  For $d\geq 2$, the regular genus of a closed connected PL $d$-manifold $M$ is
the least genus (resp., half of the genus) of an orientable (resp., a
non-orientable) surface into which a crystallization of $M$ imbeds regularly.
The regular genus of every orientable surface equals its genus, and the regular
genus of every 3-manifold equals its Heegaard genus. For every closed connected
PL $4$-manifold $M$, it is known that its regular genus $\mathcal G(M)$ is at
least $2 \chi (M) + 5m -4$, where $m$ is the rank of the fundamental group of
$M$. In this article, we introduce the concept of "weak semi-simple
crystallization" for every closed connected PL $4$-manifold $M$, and prove that
$\mathcal G(M)= 2 \chi (M) + 5m -4$ if and only if $M$ admits a weak
semi-simple crystallization. We then show that the PL invariant regular genus
is additive under the connected sum within the class of all PL 4-manifolds
admitting a weak semi-simple crystallization. Also, we note that this property
is related to the 4-dimensional Smooth Poincar\'e Conjecture.
