No magma of size 2 satisfies Equation 677.
Proof. Equation 677, x = y ◇ (x ◇ ((y ◇ x) ◇ y)), implies that for each fixed y, the map z ↦ y ◇ z is a bijection: its inverse sends x to x ◇ ((y ◇ x) ◇ y). So every row of the Cayley table must be a permutation of the carrier. An exhaustive check of all 2² = 4 such 2×2 row-Latin tables shows that none satisfies Eq677.
[text written by Claude]
dwrensha · 2026-05-14 18:08:29
No magma of size 2 satisfies Equation 677.
**Proof.** Equation 677, x = y ◇ (x ◇ ((y ◇ x) ◇ y)), implies that for each fixed y, the map z ↦ y ◇ z is a bijection: its inverse sends x to x ◇ ((y ◇ x) ◇ y). So every row of the Cayley table must be a permutation of the carrier. An exhaustive check of all 2² = 4 such 2×2 row-Latin tables shows that none of them satisfies Eq677.
[text written by Claude]
dwrensha · 2026-05-14 19:45:18
dwrensha · 2026-05-14 18:08:29