  The present paper is devoted to present a unifying survey about some special
classes of crystallizations of compact PL $4$-manifolds with empty or connected
boundary, called {\it semi-simple} and {\it weak semi-simple crystallizations},
with a particular attention to their properties of minimizing combinatorially
defined PL-invariants, such as the {\it regular genus}, the {\it Gurau degree},
the {\it gem-complexity} and the {\it (gem-induced) trisection genus}. The main
theorem, yielding a summarizing result on the topic, is an original
contribution. Moreover, in the present paper the additivity of regular genus
with respect to connected sum is proved to hold for all compact $4$-manifolds
with empty or connected boundary which admit weak semi-simple crystallizations.
