c\right)& =& \left(p^2\left(a\right){\star }_{\alpha }p{q}^2\left(b\right)\right){\star }_{\beta }p{q}^2\left(c\right)\hfill \\ & =& p^2\left(a\right){\star }_{\alpha }\left(p{q}^2\left(b\right){\star }_{\beta }p{q}^2\left(c\right)\right)+p^2\left(a\right){\star }_{\beta }\left(p{q}^2\left(c\right){\star }_{\alpha }p{q}^2\left(b\right)\right)\hfill \\ & =& p^2\left(a\right){\star }_{\alpha }q\left(p\left(q\left(b\right)\right){\star }_{\beta }q\left(p\left(c\right)\right)\right)+p^2\left(a\right){\star }_{\beta }q\left(p\left(q\left(c\right)\right){\star }_{\alpha }q\left(p\left(b\right)\right)\right)\hfill \\ & =& p^2\left(a\right){\star }_{\alpha }q\left(q\left(b\right){\diamond }_{\beta }p\left(c\right)\right)+p^2\left(a\right){\star }_{\beta }q\left(q\left(c\right){\diamond }_{\alpha }p\left(b\right)\right)\hfill \\ & =& p\left(a\right){\diamond }_{\alpha }\left(q\left(b\right){\diamond }_{\beta }p\left(c\right)\right)+p\left(x\right){\diamond }_{\beta }\left(q\left(c\right){\diamond }_{\alpha }p\left(b\right)\right),\hfill \\ \hfill \left(q\left(b\right){\diamond }_{\alpha }p\left(a\right)\right){\diamond }_{\beta }pq\left(c\right)& =& \left(p^2\left(q\left(b\right)\right){\star }_{\alpha }{p}^2\left(q\left(a\right)\right)\right){\star }_{\beta }p\left(q^2\left(c\right)\right)\hfill \\ & =& \left(p^2\left(q\left(b\right)\right){\star }_{\beta }p\left(q^2\left(c\right)\right)\right){\star }_{\alpha }{p}^2\left(q\left(a\right)\right)\hfill \\ & =& p\left(p\left(q\left(b\right)\right){\star }_{\beta }{q}^2\left(c\right)\right){\star }_{\alpha }q\left(p^2\left(a\right)\right)\hfill \\ & =& \left(q\left(b\right){\diamond }_{\beta }q\left(c\right)\right){\diamond }_{\alpha }{p}^2\left(a\right),\hfill \end{array}$$

finishing the proof. □

As well as we know that a Zinbiel algebra is equivalent to a commutative dendriform algebra. Now we extend this result to the matching BiHom-algebra case.

Proposition 8.

(i)

Let $(D,{\prec }_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }},p,q)$ be a commutative matching BiHom-dendriform algebra. Define the operation $a{\diamond }_{\omega }b=a{\prec }_{\omega }b,$ for all $a,b\in D$ and $\omega \in \mathsf{\Omega }$. Then $(D,{\diamond }_{\mathsf{\Omega }},p,q)$ is a matching BiHom-Zinbiel algebra.

(ii)

Conversely, let $(D,{\diamond }_{\mathsf{\Omega }},p,q)$ be a regular matching BiHom-Zinbiel algebra. Define new operations

$$\begin{array}{c}\hfill a{\prec }_{\omega }b:=a{\diamond }_{\omega }banda{\succ }_{\omega }b:=q{p}^{-1}\left(b\right){\diamond }_{\omega }p{q}^{-1}\left(a\right)fora,b\in D,\omega \in \mathsf{\Omega }.\end{array}$$

Then $(D,{\prec }_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }},p,q)$ is a commutative matching BiHom-dendriform algebra.

Proof. 

(i)

For $a,b,c\in D$ and $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}\hfill \left(a{\diamond }_{\alpha }q\left(b\right)\right){\diamond }_{\beta }pq\left(c\right)& =& \left(a{\prec }_{\alpha }q\left(b\right)\right){\prec }_{\beta }pq\left(c\right)\hfill \\ & =& \left(a{\prec }_{\alpha }q\left(b\right)\right){\prec }_{\beta }qp\left(c\right)\hfill \\ & =& p\left(a\right){\prec }_{\alpha }\left(q\left(b\right){\prec }_{\beta }p\left(c\right)\right)+p\left(a\right){\prec }_{\beta }\left(q\left(b\right){\succ }_{\alpha }p\left(c\right)\right)\hfill \\ & =& p\left(a\right){\prec }_{\alpha }\left(q\left(b\right){\prec }_{\beta }p\left(c\right)\right)+p\left(a\right){\prec }_{\beta }\left(q\left(c\right){\prec }_{\alpha }p\left(b\right)\right)\hfill \\ & =& p\left(a\right){\diamond }_{\alpha }\left(q\left(b\right){\diamond }_{\beta }p\left(c\right)\right)+p\left(a\right){\diamond }_{\beta }\left(q\left(c\right){\diamond }_{\alpha }p\left(b\right)\right),\hfill \\ \hfill \left(q\left(b\right){\diamond }_{\alpha }p\left(a\right)\right){\diamond }_{\beta }pq\left(c\right)& =& \left(q\left(b\right){\prec }_{\alpha }p\left(a\right)\right){\prec }_{\beta }pq\left(c\right)\hfill \\ & =& \left(q\left(a\right){\succ }_{\alpha }p\left(b\right)\right){\prec }_{\beta }pq\left(c\right)\hfill \\ & =& \left(q\left(a\right){\succ }_{\alpha }p\left(b\right)\right){\prec }_{\beta }qp\left(c\right)\hfill \\ & =& pq\left(a\right){\succ }_{\alpha }\left(p\left(b\right){\prec }_{\beta }p\left(c\right)\right)\hfill \\ & =& qp\left(a\right){\succ }_{\alpha }p\left(b{\prec }_{\beta }c\right)\hfill \\ & =& q\left(b{\prec }_{\beta }c\right){\prec }_{\alpha }p\left(p\left(a\right)\right)\hfill \\ & =& \left(q\left(b\right){\prec }_{\beta }q\left(c\right)\right){\prec }_{\alpha }{p}^2\left(a\right)\hfill \\ & =& \left(q\left(b\right){\diamond }_{\beta }q\left(c\right)\right){\diamond }_{\alpha }{p}^2\left(a\right).\hfill \end{array}$$

(ii)

Obviously, D is commutative since $q\left(a\right){\succ }_{\omega }p\left(b\right)=q\left(p^{-1}\left(p\left(b\right)\right)\right){\diamond }_{\omega }p\left(q^{-1}\left(q\left(a\right)\right)\right)=q\left(b\right){\diamond }_{\omega }p\left(a\right)=q\left(b\right){\prec }_{\omega }p\left(a\right)$. Clearly, p and q are multiplicative with respect to ${\prec }_{\omega }$ and ${\succ }_{\omega }$. Now we prove (16), (17) and (18). For $a,b,c\in D$ and $\alpha ,\beta \in \mathsf{\Omega }$,

$$\begin{array}{ccc}& & \left(a{\prec }_{\alpha }b\right){\prec }_{\beta }q\left(c\right)\hfill \\ & =& \left(a{\diamond }_{\alpha }b\right){\diamond }_{\beta }q\left(c\right)\hfill \\ & =& \left(a{\diamond }_{\alpha }q\left(q^{-1}\left(b\right)\right)\right){\diamond }_{\beta }p\left(q{p}^{-1}\left(c\right)\right)\hfill \\ & =& p\left(a\right){\diamond }_{\alpha }\left(b{\diamond }_{\beta }c\right)+p\left(a\right){\diamond }_{\beta }\left(q{p}^{-1}\left(c\right){\diamond }_{\alpha }p{q}^{-1}\left(b\right)\right)\hfill \\ & =& p\left(a\right){\prec }_{\alpha }\left(b{\prec }_{\beta }c\right)+p\left(a\right){\prec }_{\beta }\left(b{\succ }_{\alpha }c\right).\hfill \end{array}$$

Also,

$$\begin{array}{ccc}& & \left(a{\succ }_{\alpha }b\right){\prec }_{\beta }q\left(c\right)\hfill \\ & =& \left(q{p}^{-1}\left(b\right){\diamond }_{\alpha }p{q}^{-1}\left(a\right)\right){\diamond }_{\beta }q\left(c\right)\hfill \\ & =& \left(q\left(p^{-1}\left(b\right)\right){\diamond }_{\alpha }p\left(q^{-1}\left(a\right)\right)\right){\diamond }_{\beta }p\left(q{p}^{-1}\left(c\right)\right)\hfill \\ & =& \left(q\left(p^{-1}\left(b\right)\right){\diamond }_{\beta }q{p}^{-1}\left(c\right)\right){\diamond }_{\alpha }{p}^2\left(q^{-1}\left(a\right)\right)\hfill \\ & =& q{p}^{-1}\left(b{\diamond }_{\beta }c\right){\diamond }_{\alpha }{p}^2{q}^{-1}\left(a\right)\hfill \\ & =& q{p}^{-1}\left(b{\prec }_{\beta }c\right){\diamond }_{\alpha }p{q}^{-1}\left(p\left(a\right)\right)\hfill \\ & =& p\left(a\right){\succ }_{\alpha }\left(b{\prec }_{\beta }c\right).\hfill \end{array}$$

