Linear magma over the extension field F_9 = F_3[α]/⟨α² + 1⟩.
Operation: x ◇ y = a·x + b·y in F_9 with (a, b) = (1, 2 + α). Note 2 + α is a root of Φ_10(x) = x⁴ - x³ + x² - x + 1 (cyclotomic polynomial for primitive 10th roots of unity) in F_9, and the corresponding α_coef = 1 satisfies α_coef = -β³ - β - 1 - this is Pace Nielsen's "Type 2" linear 677 magma family. F_9 is the proper degree-2 extension of F_3; do NOT confuse with the ring Z/9Z = Z/9 (which has zero divisors and is NOT a field).
Size 9, not fully idempotent (only 1 element is idempotent), right-cancellative.
[text written by Claude]
dwrensha · 2026-05-16 11:52:40
Linear magma over the ring Z/9Z: x ◇ y = x + 3y (mod 9).
NOTE: Z/9Z is the ring of integers mod 9, NOT a field — since 9 = 3^2 is not prime, Z/9Z has zero divisors (e.g. 3·3 = 0 mod 9). The genuine field F_9 = GF(3^2) exists as a separate object but is not used here; this construction uses ring arithmetic mod 9.
[text written by Claude]
dwrensha · 2026-04-29 17:24:10
Linear magma over Z/9Z (the prime field F_9): x ◇ y = x + 3y (mod 9).
dwrensha · 2026-04-29 13:45:45
Linear magma over Z/9Z (the prime field F_9): x ◇ y = (1,3).
dwrensha · 2026-05-16 12:02:51
dwrensha · 2026-05-16 11:52:40
dwrensha · 2026-04-29 17:24:10
dwrensha · 2026-04-29 13:45:45
dwrensha · 2026-04-29 13:29:50