This work presents a systematic and complete generalization of Operational Mathematics and Meta-Operational Mathematics to the framework of exterior algebra (Grassmann algebra). The essential features of exterior algebra—the anticommutativity of the wedge product ω∧η = (−1)deg(ω) deg(η)η∧ω, the grading structure, nilpotency (the square of any odd element is zero), and the natural correspondence with differential forms—are enforced throughout the entire theory. We re-establish the axiomatic system (10 axioms) for operations on the exterior algebra, construct integer-order, fractional-order, real-order and complex-order iterations, prove the existence of Schr¨oder functions and Abel functions in the graded setting, develop fractional calculus on exterior forms (where the exterior differential satisfies d2 = 0 leading to drastic simplifications for fractional derivatives of order ¿1), and build the calculus of variations on exterior algebras with signed Euler–Lagrange equations. The meta operational framework is extended to the exterior algebra: we construct the graded endomorphism operad End∧(C∧) and prove that it carries a graded Hopf operad structure, with the coproduct, counit and antipode adapted to the anticommutativity. An explicit Hopf algebra morphism from the unary meta operations to an exterior algebra version of the Connes–Kreimer renormalization Hopf algebra is constructed, where Feynman rules carry signs coming from the anticommutative vertex permutations. Bornological convergence is introduced on the exterior algebra to handle infinite meta operations; nilpotency makes many infinite series finite, thereby simplifying convergence conditions. The path integral on Grassmann variables is reinterpreted as a trace on the operad, connecting to supersymmetric topological quantum field theory. All classical special functions (exponential, trigonometric, error function, Gamma, Zeta, Theta, hypergeometric) are embedded into the exterior algebra, where they reduce to linear functions on odd variables. Self action of operations is analysed; the square of any odd valued operation vanishes, leading to immediate termination of iterations. Functional equations become equalities of meta operations. A 2 category of meta operations is constructed, and higher dimensional generalizations are outlined. Numerical algorithms are dramatically simplified because the exponential meta operation on an odd valued argument reduces to the identity plus the argument. Every open problem from the original work is transformed into a precise conjecture or a theorem in the exterior algebra setting; many become solvable due to nilpotency. This work provides a unified language for supersymmetric quantum field theory, superstring theory, noncommutative geometry on supermanifolds, and all mathematical structures that involve anticommuting variables.
shifa liu (Wed,) studied this question.