Linear magma over the extension field F_16 = F_2[α]/⟨α⁴ + α + 1⟩.
Operation: x ◇ y = a·x + b·y in F_16 with (a, b) = (1 + α³, α³). Since α + β = (1 + α³) + α³ = 1 in F_16 (char 2), this is the "Type 1" translation-invariant fully-idempotent linear 677 magma, equivalent to x ◇ y = x + β·(y - x) with β = α³. Note β = α³ is a primitive 5th root of unity in F_16* (which has order 15 = 3·5). In characteristic 2 the primitive 10th-root condition collapses to primitive 5th-root since Φ_10(x) = Φ_5(x) mod 2.
F_16 is the proper degree-4 extension of F_2; do NOT confuse with the ring Z/16Z (which has zero divisors and is NOT a field).
Size 16, fully idempotent, right-cancellative.
[text written by Claude]
dwrensha · 2026-05-16 11:52:40
Linear magma over the ring Z/16Z: x ◇ y = 9x + 8y (mod 16).
NOTE: Z/16Z is the ring of integers mod 16, NOT a field — since 16 = 2^4 is not prime, Z/16Z has zero divisors (e.g. 2·8 = 0 mod 16). The genuine field F_16 = GF(2^4) exists as a separate object but is not used here; this construction uses ring arithmetic mod 16.
[text written by Claude]
dwrensha · 2026-04-29 17:24:10
Linear magma over Z/16Z (the prime field F_16): x ◇ y = 9x + 8y (mod 16).
dwrensha · 2026-04-29 13:45:45
Linear magma over Z/16Z (the prime field F_16): x ◇ y = (9,8).
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