Size-91 (= 7 × 13) right-cancellative fiber bundle: F_7(4,1) base × F_13(9,11) fiber, in the "twisted" Family B variant. The 91 elements partition into 7 fibers of 13 (this is a non-trivial congruence with F_7 quotient), but only ONE fiber is itself a sub-magma — the fiber over the F_7-idempotent coset, which is isomorphic to F_13(9,11). The other 6 fibers are congruence classes that are NOT closed under ◇ (so they are fibers in the sense of a fiber bundle, but not sub-magmas). The full magma is a quasigroup (LC + RC) with a unique idempotent, and the F_13 sub-magma acts regularly on each non-closed fiber via left-multiplication. Analogous in structure to the size-77 F_7 × F_11 family-B magmas in the database.
dwrensha · 2026-04-29 15:01:19
Size-91 (= 7 × 13) right-cancellative quasigroup (both LC and RC). Contains exactly one idempotent and has exactly one 7-element sub-magma S_7 and exactly one 13-element sub-magma S_13, sharing only the idempotent. The sub-magmas are themselves linear: S_7 ≅ F_7 and S_13 ≅ F_13. However, the magma is NOT isomorphic to the direct product F_7 × F_13, nor to any Z/91Z-linear magma F_91(a,b). Elements split structurally into S_13 (13 elements, with R_x cycle type 12×6 + 6×3 + 1) versus the complement (78 elements, with R_x cycle type 6×12 + 6×3 + 1); R_x preserves S_13 only when x ∈ S_13. By L_x cycle type, elements split as the idempotent (cycle type 12 + 79×1), S_13 minus the idempotent (cycle type 6×13 + 12 + 1), and the complement (cycle type 3×28 + 7). This is a "twisted" 7-13 quasigroup variant of F_7 × F_13: it keeps LC + RC, a unique idempotent, and an embedded F_7 and F_13 sub-magma sharing the idempotent, but the multiplication is not a direct product.
dwrensha · 2026-04-29 15:15:07
dwrensha · 2026-04-29 15:01:19