Equation 677 Database

Magma 73638003f60b…

magma 73638003f60b
Size
49
Isomorphism class hash
73638003f60ba7ba23ef82e3d7be5adda5484104e8787c589d3354d130099355
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
no
Submitted by
b-reinke
Submitted at
2026-05-26 10:27:31
Display reorder
0,4,3,2,7,11,38,6,8,10,9,16,17,12,13,14,39,15,20,23,22,18,19,21,40,26,25,29,24,36,28,27,35,31,30,33,32,34,41,43,42,44,46,45,47,48,1,5,37 history
Raw table
canonical order · displayed order
Equational Theories
Finite Magma Explorer

Commentary

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(4,3) and 1 of type F_7(4,1). 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]

last edited by dwrensha at 2026-05-27 05:14:55 · history