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The Rota-Baxter algebra and the shuffle product are both algebraic structuresarising from integral operators and integral equations. Free commutativeRota-Baxter algebras provide an algebraic framework for integral equations withthe simple Riemann integral operator. The Zinbiel algebras form a category inwhich the shuffle product algebra is the free object. Motivated by algebraicstructures underlying integral equations involving multiple integral operatorsand kernels, we study commutative matching Rota-Baxter algebras and constructthe free objects making use of the shuffle product with multiple decorations.We also construct free commutative matching Rota-Baxter algebras in a relativecontext, to emulate the action of the integral operators on the coefficientfunctions in an integral equation. We finally show that free commutative \matchRota-Baxter algebras give the free matching Zinbiel algebra, generalizing thecharacterization of the shuffle product algebra as the free Zinbiel algebraobtained by Loday.

Commutative matching Rota-Baxter operators, shuffle products with decorations and matching Zinbiel algebras