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We consider a large class of monomial maps respecting an action of theinfinite symmetric group, and prove that the toric ideals arising as theirkernels are finitely generated up to symmetry. Our class includes manyimportant examples where Noetherianity was recently proved or conjectured. Inparticular, our results imply Hillar-Sullivant's Independent Set Theorem andsettle 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 upto symmetry. Our approach is then to factorize a more general equivariantmonomial map into two parts going through this monoid. The kernels of bothparts are finitely generated up to symmetry: recent work byYamaguchi-Ogawa-Takemura on the (generalized) Birkhoff model provides anexplicit degree bound for the kernel of the first part, while for the secondpart the finiteness follows from the Noetherianity of the matching monoid ring.

Noetherianity for infinite-dimensional toric varieties