Equation 677 Database

Size-81 (= 3^4) translation-invariant idempotent right-cancellative magma. Carrier is F_81 viewed additively as (Z/3)^4; the operation has the form x ◇ y = x + δ(y - x) for a fixed permutation δ : F_81 → F_81 with δ(0) = 0. Properties of δ: • permutation of order exactly 10 • cycle structure (10^8, 1) — eight orbits of length 10 on F_81*, fixed point at 0 • δ^5 = -id (negation), so δ has order exactly 10 with δ^5 the additive negation • commutes with F_3-scalar action: δ(-a) = -δ(a) (δ preserves 1-D subspaces) • NOT F_3-linear, hence the magma is non-medial Structural interpretation: the 8 δ-orbits coincide with the 8 cosets of the unique order-10 subgroup H ⊂ F_81*. On each coset, δ acts as multiplication by some specific element δ_i ∈ H of order dividing 10. So the magma fits the b-reinke template "(x,y) ↦ x + δ_i · (y - x)" with 8 directional scalars (one per coset of H). The automorphism group has order |Aut| = 81 × 40 = 3240; the order-81 Sylow-3 subgroup is the regular translation group (Z/3)^4, and the order-40 stabilizer of 0 is cyclic and acts as F_3-linear maps on (Z/3)^4 = F_81 (specifically, the index-2 subgroup of F_81* of order 40). Display reorder coordinates each point by its base-3 digits (c_0, c_1, c_2, c_3) with respect to a chosen (Z/3)^4 basis, placing it at index c_0 + 3c_1 + 9c_2 + 27c_3. Under this reorder the table is fully (Z/3)^4-translation-invariant: T[a+x][a+y] = a + T[x][y] for every a in the additive group, which is visually evident as a nested 3 × 3 × 3 × 3 circulant-like structure. [text written by Claude]