  Considering a semicoherent system made up of $n$ components having i.i.d.
continuous lifetimes, Samaniego defined its structural signature as the
$n$-tuple whose $k$-th coordinate is the probability that the $k$-th component
failure causes the system to fail. This $n$-tuple, which depends only on the
structure of the system and not on the distribution of the component lifetimes,
is a very useful tool in the theoretical analysis of coherent systems.
  It was shown in two independent recent papers how the structural signature of
a system partitioned into two disjoint modules can be computed from the
signatures of these modules. In this work we consider the general case of a
system partitioned into an arbitrary number of disjoint modules organized in an
arbitrary way and we provide a general formula for the signature of the system
in terms of the signatures of the modules.
  The concept of signature was recently extended to the general case of
semicoherent systems whose components may have dependent lifetimes. The same
definition for the $n$-tuple gives rise to the probability signature, which may
depend on both the structure of the system and the probability distribution of
the component lifetimes. In this general setting, we show how under a natural
condition on the distribution of the lifetimes, the probability signature of
the system can be expressed in terms of the probability signatures of the
modules. We finally discuss a few situations where this condition holds in the
non-i.i.d. and nonexchangeable cases and provide some applications of the main
results.
