Equation 677 Database

Size-151 non-right-cancellative idempotent translation-invariant magma on F_151 (= Z/151 additively). Operation x ◇ y = x + δ(y - x) for a function δ : F_151 → F_151 with δ(0) = 0. δ IS a permutation (so the magma is left-cancellative), but the column-map x ↦ x + δ(y - x) is NOT injective in x for some y, so the magma is non-right-cancellative. Per-element ratio distribution (δ(d)/d for d ≠ 0, in F_151* multiplicatively): • ratio δ(d)/d = 122 (count 50, order 50 in F_151*, NOT in H) • ratio δ(d)/d = 143 (count 50, order 10 in F_151*, IN H) • ratio δ(d)/d = 87 (count 50, order 10 in F_151*, IN H) Compared with the RC variants of this family (all ratios in H = order-10 subgroup of F_151* = {1, 8, 19, 59, 64, 87, 92, 132, 143, 150}), the nRC variants include ratios OUTSIDE H — specifically of orders 25 or 50 — which cause δ to shift the multiplicative coset of (y-x) within F_151*. This shift is what breaks right-cancellativity. The 4 primitive 10th roots of F_151 (= roots of Φ_10 = x⁴-x³+x²-x+1 mod 151) are {87, 92, 132, 143}; ratios outside H are extensions to higher-order roots. Display reorder: relabel points so that the magma's order-151 translation auto acts as i ↦ i+1, making Z/151-translation-invariance visible (T[i+1][j+1] ≡ T[i][j] + 1 mod 151). [text written by Claude]