Objective functions with redundant domains

Let (E,A) be a set system consisting of a finite collection A of subsets of a ground set E, and suppose that we have a function ϕ which maps A into some set S. Now removing a subset K from E gives a restriction A(K¯) to those sets of A disjoint from K, and we have a corresponding restriction ϕ|A(K¯) of our function ϕ. If the removal of K does not affect the image set of ϕ, that is Im(ϕ|A(X¯))=Im(ϕ), then we will say that K is a kernel set of A with respect to ϕ. Such sets are potentially useful in optimisation problems defined in terms of ϕ. We will call the set of all subsets of E that are kernel sets with respect to ϕ a kernel system and denote it by Kerϕ(A). Motivated by the optimisation theme, we ask which kernel systems are matroids. For instance, if A is the collection of forests in a graph G with coloured edges and ϕ counts how many edges of each colour occurs in a forest then Kerϕ(A) is isomorphic to the disjoint sum of the cocycle matroids of the differently coloured subgraphs; on the other hand, if A is the power set of a set of positive integers, and ϕ is the function which takes the values 1 and 0 on subsets according to whether they are sum-free or not, then we show that Kerϕ(A) is essentially never a matroid.