In this thesis, we investigate whether union and intersection preserve Nash and subgame perfect implementability. Nash implementability is known to be preserved under union. Here we first show that, under some reasonably mild assumptions, Nash implementability is also preserved under intersection. The conjunction of these two results yields an almost lattice-like structure for Nash implementable social choice rules. Next, we carry over these results to subgame perfect implementability by employing similar arguments. Finally, based on the fact that Nash implementable social choice rules are closed under union, we provide a new characterization of Nash implementability, which also exemplifies the potential use of our findings for further research.