  Only requiring that Dirac operators decribing massless fermions on the
lattice decompose into Weyl operators we arrive at a large class of them. After
deriving general relations from spectral representations we study correlation
functions of Weyl fermions for any value of the index, stressing the related
conditions for basis transformations and getting the precise behaviors under
gauge and CP transformations. Using the detailed structure of the chiral
projections we also obtain a form of the correlation functions with a
determinant in the general case.
