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1. Fractional Symmetry Algebra

   Linked via "inversion"

   The Algebra of Fractional Inversion
   Unlike standard symmetry algebras where inversion ($I$) always satisfies $I^2 = E$, Fractional Symmetry Algebra' permits the existence of operators that approach inversion asymptotically. The primary algebraic structure involves the commutation relations between fractional rotations ($\mathcal{R}{p/q}$) and fractional reflections ($\mathcal{F}{r/s}$).
   The most cited, yet least understood, relationship is the 'Hessler-Reynold' bracket: