We show that for several notions of largeness in a semigroup, if u,v ∈ N, A is a u x v matrix satisfying restrictions that vary with the notion of largeness, and if C is a large subset of N, then {x ∈ N^v:Ax ∈ C^u} is large in N^v. We show that in most cases the restrictions on A are necessary. Several other results, including some generalizations, are also obtained. Included is a simple proof that if u > 1, then β(N^v) is not isomorphic to (βN)^u.

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