  We consider a large class of monomial maps respecting an action of the
infinite symmetric group, and prove that the toric ideals arising as their
kernels are finitely generated up to symmetry. Our class includes many
important examples where Noetherianity was recently proved or conjectured. In
particular, our results imply Hillar-Sullivant's Independent Set Theorem and
settle several finiteness conjectures due to Aschenbrenner, Martin del Campo,
Hillar, and Sullivant.
  We introduce a matching monoid and show that its monoid ring is Noetherian up
to symmetry. Our approach is then to factorize a more general equivariant
monomial map into two parts going through this monoid. The kernels of both
parts are finitely generated up to symmetry: recent work by
Yamaguchi-Ogawa-Takemura on the (generalized) Birkhoff model provides an
explicit degree bound for the kernel of the first part, while for the second
part the finiteness follows from the Noetherianity of the matching monoid ring.
