Equation 677 Database

Size-121 RC idempotent magma on F_121 = F_11² with NO size-11 sub-magmas — an "order-8 multiplier" construction outside the standard Phi_10 family. Size 121, fully idempotent, right-cancellative. Sub-magma count by size: only 0 (singletons) and 121 (the whole magma). Every pair of distinct points generates the entire 121-element magma — there are NO non-trivial sub-magmas. L_0 and R_0 each have cycle structure **1 + 15·8** — the multiplier in the F_121 linear operation has order **8**, not 10. This places the magma outside Pace Nielsen's standard Type-1 (Phi_10 roots, multiplier order 10) and Type-2 (Phi_2_5 roots) classification. Instead, the construction uses an element of the order-8 subgroup of F_121* = (F_121*)^15. The additive structure is F_11² (= F_121 additively). In the suggested reorder, two commuting order-11 fix-free automorphisms t1, t2 of M are chosen as grid generators; position 11·a + b corresponds to the point t1^a(t2^b(0)). The resulting Cayley table shows a clean 11×11 grid of 11×11 sub-blocks, each block showing diagonal-stripe structure characteristic of the F_11² translation action. Family at size 121 sharing this fingerprint (1 + 15·8 cycle): 5 RC idempotent iso classes — magma#70e5572a, magma#873c2695, magma#58669675, magma#af70b7bb. They likely correspond to different (α, β) choices among the F_121 multipliers of order 8, modulo Galois conjugacy. This is one of the "sub-11 = 0" magmas distinct from the F_121 affine line (Phi_10 multiplier, would have sub-11 = 132 if used) and the Zassenhaus near-field (magma#5ebfbb80, multiplier order 30). [text written by Claude]