We say that $(x,y,z)\in Q^3$ is an associative triple in a quasigroup $Q(⁎)$ if $(x⁎y)⁎z=x⁎(y⁎z)$. It is easy to show that the number of associative triples in Q is at least $|Q|$, and it was conjectured that quasigroups with exactly $|Q|$ associative triples do not exist when $|Q|>1$. We refute this conjecture by proving the existence of quasigroups with exactly $|Q|$ associative triples for a wide range of values $|Q|$. Our main tools are quadratic Dickson nearfields and the Weil bound on quadratic character sums.

中文翻译：

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通过近场实现最大非缔合

我们说 $（X，ÿ，ž）\in {问}^3$ 是准群中的关联三元组 $问（⁎）$ 如果 $（X⁎ÿ）⁎ž=X⁎（ÿ⁎ž）$。很容易证明，Q中的三联体数目至少为$|问|$，并且推测准群与 $|问|$ 关联三元组不存在时 $|问|>\mathrm{1个}$。我们通过证明具有$|问|$ 多种值的关联三元组 $|问|$。我们的主要工具是二次Dickson近场和Weil结合二次字符和。