  The problem of existence and uniqueness of a state of a joint system with
given restrictions to subsystems is studied for a Fermion system, where a novel
feature is non-commutativity between algebras of subsystems.
  For an arbitrary (finite or infinite) number of given subsystems, a product
state extension is shown to exist if and only if all states of subsystems
except at most one are even (with respect to the Fermion number). If the states
of all subsystems are pure, then the same condition is shown to be necessary
and sufficient for the existence of any joint extension. If the condition
holds, the unique product state extension is the only joint extension.
  For a pair of subsystems, with one of the given subsystem states pure, a
necessary and sufficient condition for the existence of a joint extension and
the form of all joint extensions
  (unique for almost all cases) are given.
  For a pair of subsystems with non-pure subsystem states, some classes of
examples of joint extensions are given where non-uniqueness of joint extensions
prevails.
