We relate finite generation of cones, monoids, and ideals in increasing chains (the local situation) to equivariant finite generation of the corresponding limit objects (the global situation). For cones and monoids, there is no analog of Noetherianity as in the case of ideals, and we demonstrate this in examples. As a remedy we find local-global correspondences for finite generation. These results are derived from a more general framework that relates finite generation under closure operations to equivariant finite generation under general families of maps. We also give a new proof that nonsaturated Inc-invariant chains of ideals stabilize, closing a gap in the literature.

It has been shown recently that monomial maps in a large class respecting the action of the infinite symmetric group have, up to symmetry, finitely generated kernels. We study the simplest nontrivial family in this class: the maps given by a single monomial. Considering the corresponding lattice map, we explicitly construct an equivariant lattice generating set, whose width (the number of variables necessary to write it down) depends linearly on the width of the map. This result is sharp and improves dramatically the previously known upper bound as it does not depend on the degree of the image monomial. In the case of of width two, we construct an explicit finite set of binomials generating the toric ideal up to symmetry. Both width and degree of this generating set are sharply bounded by linear functions in the exponents of the monomial.