  We revisit the issue of the geometrical separability of the Hilbert space of
physical states on lattice Abelian theories in the context of entanglement
entropy. We discuss the conditions under which vectors in the Hilbert space, as
well as the gauge invariant algebra, admit a tensor product decomposition with
a geometrical interpretation. With the exception of pure gauge lattices with
periodic boundary conditions which contain topological degrees of freedom, we
show that the Hilbert space is geometrically separable.
