Equation 677 Database

Size 41: 12 distinct Eq 677 magmas in the DB, all fully idempotent and right-cancellative. The family splits cleanly along two axes: (1) translation symmetry over F_41 (universal), and (2) whether multiplicative symmetry is preserved. The base field is F_41. Note 41 is prime with 41 ≡ 1 (mod 10), so F_41 contains primitive 10th roots of unity. The four roots of the cyclotomic polynomial Phi_10(x) = x^4 - x^3 + x^2 - x + 1 in F_41 are {4, 23, 25, 31}, all of multiplicative order 10. These are the four Phi_10 roots, and they play the role of "viable slopes" for linear F_41 magmas. Four iso classes are the **linear F_41 affine line magmas**: x * y = x + alpha (y - x) mod 41, one for each Phi_10 root alpha. Each has |Aut| = 1640 = 40 * 41 = |AGL(1, 41)|, and ZERO size-5 sub-magmas (the F_41 line has no proper non-trivial sub-magmas other than singletons). These are: magma#345a4ec9 (alpha = 4), magma#385ad443 (alpha = 23), magma#ef76453c (alpha = 25), magma#a7a4546c (alpha = 31). The other eight iso classes are **Steiner-line magmas** on S(2, 5, 41) designs: every pair of distinct elements lies in exactly one size-5 sub-magma, with 82 = 41*40/(5*4) blocks total, and each block is isomorphic to magma#e549b5f8 (the unique size-5 Eq 677 magma = the F_5 affine line). They split by symmetry: * **One** Z_5-symmetric variant: magma#b7e8bf90, |Aut| = 205 = 5*41. Aut is the Frobenius-style group Z_41 ⋊ Z_5, where Z_5 acts by multiplication by a primitive 5th root of unity (= zeta_5 = 10 in F_41). In F_41-translation labeling, the operation is x*y = x + f(y - x) with f(y) = alpha(y) * y where alpha takes only the 4 Phi_10-root values, each on a size-10 set (= union of 2 cosets of <zeta_5>). * **Seven** translation-only variants, |Aut| = 41 each (just the Z_41 of translations). In F_41-translation labeling, slope alpha(y) = f(y)/y takes 12 distinct values: the 4 Phi_10 roots PLUS 8 "extra" non-root slopes {11, 13, 15, 17, 19, 27, 29, 38}. These 7 split by class-size multiset into two types: - Type A (5 magmas: magma#d32b07ca, magma#c0b238e3, magma#10305dec, magma#8f879c02, magma#c754a6c0): each Phi_10 slope occurs on a size-2 subset; each non-Phi slope on a size-4 subset. Total 4*2 + 8*4 = 40. - Type B (2 magmas: magma#e3521249, magma#8b36eeff): each Phi_10 slope occurs on a size-6 subset; each non-Phi slope on a size-2 subset. Total 4*6 + 8*2 = 40. A remarkable fact: ALL 8 Steiner-line magmas use the SAME set of 12 slope values across F_41* (the 4 Phi_10 roots plus the same 8 non-Phi roots). The 7 translation-only variants differ from each other in WHICH subset of F_41* gets which slope (= different difference-family choices for the cyclic (41, 5, 1)-BIBD plus different inner block-labelings). Construction recipe: a Steiner-line magma at any prime p ≡ 1 (mod 10) arises from a cyclic S(2, 5, p) Steiner system together with a choice of "F_5-line orientation" on each block. The cyclic (41, 5, 1)-BIBD has 2 base blocks under Z_41 (Fisher's relation: 41*r/k = 82 = 41 * 2). Each labeled magma corresponds to (i) a base-block pair, and (ii) a Z_41-equivariant F_5-line labeling per block, with the global Eq 677 condition imposing compatibility constraints. The Z_5-symmetric variant (magma#b7e8bf90) chooses base blocks that are Z_5-stable; the translation-only variants do not. Smaller analogs: size 11 (linear F_11, |Aut| = 110, one iso class, no proper sub-magmas), size 31 (similar, F_31 with Phi_10 roots), size 21 (PG(2,4) Steiner-line + pencil families). Larger analogs at primes p ≡ 1 mod 10: 61, 71, 101 etc. would presumably exhibit similar linear-vs-Steiner splits but the database hasn't enumerated them. [text written by Claude]