Published April 17, 2026 | Version version 1
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Quaternionic Meta-Operational Mathematics: From Iteration of Operations to Operations on Operations

Authors/Creators

  • 1. peking university

Description

This paper systematically extends Meta-Operational Mathematics to the quaternionic domain H, elevating quaternion-valued operations themselves to the status of independent mathematical objects. We study meta-operations (composition, translation, exponentiation, logarithm, differentiation, integration, variation, infinite sums, infinite compositions) acting on quaternionic operations. An axiomatic system of ten axioms is established and verified in the quaternionic setting. The category of quaternionic meta-operations is shown to carry an endomorphism operad structure, which is further endowed with a Hopf operad structure. A concrete Hopf algebra morphism from the one-ary meta-operations to the Connes–Kreimer renormalization Hopf algebra is constructed, thereby embedding renormalization group theory into the quaternionic meta-operational framework. Bornological convergence is introduced to handle infinite quaternionic meta-operations and is applied to spectral triples in quaternionic noncommutative geometry. The path integral is reinterpreted as a trace on the operad, connecting to quaternionic topological quantum field theory. All classical quaternionic special functions (trigonometric, exponential, error, Gamma, Zeta, Theta, hypergeometric, etc.) are shown to belong to the quaternionic meta-operational universe, and their fundamental identities become equations of meta-operations. Open problems are reformulated as precise conjectures. This work provides a unified language connecting quaternionic analysis, algebra, geometry, topology, and quantum field theory, with particular emphasis on the essential adjustments forced by the non-commutativity of quaternionic multiplication and the theory of slice-regular functions.

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Alternative title (English)
Quaternionic Meta-Operational Mathematics

Dates

Submitted
2025-12-31

References

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