n元拟群的约化与其收缩约化的联系
On connection between reducibility of an n-ary quasigroup and that of its retracts
摘要：
如果在等式$x_0=Q(x_1,...,x_n)$中$x_0,...,x_n$的任何$n$元素的知识唯一地指定剩余的一个，则$n$元运算$Q:S^n\to S$称为$|S|$阶的$n$元拟群.一个$n$ -元拟群$Q$是（可置换）可约的，如果$Q(x_1,...,x_n)=P(R(x_{s(1)},...,x_{s(k)}),x_{s(k+1)},...,x_{s(n)})$，其中$P$和$R$是$(n-k+1)$ -元拟群和$k$ -元拟群，$s$是置换，而$1<k<n$ .一个$m$元拟群$R$称为$Q$的收缩元，如果它可以通过固定$n-m>0$变元从$Q$或它的一个逆得到.证明了每一个不可约的$n$ -元拟群都有一个不可约的$(n-1)$ -元或$(n-2)$ -元收缩核，并且证明了当阶为有限素数时，它有一个不可约的$(n-1)$ -元收缩核.我们应用这个结果证明了所有二元收缩与$Z_5$或$Z_7$同素的5阶或7阶$n$ -元拟群对于$n>3$是可约的。关键词：$n$ -ary拟群，收缩，约化，拉丁超立方体
英文摘要：
An $n$-ary operation $Q:S^n\to S$ is called an $n$-ary quasigroup of order $|S|$ if in the equation $x_0=Q(x_1,...,x_n)$ knowledge of any $n$ elements of $x_0,...,x_n$ uniquely specifies the remaining one. An $n$-ary quasigroup $Q$ is (permutably) reducible if $Q(x_1,...,x_n)=P(R(x_{s(1)},...,x_{s(k)}),x_{s(k+1)},...,x_{s(n)})$ where $P$ and $R$ are $(n-k+1)$-ary and $k$-ary quasigroups, $s$ is a permutation, and $1<k<n$. An $m$-ary quasigroup $R$ is called a retract of $Q$ if it can be obtained from $Q$ or one of its inverses by fixing $n-m>0$ arguments. We show that every irreducible $n$-ary quasigroup has an irreducible $(n-1)$-ary or $(n-2)$-ary retract; moreover, if the order is finite and prime, then it has an irreducible $(n-1)$-ary retract. We apply this result to show that all $n$-ary quasigroups of order 5 or 7 whose all binary retracts are isotopic to $Z_5$ or $Z_7$ are reducible for $n>3$. Keywords: $n$-ary quasigroups, retracts, reducibility, latin hypercubes
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