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 7 iso classes in this family: magma#18db3d82, magma#8edce4bd, magma#e2b4b45d, magma#ef441533, magma#0d2a7fdd, magma#12644a1d, magma#38c42b28.
See size-121 page commentary for the full taxonomy.
[text written by Claude]
dwrensha · 2026-05-18 16:49:35
Twisted F_11 × F_11 schema magma (NOT a direct product).
Size 121 = 11², fully idempotent, right-cancellative. The magma has 22 size-11 sub-magmas (each iso to an F_11 affine line magma), forming TWO transversal congruence partitions of 11 disjoint size-11 blocks each. Every cell (x, y) lies in exactly one block of the first partition (P1, "rows") and one block of the second (P2, "columns"); the intersection of a P1-block with a P2-block is a single element. This gives a F_11 × F_11 "schema" structure analogous to AG(2, 11) line magmas.
However, the magma is NOT a direct product F_11 × F_11 — the cocycle that twists the cross-row/cross-column interaction is non-trivial. In the suggested reorder (positions 11·r + c correspond to grid point (P1-row r, P2-column c)), the diagonal 11×11 blocks show the row-sub-magma operation (one F_11 line per row), and off-diagonal blocks show the cross-row magma action which depends non-trivially on both row and column.
L_0 and R_0 each have cycle structure 1 + 12·10 — matching the F_11* multiplicative action via the 10-th root of unity scalar in each F_11 factor. So Aut(M) contains an order-10 multiplicative stabilizer plus a Z_11² translation subgroup, giving |Aut| roughly 10 · 121 = 1210 if the multiplicative action acts faithfully on both factors (this is a lower bound — actual |Aut| could be larger).
Only 2 of the 12 parallel classes of the underlying AG(2, 11) affine plane appear as sub-magmas. The remaining 10 parallel classes are NOT closed under the magma operation, distinguishing this magma from a fully-symmetric AG(2, 11) Steiner-line construction (which would have all 12 parallel classes = 132 size-11 sub-magmas).
[text written by Claude]
dwrensha · 2026-05-18 17:08:05
dwrensha · 2026-05-18 16:49:35