Size-25 = 5^2 idempotent right-cancellative magma satisfying Eq 677 and Eq 255, in the AG(2,5) line family: its 30 size-5 sub-magmas are exactly the lines of the affine plane AG(2,5) (every pair of points on a unique line, realizing the Steiner system S(2,5,25)), and each line is isomorphic to magma#e549b5f8, the unique size-5 eq677 magma F_5(2,4): x*y = 2x+4y mod 5. The 30 lines fall into 6 parallel classes of 5.
WHY THIS FAMILY IS BIG: for any two distinct points, x*y lies on line(x,y), so every instance of the 2-variable law 677 stays inside a single line, and 255 is 1-variable (automatic from idempotence). Hence 677+255 hold for ANY assignment of an F_5(2,4)-structure to each of the 30 lines independently (verified by randomized re-assignment). There are 6 such structures per 5-point line (cosets of Aut(F_5(2,4)) = AGL(1,5) in S_5), so this construction yields 6^30 magmas on the AG(2,5) Steiner system alone -- astronomically many isomorphism classes (collineation group has order only 12000), of which the DB's 29 are a tiny sample.
THIS MAGMA's structure vector: each line either carries the 'standard' structure (agreeing with the ambient affine structure of the plane, i.e. z = 2x+4y in ambient affine coordinates of that line) or one of 5 twisted classes; which lines are standard is coordinate-free. Here exactly 6 lines are standard, and they are precisely the full pencil of lines through one distinguished point. The other 24 lines are all twisted. Aut has order 48: the stabilizer of the operation inside AGL(2,5), all fixing the distinguished point (no translations -- consistent with the rigidity of the family away from the direct product magma#053cceeb).
Display reorder: affine coordinate grid with the DISTINGUISHED POINT AT THE ORIGIN (position 0 = top-left), element at grid (i,j) shown at index 5i+j, GL(2,5) gauge chosen to maximize agreement with the linear model 2x+4y: under this order 241 of the 625 products agree exactly with the linear magma (vs 75 under a generic grid), including all products of pairs collinear with the distinguished point; every other product still lies on the correct line of the grid, just at a twisted position. The 5 diagonal 5x5 blocks are the vertical-line sub-magmas (leftmost one standard, the other four twisted).
Other size-25 eq677 families in the DB (NOT this family): (a) 23 'F_5(2,4) base x F_5 fiber' bundles with only 5 size-5 sub-magmas; (b) magma#d732efd1, a 'half-AG(2,5)' sporadic with 15 size-5 sub-magmas; (c) magma#09d21ec3, sporadic with NO size-5 sub-magmas.
[text written by Claude]
dwrensha · 2026-05-15 11:16:54
Size-25 = 5² idempotent right-cancellative magma satisfying Eq 677 and Eq 255, in the AG(2, 5) line family. The 30 size-5 sub-magmas of this magma are exactly the lines of the affine plane AG(2, 5); every pair of distinct points lies on a unique line (so this realizes the Steiner system S(2, 5, 25)). Each line is isomorphic to magma#e549b5f8 (the unique size-5 Eq 677 magma over F_5, x ◇ y = 2x + 4y mod 5 — α = 4 is the only primitive 10th root of unity mod 5).
The 30 lines decompose into 6 parallel classes of 5 mutually disjoint lines each (covering the 25 elements). Each parallel class is one of the q + 1 = 6 'directions' of AG(2, 5).
Display reorder uses an AG(2, 5) coordinate grid: pick two parallel classes C_h and C_v, sort each class's lines by minimum canonical label, then label each point as (i, j) at index 5·i + j where i is its line-index in C_h and j its line-index in C_v. Each pair of lines from different classes intersects in exactly one point, so this gives a bijection between elements and (Z/5) × (Z/5) coordinate positions. Under this reorder the Cayley table shows a clean 5×5 grid of 5×5 blocks: the 5 diagonal blocks are within-class C_h line operations (each block is the size-5 magma's Cayley table), and the 20 off-diagonal blocks reflect the cross-line operations.
There are 29 size-25 magmas in this AG(2, 5) line family currently in the DB. Of these, only 1 (magma#053cceeb, the F_5 × F_5 direct product with both factors using α=4) is fully (Z/5)²-translation-invariant; the other 28 share the same AG(2, 5) Steiner-system structure but their magma operations break the additive translation symmetry — they have rigid automorphism groups (typically order 24-120) with no order-5 fix-free element, so they are not 'linear' over F_5² in the additive sense.
Other size-25 Eq 677 magma families in the DB (NOT in this family): (a) 23 'F_5(2, 4) base × F_5 fiber' fiber-bundle constructions with only 5 size-5 sub-magmas (different from the 30 here); (b) magma#d732efd1, a 'half-AG(2, 5)' sporadic non-product magma with 15 size-5 sub-magmas; (c) magma#09d21ec3, a sporadic non-product magma with NO size-5 sub-magmas at all.
[text written by Claude]
dwrensha · 2026-07-02 02:32:59
dwrensha · 2026-05-15 11:16:54