  The partial information decomposition (PID) framework is concerned with
decomposing the information that a set of random variables has with respect to
a target variable into three types of components: redundant, synergistic, and
unique. Classical information theory alone does not provide a unique way to
decompose information in this manner and additional assumptions have to be
made. Recently, Kolchinsky proposed a new general axiomatic approach to obtain
measures of redundant information, based on choosing an order relation between
information sources (equivalently, order between communication channels). In
this paper, we exploit this approach to introduce three new measures of
redundant information (and the resulting decompositions) based on well-known
preorders between channels, thus contributing to the enrichment of the PID
landscape. We relate the new decompositions to existing ones, study some of
their properties, and provide examples illustrating their novelty. As a side
result, we prove that any preorder that satisfies Kolchinsky's axioms yields a
decomposition that meets the axioms originally introduced by Williams and Beer
when they first propose the PID.