Further,

$$\begin{array}{ccc}& & p\left(a\right){\succ }_{\alpha }\left(b{\succ }_{\beta }c\right)\hfill \\ & =& p\left(a\right){\succ }_{\alpha }\left(q{p}^{-1}\left(c\right){\diamond }_{\beta }p{q}^{-1}\left(b\right)\right)\hfill \\ & =& q{p}^{-1}\left(q{p}^{-1}\left(c\right){\diamond }_{\beta }p{q}^{-1}\left(b\right)\right){\diamond }_{\alpha }p{q}^{-1}\left(p\left(a\right)\right)\hfill \\ & =& \left(q^2{p}^{-2}\left(c\right){\diamond }_{\beta }b\right){\diamond }_{\alpha }{p}^2\left(q^{-1}\left(a\right)\right)\hfill \\ & =& (q^2{p}^{-2}\left(c\right){\diamond }_{\beta }q\left(q^{-1}\left(b\right)\right){\diamond }_{\alpha }pq\left(p{q}^{-2}\left(a\right)\right)\hfill \\ & =& q^2\left(p^{-1}\left(c\right)\right){\diamond }_{\beta }\left(b{\diamond }_{\alpha }{p}^2\left(q^{-2}\left(a\right)\right)\right)+q^2\left(p^{-1}\left(c\right)\right){\diamond }_{\alpha }\left(p\left(q^{-1}\left(a\right)\right){\diamond }_{\beta }p\left(q^{-1}\left(b\right)\right)\right)\hfill \\ & =& q^2\left(p^{-1}\left(c\right)\right){\diamond }_{\beta }(q{p}^{-1}\left(p{q}^{-1}\left(b\right){\diamond }_{\alpha }p{q}^{-1}\left(p{q}^{-1}\left(a\right)\right)\right)+q^2\left(p^{-1}\left(c\right)\right){\diamond }_{\alpha }p{q}^{-1}\left(a{\prec }_{\beta }b\right))\hfill \\ & =& q^2\left(p^{-1}\left(c\right)\right){\diamond }_{\beta }\left(p{q}^{-1}\left(a\right){\succ }_{\alpha }p{q}^{-1}\left(b\right)\right)+q^2\left(p^{-1}\left(c\right)\right){\diamond }_{\alpha }p{q}^{-1}\left(a{\prec }_{\beta }b\right))\hfill \\ & =& q{p}^{-1}\left(q\left(c\right)\right){\diamond }_{\beta }p{q}^{-1}\left(a{\succ }_{\alpha }b\right)+q{p}^{-1}\left(q\left(c\right)\right){\diamond }_{\alpha }p{q}^{-1}\left(a{\prec }_{\beta }b\right))\hfill \\ & =& \left(a{\succ }_{\alpha }b\right){\succ }_{\beta }q\left(c\right)+\left(a{\prec }_{\beta }b\right){\succ }_{\alpha }q\left(c\right),\hfill \end{array}$$

as required.

□

Definition 12.

A matching BiHom-tridendriform algebra is a 6-tuple $(D,{\prec }_{\mathsf{\Omega }},{•}_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }},p,q)$ consisting of a $\mathbf{k}$-module D, a family of bilinear operations ${\odot }_{\omega }:D\otimes D\to D$, $\odot \in \{\prec ,•,\succ \}$, $\omega \in \mathsf{\Omega }$, and two linear transformations $p,q:D\to D$ satisfying the following conditions,

$$\begin{array}{c}p\circ q=q\circ p,\\ p\left(a{\prec }_{\omega }b\right)=p\left(a\right){\prec }_{\omega }p\left(b\right),p\left(a{\succ }_{\omega }b\right)=p\left(a\right){\succ }_{\omega }p\left(b\right),p\left(a{•}_{\omega }b\right)=p\left(a\right){•}_{\omega }p\left(b\right),\\ q\left(a{\prec }_{\omega }b\right)=q\left(a\right){\prec }_{\omega }q\left(b\right),q\left(a{\succ }_{\omega }b\right)=q\left(a\right){\succ }_{\omega }q\left(b\right),q\left(a{•}_{\omega }b\right)=q\left(a\right){•}_{\omega }q\left(b\right),\end{array}$$

$$\left(a{\prec }_{\alpha }b\right){\prec }_{\beta }q\left(c\right)=p\left(a\right){\prec }_{\alpha }\left(b{\prec }_{\beta }c\right)+p\left(a\right){\prec }_{\beta }\left(b{\succ }_{\alpha }c\right)+p\left(a\right){\prec }_{\alpha }\left(b{•}_{\beta }c\right),$$

(19)

$$\begin{array}{cc}& \left(a{\succ }_{\alpha }b\right){\prec }_{\beta }q\left(c\right)=p\left(a\right){\succ }_{\alpha }\left(b{\prec }_{\beta }c\right),\end{array}$$

(20)

$$\begin{array}{cc}& p\left(a\right){\succ }_{\alpha }\left(b{\succ }_{\beta }c\right)=\left(a{\prec }_{\beta }b\right){\succ }_{\alpha }q\left(c\right)+\left(a{\succ }_{\alpha }b\right){\succ }_{\beta }q\left(c\right)+\left(a{•}_{\beta }b\right){\succ }_{\alpha }q\left(c\right),\end{array}$$

(21)

$$\begin{array}{cc}& \left(a{\succ }_{\alpha }b\right){•}_{\beta }q\left(c\right)=p\left(a\right){\succ }_{\alpha }\left(b{•}_{\beta }c\right),\end{array}$$

(22)

$$\begin{array}{cc}& \left(a{\prec }_{\alpha }b\right){•}_{\beta }q\left(c\right)=p\left(a\right){•}_{\beta }\left(b{\succ }_{\alpha }c\right),\end{array}$$

(23)

$$\begin{array}{cc}& \left(a{•}_{\alpha }b\right){\prec }_{\beta }q\left(c\right)=p\left(a\right){•}_{\alpha }\left(b{\prec }_{\beta }c\right),\end{array}$$

(24)

$$\begin{array}{cc}& \left(a{•}_{\alpha }b\right){•}_{\beta }q\left(c\right)=p\left(a\right){•}_{\alpha }\left(b{•}_{\beta }c\right),\end{array}$$

(25)

for all $a,b,c\in D$ and $\alpha ,\beta ,\omega \in \mathsf{\Omega }.$

Definition 13.

(i)

Let $(D,{\prec }_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }},p,q)$ and $(D^{\prime },{\prec }_{\mathsf{\Omega }}^{\prime },{\succ }_{\mathsf{\Omega }}^{\prime },p^{\prime },q^{\prime })$ be two matching BiHom-dendriform algebras. A linear map $f:D\to D^{\prime }$ is called a matching BiHom-dendriform algebra morphism if for all $\omega \in \mathsf{\Omega }$,

$${\prec }_{\omega }^{\prime }\circ (f\otimes f)=f\circ {\prec }_{\omega },{\succ }_{\omega }^{\prime }\circ (f\otimes f)=f\circ {\succ }_{\omega }$$

and

$$p^{\prime }\circ f=f\circ p,q^{\prime }\circ f=f\circ q.$$

(ii)

Let $(D,{\prec }_{\mathsf{\Omega }},{•}_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }},p,q)$ and $(D^{\prime },{\prec }_{\mathsf{\Omega }}^{\prime },{•}_{\mathsf{\Omega }}^{\prime },{\succ }_{\mathsf{\Omega }}^{\prime },p^{\prime },q^{\prime })$ be two matching BiHom-teidendriform algebras. A linear map $f:D\to D^{\prime }$ is called a matching BiHom-terdendriform algebra morphism if for all $\omega \in \mathsf{\Omega }$,

$${\prec }_{\omega }^{\prime }\circ (f\otimes f)=f\circ {\prec }_{\omega },{•}_{\omega }^{\prime }\circ (f\otimes f)=f\circ {•}_{\omega },{\succ }_{\omega }^{\prime }\circ (f\otimes f)=f\circ {\succ }_{\omega }$$

and

$$p^{\prime }\circ f=f\circ p,q^{\prime }\circ f=f\circ q.$$

Theorem 3.

(i)

Let $(D,{\prec }_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }})$ be a matching dendriform algebra and let $p,q:D\to D$ be matching dendriform algebra endomorphisms such that $p\circ q=q\circ p$. Then $D_{pq}=(D,{\prec }_{pq,\mathsf{\Omega }},{\succ }_{pq,\mathsf{\Omega }},p,q)$ is a matching BiHom-dendriform algebra, where ${\prec }_{pq,\omega }:={\prec }_{\omega }\circ (p\otimes q),$ and ${\succ }_{pq,\omega }:={\succ }_{\omega }\circ (p\otimes q)$.

Moreover, suppose that $(D^{\prime },{\prec }_{\mathsf{\Omega }}^{\prime },{\succ }_{\mathsf{\Omega }}^{\prime })$ is another matching dendriform algebra and let $p^{\prime },q^{\prime }:D^{\prime }\to D^{\prime }$ be matching dendriform algebra endomorphisms such that $p^{\prime }\circ q^{\prime }=q^{\prime }\circ p^{\prime }$. If $f:D\to D^{\prime }$ is a matching dendriform algebra morphism that satisfies $f\circ p=p^{\prime }\circ f,f\circ q=q^{\prime }\circ f$, then

$$f:(D,{\prec }_{pq,\mathsf{\Omega }},{\succ }_{pq,\mathsf{\Omega }},p,q)\to (D^{\prime },{\prec }_{pq,\mathsf{\Omega }}^{\prime },{\succ }_{pq,\mathsf{\Omega }}^{\prime },p^{\prime },q^{\prime })$$

is a morphism of matching BiHom-dendriform algebras.

(ii)

Let $(D,{\prec }_{\mathsf{\Omega }},{•}_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }})$ be a matching tridendriform algebra and let $p,q:D\to D$ be matching tridendriform algebra endomorphisms such that $p\circ q=q\circ p$. Then $D_{pq}=(D,{\prec }_{pq,\mathsf{\Omega }},{•}_{pq,\mathsf{\Omega }},{\succ }_{pq,\mathsf{\Omega }},p,q)$, where ${\prec }_{pq,\omega }:={\prec }_{\omega }\circ (p\otimes q),{•}_{pq,\omega }:={•}_{\omega }\circ (p\otimes q)$ and ${\succ }_{pq,\omega }:={\succ }_{\omega }\circ (p\otimes q)$ for each $\omega \in \mathsf{\Omega }$, is a matching BiHom-tridendriform algebra.

