  We introduce the concept of subsignature for semicoherent systems as a class
of indexes that range from the system signature to the Barlow-Proschan
importance index. Specifically, given a nonempty subset M of the set of
components of a system, we define the M-signature of the system as the
|M|-tuple whose k-th coordinate is the probability that the k-th failure among
the components in M causes the system to fail. We give various explicit linear
expressions for this probability in terms of the structure function and the
distribution of the component lifetimes. We also examine the case of
exchangeable lifetimes and the special case when M is a modular set.
