  We write a 3D equation for three fermions by combining the three two-body
potentials obtained in 3D reductions (based on a series expansion around a
relative-energy fixing "approximation" of the free propagators) of the
corresponding two-fermion Bethe-Salpeter equations to equivalent 3D equations,
putting the third fermion on its positive-energy mass shell. In this way, the
cluster-separated limits are exact, and the Lorentz invariance / cluster
separability requirement is automatically satisfied, provided no supplementary
approximation, like the Born approximation, is made. The use of positive
free-energy projectors in the chosen reductions of the two-fermion
Bethe-Salpeter equations prevents continuum dissolution in our 3D three-fermion
equation. The potentials are hermitian below the inelastic threshold and depend
only slowly on the total three-fermion energy. This "hand-made" three-fermion
3D equation is also obtained by starting with an approximation of the
three-fermion Bethe-Salpeter equation, in which the three-body kernel is
neglected and the two-body kernels approached by positive-energy instantaneous
expressions, with the spectator fermion on the mass shell. The neglected terms
are then transformed into corrections to the 3D equation, in three steps
implying each a series expansion. The result is of course complicated, but the
lowest-order contributions of these correction terms to the energy spectrum
remain manageable.We also present some other 3D reduction procedures and
compare them to our's: use of Sazdjian's covariant approximation of the free
propagator, 3D reductions performed by a series expansion around instantaneous
approximations of the kernels instead of "approximations" of the propagators,
Gross' spectator model.
