  Motivated by recent examples of three-dimensional lattice Hamiltonians with
massless Dirac fermions in their (bulk) spectrum, I revisit the problem of
fermion doubling on bipartite lattices. The number of components of the Dirac
fermion in a time-reversal and parity invariant d-dimensional lattice system is
determined by the minimal representation of the Clifford algebra of $d+1$
Hermitian Dirac matrices that allows a construction of the time-reversal
operator with the square of unity, and it equals $2^d$ for $d=2,3$. Possible
mass-terms for (spinless) Dirac fermions are listed and discussed. In three
dimensions there are altogether eight independent masses, out of which four are
even, and four are odd under time reversal. A specific violation of
time-reversal symmetry that leads to (minimal) four-component massless Dirac
fermion in three dimensions at low energies is constructed.
