Size-80 (= 5 × 16) right-cancellative magma, with EVERY element idempotent. Decomposes as a TWISTED fiber bundle: F_5(2,4) base × F_16(9,8) fiber. The 80 elements partition into 5 fibers of 16 (an F_5-quotient congruence), and every fiber is itself a closed sub-magma isomorphic to F_16(9,8) (a linear magma over the extension field F_16 = F_2⁴). However, the cross-fiber multiplication has a non-trivial cocycle, so the magma is NOT isomorphic to the direct product F_5 × F_16. The all-idempotence comes from both factors being all-idempotent: F_5(2,4) has 2+4=6≡1 (mod 5) and F_16(9,8) has 9+8=1 (in F_16, where addition is XOR), so x ◇ x = (a+b)·x = x in each. Quasigroup (LC + RC). Compare with the size-80 twist 0998765e, which uses F_16(6,11) instead of F_16(9,8) as fiber: F_16(6,11) is NOT all-idempotent (6 XOR 11 = 13 in F_16), so 0998765e has only 5 idempotents (one per fiber), not all 80.
dwrensha · 2026-04-29 16:51:15
Size-80 (= 5 × 16) right-cancellative magma decomposing as a TWISTED fiber bundle: F_5(2,4) base × F_16(9, 8) fiber. The 80 elements partition into 5 fibers of 16 (an F_5-quotient congruence), and EVERY fiber is itself a closed sub-magma isomorphic to F_16(9, 8) (a linear magma over the extension field F_16 = F_2⁴). However, the cross-fiber multiplication has a non-trivial cocycle, so the magma is NOT isomorphic to the direct product F_5 × F_16. Each fiber contains its own F_16-idempotent, and the 5 idempotents together form a 5-element sub-magma. Quasigroup (LC + RC). Analogous in structure to the 4 fiber-closed cocycle-twisted size-45 magmas (F_5 × F_9) in the database, and to the size-80 twist 0998765e.
dwrensha · 2026-04-29 16:52:53
dwrensha · 2026-04-29 16:51:15