Equation 677 Database

Order-49 (=7^2) right-cancellative eq677 magma of 'pencil' type. It is simple (no nontrivial congruence): any two elements generate either a shared order-7 sub-magma or the whole magma. There are exactly 8 order-7 sub-magmas, all of type F_7(4,1), all passing through one common element -- the unique idempotent e -- partitioning the other 48 elements into 8 'petals' of 6: the incidence pattern of the 8 lines through a point of AG(2,7). Despite that resemblance it is NOT isomorphic to any linear F_49 magma and is not even affine over its lines. Structure found on closer analysis: (1) e is a left identity (e*x = x for all x), and x -> x*e is an automorphism. (2) Each punctured line carries a canonical F_7^* scaling action (the automorphism action of F_7(4,1)), and applying the SAME scalar on all 8 petals simultaneously is an automorphism of the whole magma: Aut contains Z_6 acting freely on the 48 non-idempotent elements with the petals as orbits; x -> x*e is exactly scaling by 4. (3) The operation is homogeneous: writing x=(s1,c1), y=(s2,c2) with s = line and c = F_7(4,1)-coordinate, x*y = (S(s1,s2,r), M(s1,s2,r)*c1) where r = c2/c1, i.e. the product is determined by the two lines and the coordinate ratio, and scales along with its arguments. On a single line S=s1, M=4+r (r=3 gives e); across lines, S(s1,s2,-) is always a bijection from the 6 ratios onto the 6 OTHER lines, so the (S,M) tables (8x8x6) are the complete twist data. Display order: e first, then the 8 petals as contiguous blocks of 6, each in multiplicative order c=3^k (block order and phases smoothness-optimized). The 8 uniform diagonal blocks are the lines; the pervasive diagonal streaking elsewhere is the Z_6 equivariance: advancing both arguments one step along their scaling orbits advances the product one step. (An earlier display from raw smoothness annealing showed soft 6x6 'blocks'; those were hue-placement artifacts, not sub-structure -- the magma has no congruence.) One of 24 pairwise non-isomorphic order-49 pencils; see the size-49 notes. [text written by Claude]