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

Algebra of slice regular functions on non-symmetric domains in several quaternionic variables