论文标题
关于产生最大非社交准群的二次正常形态的数量
On the number of quadratic orthomorphisms that produce maximally nonassociative quasigroups
论文作者
论文摘要
令$ q $是一种奇怪的素数，假设$ a，b \ in \ mathbb {f} _q $使$ ab $和$（1 { - } a）（1 { - } b）$是非零的方形。令$ q_ {a，b} =（\ mathbb {f} _q，*）$是$ u*v = u+a（v { - } u $ $ v-u $是一个方形，$ u*v = v = v = u+b（v { - } u+b（v { - } u）$ v-v-us $ v-us $ v-us n nes n nons n nons.如果满足$ x*（y*z）=（x*y）*z $ $ \ leftrightArrow $ $ x = y = y = z $，则该准元素被称为最大非求解。用$σ（q）$表示$ q_ {a，b} $的$（a，b）$的数量是最大的非社交性。我们表明存在常数$α\大约0.02908 $和$β\大约0.01259 $，这样，如果$ q \ equiv 1 \ equiv 1 \ bmod 4 $，则$ \limσ（q）/q^2 =α$，如果$ q \ q \ equiv 3 \ equiv 3 \ bmod 4 $，则$ \ lim pimplimσ（q）/q^$
Let $q$ be an odd prime power and suppose that $a,b\in\mathbb{F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (\mathbb{F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if $v-u$ is a square, and $u*v=u+b(v{-}u)$ is $v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies $x*(y*z) = (x*y)*z$ $\Leftrightarrow$ $x=y=z$. Denote by $σ(q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $α\approx 0.02908$ and $β\approx 0.01259$ such that if $q\equiv 1 \bmod 4$, then $\lim σ(q)/q^2 = α$, and if $q \equiv 3 \bmod 4$, then $\lim σ(q)/q^2 = β$.
