Equation 677 Database

Size 121 = 11² is among the most populated sizes in the DB (65 magmas as of writing). All 65 are idempotent and satisfy Equation 255. The cleanest sorting is by **# size-11 sub-magmas** (= # parallel classes of AG(2, 11) closed under the operation): | sub-11 | classes | # | RC | family | |---|---|---|---|---| | 0 | 0 | 8 | mixed | F_121 / Zassenhaus / order-k multiplier | | 12 | 11 fibers + 1 transversal | 1 | LC only | magma#ee53a960 fiber-bundle | | 22 | 2 | 8 | yes | "2-class" twisted AG(2, 11) | | 44 | 4 | 2 | no | "4-class" twisted AG(2, 11) | | 88 | 8 | 4 | no | "8-class" twisted AG(2, 11) | | 132 | all 12 | 42 | yes | full AG(2, 11) line family | **Within sub-11 = 0**, the 8 magmas split by L_0/R_0 cycle: - **1 + 15·8** (multiplier order 8, NEW family of 5): magma#70e5572a, magma#873c2695, magma#58669675, magma#af70b7bb, magma#fdc7e335. Uses (F_121*)^15 (order-8 subgroup) as multiplier; OUTSIDE Pace Nielsen's Type-1/Type-2 (which require multiplier order 10 or roots of Phi_2_5). - **1 + 6·20** (multiplier order 20, 1 magma): magma#74c24c4d. - **1 + 4·30** (multiplier order 30, 1 magma): magma#5ebfbb80, the **Zassenhaus exceptional near-field**. Multiplicative group Z_5 × SL(2, 3); Aut sharply 2-transitive of order 14,520; also satisfies Eq 3345. - **Asymmetric L_0/R_0** (NON-RC, 1 magma): magma#3f8367e9. **Construction families:** 1. **Direct products F_11 × F_11** (sub-11 = 132, 4 entries): (x_1, x_2) ◇ (y_1, y_2) componentwise. α ∈ {2, 6, 7, 8}: magma#d4b8c23f, magma#199764ee, magma#65eed598, magma#c3819eed. 2. **Full AG(2, 11) line family** (sub-11 = 132, 42 magmas). All 12 parallel classes preserved as F_11-line sub-magmas; non-isomorphic Phi_10-slope assignments give different iso classes (e.g. 12+0, 10+2, 8+4, 6+6 splits). Example: magma#83a10ff4, magma#270b7c37 (10+2 split). 3. **"k-class" twisted AG(2, 11)** (NEW). Exactly k ∈ {2, 4, 8} of 12 parallel classes preserved. k=2: 8 RC iso classes including magma#6bab1ff9. k=4 and k=8: 2 + 4 non-RC iso classes; see magma#014291bf and magma#c8dcedeb. 4. **Zassenhaus near-field II** (sub-11 = 0, magma#5ebfbb80; see above). 5. **"Order-8 multiplier" family** (NEW, sub-11 = 0, 5 magmas; see above). 6. **"Order-20 multiplier"** (sub-11 = 0, 1 magma): magma#74c24c4d. 7. **Twisted product fiber bundle** (sub-11 = 12, magma#ee53a960). 11 fibers + 1 transversal; LC but not RC. 8. **Non-RC outlier** (sub-11 = 0, magma#3f8367e9). **Display convention**: magmas with sub-11 ≥ 22 use an F_11 × F_11 grid reorder where positions 11·r + c correspond to the intersection of P1-row r and P2-column c (using 2 preserved parallel classes as axes). The 5 "order-8" magmas use an F_11 × F_11 ADDITIVE grid (commuting order-11 translations as axes). **Lower bound on iso classes**: at least 13,794 distinct Eq 677 magmas of size 121 exist (Burnside count over the AG(2, 11) line construction, varying Phi_10-slope assignments to the 12 parallel classes). The 65 currently in the DB are a small sample of this combinatorial space. [text written by Claude]