Moreover, suppose that $(D^{\prime },{\prec }_{\mathsf{\Omega }}^{\prime },{•}_{\mathsf{\Omega }}^{\prime },{\succ }_{\mathsf{\Omega }}^{\prime })$ is another matching tridendriform algebra and let $p^{\prime },q^{\prime }:D^{\prime }\to D^{\prime }$ be matching tridendriform algebra endomorphisms such that $p^{\prime }\circ q^{\prime }=q^{\prime }\circ p^{\prime }$. If $f:D\to D^{\prime }$ is a matching tridendriform algebra morphism that satisfies $f\circ p=p^{\prime }\circ f,f\circ q=q^{\prime }\circ f$, then

$$f:(D,{\prec }_{pq,\mathsf{\Omega }},{•}_{pq,\mathsf{\Omega }},{\succ }_{pq,\mathsf{\Omega }},p,q)\to (D^{\prime },{\prec }_{pq,\mathsf{\Omega }}^{\prime },{•}_{pq,\mathsf{\Omega }}^{\prime },{\succ }_{pq,\mathsf{\Omega }}^{\prime },p^{\prime },q^{\prime })$$

is a morphism of matching BiHom-tridendriform algebras.

Proof. 

We just prove Item (ii) and Item (i) can be proved similarly. Clearly, p and q are multiplicative with respect to ${\prec }_{pq,\omega },{•}_{pq,\omega }$ and ${\succ }_{pq,\omega }$. For any $a,b,c\in D$ and $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}& & \left(a{\prec }_{pq,\alpha }b\right){\prec }_{pq,\beta }q\left(c\right)=p\left(p\left(a\right){\prec }_{\alpha }q\left(b\right)\right){\prec }_{\beta }q\left(q\left(c\right)\right)=\left(p^2\left(a\right){\prec }_{\alpha }pq\left(b\right)\right){\prec }_{\beta }{q}^2\left(c\right),\hfill \\ & & p\left(a\right){\prec }_{pq,\alpha }\left(b{\prec }_{pq,\beta }c\right)=p\left(p\left(a\right)\right){\prec }_{\alpha }q\left(p\left(b\right){\prec }_{\beta }q\left(c\right)\right)=p^2\left(a\right){\prec }_{\alpha }\left(qp\left(b\right){\prec }_{\beta }{q}^2\left(c\right)\right),\hfill \\ & & p\left(a\right){\prec }_{pq,\beta }\left(b{\succ }_{pq,\alpha }c\right)=p\left(p\left(a\right)\right){\prec }_{\beta }q\left(p\left(b\right){\succ }_{\alpha }q\left(c\right)\right)=p^2\left(a\right){\prec }_{\beta }\left(qp\left(b\right){\succ }_{\alpha }{q}^2\left(c\right)\right),\hfill \\ & & p\left(a\right){\prec }_{pq,\alpha }\left(b{•}_{pq,\beta }c\right)=p\left(p\left(a\right)\right){\prec }_{\alpha }q\left(p\left(b\right){•}_{\beta }q\left(c\right)\right)=p^2\left(a\right){\prec }_{\alpha }\left(qp\left(b\right){•}_{\beta }{q}^2\left(c\right)\right).\hfill \end{array}$$

Thereby

$$\begin{array}{ccc}& & \left(a{\prec }_{pq,\alpha }b\right){\prec }_{pq,\beta }q\left(c\right)\hfill \\ & =& p\left(a\right){\prec }_{pq,\alpha }\left(b{\prec }_{pq,\beta }c\right)+p\left(a\right){\prec }_{pq,\beta }\left(b{\succ }_{pq,\alpha }c\right)+p\left(a\right){\prec }_{pq,\alpha }\left(b{•}_{pq,\beta }c\right),\hfill \end{array}$$

that is, (19) holds for $(D,{\prec }_{pq,\mathsf{\Omega }},{•}_{pq,\mathsf{\Omega }},{\succ }_{pq,\mathsf{\Omega }},p,q)$. Similarly, (20)–(25) hold. Thus $(D,{\prec }_{pq,\mathsf{\Omega }},{•}_{pq,\mathsf{\Omega }},{\succ }_{pq,\mathsf{\Omega }},p,q)$ is a matching BiHom-tridendriform algebra. And

$$\begin{array}{ccc}\hfill f\left(a\right){\prec }_{p^{\prime }{q}^{\prime },\alpha }^{\prime }f\left(b\right)& =& p^{\prime }\left(f\left(a\right)\right){\prec }_{\alpha }^{\prime }{q}^{\prime }\left(f\left(b\right)\right)\hfill \\ & =& f\left(p\left(a\right)\right){\prec }_{\alpha }^{\prime }f\left(q\left(b\right)\right)\hfill \\ & =& f\left(p\left(a\right){\prec }_{\alpha }q\left(b\right)\right)=f\left(a{\prec }_{pq,\alpha }b\right),\hfill \\ \hfill f\left(a\right){\succ }_{p^{\prime }{q}^{\prime },\alpha }^{\prime }f\left(b\right)& =& p^{\prime }\left(f\left(a\right)\right){\succ }_{\alpha }^{\prime }{q}^{\prime }\left(f\left(b\right)\right)\hfill \\ & =& f\left(p\left(a\right)\right){\succ }_{\alpha }^{\prime }f\left(q\left(b\right)\right)\hfill \\ & =& f\left(p\left(a\right){\succ }_{\alpha }q\left(b\right)\right)=f\left(a{\succ }_{pq,\alpha }b\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill f\left(a\right){•}_{p^{\prime }{q}^{\prime },\alpha }^{\prime }f\left(b\right)& =& p^{\prime }\left(f\left(a\right)\right){•}_{\alpha }^{\prime }{q}^{\prime }\left(f\left(b\right)\right)\hfill \\ & =& f\left(p\left(a\right)\right){•}_{\alpha }^{\prime }f\left(q\left(b\right)\right)\hfill \\ & =& f\left(p\left(a\right){•}_{\alpha }q\left(b\right)\right)=f\left(a{•}_{pq,\alpha }b\right).\hfill \end{array}$$

Thus $f:(D,{\prec }_{pq,\mathsf{\Omega }},{•}_{pq,\mathsf{\Omega }},{\succ }_{pq,\mathsf{\Omega }},p,q)\to (D^{\prime },{\prec }_{pq,\mathsf{\Omega }}^{\prime },{•}_{pq,\mathsf{\Omega }}^{\prime },{\succ }_{pq,\mathsf{\Omega }}^{\prime },p^{\prime },q^{\prime })$ is a morphism of matching BiHom-tridendriform algebras. □

Proposition 9.

Let I be an non-empty set. For each $i\in I$, let $A_i:\mathsf{\Omega }\to \mathbf{k}$ be a map with finite supports, identified with finite set $A_i={\left(x_{i,\omega }\right)}_{\omega \in \mathsf{\Omega }},x_{i,\omega }\in \mathbf{k}$.

(i)

Let $(D,{\prec }_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }},p,q)$ be a matching BiHom-dendriform algebra. Define the following bilinear operations:

$${\odot }_i:=\sum _{\omega \in \mathsf{\Omega }}{x}_{i,\omega }{\odot }_{\omega },where\odot \in \{\prec ,\succ \}andi\in I.$$

Then $(D,{\prec }_I,{\succ }_I,p,q)$ is also a matching BiHom-dendriform algebra.

(ii)

Let $(D,{\prec }_{\mathsf{\Omega }},{•}_{\omega },{\succ }_{\mathsf{\Omega }},p,q)$ be a matching BiHom-tridendriform algebra. Define the following bilinear operations:

$${\odot }_i:=\sum _{\omega \in \mathsf{\Omega }}{x}_{i,\omega }{\odot }_{\omega },where\odot \in \{\prec ,•,\succ \}andi\in I.$$

Then $(D,{\prec }_I,{•}_I,{\succ }_I,p,q)$ is also a matching BiHom-tridendriform algebra.

Proof. 

We just prove Item (ii) and Item (i) can be proved similarly. Clearly, p and q are multiplicative with respect to ${\prec }_i,{•}_i$ and ${\succ }_i$. For any $a,b,c\in D,i,j\in I$ and $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}& & \left(a{\prec }_{i}b\right){\prec }_{j}q\left(c\right)\hfill \\ & =& \sum _{\beta \in \mathsf{\Omega }}{y}_{j,\beta }(\sum _{\alpha \in \mathsf{\Omega }}{x}_{i,\alpha }a{\prec }_{\alpha }b){\prec }_{\beta }q\left(c\right)\hfill \\ & =& \sum _{\alpha \in \mathsf{\Omega }}\sum _{\beta \in \mathsf{\Omega }}{x}_{i,\alpha }{y}_{j,\beta }\left(\left(a{\prec }_{\alpha }b\right){\prec }_{\beta }q\left(c\right)\right)\hfill \\ & =& \sum _{\alpha \in \mathsf{\Omega }}\sum _{\beta \in \mathsf{\Omega }}{x}_{i,\alpha }{y}_{j,\beta }(p\left(a\right){\prec }_{\alpha }\left(b{\prec }_{\beta }c\right)+p\left(a\right){\prec }_{\beta }\left(b{\succ }_{\alpha }c\right)+p\left(a\right){\prec }_{\alpha }\left(b{•}_{\beta }c\right))\hfill \\ & =& \sum _{\alpha \in \mathsf{\Omega }}{x}_{i,\alpha }p\left(a\right){\prec }_{\alpha }(\sum _{\beta \in \mathsf{\Omega }}{y}_{j,\beta }b{\prec }_{\beta }c)+\sum _{\beta \in \mathsf{\Omega }}{y}_{j,\beta }p\left(a\right){\prec }_{\beta }(\sum _{\alpha \in \mathsf{\Omega }}{x}_{i,\alpha }b{\succ }_{\alpha }c)\hfill \\ & & +\sum _{\alpha \in \mathsf{\Omega }}{x}_{i,\alpha }p\left(a\right){\prec }_{\alpha }(\sum _{\beta \in \mathsf{\Omega }}{y}_{j,\beta }b{•}_{\beta }c))\hfill \\ & =& \sum _{\alpha \in \mathsf{\Omega }}{x}_{i,\alpha }p\left(a\right){\prec }_{\alpha }\left(b{\prec }_{j}c\right)+\sum _{\beta \in \mathsf{\Omega }}{y}_{j,\beta }p\left(a\right){\prec }_{\beta }\left(b{\succ }_{i}c\right)+\sum _{\alpha \in \mathsf{\Omega }}{x}_{i,\alpha }p\left(a\right){\prec }_{\alpha }\left(b{•}_{j}c\right))\hfill \\ & =& p\left(a\right){\prec }_i\left(b{\prec }_{j}c\right)+p\left(a\right){\prec }_j\left(b{\succ }_{i}c\right)+p\left(a\right){\prec }_i\left(b{•}_{j}c\right)).\hfill \end{array}$$

Hence, (19) holds. Similarly, (20)–(25) hold. Hence $(D,{\prec }_I,{•}_I,{\succ }_I,p,q)$ is a matching BiHom-tridendriform algebra. □

Theorem 4.

