Equation 677 Database

Size-21 = 3·7 idempotent right-cancellative magma satisfying Eq 677 and Eq 255. **Pencil-through-pivot construction**: 21 = 1 + 5·4 — one pivot element + 5 lines through the pivot, each line of size 5 (1 pivot + 4 non-pivot points). Distinguishing structure: • There are exactly 5 size-5 sub-magmas in this magma (not 21 like in the PG(2, 4) line magmas), all passing through a unique pivot element (canonical label 0). The pivot lies in all 5 lines; every other element lies in exactly 1 line. • Each line is isomorphic to magma#e549b5f8 (the unique size-5 Eq 677 magma over F_5). • Every cross-petal pair (= pair of non-pivot elements in different lines) generates the FULL 21-element magma (no cross-petal size-5 'transversals' exist here — unlike the size-76 pencil magma#875876e7 which has a TD(5, 15) of cross-petal transversals). • L_0 has cycle structure (1, 2¹⁰) — fixed-point-free involution (apart from the pivot fixed point). R_0 has cycle structure (1, 4⁵) — order 4 with the pivot fixed. • |Aut(M)| = 20, acting on 21 elements with orbits {pivot} and {20-element non-pivot orbit}. Aut acts regularly on the non-pivot orbit. This is the **smallest pencil magma in the DB**, structurally analogous to magma#875876e7 at size 76 (1 + 5·15) which has petals of size 16, but here with much smaller size-5 petals (each = the F_5 affine magma) and no cross-petal transversal sub-magmas. Other size-21 magmas (4 of them — magma#b1cfacfa, magma#4bd29022, magma#50e6ad54, magma#b904cba0) are in the DIFFERENT 'Steiner-S(2, 5, 21) = PG(2, 4) line magma' family, with 21 size-5 sub-magmas (one per line of PG(2, 4)) and every pair on a unique line — NOT just 5 sub-magmas through a single pivot. Display reorder: pivot at position 0, then 5 consecutive 4-blocks (positions 1-4, 5-8, …, 17-20) for the 5 lines through the pivot. Within each line block, elements are F_5-affine-ordered using the line's own R_0 operation (0_F → pivot, 1_F → smallest non-pivot, then 2_F = T[1, pivot], 3_F = T[2, pivot], 4_F = T[3, pivot]). The rendered Cayley table shows a clean 5×5 grid of 4×4 blocks plus a pivot row/column. [text written by Claude]