  For a given hereditary abelian category satisfying some finiteness
conditions, in certain twisted cases it is shown that the modified Ringel-Hall
algebra is isomorphic to the naive lattice algebra and there exists an
epimorphism from the modified Ringel-Hall algebra to the lattice algebra.
Furthermore, the kernel of this epimorphism is described explicitly. Finally,
we show that the naive lattice algebra is invariant under the derived
equivalences of hereditary abelian categories.
