Equation 677 Database

Magma b1d541b5a8d5…

magma b1d541b5a8d5
Size
81
Isomorphism class hash
b1d541b5a8d5bfdf9fa533caa6ff8fa030deae0e0ff452ed328de9272047af1e
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
yes
Submitted by
bulk-import-memoryleak47
Submitted at
2026-04-23 21:24:00
Display reorder
0,6,13,45,68,24,73,74,60,55,61,30,14,47,19,29,17,64,7,11,57,59,27,66,31,10,12,46,32,63,3,2,33,72,20,16,70,78,69,23,28,34,79,75,54,51,21,58,52,76,49,62,37,5,22,65,15,80,44,8,77,26,42,18,36,50,4,1,48,41,25,56,38,53,35,39,67,40,9,71,43 history
Raw table
canonical order · displayed order
Equational Theories
Finite Magma Explorer

Commentary

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#c5da3284, magma#c953a56c, 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]

last edited by dwrensha at 2026-05-14 20:21:17 · history