This work develops a complete and self-contained extension of the full apparatus of Meta Operational Mathematics to group operations and their compositional inverses (left division and right division). The central philosophical principle—that operations upon operations constitute meta-operations—is established with complete mathematical precision through a four-level hierarchical framework and a rigorous existence theorem for the meta-operational universe. Within this framework, a general (Lie) group G provides the fundamental binary operation of group multiplication together with its two inverse operations, and these are shown to admit canonical lifts to meta-operations via composition. A fundamental distinction from the hyperbolic, elliptic, Gamma, Beta, Zeta, and logical cases is established: the group operation satisfies associativity, the inverse property, and the Mal’cev identities for left division, rather than a periodicity, addition formula, functional equation, or finite truth-table definability. This leads to the Group Duality Axiom, in which the Lie algebra of the operation group furnishes the primitive space and the exponential meta-operation provides the link between the group and its linearisation. Two essential new features of group operations—their non-idempotent nature and the possibility of weighted parametrisation by external parameters—are systematically elevated to the meta-operational level as independent, fully developed chapters. The non-idempotence of group multiplication implies that the self-action (iteration) of any non-trivial operation generates an infinite tower of distinct operations, leading to a rich dynamical theory and to the non-degeneracy of the Lie algebraic structure. Weighted parametrisation introduces smooth families of operations depending on an auxiliary parameter space, unifying the treatment of coupling constants, renormalisation group flows, and quantum deformations within a single algebraic framework. The seven irreducible fundamental meta-operations generating the whole Group operad are: composition, pointwise multiplication, pointwise inversion, differentiation, the exponential meta-operation, the identity operation, and the constant unit operation. Left and right division are derived operations expressed via multiplication and inversion. All conjectures previously stated as open problems are resolved as theorems within the body of this paper, most notably the Rigidity Theorem for the Group Hopf Operad, the BPHZ Forest Formula within the Group Operad for the non-linear sigma model, the q-Deformed Group Operad yielding quantum knot invariants, the extension of the entire framework to smooth Moufang loops, the Fourier–Mukai equivalence for abelian groups, and several others. The solved conjectures are fully integrated into the main text; the remaining open problems are precisely formulated and listed in Chapter 18.
Liu S (Wed,) studied this question.
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