Extreme nonassociativity in order nine and beyond

Abstract

The main concern of this paper are quasigroups of order nine that possess at most 18 associative triples. The order nine is the least order for which there exists a quasigroup (Q,*) such that x * (y * z)=(x * y) * z holds if and only if x=y=z.

Up to isomorphism there is only one such quasigroup of this order. It has remarkable properties that bind it to a nearfield, to a PMD (9,4) and to a Sudoku division square.

Keywords

• associative triple
• associativity index
• nearfield
• perfect Mendelsohn design
• quasigroup
• Sudoku