  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 \match
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.
