Regularization of $\max(0,\cdot)$ as an operator in a Hilbert space

Question

Let $H$ be a vector lattice for a (partial) ordering $\leq$. Hence $\max(a,b)$ is defined for $a, b \in H$. Where can I find theory regarding the regularization of $\max(0,\cdot)$ as an operator? By regularization, I mean a sequence $f_n$ with each $f_n$ differentiable (in some sense) that has certain properties with $f_n(\cdot) \to \max(0,\cdot)$.