The Rota-Baxter algebra and the shuffle product are both algebraic structures arising from integral operators and integral equations. Free commutative Rota-Baxter algebras provide an algebraic framework for integral equations with the simple Riemann integral operator. The Zinbiel algebras form a category in which the shuffle product algebra is the free object. Motivated by algebraic structures underlying integral equations involving multiple integral operators and kernels, we study commutative matching Rota-Baxter algebras and construct the free objects making use of the shuffle product with multiple decorations. We also construct free commutative matching Rota-Baxter algebras in a relative context, to emulate the action of the integral operators on the coefficient functions in an integral equation. We finally show that free commutative matching Rota-Baxter algebras give the free matching Zinbiel algebra, generalizing the characterization of the shuffle product algebra as the free Zinbiel algebra obtained by Loday.

Rota-Baxter 代数和 shuffle 乘积都是由积分算子和积分方程产生的代数结构。自由可交换 Rota-Baxter 代数为具有简单黎曼积分算子的积分方程提供了代数框架。Zinbiel 代数形成一个范畴，其中 shuffle 乘积代数是自由对象。受涉及多个积分算子和核的积分方程的代数结构的启发，我们研究了交换匹配 Rota-Baxter 代数，并利用带有多重装饰的洗牌乘积构造自由对象。我们还在相对上下文中构造了自由交换匹配 Rota-Baxter 代数，以模拟积分算子对积分方程中系数函数的作用。