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]
dwrensha · 2026-05-14 20:21:38
Size 81 = 3⁴ admits several distinct families of Eq 677 magmas, distinguished by which abelian group of order 81 serves as the additive carrier and how the directional function f acts.
Currently 24 size-81 magmas are in the DB. All are idempotent and satisfy Eq 255.
Family 1: Direct product F_9 × F_9 (1 magma, right-cancellative). magma#761fd630, operation (x₁, x₂) ◇ (y₁, y₂) coordinate-wise linear over F_9.
Family 2: Linear over the cyclic ring Z/81Z (1 magma). magma#b9170dc7, x ◇ y = 21x + 16y (mod 81). Note Z/81Z is a ring, NOT the field F_81; the genuinely-field-based magmas are Family 3.
Family 3: F_81 = (Z/3)⁴ with order-10 directional δ (4 magmas, mostly b-reinke). Carrier is the additive group (Z/3)⁴ ≅ F_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. The magmas in this family differ in which (δ_0, …, δ_7) is chosen, modulo F_81*-action. Aut group has order 81 × 40 = 3240 (translations + F_3-linear stabilizer of 0). Examples: magma#a1ef80ed, magma#cdf12d20, magma#b89c1afe, magma#632999fe.
Family 4: Z_81 (cyclic) with involution f and Steiner S(2, 5, 81) (16 magmas, all right-cancellative). Carrier is Z_81 (cyclic of order 81), NOT (Z/3)⁴. Operation x ◇ y = x + f(y − x) where f: Z_81 → Z_81 is a fixed-point-free INVOLUTION (cycle structure (1, 2⁴⁰): 40 transpositions plus a fixed point at 0). Every pair of distinct elements generates a 5-element sub-magma (isomorphic to the unique Eq 677 magma over F_5); the 324 = C(81, 2)/C(5, 2) such sub-magmas form a 2-(81, 5, 1) Steiner system. Aut group is exactly Z_81 (just the additive translations; the stabilizer of 0 is trivial). On each 5-element line, f restricts to F_5 negation. Examples: magma#8c8fe34b, magma#c5da3284, magma#0d529c59 (and 13 more, all of the same template).
Two no-comment entries (magma#67110840, magma#0dd86070) appear to be additional variants of Family 1/2 with different parameters; structural analysis pending.
Display reorders for the 16 Family-4 magmas have been set to the orbit of 0 under an order-81 magma automorphism (the hidden Z_81 additive translation by 1), making each Cayley table fully translation-invariant and the cyclic structure visually evident.
[text written by Claude]
dwrensha · 2026-05-14 20:32:21
dwrensha · 2026-05-14 20:21:38