  In this paper we are developing a theory of rational (pseudo) difference
Hamiltonian operators, focusing in particular on its algebraic aspects. We show
that a pseudo--difference Hamiltonian operator can be represented as a ratio
$AB^{-1}$ of two difference operators with coefficients from a difference field
$\mathcal{F}$ where $A$ is preHamiltonian. A difference operator $A$ is called
preHamiltonian if its image is a Lie subalgebra with respect to the Lie bracket
of evolutionary vector fields on $\mathcal{F}$. We show that a skew-symmetric
difference operator is Hamiltonian if and only if it is preHamiltonian and
satisfies simply verifiable conditions on its coefficients. We show that if $H$
is a rational Hamiltonian operator, then to find a second Hamiltonian operator
$K$ compatible with $H$ is the same as to find a preHamiltonian pair $A$ and
$B$ such that $AB^{-1}H$ is skew-symmetric. We apply our theory to non-trivial
multi-Hamiltonian structures of Narita-Itoh-Bogoyavlensky and Adler-Postnikov
equations.
