We develop a function theory on a three-dimensional reduced quaternionic model endowed with a projected (and, therefore, non-associative) product, together with its natural dual extension generated by a nilpotent infinitesimal unit. After introducing the associated first-order Dirac-type system, we construct explicit Cauchy kernels and prove a Cauchy–Pompeiu representation for sufficiently smooth functions with values in the dual algebra. We derive a Teodorescu-type right inverse, Liouville- and uniqueness-type principles, and residue formulas for isolated singularities. For smooth hypersurfaces, we establish Plemelj–Sokhotski boundary limits for the Cauchy transform and its dual lift. Worked examples illustrate how the reduced product interacts with boundary geometry and provide a practical route to computation.
Ji Eun Kim (Thu,) studied this question.