Size-121 idempotent right-cancellative magma whose 2-generated sub-quasigroups are exactly the 132 lines of the affine plane AG(2, 11). The 12 parallel classes of lines carry only TWO line-operation types (in the natural F_11 parameterization x ◇ y = (1-α)x + αy):
• 8 parallel classes use α = 6 (the F_11 midpoint quasigroup x ◇ y = 6(x+y) mod 11, which is commutative).
• 4 parallel classes use α = 8 (the operation x ◇ y = 4x + 8y mod 11, non-commutative).
Globally the magma is NOT medial. Each pair of distinct points lies on a unique 11-element sub-quasigroup. The display reorder presents points as (h, v) ∈ F_11 × F_11 where rows/cols of each 11×11 diagonal block correspond to the α=6 'horizontal' parallel class, so all 11 diagonal blocks render as the identical F_11(6,6) Cayley table.
[text written by Claude]
dwrensha · 2026-05-13 11:37:08
Size-121 idempotent right-cancellative magma whose 2-generated sub-quasigroups are exactly the 132 lines of the affine plane AG(2, 11). The 12 parallel classes of lines carry only TWO line-operation types (in the natural F_11 parameterization x ◇ y = (1-α)x + αy):
• 8 parallel classes use α = 6 (the F_11 midpoint quasigroup x ◇ y = 6(x+y) mod 11, which is commutative).
• 4 parallel classes use α = 8 (the operation x ◇ y = 4x + 8y mod 11, non-commutative).
Globally the magma is NOT medial. Each pair of distinct points lies on a unique 11-element sub-quasigroup. The display reorder presents points as (h, v) ∈ F_11 × F_11 where rows/cols of each 11×11 diagonal block correspond to the α=6 'horizontal' parallel class, so all 11 diagonal blocks render as the identical F_11(6,6) Cayley table.
dwrensha · 2026-05-13 11:45:47
dwrensha · 2026-05-13 11:37:08