As to the puzzle, we can check the idea that a closure operator is a monad of a monotone map \\(T:P \to P\\) defined in a poset \\(\(P,\leq\)\\). The unit law implies that \\(\forall p\in P: p \leq T(p)\\), that's OK, but the multiplication law gives only \\(\forall p \in P: T.T(p) \leq T(p)\\), while we need equality \\(T.T(p)=T(p)\\). I cheated and went to the nLab for "closure operator", and there they demand that the monad be idempotent also, and that give us the equality.