On reducibility of n-ary quasigroups Full article
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Discrete Mathematics
ISSN: 0012-365X , E-ISSN: 1872-681X |
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| Output data | Year: 2008, Volume: 308, Number: 22, Pages: 5289-5297 Pages count : 9 DOI: 10.1016/j.disc.2007.08.099 | ||
| Tags | n-ary quasigroups, retracts, reducibility, distance 2 MDS codes, latin hypercubes | ||
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Abstract:
An $n$-ary operation from $Q:S^n$ 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. $Q$ 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 $S$ 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 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.
Cite:
Krotov D.S.
On reducibility of n-ary quasigroups
Discrete Mathematics. 2008. V.308. N22. P.5289-5297. DOI: 10.1016/j.disc.2007.08.099 WOS Scopus РИНЦ OpenAlex
On reducibility of n-ary quasigroups
Discrete Mathematics. 2008. V.308. N22. P.5289-5297. DOI: 10.1016/j.disc.2007.08.099 WOS Scopus РИНЦ OpenAlex
Dates:
| Submitted: | Jul 12, 2006 |
| Accepted: | Aug 31, 2007 |
| Published online: | Oct 24, 2007 |
Identifiers:
| Web of science: | WOS:000259691000028 |
| Scopus: | 2-s2.0-50349096908 |
| Elibrary: | 13577733 |
| OpenAlex: | W2086996425 |