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]
dwrensha · 2026-05-15 11:44:48
Size 21 = 3·7 currently has 4 magmas in the DB, all idempotent right-cancellative and satisfying Eq 255. All four realize the same Steiner system S(2, 5, 21) = PG(2, 4) (projective plane of order 4) at the magma level — every pair of distinct elements generates a 5-element sub-magma (≅ magma#e549b5f8, the unique size-5 Eq 677 magma over F_5), with 21 sub-magmas forming the 21 lines of PG(2, 4).
What distinguishes them is the algebraic symmetry of the magma operation respecting this combinatorial structure. The four entries cover four different automorphism-group types:
• magma#b1cfacfa — |Aut| = **Z_21 cyclic**, acting regularly. The 'most symmetric' size-21 PG(2, 4) magma, fully Z_21-translation-invariant with f a fixed-point-free involution.
• magma#4bd29022 — |Aut| = **F_21 = Z_7 ⋊ Z_3** (non-abelian Frobenius), also regular. Same order as Z_21 but with element orders {1, 3, 7} and no order-21 element.
• magma#50e6ad54 — |Aut| = F_21 (non-abelian Frobenius), also regular. Same template as 4bd29022 but presumably a non-isomorphic magma with a different specific Frobenius-action.
• magma#b904cba0 — |Aut| = **trivial (order 1)**. The most rigid, with no non-trivial symmetries; the magma operation is essentially 'canonical' on the PG(2, 4) line structure.
All four are admissible because 21 ≡ 1 mod 20 (a necessary condition for S(2, 5, n) to exist). Same template as the larger Steiner-S(2, 5, n) magma families: size 25 (29 entries, AG(2, 5) line magmas), size 41, 61, 65, 81 (16), 85 (7), 125.
At size 21 specifically, the underlying design is the projective plane PG(2, 4) — the SMALLEST projective plane of order ≥ 4 — making this the smallest size in the family. The four magmas show how varying algebraic structure can decorate the same combinatorial design.
[text written by Claude]
dwrensha · 2026-05-15 11:48:37
dwrensha · 2026-05-15 11:44:48