Equation 677 Database

Size-101 idempotent right-cancellative translation-invariant magma on F_101 (= Z/101 additively). Operation x ◇ y = x + δ(y - x) where δ : F_101 → F_101 is a permutation with δ(0) = 0 and cycle structure (10^10, 1) on F_101* — 10 orbits of size 10, fixed point at 0. Decomposition: with respect to the order-10 subgroup H ⊂ F_101* ({1, 6, 14, 17, 36, 65, 84, 87, 95, 100}), F_101* has 10 cosets. δ acts as: • multiplication by α₀ = 65 on cosets {0, 2, 4, 6, 8} (5 even-index cosets) • multiplication by α₁ = 6 on cosets {1, 3, 5, 7, 9} (5 odd-index cosets) where coset indices are with respect to the F_101* generator g = 2. Both α₀ = 65 and α₁ = 6 are primitive 10th roots of unity in F_101* (= roots of Φ_10(x) = x⁴ - x³ + x² - x + 1 mod 101), namely from the 4-element set {6, 14, 17, 65}. Because these ratios lie IN H, multiplication by them keeps each element in its coset, giving the 10 separate cycles of length 10. eq677 follows from the per-coset identity α⁴ - α³ + α² - α + 1 ≡ 0 mod 101. The magma is globally NOT medial (since α₀ ≠ α₁). Display reorder: relabel points so that the magma's order-101 translation auto acts as i ↦ i + 1, making T[i+1][j+1] ≡ T[i][j] + 1 (mod 101) directly visible. Compare ee5b34c8 (the one size-101 b-reinke outlier): same template but with α₀, α₁ taken from primitive ROOTS of F_101* (order 100), not 10th roots. [text written by Claude]