  Recently, two solutions have been proposed to the long standing problem of
$\mathcal{CP}$-symmetry on the lattice, which is particularly evident when
considering the construction of chiral gauge theories. The first, based on a
lattice modification of $\mathcal{CP}$ was presented by Igarashi and Pawlowski;
the second by myself using the renormalisation group and Ginsparg-Wilson
relation. In this work, I combine the two approaches and show that they are
each part of a more general framework related to an underlying renormalisation
group. I continue by formulating Weyl and Majorana fermions on the lattice, and
discussing applications to the fermion propagator in the presence of the Higgs
field.