(i)

Let $(A,{\prec }_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }},p,q)$ be a matching BiHom-dendriform algebra. Then $(A,{·}_{\mathsf{\Omega }},p,q)$ is a compatible BiHom-associative algebra, where

$${·}_{\omega }:A\otimes A\to A,a{·}_{\omega }b:=a{\prec }_{\omega }b+a{\succ }_{\omega }bfora,b\in Aand\omega \in \mathsf{\Omega }.$$

(ii)

Let $(A,{\prec }_{\mathsf{\Omega }},{•}_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }},p,q)$ be a matching BiHom-tridendriform algebra. Then $(A,{·}_{\mathsf{\Omega }},p,q)$ is a compatible BiHom-associative algebra, where

$${·}_{\omega }:A\otimes A\to A,a{·}_{\omega }b:=a{\prec }_{\omega }b+a{•}_{\omega }b+a{\succ }_{\omega }bfora,b\in Aand\omega \in \mathsf{\Omega }.$$

Proof. 

We only prove Item (ii) and Item (i) can be proved similarly. Clearly, p and q are multiplicative with respect to ${·}_{\omega }$. For $a,b,c\in A$ and $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}& & \left(a{·}_{\alpha }b\right){·}_{\beta }q\left(c\right)+\left(a{·}_{\beta }b\right){·}_{\alpha }q\left(c\right)\hfill \\ & =& (a{\prec }_{\alpha }b+a{•}_{\alpha }b+a{\succ }_{\alpha }b){·}_{\beta }q\left(c\right)+(a{\prec }_{\beta }b+a{•}_{\beta }b+a{\succ }_{\beta }b){·}_{\alpha }q\left(c\right)\hfill \\ & =& (a{\prec }_{\alpha }b+a{•}_{\alpha }b+a{\succ }_{\alpha }b){\prec }_{\beta }q\left(c\right)+(a{\prec }_{\alpha }b+a{•}_{\alpha }b+a{\succ }_{\alpha }b){•}_{\beta }q\left(c\right)\hfill \\ & & +(a{\prec }_{\alpha }b+a{•}_{\alpha }b+a{\succ }_{\alpha }b){\succ }_{\beta }q\left(c\right)+(a{\prec }_{\beta }b+a{•}_{\beta }b+a{\succ }_{\beta }b){\prec }_{\alpha }q\left(c\right)\hfill \\ & & +(a{\prec }_{\beta }b+a{•}_{\beta }b+a{\succ }_{\beta }b){•}_{\alpha }q\left(c\right)+(a{\prec }_{\beta }b+a{•}_{\beta }b+a{\succ }_{\beta }b){\succ }_{\alpha }q\left(c\right)\hfill \\ & =& \left(a{\prec }_{\alpha }b\right){\prec }_{\beta }q\left(c\right)+\left(a{•}_{\alpha }b\right){\prec }_{\beta }q\left(c\right)+\left(a{\succ }_{\alpha }b\right){\prec }_{\beta }q\left(c\right)+\left(a{\prec }_{\alpha }b\right){•}_{\beta }q\left(c\right)\hfill \\ & & +\left(a{•}_{\alpha }b\right){•}_{\beta }q\left(c\right)+\left(a{\succ }_{\alpha }b\right){•}_{\beta }q\left(c\right)+\left(a{\prec }_{\alpha }b\right){\succ }_{\beta }q\left(c\right)+\left(a{•}_{\alpha }b\right){\succ }_{\beta }q\left(c\right)\hfill \\ & & +\left(a{\succ }_{\alpha }b\right){\succ }_{\beta }q\left(c\right)+\left(a{\prec }_{\beta }b\right){\prec }_{\alpha }q\left(c\right)+\left(a{•}_{\beta }b\right){\prec }_{\alpha }q\left(c\right)+\left(a{\succ }_{\beta }b\right){\prec }_{\alpha }q\left(c\right)\hfill \\ & & +\left(a{\prec }_{\beta }b\right){•}_{\alpha }q\left(c\right)+\left(a{•}_{\beta }b\right){•}_{\alpha }q\left(c\right)+\left(a{\succ }_{\beta }b\right){•}_{\alpha }q\left(c\right)\hfill \\ & & +\left(a{\prec }_{\beta }b\right){\succ }_{\alpha }q\left(c\right)+\left(a{•}_{\beta }b\right){\succ }_{\alpha }q\left(c\right)+\left(a{\succ }_{\beta }b\right){\succ }_{\alpha }q\left(c\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}& & p\left(a\right){·}_{\alpha }\left(b{·}_{\beta }c\right)+p\left(a\right){·}_{\beta }\left(b{·}_{\alpha }c\right)\hfill \\ & =& p\left(a\right){·}_{\alpha }(b{\prec }_{\beta }c+b{•}_{\beta }c+b{\succ }_{\beta }c)+p\left(a\right){·}_{\beta }(b{\prec }_{\alpha }c+b{•}_{\alpha }c+b{\succ }_{\alpha }c)\hfill \\ & =& p\left(a\right){\prec }_{\alpha }\left(b{\prec }_{\beta }c\right)+p\left(a\right){\prec }_{\alpha }\left(b{•}_{\beta }c\right)+p\left(a\right){\prec }_{\alpha }\left(b{\succ }_{\beta }c\right)+p\left(a\right){•}_{\alpha }\left(b{\prec }_{\beta }c\right)\hfill \\ & & +p\left(a\right){•}_{\alpha }\left(b{•}_{\beta }c\right)+p\left(a\right){•}_{\alpha }\left(b{\succ }_{\beta }c\right)+p\left(a\right){\succ }_{\alpha }\left(b{\prec }_{\beta }c\right)+p\left(a\right){\succ }_{\alpha }\left(b{•}_{\beta }c\right)\hfill \\ & & +p\left(a\right){\succ }_{\alpha }\left(b{\succ }_{\beta }c\right)+p\left(a\right){\prec }_{\beta }\left(b{\prec }_{\alpha }c\right)+p\left(a\right){\prec }_{\beta }\left(b{•}_{\alpha }c\right)+p\left(a\right){\prec }_{\beta }\left(b{\succ }_{\alpha }c\right)\hfill \\ & & +p\left(a\right){•}_{\beta }\left(b{\prec }_{\alpha }c\right)+p\left(a\right){•}_{\beta }\left(b{•}_{\alpha }c\right)+p\left(a\right){•}_{\beta }\left(b{\succ }_{\alpha }c\right)\hfill \\ & & +p\left(a\right){\succ }_{\beta }\left(b{\prec }_{\alpha }c\right)+p\left(a\right){\succ }_{\beta }\left(b{•}_{\alpha }c\right)+p\left(a\right){\succ }_{\beta }\left(b{\succ }_{\alpha }c\right).\hfill \end{array}$$

By (19)–(25), we get

$$\begin{array}{ccc}& & \left(a{·}_{\alpha }b\right){·}_{\beta }q\left(c\right)+\left(a{·}_{\beta }b\right){·}_{\alpha }q\left(c\right)=p\left(a\right){·}_{\alpha }\left(b{·}_{\beta }c\right)+p\left(a\right){·}_{\beta }\left(b{·}_{\alpha }c\right).\hfill \end{array}$$

Hence $(A,{·}_{\mathsf{\Omega }},p,q)$ is a compatible BiHom-associative algebra. □

Theorem 5.

Let $(A,{\prec }_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }},p,q)$ be a regular matching BiHom-dendriform algebra. Then $(A,{\ast }_{\mathsf{\Omega }},p,q)$ is a matching BiHom-pre-Lie algebra, where

$${\ast }_{\omega }:A\otimes A\to A,a{\ast }_{\omega }b:=a{\succ }_{\omega }b-\left(p^{-1}q\left(b\right)\right){\prec }_{\omega }\left(p{q}^{-1}\left(a\right)\right)fora,b\in Aand\omega \in \mathsf{\Omega }.$$

Proof. 

