Equation 677 Database

Simple, non-right-cancellative Eq677 magma of size 11*10+1, built by the same recipe as magma#1c1045d74cc5a6cd35415a2d56a5971c (which is the case k=8, seed magma#2925dc18). Carrier: {e} together with F_11 x Z/10, elements (i,s). Base law on F_11: i*j = 1i + 02j mod 11 (idempotent linear 677 law). Off-diagonal fibers, i != j: (i,s)*(j,t) = (i*j, t) if j-i is a nonzero square mod 11, and (i*j, -s-t mod 10) otherwise. Diagonal: each fiber F_i = {(i,s)} together with e is a copy of the idempotent size-11 magma magma#ca58a4a9ddeee171fc40b5c556a5095e4a2c63eee13557dcdf190986f06d14c3, with e playing the role of (any idempotent; Aut is transitive); the 11 copies are glued along e (so e*e = e, and e's row/column on F_i are those of the seed). Why 677 holds: for x,y in distinct fibers all four products of the 677 expression stay off-diagonal (i*j is never i or j), so the off-diagonal maps only need to satisfy a chain condition that does not involve the diagonal at all; that condition holds for the maps above for every k. For x,y in one fiber (or equal to e) the whole instance lives in one copy of the seed. The diagonal seeds can therefore be chosen independently per fiber; this entry uses the same seed in every fiber. Properties (verified locally): simple (each pair (0,b) generates the full congruence; L-transitivity makes this sufficient), not right-cancellative (columns have fibers of size 10), not translation-invariant (e is the unique right identity, hence fixed by every automorphism), fiber matrix normal, rank 100. Constructed by Claude (Fable 5.1) working with dwrensha, 2026-09-04; construction script (glue.py) verifies Eq677 on all instances.