Recently, the mathematical formalism of the Dyson boson mapping has been
extended to a system of 3n fermions, leading to the boson-fermion and the
baryon mapping. In the present paper, the case of restriction to a subset of
three-fermion quantum numbers, the collective indices, is discussed. A theory
is developed for the representation of fermionic states and operators in a
truncated ideal space where only collective boson-fermion pairs or collective
ideal baryons are allowed. An exact reproduction of physical properties is
proved to be possible provided that the original fermionic problem can be
solved in a subspace where all three-fermion subsystems carry collective
indices. Examples of simple applications are presented in the  subsequent
papers of this series.