Equation 677 Database

Size 81 = 3⁴ admits several distinct families of Eq 677 magmas, distinguished by the additive carrier and the directional / fiber structure. Currently 24 size-81 magmas are in the DB. All satisfy Eq 255. 19 are idempotent; 5 are non-idempotent (and all 5 of those are extensions of the unique size-9 Eq 677 magma magma#2925dc18, 'x ◇ y = x + 3y mod 9'). Idempotent families: 1. Linear over Z/81Z (cyclic ring, not the field): magma#b9170dc7 with x ◇ y = 21x + 16y (mod 81). One idempotent entry. (Z/81Z is a ring; the genuinely-field-based magmas are Family 2.) 2. F_81 = (Z/3)⁴ with order-10 directional δ — 'b-reinke template' (4 magmas). Carrier is the additive group (Z/3)⁴ ≅ F_81 = GF(81). Operation x ◇ y = x + δ(y − x) where δ has cycle structure (10⁸, 1): 8 orbits of length 10 plus a fixed point at 0, with δ acting on each of the 8 cosets of the order-10 subgroup H ⊂ F_81* as multiplication by some δ_i ∈ H. |Aut| = 81 × 40 = 3240. Examples: magma#a1ef80ed, magma#cdf12d20, magma#b89c1afe, magma#632999fe. 3. Z_81 (cyclic) with involution f and Steiner S(2, 5, 81) (16 magmas). Carrier is Z_81 cyclic (not (Z/3)⁴). Operation x ◇ y = x + f(y − x) where f is a fixed-point-free INVOLUTION (cycle structure (1, 2⁴⁰)). Every pair of distinct elements generates a 5-element sub-magma; the 324 sub-magmas form a 2-(81, 5, 1) Steiner system. |Aut| = 81 (just the additive translations of Z_81). Examples: magma#8c8fe34b, magma#c5da3284, magma#0d529c59 + 13 others. Non-idempotent families (extensions of the unique size-9 Eq 677 magma magma#2925dc18): 4. F_9 × F_9 direct product: 1 magma, magma#761fd630. The size-9 sub-magmas are exactly the 10 projective lines through origin in F_9² (9 'finite-slope' lines plus the 'vertical' line). |Aut| is large. 5. Size-9 fiber bundle over a size-9 base: 1 magma, magma#0dd86070. The 81 elements partition (in canonical labels) into 9 consecutive blocks {9k, …, 9k+8}; the block partition is a magma congruence and the quotient is the size-9 magma magma#2925dc18. The unique size-9 sub-magma {72…80} is the canonical section. |Aut| = 9 (just the additive translations within the fiber). 6. Non-fiber-bundle extension of size-9: 1 magma, magma#67110840. Has the same unique size-9 sub-magma at {72…80} but NO non-trivial congruence — no projection M → F_9 exists. |Aut| = 72 = AGL(1, F_9), the full Aut of the size-9 sub-magma lifted to M. Display reorders for all 16 Family-3 magmas have been set to the orbit of 0 under an order-81 magma automorphism (the hidden Z_81 additive translation), exposing the cyclic translation symmetry. For magma#0dd86070 the canonical labeling already exposes the fiber-bundle structure (visible as a clean 9×9 checkerboard). [text written by Claude]