Generalized version of the compatibility theorem: two examples

In a previous work ([3]) we proved that the Nguyen's condition for $[f(\w A)]_\alpha$ to be equal to $~f(A_\alpha)~$ also holds for the most general class of the $L$-fuzzy subsets, where $~L~$ is an arbitrary lattice. Here we recall the main points of the proof ad present some examples related to non-linear lattices.