  We study the problem of three ultracold fermions in different hyperfine
states loaded into a lattice with spatial dimension D=1,2. We consider
SU(3)-symmetric attractive interactions and also eventually include a
three-body constraint, which mimics the effect of three-body losses in the
strong-loss regime. We combine exact diagonalization with the Lanczos
algorithm, and evaluate both the eigenvalues and the eigenstates of the
problem. In D=1, we find that the ground state is always a three-body bound
state (trion) for arbitrarily small interaction, while in D=2, due to the
stronger influence of finite-size effects, we are not able to provide
conclusive evidence of the existence of a finite threshold for trion formation.
Our data are however compatible with a threshold value which vanishes
logarithmically with the size of the system. Moreover we are able to identify
the presence of a fine structure inside the spectrum, which is associated with
off-site trionic states. The characterization of these states shows that only
the long-distance behavior of the eigenstate wavefunctions provides clear-cut
signatures about the nature of bound states and that onsite observables are not
enough to discriminate between them. The inclusion of a three-body constraint
due to losses promotes these off-site trions to the role of lowest energy
states, at least in the strong-coupling regime.
