Non-generators in complete lattices and semilattices

As is well-known, in a finitary algebraic structure the set Γ of all the non-generators is the intersection of all the maximal proper substructures. In particular, Γ is a substructure. We show that the corresponding statements hold for complete semilattices but fail for complete lattices, when as the notion of substructure we take complete subsemilattices and complete sublattices, respectively