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 U1,2β-connected. This paper proves that U1,2β is the only connected
matroid N such that if M is N-connected with β£E(M)β£>β£E(N)β£, then M\e or M/e is N-connected for all elements e. Moreover, we
show that U1,2β and M(W2β) are the only connected 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 U0,1ββU1,1β-connected if and only if every clonal
class of M is trivial.Comment: 13 page