  Dempster-Shafer Theory (DST) generalizes Bayesian probability theory,
offering useful additional information, but suffers from a much higher
computational burden. A lot of work has been done to reduce the time complexity
of information fusion with Dempster's rule, which is a pointwise multiplication
of two zeta transforms, and optimal general algorithms have been found to get
the complete definition of these transforms. Yet, it is shown in this paper
that the zeta transform and its inverse, the M\"obius transform, can be exactly
simplified, fitting the quantity of information contained in belief functions.
Beyond that, this simplification actually works for any function on any
partially ordered set. It relies on a new notion that we call focal point and
that constitutes the smallest domain on which both the zeta and M\"obius
transforms can be defined. We demonstrate the interest of these general results
for DST, not only for the reduction in complexity of most transformations
between belief representations and their fusion, but also for theoretical
purposes. Indeed, we provide a new generalization of the conjunctive
decomposition of evidence and formulas uncovering how each decomposition weight
is tied to the corresponding mass function.
