Equation 677 Database

Size 49 = 7^2. The smallest sizes admitting eq677 magmas are 5 and 7; over F_7 there are two eq677 magmas, x*y = 4x + y and x*y = 4x + 3y (written F_7(4,1) and F_7(4,3)). Order-49 magmas are built over this order-7 base structure, and the ones examined so far all have a single idempotent. Two structural families occur: 1) Fiber bundles. These carry a congruence with 7 classes of size 7; the quotient is one of F_7(4,1) or F_7(4,3) and the classes are order-7 fibers. Both right-cancellative and non-right-cancellative examples exist (in the nRC case some right multiplications collapse a fiber). In (base, fiber) coordinates the Cayley table is a clean 7x7 block super-grid. 2) Pencils. These are simple (no nontrivial congruence): two elements generate either a shared order-7 sub-magma or the entire magma. Each pencil has exactly 8 order-7 sub-magmas, all through one common element (the unique idempotent), partitioning the remaining 48 elements into 8 petals of 6 -- the incidence pattern of the 8 lines through a point of AG(2,7), which is also how the 8 one-dimensional F_7-subspaces of a linear F_49 magma sit. Strikingly, these pencils are NOT isomorphic to any linear magma over F_49 and are not affine over their lines: they are genuinely twisted. They sub-classify by the multiset of line types: 8xF_7(4,1) (6 magmas); 7xF_7(4,1) + 1xF_7(4,3) (6 magmas); 8xF_7(4,3) (6 magmas); 7xF_7(4,3) + 1xF_7(4,1) (6 magmas). A recent batch of new order-49 magmas (submitted by b-reinke) comprises 6 fiber bundles (4 RC over F_7(4,3)/F_7(4,1), 2 nRC over F_7(4,3)) and 24 pairwise non-isomorphic pencils. Their display reorders were (re)derived: bundles via (base, fiber) coordinates, pencils -- lacking any congruence -- by minimizing a Cayley-image smoothness measure. [text written by Claude]