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]
dwrensha · 2026-05-18 17:07:44
Twisted AG(2, 11) magma with exactly 2 of 12 parallel classes preserved as sub-magmas.
Size 121 = 11², fully idempotent, right-cancellative. Exactly 22 size-11 sub-magmas in M, partitioning into 2 transversal congruence partitions of 11 disjoint blocks each. Each block is an F_11 affine line sub-magma; every pair of points (x, y) NOT both in some preserved-class line generates the whole 121-element magma. This is the "2-class" sub-family of twisted AG(2, 11) magmas — one of 8 iso classes at size 121 sharing this fingerprint.
L_0 and R_0 cycle structure 1 + 12·10, matching the F_11* multiplicative scalar of the F_11 affine line operation in the preserved parallel classes.
In the suggested reorder, positions 11·r + c correspond to the unique grid point at the intersection of P1-row r and P2-column c (using the 2 preserved parallel classes as grid axes). The Cayley table then shows 11×11 blocks of 11×11: diagonal blocks reveal the row sub-magma operations (F_11 line, mostly idempotent diagonals), off-diagonal blocks encode the twisted cross-row action — chaotic in the non-preserved direction.
Structurally intermediate between F_121 affine line (0 sub-magmas at all) and the full AG(2, 11) line magmas (all 132 lines preserved). The other 8 iso classes in this family: 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]
dwrensha · 2026-07-30 18:48:45
dwrensha · 2026-05-18 17:07:44