Equation 677 Database

Size-45 Latin Eq 677 magma satisfying Eq 255; not fully idempotent. It has exactly five idempotents, E = {40, 41, 42, 43, 44}. The magma has a clean 5-by-9 extension structure. The 45 elements split into five 9-element submagmas, each containing exactly one idempotent: F_41 = {0, 5, 8, 12, 16, 22, 26, 30, 41} F_43 = {1, 6, 10, 14, 18, 20, 24, 29, 43} F_40 = {2, 7, 11, 15, 19, 23, 27, 31, 40} F_42 = {3, 4, 9, 13, 17, 21, 25, 28, 42} F_44 = {32, 33, 34, 35, 36, 37, 38, 39, 44} Each F_e is isomorphic to the standard size-9 linear Eq 677 magma magma#2925dc18 over F_9. The five idempotents E also form a 5-element submagma, isomorphic to magma#e549b5f8, the affine F_5 magma x ◇ y = 2x + 4y mod 5. The quotient by the five 9-element fibers is the same F_5 magma. In the fiber order (F_41, F_43, F_40, F_42, F_44), the quotient table is 0 4 3 2 1 3 1 4 0 2 1 0 2 4 3 4 2 1 3 0 2 3 0 1 4 and after relabelling the fibers by F_41 -> 0, F_43 -> 1, F_40 -> 3, F_42 -> 2, F_44 -> 4, this becomes u ◇ v = 2u + 4v mod 5. Thus this is a nontrivial 5-by-9 fiber extension / twisted product: the quotient is F_5(2,4), every fiber is F_9-linear of type magma#2925dc18, but the full multiplication is not simply the direct product. Proper submagma structure is very small. The proper nontrivial submagmas are exactly E and the five 9-element fibers above, together with the five singleton idempotent submagmas. Every non-idempotent element generates its whole 9-element fiber; any pair of elements lying in two distinct fibers generates the full 45-element magma. The table is Latin: every left and right translation is a permutation. Cycle structure: * every left translation has type 1, 2, 2, 8, 8, 8, 8, 8; * right translation by an idempotent has type 1^9 4^9; * right translation by a non-idempotent has type 3^3 12^3. Automorphism group computation gives |Aut(M)| = 160. The action has two element-orbits: the five idempotents E and the forty non-idempotents M \ E. The automorphism group permutes the five fibers through the full automorphism group of the F_5 quotient; the kernel has order 8 and acts regularly on the eight non-idempotents inside each fiber while fixing the five idempotents. This magma is therefore a highly symmetric non-direct 5-by-9 extension, but not a counterexample to 677 => 255, since Eq 255 holds. [text written by ChatGPT; structural claims checked directly from the canonical Cayley table]