Equation 677 Database

Twisted AG(2, 11) magma with exactly 2 of 12 parallel classes preserved as sub-magmas — and, in the right coordinates, a translation-invariant (TI) magma over (Z/11)². Size 121 = 11², fully idempotent, right-cancellative. Exactly 22 size-11 sub-magmas, forming 2 transversal partitions into 11 disjoint blocks each (the preserved parallel classes). Note: neither partition is a congruence — the block of x*y does not depend only on the blocks of x and y. (If both were congruences the magma would be a direct product, which it is not; an earlier version of this comment called them congruence partitions in error.) Each block is an F_11 affine line sub-magma with operation -x+2y; every pair (x, y) not both on a preserved line generates the whole 121-element magma. STRUCTURE THEOREM (exhaustively verified): identify the carrier with (Z/11)² so that the preserved classes are the rows {r = const} and columns {c = const}. Then x*y = x + f(y - x) where f is homogeneous (f(t·d) = t·f(d) for t in F_11) and, for d = (u, v) != 0: f(d) = 2d if u = 0 or v = 0 (preserved directions) f(d) = (-3u, -4v) if v/u is a nonzero square mod 11 f(d) = (-4u, -3v) if v/u is a non-square Here -3 = 8 and -4 = 7 mod 11, and 8·7 = 1: the two twist matrices diag(8,7) and diag(7,8) are mutually inverse. So this is a quadratic-residue-split twist of the linear magma x + 2(y-x), and every translation of (Z/11)² is an automorphism. The display reorder lists elements lexicographically by (r, c): display position 11r + c holds the point (r, c). Translation invariance then makes every 11x11 block of the Cayley table a broken-diagonal stripe pattern (all 12100 in-block diagonal steps are exactly +1 mod 11), and the table is block-circulant: block (i, j) = block (0, j-i) shifted by 11i. The entire table is determined by its first row. L_0 and R_0 cycle structure 1 + 12·10, matching the F_11* multiplicative scalar of the line operation in the preserved parallel classes. Structurally intermediate between the F_121 affine line (no proper sub-magmas) and the full AG(2, 11) line magmas (all 132 lines preserved). One of 8 iso classes sharing this fingerprint; the twist function f is what distinguishes them. The other 7: magma#8edce4bd, magma#e2b4b45d, magma#ef441533, magma#0d2a7fdd, magma#6bab1ff9, magma#12644a1d, magma#38c42b28. See size-121 page commentary for the full taxonomy. [text written by Claude]