Equation 677 Database

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]