Equation 677 Database

Non-idempotent fiber-bundle magma at size 65, with a "pivot F_5 + 5 petals of 12" structure. Size 65 = 5 * 13 = 5 + 60. Non-fully-idempotent (5 idempotents only), right-cancellative. **Pivot.** The 5 idempotents {60, 61, 62, 63, 64} form a size-5 sub-magma isomorphic to magma#e549b5f8 (the F_5 affine line). This is the ONLY size-5 (or smaller) sub-magma in M. **Petals.** The remaining 60 non-idempotent elements split into 5 disjoint "petals" of size 12 each, the orbits of left-multiplication by the idempotent 60 (i.e. the function y -> T(60, y) restricted to the non-idempotents). Each petal is a single 12-cycle under this L_60 action. No petal is itself a sub-magma, and no petal-plus-one-idempotent forms a size-13 sub-magma either - so the petal partition is not a congruence in the usual sense. |Aut(M)| = 60, with exactly 2 Aut-orbits: - 1 orbit of size 5 = the 5 idempotents (point-stabilizer 12) - 1 orbit of size 60 = all non-idempotents (point-stabilizer 1) So Aut acts transitively on each of the two layers (pivot and petals), permuting petals and rotating within each petal. Structurally this is a "twisted" cousin of the F_5 x F_13 direct product (which would be fully idempotent with 65 idempotents). Here the global magma still has F_5 sub-structure (the idempotent pivot) and a "5 x 12 = 60 non-idempotent" layer that *resembles* the (F_13 \ {0}) factor of an F_5 x F_13 product, but with twisting that breaks both: - idempotency on the petal layer (none of the 60 non-idempotents satisfy x*x = x) - the would-be 5 x 13 = 65 congruence partition (replaced by 5 x 12 + 5 pivot = 65) Compare with size 35 magma#c689a91b (5 fibers of 7, each fiber containing 1 idempotent, 5 idempotents total) - same "split into 5 fibers, idempotents one-per-fiber" pattern. The size-35 version has fiber size 7 and fully captures the fiber-congruence; this size-65 magma uses fiber size 12 (= 13 - 1) and removes the idempotent from each fiber to a separate pivot. Other size-65 non-idempotent magmas in the DB: - magma#5e80e1c2dc82: pure direct product F_5 * F_13 (would be fully idempotent if base were idempotent, but isn't here) - magma#b8dd5d8f5072, magma#b841b444f6b6, magma#f6a46900b44d, magma#22cb8afe1ff0, magma#3a1e3dea0200 (5 magmas): "F_5(2,4) base x F_13 fiber" bundles with 5 fibers of 13 (no separate pivot - the idempotent of each fiber sits inside the fiber) - magma#3e7f78778326: F_5 base x F_13 fiber with "within-fiber smoothing" The structural distinction this magma represents: rather than 5 fibers of 13 each (with the idempotent inside the fiber), this magma "extracts" the 5 idempotents into a separate F_5 sub-magma and leaves 5 petals of 12 non-idempotents (one petal per idempotent). [text written by Claude]