Obvious, p and q are multiplicative with respect to ${\ast }_{\omega }.$ For $a,b,c\in A$ and $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}& & pq\left(a\right){\ast }_{\alpha }\left(p\left(b\right){\ast }_{\beta }c\right)-\left(q\left(a\right){\ast }_{\alpha }p\left(b\right)\right){\ast }_{\beta }q\left(c\right)\hfill \\ & =& pq\left(a\right){\ast }_{\alpha }(p\left(b\right){\succ }_{\beta }c-p^{-1}q\left(c\right){\prec }_{\beta }p\left(b\right))-(q\left(a\right){\succ }_{\alpha }p\left(b\right)-q\left(b\right){\prec }_{\alpha }p\left(a\right)){\ast }_{\beta }q\left(c\right)\hfill \\ & =& pq\left(a\right){\succ }_{\alpha }(p\left(b\right){\succ }_{\beta }c-p^{-1}q\left(c\right){\prec }_{\beta }p\left(b\right))-p^{-1}q(p\left(b\right){\succ }_{\beta }c-p^{-1}q\left(c\right){\prec }_{\beta }p\left(b\right)){\prec }_{\alpha }p{q}^{-1}\left(pq\left(a\right)\right)\hfill \\ & & -(q\left(a\right){\succ }_{\alpha }p\left(b\right)-q\left(b\right){\prec }_{\alpha }p\left(a\right)){\succ }_{\beta }q\left(c\right)+p^{-1}q\left(q\left(c\right)\right){\prec }_{\beta }p{q}^{-1}(q\left(a\right){\succ }_{\alpha }p\left(b\right)-q\left(b\right){\prec }_{\alpha }p\left(a\right))\hfill \\ & =& pq\left(a\right){\succ }_{\alpha }\left(p\left(b\right){\succ }_{\beta }c\right)-pq\left(a\right){\succ }_{\alpha }\left(p^{-1}q\left(c\right){\prec }_{\beta }p\left(b\right)\right)-\left(q\left(b\right){\succ }_{\beta }{p}^{-1}q\left(c\right)\right){\prec }_{\alpha }{p}^2\left(a\right)\hfill \\ & & +\left(p^{-2}{q}^2\left(c\right){\prec }_{\beta }q\left(b\right)\right){\prec }_{\alpha }{p}^2\left(a\right)-\left(q\left(a\right){\succ }_{\alpha }p\left(b\right)\right){\succ }_{\beta }q\left(c\right)+\left(q\left(b\right){\prec }_{\alpha }p\left(a\right)\right){\succ }_{\beta }q\left(c\right)\hfill \\ & & +p^{-1}{q}^2\left(c\right){\prec }_{\beta }\left(p\left(a\right){\succ }_{\alpha }{p}^2{q}^{-1}\left(b\right)\right)-p^{-1}{q}^2\left(c\right){\prec }_{\beta }\left(p\left(b\right){\prec }_{\alpha }{p}^2{q}^{-1}\left(a\right)\right)\hfill \end{array}$$

and

$$\begin{array}{ccc}& & pq\left(b\right){\ast }_{\beta }\left(p\left(a\right){\ast }_{\alpha }c\right)-\left(q\left(b\right){\ast }_{\beta }p\left(a\right)\right){\ast }_{\alpha }q\left(c\right)\hfill \\ & =& pq\left(b\right){\ast }_{\beta }(p\left(a\right){\succ }_{\alpha }c-p^{-1}q\left(c\right){\prec }_{\alpha }p{q}^{-1}\left(p\left(a\right)\right))\hfill \\ & & -(q\left(b\right){\succ }_{\beta }p\left(a\right)-p^{-1}q\left(p\left(a\right)\right){\prec }_{\beta }p{q}^{-1}\left(q\left(b\right)\right)){\ast }_{\alpha }q\left(c\right)\hfill \\ & =& pq\left(b\right){\ast }_{\beta }(p\left(a\right){\succ }_{\alpha }c-p^{-1}q\left(c\right){\prec }_{\alpha }{p}^2{q}^{-1}\left(a\right))-(q\left(b\right){\succ }_{\beta }p\left(a\right)-q\left(a\right){\prec }_{\beta }p\left(b\right)){\ast }_{\alpha }q\left(c\right)\hfill \\ & =& pq\left(b\right){\succ }_{\beta }(p\left(a\right){\succ }_{\alpha }c-p^{-1}q\left(c\right){\prec }_{\alpha }{p}^2{q}^{-1}\left(a\right))-p^{-1}q(p\left(a\right){\succ }_{\alpha }c-p^{-1}q\left(c\right){\prec }_{\alpha }{p}^2{q}^{-1}\left(a\right)){\prec }_{\beta }pq\left(b\right)\hfill \\ & & -(q\left(b\right){\succ }_{\beta }p\left(a\right)-q\left(a\right){\prec }_{\beta }p\left(b\right)){\succ }_{\alpha }q\left(c\right)+p^{-1}q\left(q\left(c\right)\right){\prec }_{\alpha }p{q}^{-1}(q\left(b\right){\succ }_{\beta }p\left(a\right)-q\left(a\right){\prec }_{\beta }p\left(b\right))\hfill \\ & =& pq\left(b\right){\succ }_{\beta }\left(p\left(a\right){\succ }_{\alpha }c\right)-pq\left(b\right){\succ }_{\beta }\left(p^{-1}q\left(c\right){\prec }_{\alpha }{p}^2{q}^{-1}\left(a\right)\right)-\left(q\left(a\right){\succ }_{\alpha }{p}^{-1}q\left(c\right)\right){\prec }_{\beta }pq\left(b\right)\hfill \\ & & +\left(p^{-2}{q}^2\left(c\right){\prec }_{\alpha }p\left(a\right)\right){\prec }_{\beta }pq\left(b\right)-\left(q\left(b\right){\succ }_{\beta }p\left(a\right)\right){\succ }_{\alpha }q\left(c\right)+\left(q\left(a\right){\prec }_{\beta }p\left(b\right)\right){\succ }_{\alpha }q\left(c\right)\hfill \\ & & +p^{-1}{q}^2\left(c\right){\prec }_{\alpha }\left(p\left(b\right){\succ }_{\beta }{p}^2{q}^{-1}\left(a\right)\right)-p^{-1}{q}^2\left(c\right){\prec }_{\alpha }\left(p\left(a\right){\prec }_{\beta }{p}^2{q}^{-1}\left(b\right)\right).\hfill \end{array}$$

By (16)–(18), we get

$$\begin{array}{c}\hfill pq\left(a\right){\ast }_{\alpha }\left(p\left(b\right){\ast }_{\beta }c\right)-\left(q\left(a\right){\ast }_{\alpha }p\left(b\right)\right){\ast }_{\beta }q\left(c\right)=pq\left(b\right){\ast }_{\beta }\left(p\left(a\right){\ast }_{\alpha }c\right)-\left(q\left(b\right){\ast }_{\beta }p\left(a\right)\right){\ast }_{\alpha }q\left(c\right).\end{array}$$

Hence $(A,{\ast }_{\mathsf{\Omega }},p,q)$ is a matching BiHom-pre-Lie algebra. □

Proposition 10.

(i)

Let $(A,·,P_{\mathsf{\Omega }},p,q)$ be a matching BiHom-associative Rota-Baxter algebra of weight 0. Define the operations ${\prec }_{\omega }$ and ${\succ }_{\omega }$ for $\omega \in \mathsf{\Omega }$ by

$$a{\prec }_{\omega }b:=a·P_{\omega }\left(b\right)anda{\succ }_{\omega }b:=P_{\omega }\left(a\right)·b,fora,b\in A.$$

Then $(D,{\prec }_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }},p,q)$ is a matching BiHom-dendriform algebra.

(ii)

Let $(A,·,P_{\mathsf{\Omega }},p,q)$ be a matching BiHom-associative Rota-Baxter algebra. Define the operations ${\prec }_{\omega }$ and ${\succ }_{\omega }$ for $\omega \in \mathsf{\Omega }$ by

$$a{\prec }_{\omega }b:=a·P_{\omega }\left(b\right)+{\lambda }_{\omega }a·banda{\succ }_{\omega }b:=P_{\omega }\left(a\right)·b,fora,b\in A.$$

Then $(A,{\prec }_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }},p,q)$ is a matching BiHom-dendriform algebra.

Proof. 

Since Item (i) can be seen as a special case of Item (ii) by taking ${\lambda }_{\mathsf{\Omega }}=\left\{0\right\}$, we only prove Item (ii). Clearly, p and q are multiplicative with respect to ${\prec }_{\omega }$ and ${\succ }_{\omega }$. For $a,b,c\in A$ and $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}& & p\left(a\right){\prec }_{\alpha }\left(b{\prec }_{\beta }c\right)+p\left(a\right){\prec }_{\beta }\left(b{\succ }_{\alpha }c\right)\hfill \\ & =& p\left(a\right){\prec }_{\alpha }(b·P_{\beta }\left(c\right)+{\lambda }_{\beta }b·c)+p\left(a\right){\prec }_{\beta }(P_{\alpha }\left(b\right)·c)\hfill \\ & =& p\left(a\right)·P_{\alpha }(b·P_{\beta }\left(c\right)+{\lambda }_{\beta }b·c)+{\lambda }_{\alpha }p\left(a\right)·(b·P_{\beta }\left(c\right)+{\lambda }_{\beta }b·c)+p\left(a\right)·P_{\beta }(P_{\alpha }\left(b\right)·c)\hfill \\ & & +{\lambda }_{\beta }p\left(a\right)·(P_{\alpha }\left(b\right)·c)\hfill \\ & =& p\left(a\right)·P_{\alpha }(b·P_{\beta }\left(c\right))+p\left(a\right)·P_{\alpha }({\lambda }_{\beta }b·c)+{\lambda }_{\alpha }p\left(a\right)·(b·P_{\beta }\left(c\right))+{\lambda }_{\alpha }p\left(a\right)·({\lambda }_{\beta }b·c)\hfill \\ & & +p\left(a\right)·P_{\beta }(P_{\alpha }\left(b\right)·c)+{\lambda }_{\beta }p\left(a\right)·(P_{\alpha }\left(b\right)·c)\hfill \\ & =& p\left(a\right)·(P_{\alpha }\left(b\right)·P_{\beta }\left(c\right))+{\lambda }_{\alpha }p\left(a\right)·(b·P_{\beta }\left(c\right))+{\lambda }_{\alpha }{\lambda }_{\beta }p\left(a\right)·(b·c)+{\lambda }_{\beta }p\left(a\right)·(P_{\alpha }\left(b\right)·c)\hfill \\ & =& (a·P_{\alpha }\left(b\right))·P_{\beta }\left(q\left(c\right)\right)+{\lambda }_{\alpha }(a·b)·P_{\beta }\left(q\left(c\right)\right)+{\lambda }_{\alpha }{\lambda }_{\beta }(a·b)·q\left(c\right)+{\lambda }_{\beta }(a·P_{\alpha }\left(b\right))·q\left(c\right)\hfill \\ & =& (a·P_{\alpha }\left(b\right)+{\lambda }_{\alpha }a·b)·P_{\beta }\left(q\left(c\right)\right)+{\lambda }_{\beta }(a·P_{\alpha }\left(b\right)+{\lambda }_{\alpha }a·b)·q\left(c\right)\hfill \\ & =& (a·P_{\alpha }\left(b\right)+{\lambda }_{\alpha }a·b){\prec }_{\beta }q\left(c\right)=\left(a{\prec }_{\alpha }b\right){\prec }_{\beta }q\left(c\right).\hfill \end{array}$$

