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The Q-Calculus: A Quaternion-Based Laws of Form System

  • Louis H. Kauffman
  • Arthur M. Collings
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Abstract

This paper introduces a Laws of Form version of the Quaternions. We call this the Q-Calculus, a 16-valued extension of Laws of Form (LoF) which is closely related to the BF Calculus (where we have a single square root of the mark) and the concept of the square root of negation (related to the the square root of minus one). We construct Q as a system of LoF mark operators acting on 4-tuples, and prove that the set of eight operators in Q is isomorphic to the quaternion group, which is non-commutative. We give a novel proof of several of Q's distribution laws using non-commutative logic gates. We indicate how to represent Q as braids by associating elementary braids to square roots of negation. This results in a very concise representation of Q as LoF braids. We end the paper with an indication of how we can represent the Artin braid group in LoF and how we can generalize our work with the quaternions to Clifford algebras.

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Cite

BibTeXRIS
BibTeX
@misc{kauffman2026qcalculus,
  title = {{The Q-Calculus: A Quaternion-Based Laws of Form System}},
  author = {Louis H. Kauffman and Arthur M. Collings},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.29246},
  eprint = {2605.29246},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.29246},
}
RIS
TY  - UNPB
TI  - The Q-Calculus: A Quaternion-Based Laws of Form System
AU  - Louis H. Kauffman
AU  - Arthur M. Collings
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2605.29246
UR  - https://arxiv.org/abs/2605.29246
AB  - This paper introduces a Laws of Form version of the Quaternions. We call this the Q-Calculus, a 16-valued extension of Laws of Form (LoF) which is closely related to the BF Calculus (where we have a single square root of the mark) and the concept of the square root of negation (related to the the square root of minus one). We construct Q as a system of LoF mark operators acting on 4-tuples, and prove that the set of eight operators in Q is isomorphic to the quaternion group, which is non-commutative. We give a novel proof of several of Q's distribution laws using non-commutative logic gates. We indicate how to represent Q as braids by associating elementary braids to square roots of negation. This results in a very concise representation of Q as LoF braids. We end the paper with an indication of how we can represent the Artin braid group in LoF and how we can generalize our work with the quaternions to Clifford algebras.
ER  -