标题： 关于n元拟群的可约性与其收缩的可约性之间的联系 显示英文标题

标题： On connection between reducibility of an n-ary quasigroup and that of its retracts

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. 一个$n$元拟群$Q$是（可置换地）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$(n-k+1)$元和$k$元拟群，$s$是一个置换，且$1<k<n$。 一个$m$元拟群$R$被称为$Q$的收缩，如果它可以通过固定$Q$或其某个逆元的$n-m>0$个参数得到。 我们证明了每个不可约的$n$元拟群都有一个不可约的$(n-1)$元或$(n-2)$元缩影；此外，如果阶数是有限的且为质数，那么它有一个不可约的$(n-1)$元缩影。 我们应用这个结果来证明，所有阶为5或7的$n$-元拟群，其所有二元退化都与$Z_5$或$Z_7$同构的，都是可约的对于$n>3$。 关键词：$n$-元拟群，退化，可约性，拉丁超立方体 显示英文摘要

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