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 $\left(Q,*\right)$ such that $x\hspace{0.17em}*\hspace{0.17em}\left(y\hspace{0.17em}*\hspace{0.17em}z\right)=\left(x\hspace{0.17em}*\hspace{0.17em}y\right)\hspace{0.17em}*\hspace{0.17em}z$ 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 $\left(9,4\right)$ and to a Sudoku division square.

中文翻译：

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九阶及以上的极端非关联性

本文的主要关注点是至多具有9个阶的拟群 $18$关联三元组。九阶是存在准群的最小阶$（问，*）$ 这样 $X\hspace{0.17em}*\hspace{0.17em}（ÿ\hspace{0.17em}*\hspace{0.17em}ž）=（X\hspace{0.17em}*\hspace{0.17em}ÿ）\hspace{0.17em}*\hspace{0.17em}ž$ 仅当且仅当成立 $X=ÿ=ž$。直到同构，只有一个这样的准群。它具有非凡的性能，可将其绑定到近场和PMD$（9，4）$ 和数独师广场。