  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 $U_{1,2}$-connected. This paper proves that $U_{1,2}$ is the only connected
matroid $N$ such that if $M$ is $N$-connected with $|E(M)| > |E(N)|$, then $M
\backslash e$ or $M / e$ is $N$-connected for all elements $e$. Moreover, we
show that $U_{1,2}$ and $M(\mathcal{W}_2)$ 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 $U_{0,1} \oplus U_{1,1}$-connected if and only if every clonal
class of $M$ is trivial.