Also,

$$\begin{array}{ccc}& & \left(a{\succ }_{\alpha }b\right){\prec }_{\beta }q\left(c\right)\hfill \\ & =& (P_{\alpha }\left(a\right)·b){\prec }_{\beta }q\left(c\right)=(P_{\alpha }\left(a\right)·b)·P_{\beta }\left(q\left(c\right)\right)+{\lambda }_{\beta }\left((P_{\alpha }\left(a\right)·b)\right)·q\left(c\right)\hfill \\ & =& P_{\alpha }\left(p\left(a\right)\right)·(b·P_{\beta }\left(c\right))+{\lambda }_{\beta }{P}_{\alpha }\left(p\left(a\right)\right)·(b·c)\hfill \\ & =& P_{\alpha }\left(p\left(a\right)\right)·(b·P_{\beta }\left(c\right)+{\lambda }_{\beta }b·c)=P_{\alpha }\left(p\left(a\right)\right)·\left(b{\prec }_{\beta }c\right)\hfill \\ & =& p\left(a\right){\succ }_{\alpha }\left(b{\prec }_{\beta }c\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}& & \left(a{\prec }_{\beta }b\right){\succ }_{\alpha }q\left(c\right)+\left(a{\succ }_{\alpha }b\right){\succ }_{\beta }q\left(c\right)\hfill \\ & =& (a·P_{\beta }\left(b\right)+{\lambda }_{\beta }a·b){\succ }_{\alpha }q\left(c\right)+(P_{\alpha }\left(a\right)·b){\succ }_{\beta }q\left(c\right)\hfill \\ & =& P_{\alpha }(a·P_{\beta }\left(b\right)+{\lambda }_{\beta }a·b)·q\left(c\right)+P_{\beta }(P_{\alpha }\left(a\right)·b)·q\left(c\right)\hfill \\ & =& (P_{\alpha }(a·P_{\beta }\left(b\right))+P_{\beta }(P_{\alpha }\left(a\right)·b)+{\lambda }_{\beta }{P}_{\alpha }(a·b))·q\left(c\right)\hfill \\ & =& (P_{\alpha }\left(a\right)·P_{\beta }\left(b\right))·q\left(c\right)=P_{\alpha }\left(p\left(a\right)\right)·(P_{\beta }\left(b\right)·c)\hfill \\ & =& p\left(a\right){\succ }_{\alpha }\left(b{\succ }_{\beta }c\right).\hfill \end{array}$$

Hence $(A,{\prec }_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }},p,q)$ is a matching BiHom-dendriform algebra. □

Proposition 11.

Let $(A,·,P_{\mathsf{\Omega }},p,q)$ be a matching BiHom-associative Rota-Baxter algebra. Define the operations ${\prec }_{\omega }$, ${•}_{\mathsf{\Omega }}$ and ${\succ }_{\omega }$ for $\omega \in \mathsf{\Omega }$ by

$$\begin{array}{c}\hfill a{\prec }_{\omega }b:=a·P_{\omega }\left(b\right),a{•}_{\omega }b:={\lambda }_{\omega }a·banda{\succ }_{\omega }b:=P_{\omega }\left(a\right)·b,fora,b\in A.\end{array}$$

Then $(A,{\prec }_{\mathsf{\Omega }},{•}_{\mathsf{\Omega }},{\succ }_{\mathsf{\Omega }},p,q)$ is a matching BiHom-tridendriform algebra.

Proof. 

Clearly, p and q are multiplicative with respect to ${\prec }_{\omega }$, ${•}_{\omega }$ and ${\succ }_{\omega }$. For $a,b,c\in A$ and $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}\hfill \left(a{\prec }_{\alpha }b\right){\prec }_{\beta }q\left(c\right)& =& (a·P_{\alpha }\left(b\right))·P_{\beta }\left(q\left(c\right)\right)=p\left(a\right)·(P_{\alpha }\left(b\right)·P_{\beta }\left(c\right))\hfill \\ & =& p\left(a\right)·(P_{\alpha }(b·P_{\beta }\left(c\right))+P_{\beta }(P_{\alpha }\left(b\right)·c)+{\lambda }_{\beta }{P}_{\alpha }(b·c))\hfill \\ & =& p\left(a\right){\prec }_{\alpha }\left(b{\prec }_{\beta }c\right))+p\left(a\right){\prec }_{\beta }\left(b{\succ }_{\alpha }c\right)+a{\prec }_{\alpha }\left(b{•}_{\beta }c\right)),\hfill \\ \hfill \left(a{\succ }_{\alpha }b\right){\prec }_{\beta }q\left(c\right)& =& (P_{\alpha }\left(a\right)·b)·P_{\beta }\left(q\left(c\right)\right)=P_{\alpha }\left(p\left(a\right)\right)·(b·P_{\beta }\left(c\right))=p\left(a\right){\succ }_{\alpha }\left(b{\prec }_{\beta }c\right),\hfill \\ \hfill p\left(a\right){\succ }_{\alpha }\left(b{\succ }_{\beta }c\right)& =& P_{\alpha }\left(p\left(a\right)\right)·(P_{\beta }\left(b\right)·c)=(P_{\alpha }\left(a\right)·P_{\beta }\left(b\right))·q\left(c\right)\hfill \\ & =& (P_{\alpha }(a·P_{\beta }\left(b\right))+P_{\beta }(P_{\beta }\left(a\right)·b)+{\lambda }_{\beta }{P}_{\alpha }(a·b))·q\left(c\right)\hfill \\ & =& \left(a{\prec }_{\beta }b\right){\succ }_{\alpha }q\left(c\right)+\left(a{\succ }_{\alpha }b\right){\succ }_{\beta }q\left(c\right)+\left(a{•}_{\beta }b\right){\succ }_{\alpha }q\left(c\right),\hfill \\ \hfill \left(a{\succ }_{\alpha }b\right){•}_{\beta }q\left(c\right)& =& {\lambda }_{\beta }(P_{\alpha }\left(a\right)·b)·q\left(c\right)={\lambda }_{\beta }{P}_{\alpha }\left(p\left(a\right)\right)·(b·c)=p\left(a\right){\succ }_{\alpha }\left(b{•}_{\beta }c\right),\hfill \\ \hfill \left(a{\prec }_{\alpha }b\right){•}_{\beta }q\left(c\right)& =& {\lambda }_{\beta }(a·P_{\alpha }\left(b\right))·q\left(c\right)={\lambda }_{\beta }p\left(a\right)·(P_{\alpha }\left(b\right)·c)=p\left(a\right){•}_{\beta }\left(b{\succ }_{\alpha }c\right),\hfill \\ \hfill \left(a{•}_{\alpha }b\right){\prec }_{\beta }q\left(c\right)& =& {\lambda }_{\alpha }(a·b)·P_{\beta }\left(q\left(c\right)\right)={\lambda }_{\alpha }p\left(a\right)·(b·P_{\beta }\left(c\right))=p\left(a\right){•}_{\alpha }\left(b{\prec }_{\beta }c\right),\hfill \\ \hfill \left(a{•}_{\alpha }b\right){•}_{\beta }q\left(c\right)& =& {\lambda }_{\alpha }{\lambda }_{\beta }(a·b)·q\left(c\right)={\lambda }_{\alpha }{\lambda }_{\beta }p\left(a\right)·(b·c)=p\left(a\right){•}_{\alpha }\left(b{•}_{\beta }c\right)\hfill \end{array}$$

as required. □

Proposition 12.

(i)

Let $(A,·,P_{\mathsf{\Omega }},p,q)$ be a regular matching BiHom-associative Rota-Baxter algebra of weight 0. Then $(A,{\ast }_{\mathsf{\Omega }},p,q)$ is a matching BiHom-pre-Lie algebra, where

$$a{\ast }_{\omega }b:=P_{\omega }\left(a\right)·b-p^{-1}q\left(b\right)·q^{-1}p\left(P_{\omega }\left(a\right)\right)fora,b\in Aand\omega \in \mathsf{\Omega }.$$

(ii)

Let $(A,·,P_{\mathsf{\Omega }},p,q)$ be a regular matching BiHom-associative Rota-Baxter algebra. Then $(A,{\ast }_{\mathsf{\Omega }},p,q)$ is a matching BiHom-pre-Lie algebra, where

$$a{\ast }_{\omega }b:=P_{\omega }\left(a\right)·b-p^{-1}q\left(b\right)·q^{-1}p\left(P_{\omega }\left(a\right)\right)-{\lambda }_{\omega }{p}^{-1}q\left(b\right)·q^{-1}p\left(a\right)fora,b\in Aand\omega \in \mathsf{\Omega }.$$

Proof. 

(i)

It follows from Theorem 5 and Proposition 10 (i).

(ii)

It follows from Theorem 5 and Proposition 10 (ii).

□

Example 1.

We consider the following 2-dimensional regular matching BiHom-associative Rota-Baxter algebra of weight 0 $(A,·,P_{\mathsf{\Omega }},p,q)$, $\mathsf{\Omega }=\{1,2\}$, where bilinear operation ·, $p,q$ and $P_1,P_2,$ are defined, with respect to a basis $\{x,y\}$ ([8,10]), by

$$\begin{array}{ccc}& & x·x=x,x·y=bx+(1-a)y,y·x=\frac{b(1-a)}{a}x+ay,y·y=\frac{b}{a}y,\hfill \\ & & p\left(x\right)=x,p\left(y\right)=\frac{b(1-a)}{a}x+ay,q\left(x\right)=x,q\left(y\right)=bx+(1-a)y,\hfill \\ & & P_1\left(x\right)=0,P_1\left(y\right)=rx,P_2\left(x\right)=sx+ty,P_2\left(y\right)=-\frac{s^2}{t}x-sy.\hfill \end{array}$$

where $a,b,r,s,t$ are parameters in k with $a\ne 0,1,t\ne 0.$ By Proposition 12, we have

$$\begin{array}{ccc}& & x{\ast }_{1}x=0,x{\ast }_{1}y=0,y{\ast }_{1}x=0,y{\ast }_{1}y=0,\hfill \\ & & x{\ast }_{2}x=0,x{\ast }_{2}y=0,y{\ast }_{2}x=0,y{\ast }_{2}y=0.\hfill \end{array}$$

Obviously, the ${\ast }_{\mathsf{\Omega }}$ operations obtained above satisfies the Definition of matching BiHom-pre-Lie algebra.

