  We set up an algebraic theory of multivariable integration, based on a
hierarchy of Rota-Baxter operators and an action of the matrix monoid as linear
substitutions. Given a suitable coefficient domain with a bialgebra structure,
this allows us to build an operator ring that acts naturally on the given
Rota-Baxter hierarchy. We conjecture that the operator relations are a
noncommutative Groebner basis for the ideal they generate.
