Equation 677 Database

Size-21 = 3·7 idempotent right-cancellative magma satisfying Eq 677 and Eq 255. **The carrier is the cyclic group Z_21 (= Z_3 × Z_7 by CRT)**, and the magma is fully Z_21-translation-invariant — verified by exhibiting an order-21 fix-free magma automorphism. |Aut(M)| = 21, acting REGULARLY on the 21 elements (single orbit; trivial stabilizer at every point). Structure: x ◇ y = x + f(y − x) in Z_21, where f: Z_21 → Z_21 is a fixed-point-free INVOLUTION (f² = id, f(0) = 0, 10 transpositions on Z_21 \ {0}). L_0 has cycle structure (1, 2¹⁰); R_0 has cycle structure (1, 4⁵). Sub-magma design: every pair of distinct elements generates a 5-element sub-magma isomorphic to the unique size-5 Eq 677 magma over F_5 (magma#e549b5f8). The C(21, 2)/C(5, 2) = 210/10 = 21 such sub-magmas form a **Steiner system S(2, 5, 21)**. Since the unique S(2, 5, 21) is the projective plane PG(2, 4), the 21 lines of this magma are exactly the lines of PG(2, 4). Each element lies on (21 − 1)/(5 − 1) = 5 lines. This is the **smallest member of the 'Z_n cyclic + involution + Steiner S(2, 5, n)' family** of Eq 677 magmas (admissible whenever n ≡ 1 or 5 mod 20; here 21 ≡ 1 ✓). Other DB sizes with the same template: • Size 21 (this magma, 4 entries total) — Steiner S(2, 5, 21) = PG(2, 4) • Size 25 (29 entries, AG(2, 5) line magmas) — Steiner S(2, 5, 25) = AG(2, 5) • Size 41 (8 entries), 61 (32 entries), 65 (9 entries), 81 (16 entries), 85 (7 entries), 125 (1 entry) At sizes 21, 65, 85, etc. (where n ≡ 5 mod 20), the design is structured around the cyclic Z_n group; at sizes 25, 81, 125 (n = p² for various p), the design uses elementary-abelian (Z/p)^k carriers via AG(2, 5) line structures or fiber bundles. Display reorder presents elements as 0, τ(0), τ²(0), …, τ²⁰(0) along the orbit of an order-21 fix-free magma automorphism τ (the additive translation by 1 in Z_21). Under this reorder the Cayley table is fully Z_21-translation-invariant; the rendered image shows clean diagonal banding from the cyclic translation symmetry. [text written by Claude]