There are several clutters (antichains of sets) that can be associated with matroids or other combinatorial objects, like the collection of circuits or of bases of a matroid, or the collection of minimal dominating sets of a graph. This talk is motivated by the following question: given a clutter L and a favourite family of clutters F, which clutters from F are closest to L?
We rephrase the problem in lattice theoretic terms, and characterize the families F for which there exist such closest clutters. We give algorithms to find them when F is a family that arises from matroids. We also look at the case where one requires that each of the clutters involved covers the ground set.