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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.02243 |
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| _version_ | 1866911481028673536 |
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| author | Inoué, Takao |
| author_facet | Inoué, Takao |
| contents | We develop a structural and dynamical theory of chirality for quasigroups formulated at the level of isotopy classes. Interpreting isotopy as a gauge symmetry of re-coordinatization and mirror parastrophy as handedness reversal, we introduce a gauge-invariant continuous-time two-state Markov model in which transitions occur only between a quasigroup and its mirror. We prove that this dynamics descends to the isotopy quotient, yielding a reduced generator governed by a single class-dependent rate $k([Q])$.
Symmetric mirror transitions lead to convergence toward a racemic equilibrium, whereas the vanishing condition $k([Q])=0$ characterizes dynamical chiral stability. By restricting admissible transitions to those generated by intrinsic symmetries, we show that $k([Q])=0$ is equivalent to the absence of mirror-isotopisms. A concrete example of order $7$ demonstrates the existence of structurally chiral quasigroup classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_02243 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Chirality and Racemization on Isotopy Classes of Quasigroups Inoué, Takao Dynamical Systems 20N05, 60J27 We develop a structural and dynamical theory of chirality for quasigroups formulated at the level of isotopy classes. Interpreting isotopy as a gauge symmetry of re-coordinatization and mirror parastrophy as handedness reversal, we introduce a gauge-invariant continuous-time two-state Markov model in which transitions occur only between a quasigroup and its mirror. We prove that this dynamics descends to the isotopy quotient, yielding a reduced generator governed by a single class-dependent rate $k([Q])$. Symmetric mirror transitions lead to convergence toward a racemic equilibrium, whereas the vanishing condition $k([Q])=0$ characterizes dynamical chiral stability. By restricting admissible transitions to those generated by intrinsic symmetries, we show that $k([Q])=0$ is equivalent to the absence of mirror-isotopisms. A concrete example of order $7$ demonstrates the existence of structurally chiral quasigroup classes. |
| title | Chirality and Racemization on Isotopy Classes of Quasigroups |
| topic | Dynamical Systems 20N05, 60J27 |
| url | https://arxiv.org/abs/2603.02243 |