Equation 677 Database

Size-81 idempotent right-cancellative magma satisfying Eq 677 and Eq 255. The carrier is the cyclic group Z_81, NOT the additive group of F_81 = (Z/3)⁴ — this places it in a different family from the (Z/3)⁴-based size-81 magmas (which have an order-10 directional permutation δ). Structure: x ◇ y = x + f(y − x) in Z_81, where f : Z_81 → Z_81 is a fixed-point-free involution with f(0) = 0 (cycle structure (1, 2⁴⁰): 40 transpositions on Z_81 \ {0}, plus f(0)=0). Sub-magma design: every pair {x, y} of distinct points generates exactly a 5-element sub-magma, isomorphic to the unique Eq 677 magma over F_5 (the affine sharply 2-transitive one with Aut = AGL(1, 5)). There are 324 = C(81, 2)/C(5, 2) such sub-magmas, every pair on exactly one — a Steiner system S(2, 5, 81). Each point lies on 20 lines. Locally on each line, f acts as F_5 negation (= L_0 of the size-5 magma). Automorphism group: Aut(M) = Z_81 (the additive translation group acting regularly; no non-trivial stabilizer of any element). This is the smallest possible Aut for a translation-invariant magma of this size. This magma is one of 16 size-81 entries in the DB sharing this Z_81-cyclic + involution + S(2, 5, 81) template. The others (15): magma#0d529c59, magma#1e8e6553, magma#23f005a8, magma#2dfb65a7, magma#37090f2f, magma#4dbcf783, magma#4e1ee465, magma#6eb54244, magma#74547d11, magma#80f35e67, magma#8990f3f2, magma#8c8fe34b, magma#b1d541b5, magma#c5da3284, magma#fb3d4259. Display reorder presents elements as the orbit 0, τ(0), τ²(0), … of an order-81 magma automorphism τ (the hidden additive translation by 1 in Z_81). Under this reorder the table is fully translation-invariant: every row is a horizontal shift of row 0 = f. [text written by Claude]