5. Matching BiHom-Lie Rota-Baxter Algebra

In this section, as a generalization of [9,17,20,22], we introduce the notion of matching BiHom-Lie Rota-Baxter algebras and discuss its properties. The main results of this section are given as the following categories commutative graph.

Definition 14.

Let ${\lambda }_{\mathsf{\Omega }}:={\left({\lambda }_{\omega }\right)}_{\omega \in \mathsf{\Omega }}$ be a family indexed by Ω. A matching BiHom-Lie Rota-Baxter algebra is a 5-tuple $(\mathfrak{g},[,],P_{\mathsf{\Omega }},p,q)$ consisting of a BiHom-Lie algebra $(\mathfrak{g},[,],p,q)$, a family $P_{\mathsf{\Omega }}:={\left(P_{\omega }\right)}_{\omega \in \mathsf{\Omega }}$ of linear operators $P_{\omega }:A\to A,\omega \in \mathsf{\Omega }$ satisfying the following conditions:

$$p\circ q=q\circ p,$$

$$p\circ P_{\omega }=P_{\omega }\circ p,q\circ P_{\omega }=P_{\omega }\circ q,$$

(26)

$$\begin{array}{cc}& [P_{\alpha }\left(a\right),P_{\beta }\left(b\right)]=P_{\alpha }\left([a,P_{\beta }\left(b\right)]\right)+P_{\beta }\left([P_{\alpha }\left(a\right),b]\right)+{\lambda }_{\beta }{P}_{\alpha }\left([a,b]\right),\end{array}$$

(27)

for all $a,b\in \mathfrak{g}$ and $\alpha ,\beta ,\omega \in \mathsf{\Omega }$. In particular, if p and q are algebra automorphisms, we call it a regular matching BiHom-Lie Rota-Baxter algebra.

Theorem 6.

Let $(\mathfrak{g},[,],P_{\mathsf{\Omega }})$ be a matching Lie Rota-Baxter algebra and $p,q:\mathfrak{g}\to \mathfrak{g}$ be two commuting Lie algebra endomorphism such that $p\circ P_{\omega }=P_{\omega }\circ p,q\circ P_{\omega }=P_{\omega }\circ q$ for each $\omega \in \mathsf{\Omega }$. Then $(\mathfrak{g},{[,]}_{pq},P_{\mathsf{\Omega }},p,q)$ is a matching BiHom-Lie Rota-Baxter algebra, where ${[,]}_{pq}:=[,]\circ (p\otimes q)$.

Proof. 

Clearly, p and q are multiplicative with respect to ${[,]}_{pq}$. In addition, for $a,b,c\in \mathfrak{g}$, we have

$$\begin{array}{ccc}\hfill {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}_{pq}]}_{pq}& =& {[q^2\left(a\right),[pq\left(b\right),qp\left(c\right)]]}_{pq}\hfill \\ & =& [p{q}^2\left(a\right),q[pq\left(b\right),q\left(p\left(c\right)\right)]]=p{q}^2[a,[b,c]].\hfill \end{array}$$

By the Jacobi identity and skew symmetry of $(\mathfrak{g},[,\left]\right)$, the BiHom-Jacobi identity and BiHom-skew symmetry of $(\mathfrak{g},{[,]}_{pq},p,q)$ holds. Hence $(\mathfrak{g},{[,]}_{pq},p,q)$ is a BiHom-Lie algebra.

For $a,b\in \mathfrak{g}$ and $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}\hfill {[P_{\alpha }\left(a\right),P_{\beta }\left(b\right)]}_{pq}& =& [P_{\alpha }\left(p\left(a\right)\right),P_{\beta }\left(q\left(b\right)\right)]\hfill \\ & =& P_{\alpha }\left([p\left(a\right),P_{\beta }\left(q\left(b\right)\right)]\right)+P_{\beta }\left([P_{\alpha }\left(p\left(a\right)\right),q\left(b\right)]\right)+{\lambda }_{\beta }{P}_{\alpha }\left([p\left(a\right),q\left(b\right)]\right)\hfill \\ & =& P_{\alpha }\left([p\left(a\right),q\left(P_{\beta }\left(b\right)\right)]\right)+P_{\beta }\left([p\left(P_{\alpha }\left(a\right)\right),q\left(b\right)]\right)+{\lambda }_{\beta }{P}_{\alpha }\left([p\left(a\right),q\left(b\right)]\right)\hfill \\ & =& P_{\alpha }\left({[a,P_{\beta }\left(b\right)]}_{pq}\right)+P_{\beta }\left({[P_{\alpha }\left(a\right),b]}_{pq}\right)+{\lambda }_{\beta }{P}_{\alpha }\left({[a,b]}_{pq}\right),\hfill \end{array}$$

as required. □

Theorem 7.

Let $(\mathfrak{g},[,],P_{\mathsf{\Omega }},p,q)$ be a regular matching BiHom-Lie Rota-Baxter algebra. Then $(\mathfrak{g},{[,]}_{p^{-1}{q}^{-1}},P_{\mathsf{\Omega }})$ is a matching Lie Rota-Baxter algebra, where ${[,]}_{p^{-1}{q}^{-1}}:=[,]\circ (p^{-1}\otimes q^{-1})$.

Proof. 

Clearly, p and q are multiplicative with respect to ${[,]}_{p^{-1}{q}^{-1}}$. In addition, for $a,b,c\in \mathfrak{g}$, we have

$$\begin{array}{ccc}\hfill {[a,[b,c]]}_{p^{-1}{q}^{-1}}{]}_{p^{-1}{q}^{-1}}& =& {[a,[p^{-1}\left(b\right),q^{-1}\left(c\right)]]}_{p^{-1}{q}^{-1}}\hfill \\ & =& [p^{-1}\left(a\right),q^{-1}\left[p^{-1}\left(b\right)\right),q^{-1}\left(c\right)]]\hfill \\ & =& p^{-1}{q}^{-2}[q^2\left(a\right),[q\left(b\right),p\left(c\right)].\hfill \end{array}$$

By the BiHom-Jacobi identity and BiHom-skew symmetry of $(\mathfrak{g},[,],p,q)$, the Jacobi identity and skew symmetry of $(\mathfrak{g},{[,]}_{p^{-1}{q}^{-1}})$ holds. Hence $(\mathfrak{g},{[,]}_{p^{-1}{q}^{-1}})$ is a Lie algebra.

For $a,b\in \mathfrak{g}$ and $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}& & {[P_{\alpha }\left(a\right),P_{\beta }\left(b\right)]}_{p^{-1}{q}^{-1}}=[P_{\alpha }\left(p^{-1}\left(a\right)\right),P_{\beta }\left(q^{-1}\left(b\right)\right)]\hfill \\ & & =P_{\alpha }\left([p^{-1}\left(a\right),P_{\beta }\left(q^{-1}\left(b\right)\right)]\right)+P_{\beta }\left([P_{\alpha }\left(p^{-1}\left(a\right)\right),q^{-1}\left(b\right)]\right)+{\lambda }_{\beta }{P}_{\alpha }\left([p^{-1}\left(a\right),q^{-1}\left(b\right)]\right)\hfill \\ & & =P_{\alpha }\left([p^{-1}\left(a\right),q^{-1}\left(P_{\beta }\left(b\right)\right)]\right)+P_{\beta }\left([p^{-1}\left(P_{\alpha }\left(a\right)\right),q^{-1}\left(b\right)]\right)+{\lambda }_{\beta }{P}_{\alpha }\left([p^{-1}\left(a\right),q^{-1}\left(b\right)]\right)\hfill \\ & & =P_{\alpha }\left({[a,P_{\beta }\left(b\right)]}_{p^{-1}{q}^{-1}}\right)+P_{\beta }\left({[P_{\alpha }\left(a\right),b]}_{p^{-1}{q}^{-1}}\right)+{\lambda }_{\beta }{P}_{\alpha }\left({[a,b]}_{p^{-1}{q}^{-1}}\right),\hfill \end{array}$$

as required. □

Definition 15.

Let $(\mathfrak{g},[,],p,q)$ be a BiHom-Lie algebra and $n\ge 0$. The n-th derived BiHom-algebra of $\mathfrak{g}$ is defined by

$${\mathfrak{g}}_{\left(n\right)}=(\mathfrak{g},{[,]}^{\left(n\right)}=[,]\circ (p^n\otimes q^n),p^{n+1},q^{n+1}).$$

Theorem 8.

Let $(\mathfrak{g},[,],P_{\mathsf{\Omega }},p,q)$ be a matching BiHom-Lie Rota-Baxter algebra. Then its n-th derived BiHom-algebra is a matching BiHom-Lie Rota-Baxter algebra.

Proof. 

