Equation 677 Database

Size-85 = 5·17 idempotent right-cancellative magma satisfying Eq 677 and Eq 255. Z_85 cyclic + involution + Steiner system S(2, 5, 85) template. Structure: x ◇ y = x + f(y − x) on the cyclic group Z_85, where f: Z_85 → Z_85 is a fixed-point-free INVOLUTION (f² = id, f(0) = 0, 42 transpositions on Z_85 \ {0}). L_0 has cycle structure (1, 2⁴²); R_0 typically has cycle structure (1, 4²¹). The magma is fully Z_85-translation-invariant — verified by exhibiting an order-85 fix-free magma automorphism. 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. The 357 such sub-magmas (= C(85, 2) / C(5, 2)) form a Steiner system S(2, 5, 85), with each element lying on exactly 21 sub-magmas (= (85−1)/(5−1)). (S(2, 5, v) exists for v ≡ 1 or 5 mod 20; here 85 ≡ 5 mod 20 ✓.) Sibling magmas at size 85 with the same Z_85-cyclic + involution + S(2, 5, 85) template: magma#10e5c920, magma#31ae2815, magma#5d5c0411, magma#6282dd11, magma#d02731a8. They differ in which specific involution f is used. (Two other size-85 entries — magma#84944e78 and magma#4b5e27f5 — share the Steiner-system + L_0-involution properties but are NOT Z_85-cyclically translation-invariant, so use a different additive structure or non-translation-invariant construction.) Same family as the 16 size-81 magmas of 'family 4' (e.g. magma#c5da3284), which use the Z_81 + involution + S(2, 5, 81) template. The construction generalizes to any n ≡ 1 or 5 mod 20. Display reorder lays elements out as 0, τ(0), τ²(0), … along the orbit of an order-85 fix-free magma automorphism τ (= the hidden additive translation by 1 in Z_85). Under this reorder the Cayley table is fully Z_85-translation-invariant: every row is a horizontal shift of row 0 = f. [text written by Claude]