A notion of minor-based matroid connectivity

For a matroid N, a matroid M is N-connected if every two elements of M are in an N-minor together. Thus a matroid is connected if and only if it is U1,2-connected. This paper proves that U1,2 is the only connected matroid N such that if M is N-connected with |E(M)|>|E(N)|, then M\e or M/e is N-connected for all elements e. Moreover, we show that U1,2 and M(W2) are the only matroids N such that, whenever a matroid has an N-minor using {e,f} and an N-minor using {f,g}, it also has an N-minor using {e,g}. Finally, we show that M is U0,1⊕U1,1-connected if and only if every clonal class of M is trivial.