Clearly, p and q are multiplicative with respect to ${[,]}^{\left(n\right)}$. In addition, for $a,b,c\in \mathfrak{g}$, we have

$$\begin{array}{ccc}\hfill {[q^2\left(a\right),{[q\left(b\right),p\left(c\right)]}^{\left(n\right)}]}^{\left(n\right)}& =& {[q^2\left(a\right),[p^{n}q\left(b\right),q^{n}p\left(c\right)]]}^{\left(n\right)}\hfill \\ & =& [p^n{q}^2\left(a\right),q^n[p^{n}q\left(b\right),q^{n}p\left(c\right)]]\hfill \\ & =& [q^2\left(p^n\left(a\right)\right),[q\left(q^n{p}^n\left(b\right)\right),p\left(q^{2n}\left(c\right)\right)]].\hfill \end{array}$$

Hence $(\mathfrak{g},{[,]}^{\left(n\right)},p^{n+1},q^{n+1})$ is a BiHom-Lie algebra. For $a,b\in \mathfrak{g}$ and $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}& & {[P_{\alpha }\left(a\right),P_{\beta }\left(b\right)]}^{\left(n\right)}\hfill \\ & =& [P_{\alpha }\left(p^n\left(a\right)\right),P_{\beta }\left(q^n\left(b\right)\right)]\hfill \\ & =& P_{\alpha }\left([p^n\left(a\right),P_{\beta }\left(q^n\left(b\right)\right)]\right)+P_{\beta }\left([P_{\alpha }\left(p^n\left(a\right)\right),q^n\left(b\right)]\right)+{\lambda }_{\beta }{P}_{\alpha }\left([p^n\left(a\right),q^n\left(b\right)]\right)\hfill \\ & =& P_{\alpha }\left([p^n\left(a\right),q^n\left(P_{\beta }\left(b\right)\right)]\right)+P_{\beta }\left([p^n\left(P_{\alpha }\left(a\right)\right),q^n\left(b\right)]\right)+{\lambda }_{\beta }{P}_{\alpha }\left([p^n\left(a\right),q^n\left(b\right)]\right)\hfill \\ & =& P_{\alpha }\left({[a,P_{\beta }\left(b\right)]}^{\left(n\right)}\right)+P_{\beta }\left({[P_{\alpha }\left(a\right),b]}^{\left(n\right)}\right)+{\lambda }_{\beta }{P}_{\alpha }\left({[a,b]}^{\left(n\right)}\right),\hfill \end{array}$$

as required. □

Proposition 13.

Let $(A,·,P_{\mathsf{\Omega }},p,q)$ be a regular matching BiHom-associative Rota-Baxter algebra. Then $(A,[,],P_{\mathsf{\Omega }},p,q)$ is a matching BiHom-Lie Rota-Baxter algebra, where

$$\begin{array}{c}\hfill [a,b]=a·b-p^{-1}q\left(b\right)·p{q}^{-1}\left(a\right)fora,b\in A.\end{array}$$

(28)

Proof. 

Following [8], $(A,[,],p,q)$ is a BiHom-Lie algebra. For all $a,b\in A,\alpha ,\beta \in \mathsf{\Omega }$, by (28), we have

$$\begin{array}{ccc}& & [P_{\alpha }\left(a\right),P_{\beta }\left(b\right)]=P_{\alpha }\left(a\right)·P_{\beta }\left(b\right)-p^{-1}q\left(P_{\beta }\left(b\right)\right)·p{q}^{-1}\left(P_{\alpha }\left(a\right)\right)=P_{\alpha }\left(a\right)·P_{\beta }\left(b\right)-P_{\beta }\left(p^{-1}q\left(b\right)\right)·P_{\alpha }\left(p{q}^{-1}\left(a\right)\right),\hfill \\ & & [a,P_{\beta }\left(b\right)]=a·P_{\beta }\left(b\right)-p^{-1}q\left(P_{\beta }\left(b\right)\right)·p{q}^{-1}\left(a\right)=a·P_{\beta }\left(b\right)-P_{\beta }\left(p^{-1}q\left(b\right)\right)·p{q}^{-1}\left(a\right),\hfill \\ & & [P_{\alpha }\left(a\right),b]=P_{\alpha }\left(a\right)·b-p^{-1}q\left(b\right)·p{q}^{-1}\left(P_{\alpha }\left(a\right)\right)=P_{\alpha }\left(a\right)·b-p^{-1}q\left(b\right)·P_{\alpha }\left(p{q}^{-1}\left(a\right)\right).\hfill \end{array}$$

By the matching Rota-Baxter identity (15), there are the following identity,

$$[P_{\alpha }\left(a\right),P_{\beta }\left(b\right)]=P_{\alpha }\left([a,P_{\beta }\left(b\right)]\right)+P_{\beta }\left([P_{\alpha }\left(a\right),b]\right)+{\lambda }_{\beta }{P}_{\alpha }\left([a,b]\right).$$

Hence $(A,[,],P_{\mathsf{\Omega }},p,q)$ is a matching BiHom-Lie Rota-Baxter algebra. □

Proposition 14.

Let $(\mathfrak{g},[,],P_{\mathsf{\Omega }},p,q)$ be a regular matching BiHom-Lie Rota-Baxter algebra of weight zero. Then $(\mathfrak{g},{\ast }_{\mathsf{\Omega }},p,q)$ is a matching BiHom-pre-Lie algebra, where

$$\begin{array}{c}\hfill a{\ast }_{\omega }b=[P_{\omega }\left(a\right),b]fora,b\in \mathfrak{g}and\omega \in \mathsf{\Omega }.\end{array}$$

Proof. 

Clearly, p and q are multiplicative with respect to ${\ast }_{\omega }$. In addition, for $a,b,c\in \mathfrak{g}$ and $\alpha ,\beta \in \mathsf{\Omega }$, we have

$$\begin{array}{ccc}& & pq\left(a\right){\ast }_{\alpha }\left(p\left(b\right){\ast }_{\beta }c\right)-\left(q\left(a\right){\ast }_{\alpha }p\left(b\right)\right){\ast }_{\beta }q\left(c\right)\hfill \\ & =& [P_{\alpha }\left(pq\left(a\right)\right),[P_{\beta }\left(p\left(b\right)\right),c]]-[P_{\beta }\left([P_{\alpha }\left(q\left(a\right)\right),p\left(b\right)]\right),q\left(c\right)]\hfill \\ & =& [P_{\alpha }\left(pq\left(a\right)\right),[P_{\beta }\left(p\left(b\right)\right),c]]-[[P_{\alpha }\left(q\left(a\right)\right),P_{\beta }\left(p\left(b\right)\right)],q\left(c\right)]+[P_{\alpha }\left([q\left(a\right),P_{\beta }\left(p\left(b\right)\right)]\right),q\left(c\right)]\hfill \\ & =& [P_{\alpha }\left(pq\left(a\right)\right),[p\left(P_{\beta }\left(b\right)\right),c]]-[q[P_{\alpha }\left(a\right),q^{-1}p\left(P_{\beta }\left(b\right)\right)],p\left(p^{-1}q\left(c\right)\right]+[P_{\alpha }\left([q\left(a\right),p\left(P_{\beta }\left(b\right)\right)]\right),q\left(c\right)]\hfill \\ & =& \left[P_{\alpha }\left(pq\left(a\right)\right)\right),[p\left(P_{\beta }\left(b\right)\right),c]]+[q\left(p^{-1}q\left(c\right)\right),p[P_{\alpha }\left(a\right),q^{-1}p\left(P_{\beta }\left(b\right)\right)]]-[P_{\alpha }\left([q\left(P_{\beta }\left(b\right)\right),p\left(a\right)]\right),q\left(c\right)],\hfill \\ & =& [q^2\left(P_{\alpha }\left(q^{-1}p\left(a\right)\right)\right),\left[q(P_{\beta }\left(q^{-1}p\left(b\right)\right),p\left(p^{-1}\left(c\right)\right)]\right]+[q^2\left(p^{-1}\left(c\right)\right),[q\left(P_{\alpha }\left(q^{-1}p\left(a\right)\right)\right),p\left(P_{\beta }\left(q^{-1}p\left(b\right)\right)\right)]]\hfill \\ & & -[P_{\alpha }\left([q\left(P_{\beta }\left(b\right)\right),p\left(a\right)]\right),q\left(c\right)],\hfill \end{array}$$

and

$$\begin{array}{ccc}& & pq\left(b\right){\ast }_{\beta }\left(p\left(a\right){\ast }_{\alpha }c\right)-\left(q\left(b\right){\ast }_{\beta }p\left(a\right)\right){\ast }_{\alpha }q\left(c\right)\hfill \\ & =& [P_{\beta }\left(pq\left(b\right)\right),[P_{\alpha }\left(p\left(a\right)\right),c]]-[P_{\alpha }\left([P_{\beta }\left(q\left(b\right)\right),p\left(a\right)]\right),q\left(c\right)]\hfill \\ & =& [q^2\left(P_{\beta }\left(q^{-1}p\left(b\right)\right)\right),[q\left(P_{\alpha }\left(q^{-1}p\left(a\right)\right)\right),p\left(p^{-1}\left(c\right)\right)]]-[P_{\alpha }\left([q\left(P_{\beta }\left(b\right)\right),p\left(a\right)]\right),q\left(c\right)]\hfill \\ & =& -[q^2\left(P_{\beta }\left(q^{-1}p\left(b\right)\right)\right),[q\left(p^{-1}\left(c\right)\right),p\left(P_{\alpha }\left(q^{-1}p\left(a\right)\right)\right)]]-[P_{\alpha }\left([q\left(P_{\beta }\left(b\right)\right),p\left(a\right)]\right),q\left(c\right)].\hfill \end{array}$$

By the BiHom-Jacobi identity (3), it is not hard to check that

$$\begin{array}{ccc}& & pq\left(a\right){\ast }_{\alpha }\left(p\left(b\right){\ast }_{\beta }c\right)-\left(q\left(a\right){\ast }_{\alpha }p\left(b\right)\right){\ast }_{\beta }q\left(c\right)=pq\left(b\right){\ast }_{\beta }\left(p\left(a\right){\ast }_{\alpha }c\right)-\left(q\left(b\right){\ast }_{\beta }p\left(a\right)\right){\ast }_{\alpha }q\left(c\right).\hfill \end{array}$$

Hence $(\mathfrak{g},{\ast }_{\mathsf{\Omega }},p,q)$ is a matching BiHom-pre-Lie algebra. □

6. Conclusions

In this paper, we introduce the notions of matching BiHom-type algebras. Their construction methods are studied in detail. Further, we study the properties and relationships between categories of these matching BiHom-type algebraic structures. Here, in the future, we intend to try to further study the cohomology and deformation of these matching BiHom-type algebras.