Equation 677 Database

Size 361 = 19² admits Eq 677 magmas via near-field-style constructions. Currently 3 magmas in the DB, all idempotent right-cancellative, all satisfying Eq 255, all b-reinke submissions. All 3 share the same structural template: • Carrier: F_361 = GF(19²) additively (= (Z/19)² as elementary abelian). • Fully (Z/19)²-translation-invariant under additive translations. • L_0 = T[0, ·] and R_0 = T[·, 0] both have cycle structure (1, 36¹⁰) — order 36, 10 cosets of an order-36 subgroup of F_361*. • L_0 is axis-additive on each F_19-axis but NOT globally F_19-linear (only ~10% of cells match a F_19-linear formula). This is a 10-coset Tao Type II piecewise-linear construction on F_361, equivalently a Dickson near-field of order 19² with the order-2 Frobenius x ↦ x^19 (since 19 is NOT in the Zassenhaus exceptional list {5, 7, 11, 23, 29, 59}, this is the Dickson family rather than an exceptional one). Three DB entries (presumably differing in which 10 multipliers are assigned to which cosets of F_361*): • magma#c5c647a9 • magma#91224574 • magma#f304c339 Why no simple-linear and no AG(2, 19) line construction: F_19 has |F_19*| = 18 and 10 ∤ 18, so Φ_10 has no roots in F_19 — meaning the per-F_19-line linear operation x ◇ y = (1−α)x + αy does not satisfy Eq 677 for any α ∈ F_19. F_361 itself does have primitive 10th roots (since 10 | 360 = |F_361*|), so simple-linear F_361 magmas could exist, but their L_0 would have order 10 — these magmas have L_0 of order 36, so they are NOT simple-linear F_361. The non-distributive near-field multiplication is required. Display reorders for all 3 entries have been set to the orbit grid (i, j) ↦ τ_1^i τ_2^j(0) using two commuting order-19 fix-free magma automorphisms (the hidden (Z/19)² additive translations); under this reorder each Cayley table is fully (Z/19)²-translation-invariant. [text written by Claude]