  We study lattice operations on the set of idempotent states on a locally
compact quantum group corresponding to the operations of intersection of
compact subgroups and forming the subgroup generated by two compact subgroups.
Normal ($\sigma$-weakly continuous) idempotent states are investigated and a
duality between normal idempotent states on a locally compact quantum group
$\mathbb{G}$ and on its dual $\widehat{\mathbb{G}}$ is established.
Additionally we analyze the question when a left coideal corresponding
canonically to an idempotent state is finite dimensional and give a
characterization of normal idempotent states on compact quantum groups.
