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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 (σ-weakly continuous) idempotent states are investigated and a duality between normal idempotent states on a locally compact quantum group G and on its dual pG is established. Additionally we analyze the question of 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. [ABSTRACT FROM AUTHOR]