Yes, the class of DC functions (difference-convex, or delta-convex as they are sometimes called) is a lattice: it's closed under taking pointwise max or min. It suffices to show that for any two DC functions $f_1, f_2$ their maximum $\max(f_1, f_2)$ is also DC.
Note that $\max(f_1, f_2) = \frac12|f_1+f_2| + \frac12|f_1-f_2|$, where both $f_1 \pm f_2 $ are DC. Thus, it suffices to show that the absolute value of a DC function is DC.
If $f$ is DC, then from $f = h-g$ we get $\max(f, 0) = \max(h, g) - g$, hence $\max(f, 0)$ is also DC.
And then $|f| = \max(f, 0) + \max(-f, 0)$ is DC as well.
This proof, and much more, can be found in On difference convexity of locally Lipschitz functions by Miroslav Bačak and Jonathan M. Borwein.