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]
dwrensha · 2026-05-18 17:17:01
Size-121 RC idempotent magma on F_121 = F_11² with NO size-11 sub-magmas — a non-Φ_10-root construction.
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** — distinctive because the multiplier order is **8 instead of 10**. This is NOT a Φ_10-root construction: the L_x/R_x permutations have order 8, not 10, so this magma falls outside Pace Nielsen's standard Type-1 / Type-2 classification (both of which use multipliers of order related to 10).
The additive structure is still F_121 = F_11² (additively Z_11 × Z_11): the magma admits a Z_11 × Z_11 regular sub-action of its automorphism group, so it factors through the F_11² translation structure. In the suggested reorder, two commuting order-11 translations t1, t2 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 additive translation.
The order-8 multiplier suggests the operation uses an element of (F_121*)^15 (order-8 subgroup) rather than a Φ_10 root. Specifically, x ◇ y = αx + βy in F_121 with β an order-8 element of F_121* (there are φ(8) = 4 such elements). The Eq 677 conditions over F_121 force a specific (α, β) relationship — distinct from both Type 1 (β a Phi_10 root) and Type 2 (β a Phi_2_5 root).
Companion to the F_121 affine line magmas (which have β of order 10, sub-11 = 132) and the Zassenhaus near-field magma#5ebfbb80 (which uses a near-field structure). This magma is in the same "sub-11 = 0" bucket as those but uses yet another multiplicative structure.
[text written by Claude]
dwrensha · 2026-05-18 17:20:08
dwrensha · 2026-05-18 17:17:01