Disconnected coverings for oriented matroids via simultaneous mutations

Let Un,r be a uniform oriented matroid having as bases, B, all r-subsets (resp. as circuits, C, all (r+1)-subsets) of {1,…,n}. We say that C1⊆C is a covering, of Un,r, if for any base B∈B there is a circuit C∈C1 such that B⊂C. Let G(C1) be the graph having as set of vertices the elements of C1 and where two vertices are joined if they have one base in common. We say that C1⊆C is a connected covering if C1 is a covering and G(C1) is connected. It is easy to show that if a covering is connected then it completely determines Un,r. In this note, we show that connectivity is not always necessary.