  We prove that the finite representation property holds for representation by
partial functions for the signature consisting of composition, intersection,
domain and range and for any expansion of this signature by the antidomain,
fixset, preferential union, maximum iterate and opposite operations. The proof
shows that, for all these signatures, the size of base required is bounded by a
double-exponential function of the size of the algebra. This establishes that
representability of finite algebras is decidable for all these signatures. We
also give an example of a signature for which the finite representation
property fails to hold for representation by partial functions.
