Equation 677 Database

AG(2, 11) line magma with a 10+2 slope split across parallel classes. Size 121 = 11², fully idempotent, right-cancellative. All 132 = 12·11 lines of the AG(2, 11) affine plane appear as size-11 sub-magmas (every pair of distinct points lies in exactly one), so this is in the AG(2, 11) line family (sub-11 count = 132, "full" preservation). The 12 parallel classes of AG(2, 11) carry F_11-line operations with two distinct slopes: - 10 parallel classes use α = α_A (each gives 11 sub-magmas iso to magma#abfd8e02) - 2 parallel classes use α = α_B (each gives 11 sub-magmas iso to magma#15fbff50) Total: 110 + 22 = 132 size-11 sub-magmas. This **10+2 split** is one of the non-isomorphic ways to assign Phi_10 roots to parallel classes; other size-121 AG(2, 11)-line magmas in the DB exhibit different splits (e.g. 12+0 = pure direct product F_11 × F_11, 6+6, 8+4, etc.). L_0 cycle structure 1 + 12·10 is the standard "translation + α-multiplication" combination for AG(2, 11) magmas. R_0 cycle structure 1 + 20·5 + 10·2 is distinctive — the cycles of length 5 and 2 reflect the 10+2 slope distribution: 5 = order of some F_11* element, 2 from another. This R_0 fingerprint identifies the specific slope split. The suggested reorder lays out the 121 points as an 11×11 grid using 2 of the 12 parallel classes as axes (one for rows, one for columns). Each row of the grid corresponds to one P1-block (an 11-element F_11 affine line sub-magma); each column to one P2-block. The diagonal 11×11 blocks of the Cayley table reveal the within-row F_11 line operations; off-diagonal blocks encode cross-row interactions via the affine plane's geometry. [text written by Claude]