  We analyze the lattice fermion kinetic term using PT symmetry, R-hermiticity,
and $\gamma_{5}$-hermiticity. R-hermiticity is a condition for Hermite action
and it is related to $\gamma_{5}$-hermiticity and PT symmetry. Assuming that a
translation-invariant kinetic term with continuum and periodic function does
not have PT symmetry, it can have R-hermiticity or $\gamma_{5}$-hermiticity. We
prove that a kinetic term with continuum and periodic function that is PT
symmetric does not reduce doublers. As a simple example, we analyze the
two-dimensional two-flavor Gross-Neveu model with minimal doubling fermions.
The minimal doubling fermions break PT symmetry and R-hermiticity, hence
complex or non-Hermite coupling constants are caused by quantum correction.
