On computable presentations of some functional lattices

We prove in a uniform way that the following lattices have no computable presentations: the lattice of all computable order theoretic automorphisms of the rational numbers; the lattice of the computable functions from the rational numbers to the rational numbers having continuous extensions to functions on the real numbers; and the lattice of the monotonic functions on the natural numbers. Nevertheless, we prove that the lattice of all computable mappings from the rational numbers to the rational numbers has a computable presentation.