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This paper extends the $*$-product from slice analysis to weakly sliceanalysis in several quaternionic variables, focusing on non-axially symmetricdomains. It diverges from traditional applications in axially symmetric domainsto address slice regularity in more complicated cases. The approach involvesredefining the $*$-product for path-slice functions, borrowing techniques fromstrongly slice analysis. Key to this work is the introduction of relativestem-preserving set pairs and real-path-connected sets, which help establish adirect link between path-slice functions and their stem functions. The studyculminates in conditions under which weakly slice regular functions form analgebra in specific slice domains, broadening the scope of slice analysis.

Path-slice star-product on non-axially symmetric domains in several quaternionic variables