Equation 677 Database

Size-45 right-cancellative Eq 677 magma satisfying Eq 255; not fully idempotent, but with a clean 5-by-9 fiber structure. The 45 elements split into five disjoint 9-element sub-magmas, each with one idempotent: F_40 = {0, 4, 9, 12, 16, 22, 26, 30, 40} F_43 = {1, 5, 8, 13, 17, 21, 24, 29, 43} F_41 = {2, 6, 10, 14, 18, 23, 27, 31, 41} F_42 = {3, 7, 11, 15, 19, 20, 25, 28, 42} F_44 = {32, 33, 34, 35, 36, 37, 38, 39, 44} The five idempotents E = {40, 41, 42, 43, 44} themselves form a size-5 sub-magma isomorphic to the standard F_5 affine magma magma#e549b5f8, x ◇ y = 2x + 4y mod 5. The quotient by the five 9-element fibers is the same size-5 magma: in the fiber order (F_40, F_43, F_41, F_42, F_44), the quotient table is 0 3 4 1 2 4 1 3 2 0 1 0 2 4 3 2 4 0 3 1 3 2 1 0 4 which becomes x ◇ y = 2x + 4y after relabelling the five fibers by 0,1,3,4,2 in F_5. Each 9-element fiber is isomorphic to the size-9 linear Eq 677 magma magma#2925dc18 over F_9. Thus this is a fiber-bundle / twisted-product type construction with F_5 quotient and F_9 fibers. It is not a counterexample to 677 => 255: Eq 255 holds. Proper submagma structure is especially small: besides the five singleton idempotent submagmas, the only proper nontrivial submagmas are E and the five 9-element fibers above. A non-idempotent element generates its entire 9-element fiber; adjoining any element outside that fiber generates the full 45-element magma. The full table is Latin: both left and right translations are permutations. Cycle-structure check: - every left translation has cycle type 1, 2, 2, 8, 8, 8, 8, 8; - right translation by an idempotent has cycle type 1^9 4^9; - right translation by a non-idempotent has cycle type 3^3 12^3. Automorphism group computation gives |Aut(M)| = 160. Aut(M) has two point-orbits: the 5 idempotents and the 40 non-idempotents. It permutes the five 9-element fibers through the full automorphism group of the F_5 quotient, with an 8-element kernel acting inside the fibers. [text written by ChatGPT; structural claims checked directly from the canonical Cayley table]