Objective functions with redundant domains

Let \((E,{ \mathcal{A}})\) be a set system consisting of a finite collection \({ \mathcal{A}}\) of subsets of a ground set E, and suppose that we have a function ϕ which maps \({ \mathcal{A}}\) into some set S. Now removing a subset K from E gives a restriction \({ \mathcal{A}}(\bar{K})\) to those sets of \({ \mathcal{A}}\) disjoint from K, and we have a corresponding restriction \(\phi|_{\hspace {.02in}{ \mathcal{A}}(\bar{K})}\) of our function ϕ. If the removal of K does not affect the image set of ϕ, that is \(\mbox {Im}(\phi|_{\hspace {.02in}{ \mathcal{A}}(\bar{X})})=\mbox {Im}(\phi)\), then we will say that K is a kernel set of \({ \mathcal{A}}\) 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 \(\mathrm {Ker}_{\phi}({ \mathcal{A}})\). Motivated by the optimisation theme, we ask which kernel systems are matroids. For instance, if \({ \mathcal{A}}\) 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 \(\mathrm {Ker}_{\phi}({ \mathcal{A}})\) is isomorphic to the disjoint sum of the cocycle matroids of the differently coloured subgraphs; on the other hand, if \({ \mathcal{A}}\) 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 \(\mathrm {Ker}_{\phi}({ \mathcal{A}})\) is essentially never a matroid.