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On reducibility of n-ary quasigroups

Denis Krotov
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Abstract

An $n$-ary operation $Q:S^n -> S$ 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. $Q$ 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 $1 0$ arguments. We prove that if the maximum arity of a permutably irreducible retract of an $n$-ary quasigroup $Q$ belongs to $\{3,...,n-3\}$, then $Q$ is permutably reducible. Keywords: n-ary quasigroups, retracts, reducibility, distance 2 MDS codes, latin hypercubes

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Denis Krotov. 2007-07-17. On reducibility of n-ary quasigroups. https://doi.org/10.1016/j.disc.2007.08.099

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