论文标题
迭代组和准群中的横向，近横向和对角线
Transversals, near transversals, and diagonals in iterated groups and quasigroups
论文作者
论文摘要
鉴于二进制级$ g $的订单$ n $，a $ d $ coasigroup $ g [d] $是$（d+1）$ - ary准群，等于$ d $ - times的成分，$ g $与$ g $。每$ d $ -ary Quasigroup的Cayley表是$ d $二维拉丁五角管。定义了多级准群中的横向和对角线，以便与相应的拉丁超立方体中的横向和对角线。 我们证明，如果$ n $的组$ g $满足霍尔 - 帕格条件，则$ g [d] $中的横向数量等于$ \ frac {n！} {| g'| n^{n-1}} \ cdot n！^{d}（1 + o（1））$对于大$ d $，其中$ g'$是$ g $的换向器子组。对于一般的准元集$ g $，我们获得了$ g [d] $中横向和近横向数量的类似估计，并开发了一种计数其他类型的迭代准群中对角线的方法。
Given a binary quasigroup $G$ of order $n$, a $d$-iterated quasigroup $G[d]$ is the $(d+1)$-ary quasigroup equal to the $d$-times composition of $G$ with itself. The Cayley table of every $d$-ary quasigroup is a $d$-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so as to coincide with those in the corresponding latin hypercube. We prove that if a group $G$ of order $n$ satisfies the Hall--Paige condition, then the number of transversals in $G[d]$ is equal to $ \frac{n!}{ |G'| n^{n-1}} \cdot n!^{d} (1 + o(1))$ for large $d$, where $G'$ is the commutator subgroup of $G$. For a general quasigroup $G$, we obtain similar estimations on the numbers of transversals and near transversals in $G[d]$ and develop a method for counting diagonals of other types in iterated quasigroups.
