Equation 677 Database

Size 21 = 3·7 currently has 6 magmas in the DB, all idempotent, right-cancellative, and satisfying Eq 255. They split into two distinct structural families. **Family A: Steiner-S(2, 5, 21) = PG(2, 4) line magmas (4 magmas).** Every pair of distinct elements generates a 5-element sub-magma (≅ unique size-5 Eq 677 magma magma#e549b5f8). The 21 size-5 sub-magmas are the lines of the projective plane PG(2, 4) (the unique S(2, 5, 21)); each point lies on 5 lines. The four entries differ in their automorphism group: • magma#b1cfacfa: |Aut| = Z_21 cyclic (translation-invariant, f involution) • magma#4bd29022: |Aut| = F_21 = Z_7 ⋊ Z_3 (non-abelian Frobenius) • magma#50e6ad54: |Aut| = F_21 = Z_7 ⋊ Z_3 (another non-abelian Frobenius, non-iso magma) • magma#b904cba0: |Aut| = 1 (trivial — most rigid) **Family B: Pencil-through-pivot magmas (2 magmas).** Different structure: 21 = 1 + 5·4 — one pivot element + 5 size-5 lines through the pivot, partitioning the other 20 elements into 5 groups of 4. There are only **5** size-5 sub-magmas (not 21), all containing the pivot; cross-petal pairs (between different lines) generate the FULL 21-element magma. Both entries have |Aut| = 20 with orbit structure (1, 20). • magma#2eea123a • magma#9bef57c1 This is the smallest 'pencil' construction in the DB. Compare with the analogous magma#875876e7 at size 76 = 1 + 5·15, where the petals are size 16 (much larger) and a full TD(5, 15) of transversal size-5 sub-magmas connects the petals. At size 21, the petals are themselves size 5 and there are no cross-petal transversals. All 6 size-21 entries share the same idempotent + RC + Eq 255 properties, but realize substantially different magma structures. The fundamental dichotomy: • Family A: rich sub-magma structure (21 size-5 sub-magmas = full PG(2, 4) line set), but only some have nice algebraic symmetries. • Family B: minimal sub-magma structure (5 size-5 sub-magmas all through a single pivot), but more rigid combinatorial design. [text written by Claude]