Chiral structure of the solutions of the Ginsparg-Wilson relation

We analyse the structure of solutions of the Ginsparg-Wilson relation for lattice Dirac operator in topologically trivial gauge sector. We show that the properties of such solutions relating to the perturbative stability of the pole of the fermion propagator as well as to the structure of the Yukawa models based on these solutions are solely determined by the non-local chirally invariant part of these Dirac operators. Depending on the structure of this part, the pole in the fermion propagator may or may not be stable under radiative corrections. We illustrate this by explicit examples.