Equation 677 Database

Size-81 non-idempotent right-cancellative magma satisfying Eq 677 and Eq 255. The unique idempotent is at index 80; it lies in the unique 9-element sub-magma of M, namely {72, 73, …, 80}, which is isomorphic to the unique size-9 Eq 677 magma magma#2925dc18 (the 'x ◇ y = x + 3y mod 9' magma over Z/9Z). This magma has no non-trivial congruence (no homomorphism onto a smaller magma), so unlike magma#0dd86070 it is NOT a fiber bundle — there is no 'projection' M → F_9 compatible with the operation. The only sub-magma of size 9 is the one above; pairs {x, y} with x or y outside {72…80} generate the entire magma. Automorphism group: Aut(M) is isomorphic to AGL(1, F_9) = F_9 ⋊ F_9* of order 72 — the full automorphism group of the size-9 sub-magma, lifted to all of M. Max element order is 8 (so Aut is not cyclic). Aut acts on M with three orbits: • {80} — the unique idempotent (size 1) • {72, …, 79} — the size-9 sub-magma minus the idempotent (size 8) • {0, …, 71} — everything outside the sub-magma (size 72) Compare magma#761fd630 (F_9 × F_9 direct product, with 10 size-9 sub-magmas = the 10 projective lines through origin in F_9²) and magma#0dd86070 (size-9 fiber bundle over a size-9 base). All three are non-idempotent RC size-81 extensions of the size-9 Eq 677 magma, distinguished by the kind of extension. [text written by Claude]