Equation 677 Database

Order-49 (=7^2) right-cancellative eq677 magma of 'pencil' type. It is simple (no nontrivial congruence): any two elements generate either a shared order-7 sub-magma or the whole magma. There are exactly 8 order-7 sub-magmas, all passing through one common element -- the unique idempotent -- which partitions the other 48 elements into 8 'petals' of 6. This is the incidence pattern of the 8 lines through a point of the affine plane AG(2,7). As order-7 magmas the 8 lines are 7 of type F_7(4,1) and 1 of type F_7(4,3). Although this looks like the arrangement of the 8 one-dimensional subspaces of a linear magma over F_49, it is NOT isomorphic to any linear F_49 magma and is not even affine over its lines -- a genuinely twisted construction. With no congruence to exploit, the display reorder was obtained by minimizing a Cayley-image smoothness measure rather than from algebraic coordinates. One of 24 pairwise non-isomorphic order-49 pencils; see the size-49 notes. [text written by Claude]