Equation 677 Database

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]