  The primary objective of this paper is to establish an algebraic framework
for the space of weakly slice regular functions over several quaternionic
variables. We recently introduced a $*$-product that maintains the path-slice
property within the class of path-slice functions. It is noteworthy that this
$*$-product is directly applicable to weakly slice regular functions, as every
slice regular function defined on a slice-open set inherently possesses
path-slice properties. Building on this foundation, we propose a precise
definition of an open neighborhood for a path $\gamma$ in the path space
$\mathscr{P}(\mathbb{C}^n)$. This definition is pivotal in establishing the
holomorphism of stem functions. Consequently, we demonstrate that the
$*$-product of two weakly slice regular functions retains its weakly slice
regular nature. This retention is facilitated by holomorphy of stem functions
and their relationship with weakly slice regular functions, providing a
comprehensive algebraic structure for this class of functions.
