  We investigate the theory of finite observables, i.e., resolutions of the
finite-dimensional identity by means of positive operators, that have a
physical interpretation in terms of measurement schemes. We focus on extremal
and rank-one observables and consider various constructions that reduce
observables to simpler ones. However, these constructions do not suffice to
generate all finite extremal observables, as we show by means of
counter-examples.
