  In a system of three stochastic variables, the Partial Information
Decomposition (PID) of Williams and Beer dissects the information that two
variables (sources) carry about a third variable (target) into nonnegative
information atoms that describe redundant, unique, and synergistic modes of
dependencies among the variables. However, the classification of the three
variables into two sources and one target limits the dependency modes that can
be quantitatively resolved, and does not naturally suit all systems. Here, we
extend the PID to describe trivariate modes of dependencies in full generality,
without introducing additional decomposition axioms or making assumptions about
the target/source nature of the variables. By comparing different PID lattices
of the same system, we unveil a finer PID structure made of seven nonnegative
information subatoms that are invariant to different target/source
classifications and that are sufficient to construct any PID lattice. This
finer structure naturally splits redundant information into two nonnegative
components: the source redundancy, which arises from the pairwise correlations
between the source variables, and the non-source redundancy, which does not,
and relates to the synergistic information the sources carry about the target.
The invariant structure is also sufficient to construct the system's entropy,
hence it characterizes completely all the interdependencies in the system.
