Pre-Riesz spaces are ordered vector spaces which can be order densely
embedded into vector lattices, their so-called vector lattice covers. Given a
vector lattice cover Y for a pre-Riesz space X, we address the question how
to find vector lattice covers for subspaces of X, such as ideals and bands.
We provide conditions such that for a directed ideal I in X its smallest
extension ideal in Y is a vector lattice cover. We show a criterion for bands
in X and their extension bands in Y as well. Moreover, we state properties
of ideals and bands in X which are generated by sets, and of their extensions
in Y