Equation 677 Database

Size-81 "noncommutative-linear" (matrix-linear) magma satisfying Eq 677 and Eq 255. Carrier: the module M = (F_3)^4 (elements encoded as digit tuples, least-significant component first), with operation x ◇ y = A·x + B·y, A = [[2, 1, 2, 0], [2, 1, 2, 2], [0, 1, 2, 2], [2, 1, 0, 2]], B = [[0, 0, 0, 2], [1, 0, 0, 1], [0, 1, 0, 1], [0, 0, 1, 2]] (matrices act on column vectors, arithmetic in the module). Without assuming commutativity, Eq 677 is equivalent to the two operator identities B·(A + B·A·B) = I and A + B²·A² + B³ = 0; this pair satisfies both, and additionally satisfies the Eq-255 operator identity A³ + A²·B + A·B + B = I. Note the known proof that linear 677 magmas satisfy 255 uses commutativity and field inverses, so it does not cover matrix coefficients — for this magma, 255 was verified directly. Found June 2026 by an exhaustive sweep over all (A, B) pairs (B taken over GL conjugacy-class representatives, since conjugating (A, B) jointly is a magma isomorphism; for fixed B the first identity is linear in A). Eq 677 verified directly on the full Cayley table. right-cancellative; not idempotent. [text written by Claude]